The Space of Stability Conditions on the Projective Plane

The space of Bridgeland stability conditions on the bounded derived category of coherent sheaves on P2 has a principle connected component Stab^\dag(P2). We show that Stab^\dag(P2) is the union of geometric and algebraic stability conditions. As a consequence, we give a cell decomposition for Stab^\dag (P2) and show that Stab^\dag(P2) is contractible.


Introduction
Motivated by the concept of Π-stability condition on string theory by Douglas, the notion of a stability condition, σ = (P, Z), on a C-linear triangulated category T was first introduced by Bridgeland in [Br07]. In the notion, the central charge Z is a group homomorphism from the numerical Grothendieck group K 0 (T ) to C. Bridgeland proves that the space of stability conditions inherits a natural complex manifold structure via local charts of central charges in Hom Z (K 0 (T ), C). In particular, when K 0 (T ) has finite rank, the space of stability condition (satisfying support condition), Stab(T ), has complex dimension rank(K 0 (T )).
As mentioned in [Br09], Stab(T ) is expected to be related to the study of string theory and mirror symmetry. The main interesting example is to understand the space of stability conditions on a compact Calabi-Yau threefold X such as a quintic in P 4 . Yet this problem is still wildly open mainly due to some technical difficulties. Although the compact Calabi-Yau threefold case is still difficult to study, Stab(T ) of various analog categories has been very well understood, see [BSW15,BQS14,DK16,Ik14,Qi15]. While most of these examples are build from quivers or locally derived category of sub-varieties, few cases of Stab(X) for smooth compact varieties X are known. Such Stab(X) is 'well-understood' only when X is P 1 ( [Ok06]), a curve ( [Br07]), a K3 surface ( [Br08,BB13]), an abelian surface or threefold ( [BMS]). In this paper, based on some important technical results from [Ma04] and [Ma07], we make an attempt to analyze the space Stab(P 2 ).
Here Stab Geo (X) denotes the space of geometric stability conditions (Definition 1.7), at where the skyscraper sheaves are stable with the same phase. Stab Alg (P 2 ) denotes the space of algebraic stability conditions (Definition 2.3), which can be constructed from exceptional collections.
Rough description for Stab † (P 2 ): We first describe the geometric part Stab Geo (X). When X is a smooth surface, by the philosophy of [Br08] and [BB13], Stab Geo (X) can be determined once people know the Chern characters of Gieseker-stable sheaves. TheGL + (2, R)-action (see Lemma 8.2 in [Br07]) acts freely on the part of Stab Geo (X). Any point in Stab Geo (X)/GL + (2, R) is uniquely determined by the kernel of its central charge, which is a linear subspace in K R (X) of real codimension two. From the inverse side, such a linear subspace can be realized as the kernel of a central charge if and only if one can construct a quadratic form Q on K R (X) satisfying the support condition (see the definition above Definition 1.7) for this subspace.
In the case that X is of Picard number one, one may take the projectivization of K R (X), Ker Z is a point on P(K R (X)). A point on P K R (P 2 ) can be the kernel of a central charge if and only if it has an open neighborhood which is not 'below' any Gieseker stable character. Now we focus on the case that T is D b (P 2 ). The (projective) Gieseker stable characters have been completely determined in [DP85] by Drezet and Le Potier. On P K R (P 2 ) , the characters form a dense set below the Le Potier curve (see Definition 1.4) together with some isolated points of exceptional characters.
For the algebraic part Stab Alg (P 2 ), it goes back to the work [Be83] that D b (P 2 ) can be generated by an exceptional collection {O, O(1), O(2)}. One can do mutations between the exceptional objects to get other exceptional triples, such as {O(1), T P 2 , O(2)}, {O(−4), O(−3), O(−2)}, which also generates the category. For each exceptional triple E = E 1 , E 2 , E 3 , one may assign numbers z j = m j exp(iπφ j ), φ j as the central charges and phases of E j . Due to the result in [Ma07], when m j ∈ R >0 , φ 1 < φ 2 < φ 3 , and φ 1 + 1 < φ 3 , there is a unique stability condition with the given central charge and E i ∈ P(φ i ). Denote all such stability conditions by Θ E with parameters m j and φ j . The space of algebraic stability conditions Stab Alg (P 2 ) is the union of all Θ E . Note that theGL + (2, R)-action does not act freely on Stab Alg (P 2 ). Each Θ E can be divided into three parts: the head Θ Geo E ; the legs Θ + E,E 1 , Θ − E,E 3 ; and the tail Θ Pure E (see Definition 2.3). The head part is the overlap part with the geometric stability conditions, this is the only part of Θ E that 'glues' on the Stab Geo (X). The leg part overlaps with other algebraic stability conditions, we will show that any two legs of Θ E and Θ E ′ are either the same, or separated from each other (see Proposition 3.4). Each tail part Θ Pure E is a private area for Θ E , which is separated from any other Θ E ′ (see Lemma 2.4). We will show that one may contract the whole space of Stab Alg (P 2 ) by first contracting all the tails simultaneously to their boundaries with legs, and then contracting all the legs to their boundaries with heads. The union of all heads Θ Geo E is aGL + (2, R)-bundle over an open subset of Stab Geo (P 2 ), which is contractible.
Related works: Many important technical results on Stab Alg (P 2 ) have been set up in [Ma04] and [Ma07], and our result is a natural continuation of the previous work. The space Stab † (P 2 ) can be compared with some previous geometric examples such as Stab † (K3) and Stab † (local P 2 ). As described in the previous section, their geometric parts Stab Geo (X) are quite similar. In addition, each exceptional/spherical object provides two boundary sets of Stab Geo (X). But the remaining parts are very different, for a K3 surface or local P 2 , the remaining parts can be viewed as copies of the geometric part. While for Stab † (P 2 ), the remaining parts are similar to the space of stability conditions of quivers representations, see the works of [BSW15,BQS14,DK16,Ik14,Qi15,QW14]. In most of the previous quiver representation examples, the stability conditions are all of the algebraic type. Yet the quiver representation for D b (P 2 ) has a complicated relation, this leads the fact that some of the geometric stability conditions on P 2 are not of the algebraic type. In addition, it seems to the author that the contractibility of the algebraic parts Stab Alg (P 2 ) is not implied by the results in any of the previous papers. In particular, the paper [QW14], at where the authors prove the contractibility for many interesting examples, does not apply to the case Stab Alg (P 2 ), since the heart O[2], O(1)[1], O(2) is not locally finite and has infinitely many algebraic tilts, which are crucial assumptions on the t-structure in [QW14].
Open questions: It is reasonable for us to believe that Stab † (P 2 ) actually contains all the stability conditions that satisfy the support condition.
In addition, as the case of P 1 , we wish to understand the global complex structure of Stab(P 2 ). We expect that there is a period map as that of the CY quiver cases, [BQS14,Ik14], so that we may have differential forms on Stab † (P 2 ) and the central charge is neatly computed as integrations. But this seems difficult to realize because there is some 'pure geometric' part on Stab † (P 2 ). For the algebraic part Stab Alg (P 2 ), we also expect that there is a fundamental domain R on (H) 3 ≃ Θ E independent of the triples E such that all the R E 's form a disjoint cover of Stab Alg (P 2 ).
Acknowledgments. The author is grateful to Arend Bayer, Zheng Hua, Yu Qiu and Xiaolei Zhao for helpful conversations. The author is supported by ERC starting grant no. 337039 "WallXBirGeom".

