Some new categorical invariants

We introduce several notions and give examples. We prove that ${\rm Stab}(D^b(K(l)))\cong {\mathbb C}\times \mathcal H$ for $l\geq 3$, where $K(l)$ is $l$-Kronecker quiver. This is an example of SOD, where ${\rm Stab}( \langle \mathcal T_1,\mathcal T_2\rangle )\not \cong{\rm Stab}(\mathcal T_1)\times {\rm Stab}(\mathcal T_2)$. This example suggest a new notion of a norm, strictly increasing on $\{D^b(K(l))\}_{l\geq 2}$. To a triangulated category $\mathcal T$ which has property of a phase gap we attach a non-negative number $\Vert \mathcal T \Vert_{\varepsilon}$. Natural assumptions on a SOD imply $ \Vert \langle \mathcal T_1,\mathcal T_2\rangle \Vert_{\varepsilon}\geq {\rm max}\{ \Vert \mathcal T_1 \Vert_{\varepsilon}, \Vert\mathcal T_2 \Vert_{\varepsilon}\}$. Using this we define a topology on the set of equivalence classes of triangulated categories with a phase gap, where the set of discrete derived categories is a discrete subset and the rationality of a smooth surface $S$ ensures that $[D^b(point)] \in {\rm Cl}([D^b(S)])$. Viewing $D^b(K(l))$ as a non-commutative curve, we observe that it is reasonable to count non-commutative curves in any category in a small neighborhood of $D^b(K(l))$. Examples show that this idea (non-commutative curve-counting) opens directions to new categorical structures and connections to number theory and classical geometry. We give a definition, which specializes to the non-commutative curve-counting invariants. In an example arising on the A side we specialize our definition to non-commutative Calabi-Yau curve-counting, where the entities we count are a Calabi-Yau modification of $D^b(K(l))$. Finally we speculate that one might consider a holomorphic family of categories, introduced by Kontsevich, as a non-commutative extension with the norm playing a role similar to the classical notion of degree of an extension in Galois theory.


Introduction
Motivated by M. Douglas's work in string theory, and especially by the notion of Π-stability, T. Bridgeland defined in [12] a map: For a triangulated category T the associated complex manifold Stab(T ) is referred to as the space of stability conditions (or the stability space or the moduli space of stability conditions ) on T .
Bridgeland's manifolds are expected to provide a rigorous understanding of certain moduli spaces arising in string theory. Homological mirror symmetry predicts a parallel between dynamical systems and categories, which is being established in [23], [14], [30], [39], [12], [13], [35], [47]. According to this analogy the stability space plays the role of the Teichmüller space. However while the map (1) being well defined, it is hard to extract global information for the stability spaces. In the present paper we determine explicitly the entire stability spaces on a new list of examples.
The map (1) behaves well with respect to orthogonal decompositions (see Definition 5.1). This is easy to show and due to lack of an appropriate reference in the literature we have given details on this in Section 5. In particular, there is a a bijection Stab(T 1 ⊕ T 2 ⊕ · · · ⊕ T n ) ∼ = Stab(T 1 ) × Stab(T 2 ) × · · · × Stab(T n ). (2) which is biholomorphism, when the categories are with finite rank Grothendieck groups. Theorem 1.5 in this paper contains examples of semi-orthogonal decomposition, SOD, T = T 1 , T 2 where rank(K 0 (T )) = 2 however Stab(T ) is not biholomorphic to Stab(T 1 ) × Stab(T 2 ).
The behavior of the map (1) with respect to general SOD has been studied in [19]. This study is difficult and so far a formula relating Stab( T 1 , T 2 ) and Stab(T 1 ), Stab(T 2 ) has not been obtained.
In this paper using Bridgeland stability conditions we define (Definition 4.11 ) for any 0 < ε < 1 a function (the domain is explained below and it does not depend on ε):    triangulated categories with a phase gap and prove (Theorem 6.1) that if T = T 1 , T 2 is a semi-orthogonal decomposition in which T is proper, 1 rank(K 0 (T )) < ∞, T 1 and T 2 have phase gaps, then T has phase gap as well and For the proof of this inequality we employ the method for gluing of stability conditions in [19], crucial role has also [13,Lemma 4.5] which ensures certain finiteness property of a stability condition with a phase gap.
If T = T 1 ⊕ T 2 is an orthogonal decomposition with proper T and rank(K 0 (T )) < ∞, then (Corollary 5.6): The function (3) depends on ε ∈ (0, 1), however the three subsets of its domain determined by the three conditions on the first raw in the following table do not depend on ε (Lemma 4.16): Categories with: examples: for any acyclic quiver Q D b (Q) is here iff Q is Dynkin or affine, any discrete derived category is here many wild quivers as in Prop. 8.5 (a) Further examples can be obtained by using (4) and (5). In particular by blowing up the varieties in the last column one obtains other elements in this column (see Corollary 6.4).
In Section 11 using (3) we introduce a non-trivial topology on the class of triangulated categories with a phase gap up to equivalence. The function π(1 − ε) − · ε is upper semi-continuous for this topology. The class of discrete derived categories modulo equivalence (we discuss this class in Section 10) is a discrete subset w.r. to it. We show also that for any smooth complete rational surface S In view of Corollary 11.10 and Proposition 8.5 any surface which gives a positive answer of some of these two questions would not be rational and would give a counter example to a folklore Orlov's conjecture stating that a surface over an algebraically closed field admits a full exceptional collection only if it is rational.
In all examples of D b (X), where X is a smooth projective variety, for which we compute the norm, only for X = P 1 fails the condition of Corollary 8.3. This condition fails for D b (K(n)), n ≥ 1, and conjecture 1 would impy that it fails for the quivers depicted there. We would like to generalise the question 1.2 as follows: Question 1.3. Is there a smooth complete variety X, different from P 1 , and a natural number N ∈ N s. t. for any exceptional collection (E 0 , . . . , E n ) in D b (X) and any 0 ≤ i < j ≤ n we have hom min (E i , E j ) ≤ N ?
Following Kontsevich-Rosenberg [40] we denote sometimes D b (K(l + 1)) by N P l for l ≥ 0. By rescaling · 1 2 (see (220)) we define a function: Thus the invariant dim nc takes all natural numbers as values. The last column in table (7) contains categories where the invariant dim nc is ∞, and in the other columns dim nc is finite. For an acyclic quiver Q we have dim nc D b (Q) = 0 iff Q is Dynkin and dim nc D b (Q) = 1 iff Q is affine ( remark 12.1. We do not know the answer of the following: Question 1.4. Is there a category T with a phase gap s.t. dim nc (T ) ∈ Q ?
In our topology on the domain of (6), up to equivalence, whenever T is in a small neighborhood of N P l (more precisely whenever [T ] ∈ B δ (N P l ) for some real δ > 0 as defined in (222)) we have a SOD of the form T = N P l , A where A has a phase gap, and hence dim nc (T ) ≥ l. In particular if T ∈ B δ l (N P l ) for arbitrary big l, then dim nc (T ) = +∞, and this idea is used to obtain the last column of the table above.
All examples of categories considered in this paper (except D b (pt) and Fuk(S)) satisfy an incidence T ∈ B δ (N P l ), i.e. they have a SOD N P l , A , for some l and some A. Recalling that Gromow-Witten invariants count pseudo-holomorphic curves, we view such embeddings of N P l into T as analogous to a "pseudo-holomorphic curve" in the category T , and we ask a question: can we count such entities in a given T , how many are they ?
In Section 12 we show that the answer is positive. The idea is: for a linear over a field k category T , a subgroup Γ ⊂ Aut(T ), and a choice of some additional restrictions P to define and study the set of subcategories of T , which are equivalent to another chosen category A, which satisfy P , and modulo Γ. We denote this set by C Γ A,P (T ) and define it in Definition 12.7. We prefer to choose some A, which is non-trivial but well studied.
The studies in this paper naturally impose N P l as our first choice. In particular we refer to N P l as a non-commutative curve and dim nc (N P l ) = l as its "non-commutative genus" (see Remark 12.4 for further motivation). We denote C Γ N P l ,P (T ) by C Γ l,P (T ) and that's the set of non-commutative curves of genus l ≥ 0 in T satisfying P and modulo the subgroup Γ. Furthermore, by fixing a stability condition σ ∈ Stab(T ) we define the set of σ-semistable non-commutative curves of genus l ≥ 1 in T and denote it by C Γ l,P,σ (T ) (Definition 12.13). Non-commutative curve in T is just an equivalence class of exact fully faithful functors F : N P l → T with two functors F, F ′ being equivalent if the one is obtained from the other via re-parametrization, i. e. F ′ ∼ = F • α for some α ∈ Aut(N P l ). For the examples, which we consider, additional restrictions are not necessary and P = ∅. These examples are two affine quivers (Proposition 12.8) and D b (P 2 ) (Proposition 12.9), where we point out nonempty and finite sets C Γ l (T ) and their cardinalities. More precisely, Proposition 12.9 concludes that C Aut(D b (P 2 )) l D b (P 2 )) is finite for all l and non-empty iff l = 3m − 1 for some Markov number m. Furthermore Corollary 12.10 is that the famous Markov's conjecture in number theory and a conjecture by Tyurin ( [55, p. 100] or [33,Section 7.2.3 ]) are true iff for all Markov numbers m = 1, m = 2 we have # C Aut(D b (P 2 )) 3m−1 (D b (P 2 )) = 2. Via the latter Corollary in future works (starting with [26] where will be written the proofs of Propositions 12.8, 12.9, Corollary 12.10) we plan to approach Markov's conjecture using homological mirror symmetry and applying A side techniques for computing the non-commutative curve-counting invariants introduced here.
We explain also a non-trivial example, where C (T )} as σ varies in Stab(T ) (subsection 12.4). Due to lack of space we will present the full proofs of these examples in a future work [26] devoted to non-commutative curve-counting. Here we prove (see part 1.2 of the introduction below) that C {Id} l (N P k ) = δ l,k for l, k ≥ 0 and for l ≥ 1 we describe the zones in Stab(N P l ), where C {Id} l,σ (N P l ) is zero and one respectively. Section 12.5 contains an example of finite sets C Γ A,P (T ) of different origin (the proof is postponed for future work as well). Here again we don't need additional restrictions, i. e. P = ∅, and T is the so called Fukaya category of an elliptic curve, Fuk(E). In this case the role of A is played by a category, denoted by CY (l), which is very well studied by simplectic geometers, beginning with P. Seidel, and seems to be the right substitute for N P l . The question about the cardinality of C Γ CY (l) (Fuk(S)) for higher genus curves should be realted to counting geodesics on S.
Finally (Section 13), relating our norm to the notion of holomorphic family of categories introduced by Kontsevich  Let us first give some prehistory. By definition each stability condition σ ∈ Stab(T ) determines a set of non-zero objects in T (called semi-stable objects) labeled by real numbers (called phases of the semistable objects). The semi-stable objects correspond to the so called "BPS" branes in string theory. The set of semi-stable objects will be denoted by σ ss , and φ σ (X) ∈ R denotes the phase of a semi-stable X. For any σ ∈ Stab(T ) we denote by P T σ the subset of the unit circle {exp(iπφ σ (X)) : X ∈ σ ss } ⊂ S 1 . A categorical analogue of the density of the set of slopes of closed geodesics on a Riemann surface was proposed in [23]. In [23, section 3] the focus falls on constructing stability conditions for which the set P σ is dense in a non-trivial arc of the circle. The result is the following characterization of the map (1), when restricted to categories of the form D b (Rep k (Q)) (from now on Q denotes an acyclic quiver, T denotes a triangulated category linear over an algebraically closed field k): Dynkin quivers (e.g. • ✲ •) P σ is always finite Affine quivers (e.g. • ✲ ✲ •) P σ is either finite or has exactly two limit points Wild quivers (e.g. • ✲ ✲ ✲ •) P σ is dense in an arc for a family of stability conditions (7) In [22,Proposition 3.29] are constructed stability conditions σ ∈ Stab(D b (Q)) with two limit points of P σ for any affine quiver Q (by D b (Q) we mean D b (Rep k (Q))).
In [52] and [17] is proved that the stability spaces on Dynkin quivers are contractible, but the affine case is beyond the scope of these papers. For an integer l ≥ 1 the l-Kronecker quiver K(l) (the quiver with two vertices and l parallel arrows) is in the first, second, and third raw of the table for l = 1, 2, 3, respectively. In [42] are given arguments that Stab(D b (K(l))) is simply-connected for any l ≥ 1. We develop further in [24], [25] the ideas of Macrì from [42], in particular we give a description of the entire stability space on the acyclic triangular quiver and prove that it is contractible. In [51], [15] and earlier by King is shown that Stab(D b (K(1))) as a complex manifold is C 2 . Recall that D b (K(2)) ∼ = D b (P 1 ) and D b (K(l)) for l ≥ 3 is referred to as non-commutative projective space N P l−1 , introduced by Kontsevich and Rosenberg in [40] and studied further in [43]. In [49] was shown that Stab(D b (P 1 )) ∼ = C 2 , and hence Stab(D b (K(2))) ∼ = C 2 (biholomorphisms). However the question: What is Stab(D b (K(l))) for l ≥ 3 ? (8) was open after the mentioned papers.
Thus the map (1) has the same value (up to isomorphism) on all the categories {D b (K(l))} l≥3 . Stability conditions on wild quivers whose set of phases are dense in an arc were constructed in [23], however for them the set of phases is still not dense in the entire S 1 , i.e. P σ does misses a non-trivial arc, in which case we say for short that P σ has a gap. In particular all the categories in table (7) are examples of what we call in this paper a triangulated category with phase gap, this is a triangulated category T for which there exists a full 2 σ ∈ Stab(T ) whose set of phases P T σ has a gap. Stability conditions whose set of phases is not dense in S 1 and their relation to so called algebraic stability conditions have been studied in [52]. In particular the results in [52] imply that when rankK 0 (T ) < ∞, then T has a phase gap iff there exists a bounded t-structure in T whose heart is of finite length and has finitely many simple objects (Lemma 4.7). Whence the domain of the invariant (3) contains also the CY3 categories discussed in [14].
From the very definition and table (7) one easily derives that for any acyclic quiver Q: Thus, we can compose the following table, concerning only the quivers K(l), l ≥ 1: In the present paper we compute D b (K(l)) ε for any l and any 0 < ε < 1. In particular we derive the following formulas: Combining (9) and table (10) we deduce that for l ∈ N ≥1 We expect that the domains of validity of (4) and (13) can be extended. Regarding (13) we propose: Conjecture 1.6. Let 0 < ε < 1 and let Q be any acyclic quiver.
The stability space We give some criteria ensuring that T ε = π(1− ε), they imply that for many of the wild quivers Q we have D b (Q) ε = π(1 − ε) (see Proposition 8.5 (a)) and also D b (X) ε = π(1 − ε) where X is P n , n ≥ 2, P 1 × P 1 , F a , a ≥ 0 or a smooth algebraic variety obtained from these by a sequence of blow ups in finitely many points (see Proposition 8.5 (e), (f)), for n = 1 we have D b (P 1 ) ε = 0.
The criteria for T ε = π(1 − ε) obtained here do not apply to category of the form T ∼ = D b (K(l 1 )) ⊕ D b (K(l 2 )) ⊕ · · · ⊕ D b (K(l N )) and we do prove that T ε < π(1 − ε) in this case, which is a generalization of the already discussed wild Kronecker quivers (11).
We expect that the criterion in Corollary 8.3 does not apply to all wild quivers, and we do know that its corollary, Corollary 8.5 (a), cannot be applied to all of them, for example, to the following: We conjecture, that: 1.2. It follows a brief discussion (in this order) on the proof of Theorem 1.5, and of the computations of T ε .
For any T Bridgeland defined actions of GL + (2, R) (right) and of Aut(T ) (left) on Stab(T ), which commute. The strategy for determining Stab(D b (P 1 )) in [49] is to show that the quotient of Stab(D b (P 1 )) for an action of C×Z is isomorphic to C ⋆ , where the action of C on Stab(D b (P 1 )) comes from an embedding of C in GL + (2, R) and the action of Z comes from the subgroup of Aut(D b (P 1 )) generated by the functor (·) ⊗ O(1). On the one hand in [49] Okada relies on the commutative geometric nature of D b (K(2)) ( ∼ = (D b (P 1 )) and on the other hand, implicitly, he relies on the affine nature of the root system of K(2), which are obstacles to answer the question (8). In this paper we use the ideas of Okada in [49] and we go through the mentioned obstacles by observing how to apply simple geometry of the action of modular subgroups on H and by employing the interplay between exceptional collections and stability conditions developed in [42], [24], , [25], [22]. In Section 7.1 we recall facts about the action of the modular group SL(2, Z) on H, which we need.
In Section 3 we recall what are the actions of C and Aut(T ) on Stab(T ) for any T . It is known that the action of C is free and holomorphic ( [49], [14]). When K 0 (T ) has finite rank, we show that the action of C is proper on each connected component of Stab(T ), and in particular, when Stab(T ) is connected, then Stab(T ) → Stab(T )/C is a principal holomorphic C-bundle: Proposition 3.2. Section 7.2 is devoted to the exceptional objects in Stab(T l ), where T l = D b (K(l)) for l ≥ 2. We utilize here the method of helices [11]. Up to shifts, there is only one helix in T l , which follows from [20]. We denote by {s i } i∈Z the helix, for which s 1 is the object in Rep k (K(l)), which is both simple and projective. Lemma 7.4 is the place, where we invoke the action of SL(2, Z), here we give a formula relating the fractions { Z(s i+1 ) Z(s i ) } i∈Z of a central charge 3 Z : K 0 (T ) ✲ C, it is a simple but important observation for the present paper. For any two exceptional objects E 1 , E 2 in T l Lemma 7.5 describes exactly those p ∈ Z for which hom p (E 1 , E 2 ) does not vanish. In [42,Lemma 4.1] is given a statement, but no proof. The statement of Lemma 7.5 is a slight modification of [42,Lemma 3 i.e. a group homomorphism Z : K0(T ) ✲ C such that Z(X) = 0 for any X ∈ Ob(T ).

4.1]
and we give a proof here. By the equivalence D b (P 1 ) ∼ = D b (K(2)) the sequence {O(i)} i∈Z corresponds, up to translation, to the helix {s i } i∈Z . In the non-commutative case l ≥ 3 we still have the helix {s i } i∈Z and it plays the role of {O(i)} i∈Z . Corollary 7.6 is a corollary of [44, Theorem 0.1]) and it ensures existence of a functor A l ∈ Aut(T l ) for any l ≥ 2, which is analogous to the functor (·) ⊗ O(1) for l = 2, more precisely it satisfies A l (s i ) = s i+1 for all i ∈ Z. In particular from the subgroup A l ⊂ Aut(T l ) we get an action of C × A l ∼ = C × Z on Stab(T l ) for all l ≥ 2, which for l = 2 coincides with the action on Stab(D b (P 1 )) used by Okada for studying Stab(D b (P 1 )).
In sections 7.3 and 7.4, with the help of ideas and results of [42], [24], we go on adapting arguments of Okada about the C × Z-action on Stab(D b (P 1 )) to the non-commutative case, i.e. to the C × A laction on Stab(T l ) for l ≥ 3. We will explain briefly how we utilize the SL(2, Z)-action on H. [49, p. 497,498] contain arguments about choosing a representative in the C × Z-orbit of any stability condition σ for which O, O(−1) are semi-stable and These arguments rely on the fact that for any central charge Z the vectors {Z(O(i))} i∈Z lie on a line in C ∼ = R 2 (see [49, figures 3,4,5]), more precisely Z(O(i + 1)) − Z(O(i)) is the same vector for all i ∈ Z, or in other words Z(s i+1 ) − Z(s i ) does not change as i varies in Z, when l = 2. For l ≥ 3 this property of {s i } i∈Z fails and the arguments and pictures on [49, p. 497,498] cannot be applied anymore. We avoid this obstacle by translating this problem to the problem of finding a fundamental domain in H of a subgroup of the form α l ⊂ SL(2, Z). 4 This translation is encoded in formula (105) in Lemma 7.14, whose derivation spreads throughout Subsections 7.1, ..., 7.4. The matrix α l appears first in Lemma 7.4. For l = 2 this matrix is a parabolic element and for l ≥ 3 it is a hyperbolic element in SL(2, Z), 5 which determines the difference of the type of the fundamental domains of α l (see Figure (2)) in H, which in turn determines the difference between the pictures on Figure (3). The colored parts in Figures (3a) and (3b) with only one of the two boundary curves included are in 1-1 correspondence with the set Stab(T l )/C × A l for l ≥ 3 and l = 2, respectively. Further properties of the sets given in Figure (3) are derived in Corollary 7.18. In the rest of Section 7.4 is shown how these properties and the presence of the non-trivial real segment (−∆ l , ∆ l ) seen on Figure (3a) imply Theorem 1.5. In Sections 7.5, 7.6 we compute T l ε . By definition T l ε is the supremum of 6 vol P l σ /2 as σ varies in the subset Stab ε (T l ) ⊂ Stab(T l ) of those stability conditions σ for which P l σ misses at least one closed ε-arc (see Definitions 4.3).
Sections 7.2 and 7.3 are a prerequisite for the explicit determining of the set of phases P l σ for each σ ∈ Stab(T l ) and each l ≥ 2, which is done in Section 7.5 (Proposition 7.24). It turns out that for l ≥ 3 a stability condition has vol P l σ = 0 and satisfies σ ∈ Stab ε (T l ) iff there exists j ∈ Z such that s j , s j+1 ∈ σ ss and ε < φ σ (s j+1 ) − φ σ (s j ) < 1, the set P l σ for such a σ is the set of fractions {n/m : (n, m) ∈ ∆ + (K(l))} appropriately embedded in the circle via a function depending on the stability condition. In Lemma 7.23 we shed light on the structure of the set {n/m : (n, m) ∈ ∆ + (K(l))} (see formulas (130), (131)) and use it in the proof of Proposition 7.24.
We start Section 7.6 by deriving a formula expressing the non-vanishing numbers vol P l σ /2 as a smooth function depending on |Z(s j+1 )| |Z(s j )| and φ σ (s j+1 ) − φ σ (s j ) for any j ∈ Z (see Proposition 4 In our paper fundamental domain is as defined on [45, p. 20], in particular it is a closed subset of H. 5 Which are the hyperbolic and the parabolic elements in SL(2, Z) is recalled in Subsection 7.1.1 6 For a Lebesgue measurable subset X ⊂ S 1 we denote by vol(X) its Lebesgue measure with vol(S 1 ) = 2π. 7.26), which is a straightforward application of the results in Section 7.5. After computing partial derivatives of this function we find that the supremum of vol P l σ /2 as σ varies in Stab ε (T l ) is equal to vol P l σ /2 where σ has s j , s j+1 ∈ σ ss , The precise formula for T l ε is in Proposition 7.27 and it produces (11), (12). In particular it follows that Section 8 contains examples of T with T ε = π(1 − ε) (Proposition 8.5). This section is based on (14) and the observation (Proposition 8.1) that for any exceptional pair (E 1 , E 2 ) in a proper T holds E 1 , E 2 ε ≥ T l ε where l = hom min (E 1 , E 2 ). From the arguments leading to these examples it follows that the condition T ε < π(1 − ε) imposes restrictions on hom min (E i , E j ) in a full exceptional collection (E 0 , . . . , E n ) (see Corollary 8.4).
Section 9 is devoted to the proof (for any N ∈ Z ≥1 and any 0 < ε < 1) of Using the results for the sets P l σ from Sections 7.5, 7.6 we show here that, whenever P l σ is contained in C ∪ −C for an open arc C ⊂ S 1 with length less than π, then for some closed arc p l σ ⊂ C ∩ P l σ the set P l σ \ (p l σ ∪ −p l σ ) is at most countable, and furthermore, provided that the length of C is fixed, we show that when some of the end points of p l σ is very close to some of the end points of C, then p l σ itself has very small length (Corollary 9.3). Due to the fact, proven in Section 5, that for any orthogonal decomposition T = T 1 ⊕ · · · ⊕ T n and any σ ∈ Stab(T ) holds P T σ = n i=1 P T i σ i , where (σ 1 , . . . , σ n ) is the value of the map (2) at σ (see Proposition 5.2 and Corollary 5.5), the proof of (15) reduces to proving that the measure of union of arcs ∪ n i=1 p l i σ ⊂ C of the type explained above, cannot become arbitrary close to the length of C. Having proved this for one arc (in Section 7.6) we perform induction and the tool for the induction step is the already discussed Corollary 9.3.
In Section 10 we discuss the class of discrete derived categories and show that T ε = 0 for any such category. These categories were introduced by Vossieck [59], they were classified in [7] and thoroughly studied in [16], whereas the topology of the stability spaces on them were studied in [17], [52], in particular it was shown that these spaces are all contractible. This class contains the categories {D b (Q) : Q is Dynkin}, and the discrete derived categories not contained in this list are of the form D b (Λ(r, n, m)) for n ≥ r ≥ 1 and m ≥ 0, where Λ(r, n, m) is the path algebra of the quiver with relations shown on [52, Section 4.3, Figure 1]. Actually we show that if T is a category with phase gap, s.t. every heart of a bounded t-structure has finitely many indecomposable objects up to isomorphism (the discrete derived categories have this property) , then T ε = 0.
1.3. Having explained what is the helix {s j } j∈Z in D b (K(l + 1)) ∼ = N P l for l ≥ 1 we can explain what we mean by a σ-semistable non-commutative curve (see Definition 12.13 for precise statement). Let Stab(T ) = ∅ and σ ∈ Stab(T ). Recall that a non-commutative curve of genus l in T is equivalence class of fully faithful exact functors from N P l to T (equivalence is re-parametrization in the domain), we will say that the curve is σ-semistable if for infinitely many j ∈ Z the object F (s j ) ∈ T is σ-semistable object (it does not matter which functor F we take as a representative). We denote the set of σ-semistable non-commutative curves of genus l, satisfying the additional restrictions P and modulo subgroup Γ ⊂ Aut(N P l ), by C Γ l,P,σ (T ). The basic example is C {Id} l,σ (N P l ), l ≥ 1. First note that from Remark 12.6 it follows that C {Id} j (N P j ) = δ i,j for i, j ≥ 0. In Lemmas 7.8, 7.9 is shown that for any σ ∈ Stab(N P l ) we have one of the following possibilities • only two elements in the helix, of the form s j , s j+1 , are semi-stable and φ σ (s j+1 ) > φ σ (s j )+1, in particular C {Id} l,σ (N P l ) = 0 • all elements {s j } j∈Z are semistable and φ σ (s j+1 ) = φ σ (s j ) + 1 for some j ∈ Z, hence C {Id} l,σ (N P l ) = 1 • all elements {s j } j∈Z are semistable and φ σ (s j ) < φ σ (s j+1 ) < φ σ (s j ) + 1 for all j ∈ Z, hence C {Id} l,σ (N P l ) = 1 For details on this example see Proposition 12.14. Another example where we discuss the numbers C In the last section we pose some questions and conjectures about possible interplay between our norm, the notion of holomorphic family of categories, introduced by Kontsevich, and the classical notion of uni-rationality. Acknowledgements