Notations
The Picard group of P 2 is of rank one with generator H = [O(1)], and we will, by abuse of notation, identify the i-th Chern character ch i with its degree H 2−i ch i . The slope µ of a non-torsion sheaf E on P 2 is defined as ch 1 ch 0 . We denote K(P 2 ) ⊗ R by K R (P 2 ). Consider the real projective space P K R (P 2 ) with homogeneous coordinate [ch 0 , ch 1 , ch 2 ], we view the locus ch 0 = 0 as the line at infinity. The complement forms an affine real plane, which is referred to as the {1, ch 1 ch 0 , ch 2 ch 0 }−plane. We call P K R (P 2 ) the projective {1, ch 1 ch 0 , ch 2 ch 0 }−plane. For any object F in D b (P 2 ), we writẽ v(F) := ch 0 (F), ch 1 (F), ch 2 (F) as the numerical character of F, and v(F) the projection ofṽ(F) on the {1, ch 1 ch 0 , ch 2 ch 0 }−plane with locus (1, s, q). Let E, F be two objects in D b (P 2 ) with characters on the {1, ch 1 ch 0 , ch 2 ch 0 }−plane and P be a point on the projective {1, ch 1 ch 0 , ch 2 ch 0 }−plane. For the convenience of the reader, we make the list of notations and symbols that are commonly used in this article. Most of them are explicitly defined at other places of the article.
H P the right half plane with ch 1 ch 0 > s, or ch 1 ch 0 = s and ch 2 is not at infinity L EF the line on P K R (P 2 ) across v(E) and v(F) L EP the line on P K R (P 2 ) across v(E) and P l EF (l EP ) the line segment v(E)v(F) (v(E)P)on the {1, ch 1 ch 0 , ch 2 ch 0 }−plane l r EF the ray along L EF from v(F) to infinity and does not contain v(E) l + EF the ray along L EF from v(E) on the H E part l E+ the ray segment on L E(0,0,1) on the H E part l E− the ray segment on L E(0,0,−1) outside the H v(E) part E a triple of ordered exceptional objects {E 1 , E 2 , E 3 } TR E the inner points in the triangle bounded by l E i E j , for 1 ≤ i < j ≤ 3. e * i v * (E i ) as defined in section 1, * can be +, l, r or blank MZ E the inner points of region bounded by l e 1 e + 1 , l e + 1 e 2 ,l e 2 e + 3 , l e + 3 e 3 and l e 3 e 1 Let T be a C-linear triangulated category of finite type. For convenience, one may always assume that T is D b (P 2 ): the bounded derived category of coherent sheaves on the projective plane over C. The following definitions follow from [AKO06, GR87,Or92].