Notations
In this paper the letters T and A denote a triangulated category and an abelian category, respectively, linear over an algebraically closed field k. The shift functor in T is designated by [1].
A triangulated category T is called proper if i∈Z hom i (X, Y ) < +∞ for any two objects X, Y in T . For X, Y ∈ T in a proper T , we denote: We write S ⊂ T for the triangulated subcategory of T generated by S, when S ⊂ Ob(T ). An exceptional object is an object E ∈ T satisfying Hom i (E, E) = 0 for i = 0 and Hom(E, E) = k. We denote by A exc , resp. D b (A) exc , the set of all exceptional objects of A, resp. of D b (A).
An exceptional collection is a sequence E = (E 0 , E 1 , . . . , E n ) ⊂ T exc satisfying hom * (E i , E j ) = 0 for i > j. If in addition we have E = T , then E will be called a full exceptional collection. For a vector p = (p 0 , p 1 , . . . , p n ) ∈ Z n+1 we denote E is also an exceptional collection. The exceptional collections of the form {E[p] : p ∈ Z n+1 } will be said to be shifts of E.
If an exceptional collection E = (E 0 , E 1 , . . . , E n ) ⊂ T exc satisfies hom k (E i , E j ) = 0 for any i, j and for k = 0, then it is said to be strong exceptional collection.
For two exceptional collections E 1 , E 2 of equal length we write An abelian category A is said to be hereditary, if Ext i (X, Y ) = 0 for any X, Y ∈ A and i ≥ 2, it is said to be of finite length, if it is Artinian and Noterian.
By Q we denote an acyclic quiver and by D b (Rep k (Q)), or just D b (Q), -the derived category of the category of representations of Q.
For an integer l ≥ 1 the l-Kronecker quiver (the quiver with two vertices and l parallel arrows) will be denoted by K(l).
The real and the imaginary parts of a complex number z ∈ C will be denoted by ℜ(z) and ℑ(z), respectively, and by H we denote the upper half plane, i. e. H = {z ∈ C : ℑ(z) > 0}. For any complex number z ∈ H we denote by Arg(z) the unique φ ∈ (0, π) satisfying z = |z| exp(iφ).
The letter H will denote the upper half plane with the negative real axis included, i. e. H = {r exp(iπt) : r > 0 and 0 < t ≤ 1}.
For an element α in a group G we denote by α ⊂ G the subgroup α = {α i } i∈Z . If A ⊂ B are subsets in a top. space X, we denote by Bd B (A) the boundary of A w.r. to B, and by L B (A) the set of limit points of A w.r. to B.

On Bridgeland stability conditions
We use freely the axioms and notations on stability conditions introduced by Bridgeland in [12] and some additional notations used in [24,Subsection 3.2]. In particular, the underlying set of the manifold Stab(T ) is the set of locally finite stability conditions on T and for σ = (Z, P) ∈ Stab(T ) we denote by σ ss the set of σ-semistable objects, i. e.
Also for a hearth A of bounded t-structure in T we denote by H A ⊂ Stab(T ) the subset of the stability conditions (Z, P) ∈ Stab(T ) for which P(0, 1] = A (see [22,Definition 2.28]).
Recall that one of Bridgeland's axioms [12] is: for any nonzero X ∈ Ob(T ) there exists a diagram of distinguished triangles called Harder-Narasimhan filtration: , t 1 > t 2 > · · · > t n and A i is non-zero object for any i = 1, . . . , n (the non-vanishing condition makes the factors {A i ∈ P(t i )} n i=1 unique up to isomorphism). Following [12] we denote φ − σ (X) := t n , φ + σ (X) := t 1 , and the phase of a semistable object A ∈ P(t) \ {0} is denoted by φ σ (A) := t. The positive integer: is called the mass of X w.r. to σ([12, p.332]). We will use also the following axioms [12]: Finally we note that: which follows easily from (19)  3.1.1. The universal covering group of GL + (2, R). The universal covering group GL + (2, R) of GL + (2, R) can be constructed as follows (we point the steps without proving them). First step is to show that the following set with the specified bellow operations and metric is a topological group: unit element: Second step is to show that the following is a covering map: The subset U ε = {G ∈ GL + (2, R); sup t∈R {|G(exp(iπt)) − exp(iπt)|} < sin(πε)} 7 is evenly covered by a family of open subsets {(G, ψ); G ∈ U ε sup t∈R |ψ(t) − t − 2k| < ε} indexed by k ∈ Z for small enough ε. In particular one obtains a structure or a Lie Group on GL + (2, R) such that π is a morphism of Lie groups. Finally, one can show that GL + (2, R) is simply connected by recalling that π 1 (GL + (2, R)) ∼ = Z is generated by S 1 = SO(2) ⊂ GL + (2, R) and then by finding the lifts of this path in GL + (2, R).
The right action of GL + (2, R) on Stab(T ) is defined by (recall [12] ): Using the formula (27) determining the topology on GL + (2, R) and the basis of the topology in Stab(T ) explained on [12, p. 335] one can show that the function in (30) is continuous.
We recall also (see [12,Theorem 1.2]) that the projection Stab(T ) proj ✲ Hom(K 0 (T ), C), proj(Z, P) = Z restricts to a local biholomorphism between each connected component of Stab(T ) and a corresponding vector subspace of Hom(K 0 (T ), C) with a well defined linear topology (when rank(K 0 (T )) < +∞ this is the ordinary linear topology). Note also that the results in [12] imply that Stab(T ) is locally path connected (follows from the results in [12, Section 6] and [12, Theorem 7.1]), therefore the components and the path components of Stab(T ) coincide and they are open subsets in Stab(T ). Finally, assume for simplicity that rank(K 0 (T )) < +∞. Due to continuity of (30) it follows that for each connected component Σ of Stab(T ) the action (30) restricts to a continuous action Σ × GL + (2, R) → Σ and it is easy to show that there is a commutative diagram: where V (Σ) ⊂ Hom(K 0 (T ), C) is the corresponding to Σ vector subspace, such that the vertical arrows are local diffeomorphisms (the right arrow is local biholomorphism), and the lower horizontal arrow is an action of the form (A, G) → A • G −1 on V (Σ). Now it follows that the upper horizontal arrow is smooth, and therefore (30) is smooth as well.
3.1.2. The action of C. There is a Lie group homomorphism C → GL + (2, R) given by λ → e −λ , Id R − ℑ(λ)/π . And composing the action (30) with this homomorphism results in the action (32) below. This action is free [49, Definition 2.3, Proposition 4.1]. It is easy to show that for any X ∈ T , σ ∈ Stab(T ), z ∈ C hold the properties in (33), (34) below, and the HN filtrations of X w.r. to σ and to z ⋆ σ are the same: Proposition 3.2. Let K 0 (T ) have finite rank and let Σ ⊂ Stab(T ) be a connected component. Then the action (32) restricted to Σ is proper. In particular, Σ/C with the quotient topology carries a structure of a complex manifold, s. t. the projection pr : Σ → Σ/C is a holomorphic C-principal bundle of complex dimension dim C (Stab(T )) − 1.
Proof. The action is holomorphic (now we get a diagram (31) with C instead of GL + (2, R), C ⋆ instead of GL + (2, R), and now both the vertical arrows are local biholmorphisms, whereas the lower horizontal arrow is holomorphic) and free (see e.g. [49,Proposition 4.1]). If we show that the function is proper, then the proposition follows from [38,Proposition 1.2]. Let K i ⊂ Σ, i = 1, 2 be two compact subsets. Since Stab(T ) × Stab(T ) is locally compact, it is enough to show that γ −1 (K 1 × K 2 ) ⊂ C × Σ is compact.
Proof. Proposition 3.2 imply that pr : Stab(T ) → Stab(T )/C is a C-principal bundle and now we are given that the total space is contractible and the base is a one dimensional connected complex manifold. Now from the long sequence of the homotopy groups associated to a fibration one can deduce that Stab(T )/C is contractible one dimensional complex manifold (see e.g. [58, p. 82] for more details on the arguments). By uniformisation theorem Stab(T )/C is biholomorphic either to C or to H. Now the corollary follows from the theorem that every fiber bundle over a non-compact Riemann surface is trivial, provided the structure group G is connected. ([34], [53]).

The action of Aut(T ).
There is a left action of the group of exact auto-equivalences Aut(T ) on Stab(T ), which commutes with the action (32) [12,Lemma 8.2]. This action is determined as follows: where [Φ] : K 0 (T ) → K 0 (T ) is the induced isomorphism (we will often omit specifying the square brackets) and Φ(P(t)) is the full isomorphism closed subcategory containing Φ(P(t)).
Full stability condition on K3 surface is defined in [12,Definition 4.2]. Analogous definition can be given for any triangulated category T and locally finite stability condition whose central charge factors through a given group homomorphism ch : K 0 (T ) → Z n .
When K 0 (T ) has finite rank, we choose always the trivial homomorphism K 0 (T ) → K 0 (T ). Now the projection Stab(T ) proj ✲ Hom(K 0 (T ), C), proj(Z, P) = Z restricts to a local biholomorphism between each connected component of Stab(T ) and a corresponding vector subspace of Hom(K 0 (T ), C) (see [12,Theorem 1.2]). A stability condition σ ∈ Stab(T ) in this case is a full stability condition, if the vector subspace of Hom(K 0 (T ), C) corresponding to the connected component Σ containing σ is the entire Hom(K 0 (T ), C), which is equivalent to the equality dim C (Σ) = rank(K 0 (T )). As we will see later all stability conditions on K(l) are full, for all l ≥ 1 (see table 10). It is reasonable to hope that, whenever Stab(T ) = ∅, there are always full stability conditions on T and, to the best of our knowledge, there are no counterexamples of this statement so far.
In Definition 4.11 we will use the following subset of the set of stability conditions: Definition 4.3. For any 0 < ε < 1 and any triangulated category T we denote: It is obvious that (recall also (46)): The next simple observation is: Proof. Using (46), (29), and the fact that ψ is diffeomorphism we compute Now the lemma follows from the very Definition 4.3 and the property g −1 ε,ε ′ = g ε ′ ,ε . Corollary 4.5. Let T be any triangulated category. The following are equivalent: (47) and Lemma 4.4.
(b) ⇒ (c). It is obvious from the definitions that for any 0 < ε < 1 and any σ ∈ Stab ε (T ) the set P T σ is not dense in S 1 . (c) ⇒ (a). If P T σ is not dense, then S 1 \ P T σ contains an open arc, but then it contains a closed arc as well and then σ ∈ Stab ε (T ) for some ε ∈ (0, 1). Definition 4.6. A triangulated category T will be called a category with phase gap if P T σ is not dense in S 1 for some full σ ∈ Stab(T ) (by Corollary 4.5 then Stab ε (T ) is not empty for any 0 < ε < 1).
Lemma 4.7. If K 0 (T ) has finite rank, then T has a phase gap iff there exists a bounded t-structure in T whose heart is of finite length and has finitely many simple objects.
Remark 4.8. The elements σ ∈ Stab(T ) for which P(0, 1] is of finite length and with finitely many simple objects are called algebraic stability conditions and have been discussed extensively in [52].  The main definition of this section is: Definition 4.11. Let T be a triangulated category with phase gap. Let 0 < ε < 1. We define: Remark 4.12. For a category T which carries a full stability condition, but has no phase gap (i. e. P T σ is dense in S 1 for all full stability conditions σ) it seems reasonable to define T ε = π(1 − ε), but we will restrict our attention to categories with phase gaps in the rest.
(a) There exist 0 < m < M such that m T ε ≤ T ε ′ ≤ M T ε for any category with a phase gap T . In particular, for any category with a phase gap T we have: Proof. We will use the element g ε,ε ′ = (G, ψ) ∈ GL + (2, R) from Remark 3.1. In particular the 1]. Let us denote the inverse function by κ, then we choose m, M ∈ R as follows: With the help of [56, formula (15) on page 156], we see that for any Lebesgue measurable subset 1] we denote by χ E the function equal to 1 at the points of E and 0 elsewhere): which by (51) implies: Using Remark 4.13, Lemma 4.4, and the second equality in (46) we get: Now (a) follows from (52), (53), (54).
We combine (52) and the latter inequality to deduce the desired (55) [23, Corollary 3.28] (see [22,Corollary 3.25] for general algebraically closed field k) amounts to the following criteria for non-vanishing of T ε Proposition 4.17. Let (E 0 , E 1 , . . . , E n ) be a full exceptional collection in a k-linear proper triangulated category D. If for some i the pair Corollary 4.18. Let ε ∈ (0, 1). Then: (a) If Q is an acyclic quiver, which is neither Dynkin nor affine, then is generated by a strong exceptional collection of three elements Proof. (a) Follows from the previous proposition, [23,Proposition 3.34], and the fact that each exceptional collection in D b (Q) can be extended to a full exceptional collection (see [20]).
In Section 8 we will refine Proposition 4.17, which will help us to prove that D b (coh(X) ε = π(1 − ε) if X is P 1 × P 1 , P n with n ≥ 2 or some of these blown up in finite number of points.
Proof. If Q is affine or Dynkin, then from the first and the second raws of table (7) we see that vol P σ = 0 for any σ ∈ Stab(D b (Q)), therefore D b (Q) ε = 0, and in Corollary 4.18 we showed that D b (Q) ε > 0 for the rest quivers.

Stability conditions on orthogonal decompositions
First we recall the definition of a semi-orthogonal, resp. orthogonal, decomposition of a triangulated category: . . , T n are triangulated subcategories in it satisfying the equalities T = T 1 , T 2 , . . . T n and Hom(T j , T i ) = 0 for j > i, then we say that T = T 1 , T 2 , . . . T n is a semi-orthogonal decomposition. If in addition holds Hom(T i , T j ) = 0 for i < j, then we say that T = T 1 , T 2 , . . . T n is an orthogonal decomposition, in which case we will write sometimes The following map is a bijection: (b) For any (Z, {P(t)} t∈R ) ∈ Stab(T ) and any t ∈ R the subcategory P(t) is non-trivial iff for some j P(t) ∩ T j is non-trivial.
is the value of (56) at σ. Proof. We will give all details for the proof of (a), (b), (c) in the case n = 2. The general case follows easily by induction. (d) follows from the very definition (43) and (a), (b).
It is well known that for each X ∈ T there exists unique up to isomorphism triangle By Hom(T 1 , T 2 ) = 0 it follows that each of these triangles is actually part of a direct product diagram and pr i ( . If we apply these arguments to the last triangle in (18) and using that hom(E n−1 , A n [i]) = 0 for i ≤ 0, we immediately obtain E n−1 , A n ∈ T 1 and then by induction it follows that the entire HN filtration of X lies in T 1 , in particular A i ∈ P(t i ) ∩ T 1 for i = 1, 2, . . . , n, furthermore we have Z 1 ([X]) = Z(pr 1 ([X])) for each X ∈ P(t) ∩ T 1 and now it is obvious that is a stability condition on T 1 . The same arguments apply to the case X ∈ T 2 and show that (Z • pr 2 , {P(t) ∩ T 2 } t∈R ) = (Z 2 , P 2 ) is a stability condition on T 2 . We will show that σ i are locally finite for i = 1, 2. Indeed, since σ is locally finite stability condition on T , then there exists 1 2 > ε > 0 such that P(t − ε, t + ε) is quasi-abelian category of finite length for each t ∈ R. One easily shows that Thus the map is well defined. Since for any interval I ⊂ R the subcategory P(I) is thick (see e.g. [22,Lemma 2.20.]), it follows that P(t) = P 1 (t) ⊕ P 2 (t) for each t ∈ R and hence follows the injectivity of the map. Furthermore, using the terminology of [19, Definition before Proposition 2.2] we see that σ is glued from σ 1 and σ 2 . From the given arguments it follows also that for X ∈ T i the HN filtrations w.r. to σ and w.r. to σ i coincide, in particular: on the other hand any X ∈ T can be represented uniquely (up to isomorphism) as a biproduct , then [19, Proposition 3.5] ensures existence of a locally finite stability condition σ ∈ Stab(T ) glued from σ 1 , σ 2 and using [19, (3) in Proposition 2.2]) one easily shows that our map sends the glued σ to the pair (σ 1 , σ 2 ), hence the surjectivity of the map follows. Now we will show that if rank(K 0 (T i )) < +∞ for i = 1, 2, then the map defined above is biholomorphism. First we show that it is continuous. We denote by d, d 1 , d 2 the generalized metrics on Stab(T ), Stab(T 1 ), Stab(T 2 ) (as defined in (36)). For any σ, σ ′ ∈ Stab(T ) let (σ 1 , σ 2 ) and (σ ′ 1 , σ ′ 2 ) be the pairs assigned via the bijection. To show that the map is homeomorphism we will show that : The first (60) follows easily from (58). The second requires a bit more computations, which we will present partly. Take any X ∈ T and decompose it X ∼ = X 1 ⊕ X 2 , X i ∈ T i , then from (59) we see that where we used, besides the definition of the generalized metrics (36), the following lemma: Lemma 5.3. For any positive real numbers x 1 , x 2 , y 1 , y 2 holds the inequality: Proof. We can assume that x 1 +x 2 y 1 +y 2 ≥ 1 (otherwise take y 1 +y 2 x 1 +x 2 ). Now we consider three cases: If x 1 y 1 ≥ 1 and x 2 y 2 ≥ 1, then the desired inequality becomes log x 1 +x 2 y 1 +y 2 ≤ log x 1 y 1 + log x 2 y 2 which after exponentiating is equivalent to the latter inequality follows from x 1 ≥ y 1 , x 2 ≥ y 2 . If x 1 y 1 ≤ 1 and x 2 y 2 ≥ 1, then the desired inequality becomes log x 1 +x 2 y 1 +y 2 ≤ log y 1 x 1 + log x 2 y 2 which after exponentiating is equivalent to the latter inequality follows from y 1 ≥ x 1 , x 2 ≥ y 2 . If x 1 y 1 ≤ 1 and x 2 y 2 ≤ 1, then the desired inequality becomes log x 1 +x 2 y 1 +y 2 ≤ log y 1 x 1 + log y 2 y 2 which after exponentiating is equivalent to (59) is the same as which in turn follow from the following: Lemma 5.4. For any real numbers x 1 , x 2 , y 1 , y 2 we have: Proof. If max{x 1 , x 2 } = x i and max{y 1 , y 2 } = y i for the same i, then the inequalities follow immediately. So let max{x 1 , x 2 } = x i max{y 1 , y 2 } = y j , i = j, e.g. let i = 1, j = 2. Then x 1 ≥ x 2 , y 1 ≤ y 2 , and the lemma follows from: Thus, we have (60), (61) and they imply that (56) is homeomorphism for n = 2.
Then the following diagram (the first row is the map (56) and the second row is the assignment Z → (Z • pr 1 , Z • pr 2 )) is commutative: . From the Bridgeland's main theorem we know that proj restricts to local biholomorphisms between Σ and an m-dimensional vector subspace V ⊂ Hom(K 0 (T ), C) and proj 1 × proj 2 restricts to local biholomorphisms between Σ 1 × Σ 2 and an m-dimensional vector subspace . It follows (using that ϕ ′ is a linear isomorphism and that each open subset in a vector subset contains a basis of the space) that ϕ ′ (V ) = V 1 × V 2 . Thus, the diagram above restricts to a diagram with vertical arrows which are local biholomorphisms, the bottom arrow is biholomorphism, and the top arrow is a homeomorphism, it follows with standard arguments that the the top arrow must be biholomorphic. It follows that ϕ is biholomorphism and we proved the proposition.
If T has a phase gap and T j ε = 0 for some j, then Theorem 6.1. Let T be proper and let K 0 (T ) has finite rank. Assume 0 < ε < 1. Let T = T 1 , T 2 be a semi-orthogonal decomposition. If T 1 , T 2 are categories with phase gaps, then T is a category with phase gap and for any 0 < ε < 1 holds : By the same arguments as in the last paragraph of the proof of Lemma 4.7 it follows that P σ i (0, 1] are finite length abelian categories, therefore the simple objects in them are a basis of K 0 (T i ) for i = 1, 2, and these abelian categories are the extension closures of their simple objects. In particular the sets of simple objects are finite and it follows that for some j ∈ Z Hom ≤1 (P σ satisfying the following conditions: P σ 2 (0, 1] and P σ 2 (0, 1] are of finite length and with f.m. simples, (71) In the listed properties of σ i ∈ Stab(T i ) with the given semi-orthogonal decomposition T = T 1 , T 2 are contained the conditions of [19,Proposition 3.5 (b)]. This proposition ensures a glued (see [19,Definition ]) locally finite stability condition σ = (Z, P) ∈ Stab(T ). The glued stability condition satisfies the following (we use [19, Proposition 2.2 (3)] and write P i instead of P σ i ) We will show that Indeed, let s 11 , s 12 , . . . , s 1n and s 21 , s 22 , . . . , s 2m be the simple objects of P 1 (0, 1] and P 2 (0, 1], respectively. Then  (20) we deduce that and on the other hand by (73) , and therefore Z(X) ∈ R >0 exp(iπ(ε, 1)) , hence (20) gives φ σ (X) ∈ (ε, 1) and (76) follows. This in turn implies exp(iπ[0, ε]) ∩ P T σ = ∅ and then for obtaining σ ∈ Stab ε (T ) (recall Definition 4.3) it remains to show that σ is a full stability condition. We will prove this by showing that P(0, 1] is a finite length abelian category (then it follows that H P(0,1] ∼ = H n+m and σ is full, since σ ∈ H P(0,1] ). However [19, Proposition 3.5 (a)] claims that if 0 is an isolated point for ℑ (Z i (P i (0, 1])) for i = 1, 2 (which is satisfied due to (71) and (72)), then P(0, 1) is a finite length category, and on the other hand due to (76) holds P(0, 1] = P(0, 1). Therefore indeed P(0, 1] is finite length category and σ is a full stability condition. This ineqaullity holds for any µ > 0 and from the very definition 4.11 we deduce (68).
From the proof of Theorem 6.1 we see that if for some i = 1, 2 there exists a full σ ∈ Stab(T i ) with infinite set of phases P σ i , then there exists a full σ ∈ Stab(T i ) with infinite P σ as well. Corollary 6.3. For any exceptional collection (E 0 , E 1 , . . . , E n ) in a proper triangulated category and for any 0 ≤ i ≤ n we have: . , E n have phase gaps. All the conditions of Theorem 6.1 are satisfied for the semi-orthogonal decomposition . . , E n , hence equality (68) gives rise to (79). Corollary 6.4. Let X be a smooth algebraic variety and let Y be a smooth sub-variety so that Denote by X the smooth algebraic variety obtained by blowing up X along the center Y .
Then D b ( X) has phase gap and . . , k. Now Theorem 6.1 ensures that the inequality holds.

The space of stability conditions and the norms on wild Kronecker quivers
This section is devoted to our main example (namely D b (K(l))), where we compute both: the stability space and · ε . 7.1. Recollection on the action of SL(2, Z) on H. Let GL + (2, R) be the group of 2 × 2 matrices with positive determinant. Recall that (see e.g. [18]) for any matrix γ ∈ GL + (2, R) we have a biholomorphism: and this defines an action of GL + (2, R) on H. Let SL(2, Z) ⊂ GL + (2, R) be the subgroup of matrices with integer coefficients and determinant 1. The action of SL(2, Z) on H defined by the formula (80) is properly discontinuous (see e.g. [45, p.17 and p.20] respectively. When α is parabolic or hyperbolic, then α has no fixed points in H (see [45, p. 7]). Furthermore, if α is elliptic, parabolic, or hyperbolic and α ∈ SL(2, Z), then each non-trivial element in the subgroup α generated by α is elliptic, parabolic, or hyperbolic, respectively.