An ordered collection of exceptional objects
Definition 1.2. Let E = {E 0 , . . . , E n } be an exceptional collection. We call this collection E strong, if for all i, j and q 0. This collection E is called full, if E generates T under homological shifts, cones and direct summands.
We summarize some of the classification results of the exceptional bundles on P 2 and make some notations, see [DP85,GR87,LeP97]. There is a one-to-one correspondence between the dyadic integers p 2 m and exceptional bundles E ( p 2 m ) . Let the Chern character of the exceptional bundle corresponding to p , the characters are inductively given by the formulas: •ṽ(E (n) ) = 1, n, n 2 2 , for n ∈ Z.

Review: Exceptional objects, triples, and Le Potier curve
• When q > 0 and p ≡ 1(mod 4), the character is given bỹ Here are some first observations from the definition.
We use Chern characters [ch 0 , ch 1 , ch 2 ] for the coordinate of K R (P 2 ). Consider the real projective space P K R (P 2 ) with homogeneous coordinate [ch 0 , ch 1 , ch 2 ]. We view the locus ch 0 = 0 as the line at infinity, and call P K R (P 2 ) the projective {1, ch 1 ch 0 , ch 2 ch 0 }−plane. The complement of the line at infinity forms an affine real plane, which is referred to as the {1, ch 1 ch 0 , ch 2 ch 0 }−plane. We will define the Le Potier curve on this {1, ch 1 ch 0 , ch 2 ch 0 }−plane. Let e p 2 m be the point on the {1, ch 1 ch 0 , ch 2 ch 0 }−plane with coordinate v E ( p 2 m ) . We associate three more points e + p 2 m , e l p 2 m and e r p 2 m to E ( p 2 m ) on the {1, ch 1 ch 0 , ch 2 ch 0 }−plane. The coordinate of e + p 2 m is given as: For any real number a, let ∆ a be the parabola: . We call this curve the Le Potier curve on the {1, ch 1 ch 0 , ch 2 ch 0 }−plane, and denote it by C LP . We call the cone in K R (P 2 ) spanned by the origin and C LP as the Le Potier cone. We also make a notation for the following open region above C LP . 1, a, b) is above C LP and not on any segment l ee + }.