Fundamental domain of a hyperbolic element. The definition of a fundamental domain in
H of a subgroup Γ ⊂ SL(2, Z) which we adopt here is in [45, p. 20]. We will need to determine fundamental domains of subgroups of the form α for a non-scalar α ∈ SL(2, Z). The following arguments will be useful.
It is clear that any strip F ′ as in Figure (1a), where δ > 0, is a fundamental domain for the subgroup α ′ . Two points in F ′ lie in common orbit iff they are of the form Remark 7.1. For computing β(F ′ ) we will use the feature of β that it maps circles or lines perpendicular to the real axis to circles or lines perpendicular to the real axis (see [45, Lemma 1.1.1] and recall that β is a conformal map, which maps the real axis into itself ).
Any strip F ′ as in Figure (1b), where δ ∈ R, is a fundamental domain for subgroup α ′ . Two points in F ′ lie in a common orbit iff they are of the form 7.2. The Helix in D b (Rep k (K(l))) for l ≥ 2. From now on we assume that l ≥ 2 and denote T l = D b (Rep k (K(l))). On [42, p. 668 down] Macrì constructs a family of exceptional objects {s i } i∈Z and states that: if i < j, then hom k (s i , s j ) = 0 only if k = 0, and hom k (s j , s i ) = 0 only if k = 1, without giving a proof of this statement. In this section we view {s i } i∈Z as a helix, as defined in [11, p. 222] (and introduced earlier in [8], [32]) and using the results about geometric helices in [11, p. 222] we give a simple proof of Lemma 7.5. Lemma 7.4 is the place where the SL(2, Z)-action on H comes into play, it is a simple but important observation for the rest of the paper.
We write dim(X) = (n, m), dim 0 (X) = n, dim 1 (X) = m. for a representation of the form: Recall that Rep k (K(l)) is hereditary category in which for any two X, Y ∈ Rep k (Q) with dimension vectors dim(X) = (n x , m x ), dim(Y ) = (n y , m y ) holds the equality (the Euler Formula): Let s 0 , s 1 ∈ T l be so that s 0 [1], s 1 are the simple objects in Rep k (Q) with dim(s 0 [1]) = (1, 0) and dim(s 1 ) = (0, 1). Using (83) one easily computes hom(s 0 , s 1 ) = l, hom p (s 0 , s 1 ) = 0 for p = 0 and hom * (s 1 , s 0 ) = 0. With the terminology from Section 2 we can say that (s 0 , s 1 ) is a strong exceptional pair, furthermore it is a full exceptional pair in T l = D b (Rep k (K(l))). It follows, that for any exceptional collection E = (E 0 , E 1 , . . . , E n ) in T and for any 0 ≤ i < n the sequences In particular, from the exceptional pair (s 0 , s 1 ) we get objects L s 0 (s 1 ), R s 1 (s 0 ), denoted by s −1 , s 2 , respectively, and each two adjacent elements in the sequence s −1 , s 0 , s 1 , s 2 form a full exceptional pair. Applying iteratively left/right mutations on the left/right standing exceptional pair generates a sequence (infinite in both directions) of exceptional objects {s i } i∈Z . This is the helix induced by the exceptional pair (s 0 , s 1 ), as defined in [11, p. 222]. Any two adjacent elements in this sequence form a full exceptional pair in T . Actually the right mutation generates an action of Z on the set of equivalence classes of exceptional pairs (w.r. ∼), (the inverse is the left mutation). By the transitivity of this action shown in [20], it follows that in T l : complete lists of exceptional pairs and objects (up to shifts) are {(s i , s i+1 )} i∈Z and {s i } i∈Z .
It is shown in [11,Example 2.7] that any strong exceptional pair is geometric ([11, Definition on p. 223]), which applied to our strong exceptional collection (s 0 , s 1 ) gives the following vanishings: We will show in Lemma 7.5 that, when p = 0, the dimensions above do not vanish. When j = i + 1, we use the equalities hom p (L A (B), A) = hom −p (A, B) for any p ∈ Z, and analogously for R B (A), (see again [11,Example 2.7], or [23, p. 157 down] for details) to deduce that hom(s i , s i+1 ) = hom(s 0 , s 1 ) = l for any i. We obtained {s i } i∈Z using the triangles (84), therefore for any i ∈ Z exists a triangle in T : Using these triangles we prove: where α l = l −1 1 0 ∈ SL(2, Z) and α j l : H → H is the corresponding automorphism given in (80).
The statement of the following lemma is a slight modification of [42, Lemma 4.1] and we give a proof here: Lemma 7.5. Assume that l ≥ 2. Then no two elements in {s i } i∈Z are isomorphic and: s ≤0 [1], s ≥1 ∈ Rep k (K(l)). Furthermore, we have: Proof. The matrix α l in (88) has trace tr(α l ) = l ≥ 2 and therefore (see Subsection 7.1.1) α j l is either parabolic or hyperbolic for any j ∈ Z, in particular it has no fixed points in H. If s i ∼ = s i+j for some i ∈ Z, j ∈ N, then since (s i−1 , s i ) and (s i+j−1 , s i+j ) are both full exceptional pairs it follows that s i−1 ∼ = s i+j−1 (note that s i−1 ∼ = s i+j−1 [k] for k = 0 is impossible by the already proved (86)), and hence , which contradicts (88). Therefore no two elements in {s i } i∈Z are isomorphic, and the non-vanishings in (91) follow. Indeed, if hom(s i , s j ) = 0 for some i < j, then (s j , s i ) is a full exceptional pair, which contradicts the already proven s i−1 ∼ = s j . Since Rep k (K(l)) is hereditary category, the non-vanishings in (91) imply that s ≤0 [1], s ≥1 ∈ Rep k (K(l)).
Recall [9] that there exists an exact functor (the Serre functor) F : T l → T l , s. t. hom(A, B) = hom(B, F (A)) for any two objects A, B ∈ T . Furthermore, the formula on [11, p.223 above] (in our case n = 2) says that F (s i ) ∼ = s i−2 [1] for each i ∈ Z , hence for any three integers i, j, p we have hom p (s i , s j ) = hom 1−p (s j , s i−2 ). Now (92) follows from (91).
The following corollary of [44, Theorem 0.1] ensures existence of a functor in Aut(T l ), which plays the role of the functor (·) ⊗ O(1) for any l ∈ Z. (a) ZK(l) is isomorphic to quiver with set of vertices Z and for any (i, j) ∈ Z × Z there are l arrows from i to j iff j = i + 1 and no arrows otherwise; (b) Z × (ZK(l)) is quiver whose connected components are {i} × ZK(l) and each of them is a labeled copy of ZK(l).
(c) The translation functor of T l induces action on Γ irr which by ρ corresponds to an automorphism σ of Z × (ZK(l)) acting by increasing the first component by 1. The Serre functor shifted by [−1] induces an automorphism τ of Z × (ZK(l)), which on vertices maps (i, j) to (i, j − 2).
Taking into account (91), (92) and that the Serre functor maps s i to s i−2 [+1] (see the proof of Lemma 7.5) we see that ρ can be chosen so that ρ(i, j) ∼ = s j [i] for any i, j ∈ Z. Combining [44, Corollary 1.9; Theorem 3.7] and the last paragraph of the proof of [44,Theorem 4.3] we see that Aut(T l ) ∼ = Z × (Z ⋉ PGL n (k)) and a generator of the factor Z in Z ⋉ PGL n (k) is the desired A l ∈ Aut(T l ).

7.3.
The principal C-bundle Stab(T l ) → X l . We will denote by X l the set of orbits of the Caction on Stab(T l ), i.e. X l = Stab(T l )/C. In this section we show that X l is a complex manifold biholomorphic to either C or H (Corollary 7.11) and that the action of A l on Stab(T l ), where A l ∈ Aut(T l ) is the functor from Corollary 7.6, descends to an action by biholomorphism on X l (Corollary 7.13).
In [42,Section 3.3] are constructed stability conditions generated by a full Ext-exceptional collection. The set of stability conditions generated by a full Ext-exceptional pair (A, B) will be denoted by Θ ′ (A,B) and for any full exceptional pair 9 (A, B) the notation Θ (A,B) will denote the union of the sets Θ ′ Proposition 7.7. For i ∈ Z the subset Θ (s i ,s i+1 ) ⊂ Stab(T l ) has the following description: In particular, the set Θ (s i ,s i+1 ) is biholomorphic to the contractible set S = {(z 1 , z 2 ) ∈ C 2 ; ℑ(z 1 ) < Proof. We use [25, formulas (12), (17), (18), and Lemma 2.4] and deduce that σ is commutative. Since the horizontal arrows are local biholomorphisms and we already showed that ϕ i is homeomorphism, it follows that ϕ i is biholomorphic.
Proof. We show first the inclusion ⊃. So let (s i , s i+1 ) ⊂ σ ss and φ σ (s i ) < φ σ (s i+1 ) < φ σ (s i ) + 1. We will show below that the given inequalities imply: Then by induction we obtain the inclusion ⊂. We use [25, Proposition 2.2] first to show that s i−1 and s i+2 are semi-stable. More precisely, the given inequality is the same as The given φ σ (s i ) < φ σ (s i+1 ) < φ σ (s i ) + 1 amounts to Z(s i+1 ) Z(s i ) ∈ H and then Lemma 7.4 ensures that Thus we derived (97) and the inclusion ⊃ follows.
Assume now that t = φ(s i+1 ) = φ(s i ) + 1. Recall that s 0 [1] and s 1 are the simple ojects in Rep k (K(l)) (see after (83)). It follows that for each j we s j [k] is in the extension closure of s 0 [1] and s 1 for some k. Due to Corollary 7.6 it follows that for each j we s j [k] is in the extension closure of s i [1] and s i+1 for some k, and since s i [1], s i+1 ∈ P(t) it follows that s j [k] ∈ P(t), therefore s j ∈ σ ss . Definition 7.10. We will denote the quotient Stab(T l )/C by X l , the corresponding projection by Stab(T l ) pr ✲ X l . The intersection p∈Z Θ (sp,s p+1 ) will be denoted by Z. Due to Lemma 7.8 we have From (95) we get a disjoint union Stab(T l ) = Z ∐ ∐ i∈Z Θ (s i ,s i+1 ) \ Z .

Corollary 3.3 and Lemma 7.8 imply:
Corollary 7.11. X l is biholomorphic either to C or to H and Stab(T l ) pr ✲ X l is trivial C-principal bundle.
The action (40) descents to an action by biholomorphisms on X l . To show this and some basic properties of this action we note first: Lemma 7.12. For any i, j ∈ Z, any λ ∈ C, and A l ∈ Aut(T l ) from Corollary 7.6 hold the equalities: Proof. From (33) we see that the conditions s i , s i+1 ∈ σ ss , φ σ (s i ) < φ σ (s i+1 ) are equivalent to the conditions: s i , s i+1 ∈ (z ⋆ σ) ss , φ z⋆σ (s i ) < φ z⋆σ (s i+1 ), hence by (93) we obtain the first equality. From Lemma 7.6 by induction we get A j l (s i ) ∼ = s i+j , A j l (s i+1 ) ∼ = s i+j+1 . Now with the help of (41) and (93) we establish the second equality by a sequence of equivalences: The third and the fourth equalities in (100) follow from the already proven first and second.
Corollary 7.13. The action (40) descents to an action by biholomorphisms on X l by the formula Φ · pr(σ) = pr(Φ · σ). Denote Θ i = pr(Θ (s i ,s i+1 ) ) for i ∈ Z and Z = pr(Z). Then for any i, j ∈ Z, Proof. Since the actions (32), (40) commute and pr from Corollary 7.11 has holomorphic sections, it follows the first sentence. The rest of the corollary follows from the definition of quotient topology on X l and from Lemmas 7.12, 7.8.

7.4.
The action A l on X l is free and properly discontinuous for l ≥ 2. In this Section we complete the proof of Theorem 1.5. Corollary 7.13 gives a decomposition of X l into A l -invariant subsets Z and X l \Z and furthermore it gives a decomposition X l \ Z = ∐ i∈Z (Θ i \ Z) into subsets which A l permutes, more precisely: Now we construct biholomorphisms between H, Θ j and Z and describe the action of A l on Z.
This function, restricted to the strip {z ∈ C : 0 < ℑ(z) < π}, is a biholomorphism between this strip and Z = pr(Z). In particular, we obtain a biholomorphism: We claim that for any p ∈ Z and any z ∈ H we have: where α l is the matrix in Lemma 7.4 and α p l (z) is defined in (80). Proof. Since all composing maps in (103) are holomorphic, the composite function is also holomorphic.
The matrix α l = l −1 1 0 has tr(α l ) = l. Since l ≥ 2, then α l is either parabolic or hyperbolic (see Section 7.1.1). As a corollary following from (102), Remark 7.2 (a) and (105) we get: Corollary 7.15. The action of A l on X l is free and its restriction to Z is properly discontinuous.
The next step is to find a fundamental domain of the action of A l on Z ∼ = H. We choose j ∈ Z and use the biholomorphism ψ : H → Z (104) (depending on j) constructed in Lemma 7.14. By (105) we see that the fundamental domain we want is of the form ψ(F l ), where F l ⊂ H is a fundamental domain of the action of α l on H discussed in Section 7.1. We use now Sections 7.1.2 and 7.1.3 to find such a F l . for l ≥ 2, a 2 = 1. A fundamental domain of α l is: For l ≥ 3 the fundamental domain is shown in Figure (2a) and for l = 2 in Figure (2b). Two points in F l lie in a common orbit iff they satisfy z ± ∈ Bd H (F l ), Arg(z + ) = Arg(z − ). Proof. We need to consider two cases l ≥ 3 and l = 2, because this determines the type of α l according to the classification recalled in Subsection 7.1.1.
Assume first that l ≥ 3. The feature of a l we need, which we have for all l ≥ 3, is: a l > 1, that's why we will omit the subscript l and write just a, remembering that a > 1. Now α l is a hyperbolic element in SL(2, Z) (see Subsection 7.1.1) and we can use the method described in Subsection 7.1.2.
(f) For each q ∈ Bd Θ j (F l ) there exists an open subset U ⊂ X l , s. t. q ∈ U and {i ∈ Z : A i (U ) ∩ F l = ∅} is finite (in fact contains only two elements).

Proof. (a): It is clear that Bd
is the same set by Corollary 7.17 (a). (b): Since Θ j is open subset in X l , the interiors of F l w.r. to Θ j and w.r. to F l coincide. Hence (b) is due to the fact that (103) is biholomorphism.
(c) and (d): From Lemma 7.14 and the definition of F ′ l in Corollary 7.17 we see that Therefore by (102) and since F ′ l is fundamental domain of the action of A l on Z (as defined on [45, p. 20]) we get: (118) and (102) we see that if two points in F l lie in a common orbit, then they lie in F ′ l , and then we use the already proven (a) here and Corollary 7.17 (b) to obtain (e). (f): For q ∈ Bd Θ j (F l ) the same neighborhood U ∋ q ensured by Corollary 7.17 (c) satisfies the required property in (f) due to (102), (118), and the already proven (a) here.
Corollary 7.19. The action of A l on X l is free and properly discontinuous.
Proof. In Corollary 7.15 we already showed that the action is free. Take any two q 1 , q 2 ∈ X l . We need to find open subsets U i ∋ q i in X l , i = 1, 2, such that the set {i ∈ Z : A i l (U 1 ) ∩ U 2 = ∅} is finite (see e.g. [45, p. 17]). From (c) in Corollary 7.18 it follows that it is enough to consider the case q 1 , q 2 ∈ F l . 10 If q 1 , q 2 ∈ Bd Θ j (F l ), then q 1 , q 2 ∈ Z (see Corollary 7.18 (a)) and the neighborhoods U i ∋ q i we need exist since Z is an A l -invariant open subset and by Corollary 7.15. If q 1 , q 2 ∈ F o l , then U 1 = U 2 = F o l satisfy the condition we need by Corollary 7.18 (d). Since F l = F o l ∪ Bd Θ j (F l ), it remains to consider the case q 1 ∈ Bd Θ j (F l ), q 2 ∈ F o l . In this case we take U ∋ q 1 as in Corollary 7.18 (f) and put U 1 = U , U 2 = F o l , and the corollary is proved.
Рис. 3. (u + iv)(F l ) = F l Corollary 7.20. The orbit-space X l / A l with the quotient topology carries a structure of a one dimensional complex manifold, s. t. the projection pr : X l → X l / A l is a holomorphic covering map: a universal cover of X l / A l . In particular π 1 (X l / A l ) = Z.
Proof. In the previous corollary we showed that the action is free and proper. From these properties [38,Proposition 1.2] ensures that X l / A l has the structure of a one dimensional complex manifold, s. t. pr : X l → X l / A l is a holomorphic principal Z-bundle and recalling that X l is contractible (see Corollary 7.11) the corollary follows from the long exact sequence for the homotopy groups associated to pr.
From now on we fix j ∈ Z and let F l ⊂ Θ j be the closed in Θ j subset obtained in Corollary 7.18. By (c) in Corollary 7.18 it follows that pr(F l ) = pr(Θ j ) = X l / A l , i. e. we have the surjectivity of the restriction of the map pr from Corollary 7.20: Lemma 7.21. The restriction pr |F l : F l → X l / A l is a proper surjective map.
Proof. It remains to show the properness. Recall that Θ j is an open subset in X l , and F l is a closed subset in Θ j . Thus, the restriction Θ j pr |Θ j ✲ X l / A l is a local biholomorphism between locally compact spaces. Let K ⊂ X l / A l be compact.
For any q ∈ K we fix q ′ ∈ F l , s. t. pr(q ′ ) = q. Now we will choose an open U q ∋ q ′ subset of Θ j with certain properties for each q ∈ K. If q ′ ∈ F o l we choose U q ∋ q ′ whose closure w. r. to Θ j is compact and contained in F o l (in particular A j l (U q ) ∩ F l = ∅ for j = 0). If q ′ ∈ Bd Θ j (F l ) we choose U q ∋ q ′ with compact closure w. r. to Θ j , which is contained in Z and such that {i ∈ Z : A i l (U q ) ∩ F l = ∅} is finite (we can do this by Corollary 7.18 (f) and since By our choice of U q the union on the right hand side is finite and each element A m l (U q i ) in it is contained in a compact subset of Θ j (recall also (102)). Since F l is closed in Θ j , we deduce that ( pr | ) −1 (K) is contained in a compact subset of F l , and therefore, being closed subset of compact, ( pr | ) −1 (K) is compact. Now we can prove: Proposition 7.22. If l ≥ 3, then X l is biholomorphic to H, and Corollary 7.11 implies Theorem 1.5.
Proof. Suppose that X l is not biholomorphic to H. We will obtain a contradiction. By Corollary 7.11 we see that we have a biholomorphism X l ∼ = C, and we showed in Corollary 7.20 that pr : X l → X l / A l is a universal covering of X l / A l and π 1 (X l / A l ) = Z. [29,Prop. 27.12] ensures that X l / A l is biholomorphic to one of these: C, C ⋆ , or a torus. However π 1 (X l / A l ) = Z and we deduce that θ : X l / A l ∼ = C ⋆ for some biholomorphism θ. Let us denote by f the following composition: As in Corollary 7.18 we denote by Θ j u+iv ✲ H the inverse of (103). Let us denote F l = (u + iv)(F l ) (see Figure (3a)). From Corollary 7.18 (or Corollary 7.17) (a) we see that: where δ(t) = (∆ l − log(cos(t))).
Then from Lemma 7.14, Corollary 7.18, and Lemma 7.21 follow (a), (b), (c) below: (a) f : H → C ⋆ is a local biholomorphism; For any t ∈ (0, π/2) let us denote the segment γ t in F l defined by γ t (s) = s+it for s ∈ [−δ(t), δ(t)]. Then (see Figure (4 From (c) above it follows that f (F t+ l ), f (im(γ t )), f (F t− l ) are pairwise disjoint and we can write : Due to (c) above f • γ t is a closed Jordan curve in C ⋆ . By Jordan curve theorem (see e.g. [41,Theorem 4.13 on p. 70 1 ]) we have C\f (im(γ t )) = U t ∐V t , where U t , V t are the connected components of the complement of im(f • γ t ) in C. In particular, since f (F t± l ) are connected, by (122) we get Due to the first equality in (122) we can write: and we obtain: Furthermore, by Jordan theorem we can assume that is compact, which in turn by (122) and (b) above leads to either F t+ l ∪im(γ t ) or F t− l ∪im(γ t ) is compact, which is a contradiction (see (121)). So we see that 0 and f (F t+ l ) = V t for any t ∈ (0, π/2). We will show that we can ensure: Indeed, if for some t ∈ (0, π/2) this holds, then f for any other t ′ ∈ (0, π/2). 11 On the other hand by composing θ in (119) with 1 z we can ensure that the equalities in (124) hold for some t and then they will hold for any t ∈ (0, π/2). Next step is to show: We can assume that {z i } i∈N ⊂ F t− l for some t ∈ (0, π/2) and therefore by (124) does not contain any point with zero imaginary part, so we get a contradiction to (b) and proved (125).
). Therefore we can choose 0 < ǫ < min{∆ l , π/4} and denote (see figure (4)): Bd C (f (Q)) cannot be (image of) a closed Jordan curve, otherwise a one-to-one extension of f |Q to a homeomorphism between Cl C (Q) and Cl C (f (Q)) must exist. We will prove the theorem by deriving the contrary: Bd C (f (Q)) is an image of a closed Jordan curve. Since f |F l is proper, it follows that it is a closed map (see e.g. [29, p. 35]), therefore f (Q∐Bd H (Q)) is closed in C * , and it follows that On the other hand (125) shows that {0} ∈ Cl C (f (Q)) and f (Bd H (Q)) ⊂ Cl C (f (Q)) by continuity, hence is open in C, we obtain: Take any homeomorphism (0, 2π) ) is a homeomorphism and by (125) we have lim t→0 f (κ(t)) = lim t→2π f (κ(t)) = 0, then it is clear that the following function (128) is continuous and bijective, hence by compactness of S 1 it is homeomorphism and the theorem is proved.
The set of phases. Let l ≥ 2 be an integer. Recall that we denote In this section we write for short P l σ instead of P T l σ , and will determine P l σ (Proposition 7.24). We start by some comments on the root system of K(l).
It is well known that the real roots ∆ re l+ are exactly the dimension vectors of the exceptional representations in Rep k (K(l)) and for the imaginary roots ∆ im l+ we have formula (74) in [23]. From (85) and Lemma 7.5 we have the complete list {s ≤0 [1], s ≥1 } of exceptional representations in Rep k (K(l)). Let us denote the corresponding dimension vectors as follows: Therefore we can write: We will need later the following facts for the real roots {(n i , m i ) : i ∈ Z}: Proof. The equality for 0 ≤ i ≤ 1 follows from (a). We make the induction assumption that for some p ≥ 1 the equality holds for any 0 ≤ i ≤ p, we will make the induction step, namely that the equality for i = p + 1 follows from this induction assumption. Indeed, for i ≥ 1 from (87) we obtain the following short exact sequences in Rep k (K(l)): ✲ s i+2 therefore for i ≥ 1 we obtain: having these recursive formulas one easily caries out the inductive step.

Putting the equality from (b) in the last inequality we get
and then from (c) we deduce (d).
Now we have the necessary notations to describe P l σ : Proposition 7.24. Let σ ∈ Stab(D b (K(l))).
(a) If σ ∈ Z (see also Definition 7.10), then the set of phases P l σ is finite (has up to 4 elements). (b) If σ ∈ Z, then for any j ∈ Z we have the following formulas: where x, y, and the function (strictly increasing smooth) f : [0, ∞) → [π(φ σ (s j+1 ) − φ σ (s j )), π) are: (c) For σ ∈ Z holds the equality {± exp(iπφ σ (s i ))} i∈Z = P l σ \ L(P l σ ) (recall that by L(P l σ ) we denote the set of limit points in the circle of P l σ ).
In this section we dervie a formula, which will help us to compute other norms. To that end it is useful to extend the definition of K ε (l) in (164) by postulating K ε (0) = K ε (1) = 0. Recall that the notation hom min (E 1 , E 2 ) is explained in (16).
Proposition 8.1. Let T be a proper category, and let (E 1 , E 2 ) be any exceptional pair in it. Then Proof. We can assume that hom ≤0 (E 1 , E 2 ) = 0 and l = hom 1 (E 1 , E 2 ) = 0, and under these assumption we have to show that Let D be the triangulated subcategory E 1 , E 2 . The assumptions on (E 1 , E 2 ) are the same as in the definition of an l-Kronecker pair, [22,Definition 3.20], and we can apply [22, Lemma 3.19, Corollary 3.21] to it. In particular the extension closure A of (E 1 , E 2 ) is a heart of a bounded t-structure in D with simple objects E 1 , E 2 , and any stability condition σ = (Z, ). The arguments in the beginning of the proof of Lemma 4.7 show that for each v ∈ H 2 there exists unique σ = (Z, P) ∈ H A with v = (Z(E 1 ), Z(E 2 )) and that σ is full. For any 0 < µ such that µ + ε < 1 choose the vector (−1, exp(iπ(ε + µ))) = v µ and denote by σ µ the stability condition σ µ = (P µ , Z µ ) ∈ H A with (Z µ (E 1 ), Z µ (E 2 )) = v µ . The given arguments ensure that σ µ is full and P D σµ = R vµ,∆ l+ . Using the formula for R vµ,∆ l+ in Lemma 7.25 for the given v µ one derives: where K ε+µ (l) is in (164). Note that the arc exp (iπ[µ/2, ε + µ/2]) is in the complement of P D σµ and therefore σ µ ∈ Stab ε (D). Now from the very Definition 4.11 we see that D ε ≥ K ε+µ (l) for any small enough positive µ, letting µ → 0 we derive the desired D ε ≥ K ε (l).
. . , E n ) be an exceptional collection in a proper triangulated category T . Then for any 0 ≤ i < j ≤ n we have E ε ≥ K ε hom min (E i , E j ) .
Proof. Take 0 ≤ i < j ≤ n. By mutating the sequence E (see Remark 7.3) one can get a sequence E ′ of the form E ′ = (E i , E j , C 2 , . . . , C n ) such that E = E ′ . Corollary 6.3 implies E ε = E ′ ε ≥ E i , E j ε , and due to Proposition 8.1 we get E i , E j ε ≥ K ε (hom min (E i , E j )). Corollary 8.3. Let T be a proper triangulated category such that for each l ∈ N there exists a full exceptional collection (E 0 , E 1 , . . . , E n ) and integers 0 ≤ i < j ≤ n for which hom min (E i , E j ) ≥ l. Then T ε = π(1 − ε) for any ε ∈ (0, 1).
(a) If T ε = 0, then for any full exceptional collection E = (E 0 , E 1 , . . . , E n ) and for any 0 ≤ i < j ≤ n we have hom min (E i , E j ) ≤ 2.
(b) If T ε ≤ K ε (l), l ≥ 2, then for any full exceptional collection E = (E 0 , E 1 , . . . , E n ) and for any 0 ≤ i < j ≤ n we have hom min (E i , E j ) ≤ l.
(c) If T ε < π(1 − ε), then there exists l ∈ N such that for any full exceptional collection E = (E 0 , E 1 , . . . , E n ) and for any 0 ≤ i < j ≤ n we have hom min (E i , E j ) ≤ l.
We will aply Corollary 8. (a) D b (Q), where Q is an acyclic quiver, s.t. there exists a subset A ⊂ V (Q) such that the quiver Q A is affine and there exists a vertex v ∈ V (Q) such that v is a source or a sink in Q A∪{v} (see Definition 8.6 for the terminology) , where X is a smooth algebraic variety obtained from P n , n ≥ 2, or from P 1 × P 1 , or from F m ), m ≥ 0 by a sequence of blow ups at finite finite number of points; (f )D b (S), where S is any smooth complete rational surface 14 Definition 8.6. For any quiver Q and any subset A ⊂ V (Q) we denote by Q A the quiver whose vertices are A and whose arrows are those arrows of Q whose initial and final vertex is in  [20] is shown that (E 0 , E 1 ) can be extended to a full exceptional collection. Therefore we can apply Corollary 8.3 to D b (Q). Now we present one method (Lemma 8.8) to obtain l-Kronecker pairs with arbitrary big l as part of full exceptional collections, i.e. method to obtain the conditions of Corollary 8.3. This method relies on full exceptional collections in which a triple remains strong after certain mutations (see (c) in the statement of Lemma 8.8). In [11] a strong exceptional collection E which remains strong under all mutations is called non-degenerate. Furthermore in [11] are defined so called geometric exceptional collections and [11,Corollary 2.4] says that geometricity implies non-degeneracy. Furthermore, [11,Proposition 3.3] claims that a full exceptional collection of length m of coherent sheaves on a smooth projective variety X of dimension n is geometric if and only if m = n + 1. In particular it follows: That's why the method of Lemma 8.8 is readily applied to D b (P n ), whereas applying it to D b (P 1 × P 1 ) requires some additional arguments to ensure (c) in Lemma 8.8. Lemma 8.8. Let T be a proper triangulated category and ε ∈ (0, 1). Let E = (F 0 , F 1 , F 2 , E 3 , . . . , E n ) be a full exceptional collection with n ≥ 3. Let {F i } i∈N be a sequence starting with F 0 , F 1 , F 2 and If the following three properties hold: is strong for all i ≥ 1, then T satisfies the condition of Corollary 8.3 and T ε = π(1 − ε).

Proof. Now (84) becomes
Since the property of being full is preserved under mutations, it follows that (F 0 , F i−1 , F i , E 3 . . . , E n ) is full for each i ≥ 2. We will show that (178) holds, and then our T satisfies the conditions of 13 where Fm is the m-th Hirzebruch surface 14 in particular any somoth projective surface.
To show (178) we first note that due to (c) we have hom k (F i−1 , F i ) = 0 for each k = 0 and each i ≥ 2 and it follows that (see e.g. [11,Example 2.7] for each i ≥ 2 and then (177) has the form: In (a) we are given hom(F 0 , F i−1 ) < hom(F 0 , F i ) for i = 2 and we will show (178) by induction. Indeed, since (F 0 , F i−1 , F i ) is a strong exceptional collection for each i ≥ 2, applying Hom(F 0 , _) to (179) yields short exact sequences between finite dimensional vector spaces: The obtained exact sequences and l ≥ 2 imply: The lemma is proved. . Therefore we can apply Lemma 8.8 and the corollary follows.
Proof of Proposition 8.5 (c). Let us denote here T = D b (P 1 × P 1 ). Exceptional collections in T have been studied in [54] and [31]. In particular the full strong exceptional collection  [50, p. 3] or [8,Example 6.5]). After one mutation we get a full exceptional collection To apply Lemma 8.8 and deduce that T ε = π(1 − ε) we need to show that From [31, Proposition 5.3.1, Theorem 3.3.1.] it follows that: From the way we defined {F i } i∈N it follows (see e.g. [11,Example 2.7 is strong for all i ≥ 1 suffices to show that hom(F 0 , F i ) = 0 for each i ≥ 1. Now (84) becomes distinguished triangle We have 0 < hom(F 0 , F 1 ) < hom(F 0 , F 2 ). Assume that for some i ≥ 2 holds we will show that this implies hom(F 0 , F i ) < hom(F 0 , F i+1 ) and by induction the corollary follows. Applying Hom(F 0 , _) to (183) and since hom k (F 0 , F i−1 ) = hom k (F 0 , F i ) = 0 for k = 0 one easily deduces that hom k (F 0 , F i+1 ) = 0 for k ∈ {−1, 0}. If hom −1 (F 0 , F i+1 ) = 0, then by (182) it follows that hom(F 0 , F i+1 ) = 0 and applying Hom(F 0 , _) to (183) yields an exact sequence of vector spaces: which contradicts (184). Therefore hom −1 (F 0 , F i+1 ) = 0 and hom k (F 0 , F i+1 ) = 0 for k = 0. Now we apply Hom(F 0 , _) to (183) again and get a short exact sequence as in (179) which by the same computation as in (181) implies hom(F 0 , F i+1 ) > hom(F 0 , F i ), thus we proved the corollary. Proof. [10,Theorem 4.2] ensures that there is a semi-orthogonal decomposition . . , k, which implies that T i is generated by an exceptional object for each i. Now it is clear that the full exceptional sequences of D b (X) ensuring the conditions of Corollary 8.3 extend to full exceptional collections on D b ( X), so these conditions are satisfied in D b ( X) as well.
Proof of Proposition 8.5 (d), (e), and (f ). Since F 0 = P 1 × P 1 and F 1 is P 2 blown up at a point, then the cases a = 0, 1 are contained in Proposition 8.5 (a), (b), and Lemma 8.9. In [36] they construct families of full exceptional collections of invertible sheaves on D b (F a ) for any a. To show that their exceptional collections furnish the conditions of Corollary 8.3 we just need to combine some results in [36]. First adopt here some notations and terminology from [36]: P, Q denotes basis of P ic(F a ) (see [36,Section 4,p.1224]) s.t.
Hille and Perling study certain sequences of Cartier divisors on a rational surfaces X which they call toric systems. Furthermore a toric system A 1 , A 2 , . . . , A n is called strongly exceptional (see [36,Definition 3.6]), if they generate a sequence of invertible sheaves , which is strong exceptional. For such a toric system each divisor A j is numerically left orthogonal [36, Definition 3.1 (a)], which means that χ(−A j ) = 0. Indeed, we have for each k ∈ Z, since the sequence is exceptional, and on the other hand (see e.g. [36, the beginning of Section 3]), and hence χ(−A j ) = 0. Note also that since the sequence is strong it follows that On the other hand, having that A j is numerically left orthogonal and using [36, Lemma 3.3 (i)] we derive: [36,Proposition 5.2] proves that P, sP + Q, P, −(a + s)P + Q is a strongly exceptional toric system on F a when s ≥ −1. If we denote by E s 1 , E s 2 , E s 3 , E s 4 the corresponding strong exceptional collection, then using the formula (187) and the property of toric system, that n i=1 A i = −K X (see [36, p. 1233 down]), and also the equalities (186) we compute:  [36,Theorem 5.8.], also [36, the beginning of the proof of Theorem 8.6.] or [37, Proposition 2.1] ). Part (d) is proved. Part (e) follows by recursively applying Lemma 8.9 and the already proven cases. Part (f) reduces to part (e), since any smooth complete rational surface S can be constructed after applying a finite sequence of blow ups starting with P 2 of F a , a ≥ 0 (see e.g. [36, the beginning of Section 4, p. 1243]). 9. The inequality T l 1 ⊕ · · · ⊕ T ln ε < π(1 − ε) The goal of this seciton is to prove the following: Proposition 9.1. Let n ≥ 1, let l i ≥ 1, i = 1, 2, . . . , n be a sequence of integers, and let 0 < ε < 1. Then for any orthogonal decomposition of the form T = T l 1 ⊕ T l 2 ⊕ · · · ⊕ T ln , where T l i ∼ = D b (K(l i )), holds T ε < π(1 − ε). Furthermore T ε > 0 iff l i ≥ 3 for some 1 ≤ i ≤ n.
Proof of Proposition 9.1 From Remark 4.15 and Subsection 7.6 we see that: hence the proposition follows for n = 1.
Assume that we have already proved the proposition for 1 ≤ n ≤ N . And assume that T = T l 1 ⊕T l 2 ⊕· · · ⊕T l N ⊕T l N+1 , where T l i ∼ = D b (K(l i )) and denote by L the set L = {l 1 , l 2 , . . . , l N , l N +1 }.
If 1 ≤ l j ≤ 2 for some j, then T l j ε = D b (K(l j )) ε = 0, and the statement follows from the induction assumption, Corollary 5.6, and T l j ε = 0. Therefore we can assume that all integers in L are at least 3. From the induction assumption there exists δ > 0 such that: Note that due to Remark 4.2, Proposition 5.2 (d), and Corollary 5.5 for any sequence x 1 , x 2 , . . . , x j in L holds: Assume now that σ i ∈ Stab(D b (K(l i ))) for i = 1, . . . , N + 1 and that there exists a closed ε-arc γ satisfying ∅ = P l i σ i ∩ γ = ∅ for i = 1, . . . , N + 1. In particular we can represent the circle S 1 : where α ∈ R and β = α + π(1 − ε). If for some k the corresponding σ k ∈ Z l k ⊂ Stab(D b (K(l k ))), then by Lemma 9.2 P l k σ k is finite and taking into account (203), (204) we derive: otherwise for all i we have σ i ∈ Z l i , and then by Corollary 9.

Discrete derived categories and their norms
There are categories, in which every heart of a bounded t-structure has finitely many indecomposable objects up to isomorphism. Due to the following lemma the norm of these categories vanishes: Lemma 10.1. For any triangulated category T and any a ∈ R we have: (209) P T σ = {± exp(iπφ σ (I)) : I ∈ σ ss ∩ P(a, a + 1] and I is P(a, a + 1]-indecomposable}.
Proof. First recall that for each σ = (Z, P) ∈ Stab(T ) and for any a ∈ R the subcategory P(a, a+ 1] is a heart of a bounded t-structure. From the previous lemma P T σ is finite for each σ ∈ Stab(T ). Therefore from Definition 50 it follows that T ε = 0.
In representation theory was introduced a class of triangulated categories with a particularly discrete structure, called Discrete derived categories (Vossieck [59]), they were classified in [7] and thoroughly studied in [16], whereas the topology of the stability spaces on them were studied in [17], [52], in particular it was shown that these spaces are all contractible. This class contains the categories {D b (Q) : Q is Dynkin}, and the discrete derived categories not contained in this list are of the form D b (Λ(r, n, m)) for n ≥ r ≥ 1 and m ≥ 0, where Λ(r, n, m) is the path algebra of the quiver with relations shown on [52, Section 4.3, Figure 1]. Proposition 10.3. For any discrete derived category T (in the sense of [59], [7]) and any ε ∈ (0, 1) holds T ε = 0.
Proof. [16, Proposition 7.1] says that each heart of a bounded t-structure in T has finitely many indecomposable objects and is of finite length. In articular (see Lemma 4.7) T has a phase gap and it satisfies the conditions of Lemma 10.2, therefore T ε = 0.

Topology on the class of triangulated categories with a phase gap
In this section we denote by T ′ the set of all small triangulated categories within a certain universe (a universe which contains the derived categories or representations of algebras) and by PG ′ ⊂ T ′ we denote the subset of proper categories with finite rank Grothendieck group and with a phase gap. From Proposition 10.3 (see also its proof) it follows that each discrete category is in PG ′ . Furthermore, from [16,Proposition 7.6] it follows that each discrete derived category has a full Exceptional collection. Thus if we denote by DDK ′ the subset in T ′ of discrete derived categories, and by E ′ the subset of proper categories with a full exceptional collection, then we have the inclusions: DDK ′ ⊂ E ′ ⊂ PG ′ ⊂ T ′ . When we write A ∼ = B for A, B ∈ T ′ , we mean an equivalence between triangulated categories, and by T = T ′ / ∼ = , PG = PG ′ / ∼ =, DDK = DDK ′ / ∼ = we denote the corresponding sets of equivalence classes and then we have inclusions: We give first an example of a topology on the largest class T and give evidence that this topology is too coarse.
Proposition 11.7. (a) The function below is upper semi-continuous: For any x > 0 the subset PG ≥x = {y ∈ PG : y ε ≥ x} is a discrete subset of PG w. r. to the topology from Lemma 11.6.
Proof. (a) follows from the following application of (216): for any δ > 0, T ∈ PG ′ holds (b) follows from the same formula. Indeed, from this formula one checks that for any [T ] ∈ PG ≥x and any 0 < δ < Q is affine} has T ε = π(1 − ε), hence (since the function (215) takes values in (0, π(1 − ε))) we obtain: On the other hand from Proposition 11.7 (b) we know that PG ≥π(1−ε) is a discrete subset and the corollary follows.
Examples of non-closed points are contained in Proposition 8.5. More precisely: Proposition 11.9. The element [D b (point)] ∈ PG is in the closure of [T ] ∈ PG for any T ∈ PG ′ which satisfies the conditions of Corollary 8.3 and such that rank(K 0 (T )) ≥ 3.
Proof. We will show that [T ] ∈ B δ (D b (point))/ ∼ = for any δ > 0. Indeed, take any δ > 0. From (168) it follows that there exists N s.t. π(1 − ε) − K ε (l) < δ for l ≥ N . Since T satisfies the conditions of Corollary 8.3 and rank(K 0 (T )) ≥ 3, therefere there is a full exceptional collection E 0 , E 1 , . . . , E n−1 , E n with n ≥ 2, s. t. hom min (E i , E j ) > N for some i < j. Since we can apply mutations, we can assume that hom min (E n−1 , E n ) > N . Now let us denote A = E 0 , B = E 1 , . . . , E n . Then we have a SOD T = A, B with A ∼ = D b (point), and B ε ≤ E n−1 , E n ε ≤ π(1 − ε) − K ε (hom min (E n−1 , E n )) < δ, where in the latter chain of inequalities we used (216), Proposition 8.1. Recalling the definition of B δ (D b (point)) ( Definition 11.5) we conclude that T ∈ B δ (D b (point)) and the proposition follows.  so that all natural numbers are values. Following Kontsevich-Rosenberg [40] we denote D b (K(l + 1)) by N P l (non-commutative projective space) for l ≥ 0. Note that we include the case l = 0, and N P 0 is a non-trivial cateogry. Then if we define a function: , using (170), Proposition 7.27, and table (7) we see that Thus the invariant dim nc takes all natural numbers as values and due to Theorem 6.1 and Remark 6.2 we have whenever A, B have phase gap and A, B is a semi-orthogonal decomposition. (222) ensures that whenever T has a finite dim nc (T ) < +∞ and A ⊂ T is a good enough embedded subcategry, then A has also finite dim nc (A) ≤ dim nc (T ) < +∞. Note that due to Remark 12.1. Proposition (4.19) and table (7) imply that for an acyclic quiver Q we have dim nc D b (Q) = 0 iff Q is Dynkin and dim nc D b (Q) = 1 iff Q is affine. 12.2. The general definition and the question which motivates it. Due to (222) we see that whenever we have T ∈ B δ (N P l ) for some real δ > 0 (recall that by definition 11.5 this means that there is a SOD of the form T = N P l , A where A has a phase gap ) and some integer l ≥ 0, then dim nc (T ) ≥ l. In particular if T ∈ B δ l (N P l ) for arbitrary big l, then dim nc (T ) = +∞, and this idea was used in Section 8. Now we come to the main question of this section: Question 12.3. Let l ≥ 0. Incidences of the form T ∈ B(N P l ) 15 , are a common phenomenon for this paper. Recalling that Gromow-Witten invariants count pseudo-holomorphic curves, we view such embeddings of N P l into T as analogous to a "pseudo-holomorphic curve" in the category T , and we ask a question: can we count such entities in a given T , how many are they ?
We answer positively this question here, and give examples. We plan to develop this idea in future works (starting with [26]). Remark 12.4. Recall that the homological dimension of N P l , l ≥ 0 is one. Also due to table (10) we have Stab(N P l ) ∼ = C × C for l = 0, 1 and Stab(N P l ) ∼ = C × H for l ≥ 2. Note also that, whereas the spirals in N P 0 are periodic (up to shifts there are only three exceptional objects), for l ≥ 1 the spirals in N P l consist of pairwise non-isomorphic objects.
In view of these notes, we find it convenient to view N P l as a non-commutative curve of genus l.
Let us also note: Remark 12.5. Let l ≥ 0 and T be any triangulated cateogry linear over k, let N P l F ✲ T be any fully faithful exact functor 16 , and denote by A the isomorphism closure of the image of F in T . Then A is a triangulated subcategory of T generated by two exceptional objects, hence due to [8, Theorem 3.2] the functor F has left and right adjoints and there are SOD T = A, A ⊥ , T = ⊥ A, A .
Remark 12.6. From the previous remark it follows that, if for some integer j ≥ 0 there exists a fully faithful functor N P j F ✲ N P l , then l = j and F is equivalence (from [21] we know that any exceptional pair in N P l is full). Next we fix an equivalence relation in C ′ A,P (T ), and we will be interested in the set of equivalence classes, in particular the size of this set.
where F • α ∼ = β • F ′ means equivalence of exact functors between triangulated categories (this is so called graded equivalence).

First non-trivial examples with
A = N P l , l ≥ 0 and tree diferent targets: two quivers, and D b (P 2 ). These examples we can prove now. We postpone writing down the details of the proof of Proposition 12.9 and 12.8 for a future work [26], where we plan to study also other examples. For the statements till the end of this subsection we fix the additional properties P from (223) as follows, note that this additional restriction implies that T has a phase gap (Proposition 6.1 and Remark 12.5): Property P : the left or the right orthogonal to the image of F in T has a phase gap.
Unless otherwise specified in this subsection we take either a trivial subgroup Γ = {Id}, or the subgroup Γ = S generated by the Serre functor S of T (such exists in the discussed examples), or Γ = Aut(T ).
Remembering that in this subsection P is like above and A = N P l for some l ≥ 0 we will write C ′ l (T ), C Γ l (T ) instead of C ′ N P l ,P (T ), C Γ N P l ,P (T ). We refer to C Γ l (T ) as the set of non-commutative curves of genus l in T modulo the subgroup Γ.
This paper contains many examples of T and l ≥ 0, s. t. C Γ l (T ) = ∅, on the other hand from (222) one sees immediately: dim nc (T ) ≤ n ⇒ C Γ l (T ) = ∅ for l > n.
Furthermore, in all the examples of categories T with dim nc (T ) = ∞ given in Section 8 one has C Γ l (T ) = ∅ for infinitely many l. Using results of [24] and [25] we can completely determine the invariants C {Id} l (D b (Q)), C S l (D b (Q)) for two affine quivers: 16 recall that an exact functor is actually a a pair of a functor F ′ and a natural isomorphism between the functors F ′ • T1 and T2 • F2, where T1, T2 are the translation functors of the source and the target categories, respectively Proposition 12.8. Let T i = D b (Q i ), i = 1, 2, where: Then #(C Proof. The vanishings follow from (see Remark 12.1) dim nc (T 1 ) = dim nc (T 2 ) = 1 and (226). The rest of the proof will be written in [26].
where c 1 (E), r(E) are the first Chern class (which we consider as an integer) and the rank of E.
Proof. Coming in [26]. Proof. Coming in [26]. 17 Recall that a Markov number x is a number x ∈ N ≥1 such that there exist integers y, z with x 2 + y 2 + z 2 = 3xyz. Remark 12.11. When T = D b (P 2 ) or T = D b (Q) for an acyclic quiver Q, then for any fully faithful exact functor F : N P l → T the right and the left orthogonal to the image of F is generated by an exceptional collection, therefore any such functor automatically satisfies the additional property P fixed in the beginning of this subsection. This follows from the fact that every exceptional pair in T can be extended to a full exceptional collection in T (this is proved in [21] and [32]).
Conjecture 12.12. Let T = D b (S i ), for i = 1 or i = 2 or i = 3, where S 1 , S 2 , S 3 are the quivers from Conjecture 1.7. Remark 12.11 holds for T and we take Definition 12.7 with no additional restrictions P . We conjecture that dim nc (T ) < ∞ and C {Id} l (T ) is fnite for all l ≥ 1.
Definition 12.13. Let l ∈ Z ≥1 and let T be a triangulated category linear over k and s.t. Stab(T ) = ∅, let Γ and P be as in Definition 12.7. One approach to define, semi-stable w.r. to a stability condition non-commutative curves is as follows. Choose σ ∈ Stab(T ). Now we apply the same Definition 12.7 with A = N P l and a modified additional properties, namely they are P and one additional restriction depending on σ. More precisely, let {s j } j∈Z be a Helix in N P l (see Section 7.2), then let us denote: C ′ l,P,σ (T ) = {F ∈ C ′ N P l ,P (T ) : #{j : F (s j ) ∈ σ ss } = ∞ }, where C ′ N P l ,P , is the set (223). The equivalence relation in C ′ l,P,σ (T ) is the same as in (224). For completeness we write it again, to define σ-semistable non-commutative curves of genus l in T , satisfying properties P and modulo Γ: C Γ l,P,σ (T ) = C ′ l,P,σ (T )/∼ F ∼ F ′ ⇐⇒ F • α ∼ = β • F ′ for some α ∈ Aut(N P l ), β ∈ Γ (239) where F • α ∼ = β • F ′ means equivalence of exact functors between triangulated categories.
We will give two examples. In both of them the additional restrictions P are empty (for both of them holds Remark 12.11) and Γ = {Id}, and we will write just C Proof. The equality C {Id} k (N P l ) = δ k,l follows from Remark 12.6. For the proof of the rest we note fist that for any j the subset in Stab(N P l ) where s j is semi-stable is closed subset. From Lemma 7.8 we know that for σ ∈ Z all alements in {s j } j∈Z are semi-stable, therefore this holds also for σ ∈ Cl(Z). Thus we see that σ ∈ Cl(Z) ⇒ C l,σ (NP l ) = 1. Recalling that Stab(N P l ) = Z ∐ ∐ i∈Z Θ (s i ,s i+1 ) \ Z we see that if σ ∈ Cl(Z), then σ ∈ Θ (s i ,s i+1 ) \Z for some i ∈ Z. From Lemma 7.8 and the description of Θ (s i ,s i+1 ) in Proposition 7.7 we see that s i , s i+1 ∈ σ ss and φ(s i+1 ) > φ(s i ) + 1 and then Lemma 7.9 ensures that only s i , s i+1 are semi-stable, therefore σ ∈ Cl(Z) ⇒ C l,σ (NP l ) = 0.
Using the description of Stab(D b (Q 1 )) in [25] we have proved: Proof. Coming in a future work. 12.5. Non-commutative Calabi-Yau curve-counting. Now we give an example of a finite C Γ A,P (T ) (defined in Definition 12.7) coming from categories appearing naturally on the A-side. We first introduce a CY version of the domain category A. We pass from D b (K(l)) to the new domain by changing exceptional objects ←→ spherical objects and this amounts to considering A = CY (l), instead of A = D b (K(l)), where CY (l) is defined in [48]. The definition is based on the quiver: In the next example we take the entire Γ = Aut(T ), more precisely:

A-side interpretation and holomorphic sheaves of categories
In this section we give a different point of view on the category of representations of the Kronecker quiver and introduce the notion of holomorphic families of Kronecker quivers.
We suggest a framework in which sequences of holomorphic families of categories are viewed as sequences of extensions of non-commutative manifolds by relating our norm to the notion of holomorphic family of categories introduces by Kontsevich. Several questions and conjectures are posed.
First we sketch how to interpret D b (K(n)) as a perverse sheaf of categories. Recall that LG model of P 2 is C * 2 , w = x + y + 1 xy -see [1].

D
The category D b (K(3)) can be obtained by taking the part of the Landau Ginzburg model over a disc D which contains 2 singular fibers.
As a result we get D b (K(4)). By similar surgeries we can get all quivers from K(0) = A 1 + A 1 to K(n). To interpret D b (K(n)) as a perverse sheaf of categories one considers a locally constant sheaf of categories over a graph Γ shown on the picture below, the picture encodes also the data about the sheaf, in particular p 1 , p 2 denote spherical functors (see [27], [1]): Γ

Fuk(E)
A 3 ⊗Fuk(E) The category of global sections H 0 (Γ, F) of the sheaf F is the same as D b (K(n)). The surgeries are recorded by the changes of the spherical functors p 1 , p 2 . The category D b (K(4)) can be interpreted also as part of the LG model of P 3 , C * 3 , w = x + y + z + 1 xyz : We make a surgery on the fiber -a K3 surface:

K3 S 1
This surgery amounts to change from to K(4) K(5) The Landau Ginzburg models with K3 surfaces in the fibers can be interpreted as perverse sheafs of categories, encoded in the following picture -see [27]: (K(n)) .
Remark 13.1. The property of having a phase gap, which we require in this paper to define the norm, can also be interpret as existence of a CY form with certain properties. Namely let Y be a LG model, Ω is a CY form on Y . Let L be a Lagrangian s.t. θ 1 ≤ argΩ| L ≤ θ 2 . Assume that there exits a form β on Y s.t.
Then there are no stable lagrangians L with θ 1 ≤ argΩ| L ≤ θ 2 . In other words existence of such forms Ω and α lead to gaps in phases.
One more direction for future research is holomorphic families of categories, in particular holomorphic families of Kronecker quivers.
Holomorphic families of categories over X with fiber K(n) should be defined by homomorphisms ϕ i : O(U i ) → HH 0 (D b (K(n))) in the zero-th Hochschild cohomology of D b (K(n)) where {U i } is a covering of X by open sets. We use the following picture for such a holomorphic family of categories:

X K(n)
The holomorphic sheaves of categories are enhanced by perverse sheaves of stability conditions -see [28] for defining morphisms and the gluing between the categories on intersecting opens that defines the sheaf.
The case of holomorphic family of K(2) is the classical case of conic bundles: The global sections H 0 (X, F) are D b (Z). Similarly H 0 (X, F) with K(n) for n ≥ 3 produces a new non-commutative variety.
Iterating the procedure described above results in a family of categories over a family of categories. Some questions addressing relations between the norms of the fibers and of the gobal sections follow: Under what condition ||C|| ε ||H 0 (X, F)|| ε ? (here C is the category in the fiber) X F C Question 13.3. Let us consider a tower of families of categories and each of the fiber categories C i has non maximal (Recall the relation of · ε and · ε in Definition 11.5) · ε . Is it true that if the category in the combined fiber has a Rouquier dimension [2] equal to one then the norm of this category is non-maximal · ε ?
We summarise the proposed analogy in the table bellow.

Galois theory Norms
The sequence of finite coverings is finite Rouquier dim (C) = 1 ⇓ ||C|| ǫ < max Question 13.4. Do we have a similar theory as the classical theory of conic bundles for sheaves of categories with fibers categories of representations of Kronecker quivers or any other quiver category with a Rouquier dimension [2] equal to one?
In a certain way our norm can be seen as analogue of height function defined in [4]. We expect that some higher analogues of this norm for higher Rouquier dimensions can be defined. In fact in this paper we only scratch the surface proposing a possible approach to "noncommutative Galois theory" -representing "noncommutative manifolds" (categories) as a sequence of perverse sheaves of categories and holomorphic families of categories.
It will be interesting to study categories which can be represented as a tower of holomorphic families of categories with nonmaximal norms. One example of such category is D b (P 1 × ... × P 1 ).
Question 13.5. Characterise projective varieties X whose derived categories D b (X) can be represented as tower of holomorphic families of categories with non-maximal norms.
1) Under what conditions are these projective varieties X rational? (It is rather clear that a nontrivial condition is needed since every hyperelliptic curve can be seen as such a tower. ) 2) Can the existence of tower of holomorphic families of categories be represented in terms of modular forms?
3) We conjecture that a unirational variety U can be represented as a tower where D b (F i ) ε < π(1 − ε) (see [3]), and Z is rational. And another question is, does there exists a cohomology theory, which determines, if U is rational.
In the end we put a question coming from the interplay between towers of sheaves of categories and stability conditions. Let K(N 1 , N 2 ) be a category obtained from a family where the base is a category of representations of the Kronecker quiver K(N 1 ) and the fibers are the category of representations of the Kronecker quiver K(N 2 ). Similarly we have K (N 1 , N 2 , ..., N l ) denoting extensions of extensions.
Question 13.6. Is the moduli space of stability condition of K(N 1 , N 2 , .., N l ) a bundle over Hilbert modular variety?
It would be interesting to investigate the connection of the geometry of these Hilbert modular surfaces with the norms we have defined as well as new modular identities coming from wall-crossing formulas.