Review:Geometric stability conditions
Theorem 1.5 (Drezet, Le Potier). There exists a Gieseker semistable coherent sheaf with character (ch 0 (> 0), ch 1 , ch 2 ) ∈ K(P 2 ) if and only if either: 1. it is proportional to an exceptional character e p 2 m ; Remark 1.6. In this article, when we talk about the {1, ch 1 ch 0 , ch 2 ch 0 }−plane, we always assume the ch 1 ch 0 -axis is horizontal and the ch 2 ch 0 -axis is vertical. The phrase 'above' is translated as ' ch 2 ch 0 coordinates is greater than'. Other words such as: below, right, left can be translated in a similar way.
The full strong exceptional collections on D b (P 2 ) have been classified by Gorodentsev and Rudakov [GR87]. In particular, up to a cohomological shift, the collection consists of exceptional bundles on P 2 . In terms of dyadic numbers, their labels are of three cases: (♣)

Review:Geometric stability conditions
We briefly recall the definition of stability condition on a triangulated category from [Br07]. Let T be the bounded derived category of coherent sheaves on a smooth variety. A pre-stability condition σ = (P, Z) on T consists of a central charge Z : K 0 (T ) → C, which is an R-linear homomorphism, and a slicing P : R → (full additive subcategories of T ), satisfying the following axioms: 3. when φ 1 > φ 2 and A i ∈ obj(P(φ i )), we have Hom T (A 1 , A 2 ) = 0; 4. (Harder-Narasimhan filtration) For any object E in T , there is a sequence of real numbers φ 1 > · · · > φ n and a collection of vanishing triangles A pre-stability condition is called a stability condition if it satisfies the support condition: there exists a quadratic form Q on the vector space K R (T ) such that For the rest part of this section, we will follow the line of [Br08] and [BM11] and conclude that the space of geometric stability condition on P 2 is aGL + (2, R) fiber space over Geo LP .
Definition 1.7. A stability condition σ on D b (P 2 ) is called geometric if all skyscraper sheaves k(x) are σ-stable with the same phase. We denote the subset of all geometric stability condition by Stab Geo (P 2 ).
Let s be a real number, a torsion pair of coherent sheaves on P 2 is given by: Coh ≤s : the subcategory of Coh(P 2 ) generated by slope semistable sheaves of slope ≤ s by extension.
Coh >s : the subcategory of Coh(P 2 ) generated by slope semistable sheaves of slope > s and torsion sheaves.
Remark 1.9. This definition of the central charge Z s,q is slightly different from the usual case as that in the [ABCH13]. The imaginary parts are defined in the same way, but the real part is different from the usual case by a scalar times the imaginary part. We would like to use the version here because its kernel is clear.
In addition, if we write P for the point (1, s, q), then the phase (times π) of an object E in Coh #s is the angle spanned by the rays l + PE and l P− (for definition, see Table 1) at P on the H P half plane. Proposition 1.10. For any (s, q) ∈ Geo LP , σ s,q = (Z s,q , P s,q ) is a geometric stability condition.
For the proof, readers are referred to the arguments in [Br08] and [BM11] Corollary 4.6, which also work well in the P 2 case. Up to theGL + (2, R)-action, geometric stability conditions can only be of the form given in Proposition 1.10.
Notation 1.11. Given a point P = (1, s, q) in Geo LP , we will also write σ P , φ P , Coh P (P 2 ) and Z P for the stability condition σ s,q , the phase function φ s,q , the tilt heart Coh #s (P 2 ) and the central charge Z s,q respectively. The complex numbers a and b satisfies the following conditions: • ℑa > 0, ℑb ℑa = s; Knowing the classification result of stable characters, Theorem 1.5, the property is proved in the same way as that in the local P 2 and K3 surfaces case.

Destabilizing walls
We collect some small but useful lemmas in this section. Definition 1.13. We call a stability condition non-degenerate if the image of its central charge is not contained in a real line. We write Stab nd (P 2 ) for all the non-degenerate stability conditions.
Note that by Proposition 1.12, Stab Geo (P 2 ) ⊂ Stab nd (P 2 ). In this Picard rank 1 case, the kernel map on the central charge is well-defined on Stab nd (P 2 ).
if and only if the ray l + PE is above l + PF . Proof. By the definition of l + PE , l P− , Z P and Remark 1.9, the angle spanned by the rays l + PE and l P− at point P on the {1, ch 1 ch 0 , ch 2 ch 0 }−plane is πφ P (E). The statement is clear.
Proposition 1.18. Let E be an σ P -stable object, then one of the following cases will hold:

1.ṽ(E) is not in the open cone spanned by
Geo LP and the origin. The equality only holds when D and G are both zero, but this is not possible as else v(E) = v(F) and is inside Geo LP by Theorem 1.5. Therefore, v(F) is to the left of v(E) on the {1, ch 1 ch 0 , ch 2 ch 0 }−plane. Let P = (1, s, q), as F ∈ Coh >s , P is to the left of v(F). In addition, as φ P (E) < φ P (F), by Lemma 1.17, P is below the line L EF . Therefore, P is in the region bounded by l r EF and l F− . When ch 0 (E) < 0, let F = H −1 (E) max be the subsheaf of H −1 (E) with maximum Mumford slope. By the same argument, v(F) is to the right of v(E). As F ∈ Coh ≤s , P is to the right of v(F) or on the line L F (0,0,1) . In addition, as φ P (F[1]) < φ P (E), by Lemma 1.17, P is below L EF . As l F− does not intersect Geo LP , P is not on L F(0,0,1) . Therefore, P is in the region bounded by l r EF and l F− . For the last statement, the region ∆ <0 is bounded by a parabola and is convex. For any v(E) and P that are both in the region, l EP is also in the region which is contained in Geo LP .

Corollary 1.19. Let E be an exceptional bundle, and P = (1, s, q) be a point in Geo LP , then E is σ P -stable if s < µ(E) and l EP is contained in Geo LP . On the shifted side, E[1] is σ P -stable, if µ(E) ≤ s and l EP is contained in Geo LP .
Proof. Assume s < µ(E) and E is not σ P -stable, then there is a σ P -stable object F destabilizing E. By the exact sequence: between L P(0,0,1) and L E(0,0,1) . As φ P (F) ≥ φ P (E) by assumption, by Lemma 1.17, v(F) is in the region bounded by l P+ , l PE , and l E+ . As l EP is in Geo LP , the whole open region bounded by these three segments is also in Geo LP . The whole line segment l FP is contained in Geo LP (unless v(F) = v(E), which implies E = F). By Proposition 1.18, F is not σ P -stable, which is a contradiction. The s ≥ µ(E) case is proved in a similar way.
Remark 1.20. The condition 'l EP is contained in Geo LP ' is also a necessary condition. Any ray from v(E) only intersects the Le Potier curve once, and only intersects finitely many ee + segments. Assume we are in the s < µ(E) case and l EP intersects some ee + segments, we may choose the one (denoted by F) with minimum ch 1 ch 0 coordinate. The segment l FP is contained in Geo LP , and the φ s,q (F) > φ s,q (E). By [GR87], Hom(F, E) 0 when µ(F) < µ(E). This leads a contradiction if E is σ s,q -stable.

Algebraic stability conditions 2.1 Review: Algebraic stability conditions Definition 2.1. We call an ordered set
We will write e * i for e * (E i ) as the associated points on the {1, ch 1 ch 0 , ch 2 ch 0 }−plane, where i = 1, 2, 3 and * could be +, l, or r. By the definition of e * 's, the relation of dyadic numbers (♣), and Serre duality, the points e + 1 , e r 1 , e 2 , e 3 are collinear on the line of χ(−, E 1 ) = 0, and e + 3 , e l 3 , e 2 , e 1 are collinear on the line of χ(E 3 , −) = 0.
We are now ready to recall the construction of algebraic stability conditions with respect to exceptional triples.

Proposition 2.2 ([Ma07] Section 3)
. Let E be an exceptional triple on D b (P 2 ), for any positive real numbers m 1 , m 2 , m 3 and real numbers φ 1 , φ 2 , φ 3 such that: There is a unique stability condition σ = (Z, P) such that

Definition 2.3. Given an exceptional triple
We make the following notations for some subsets of Θ E .

Common areas of geometric and algebraic stability conditions
We denote Stab Alg as the union of Θ E for all exceptional triples on D b (P 2 ), and call it the algebraic stability conditions.
We may therefore assume a = 2 and b = 1, then F is in the form of

Common areas of geometric and algebraic stability conditions
Let E = {E 1 , E 2 , E 3 } be an exceptional triple, in this section, we will explain how does the algebraic part Θ E 'glue' on to Stab Geo . We denote TR E as the triangle region on {1, ch 1 ch 0 , ch 2 ch 0 }−plane bounded anti-clockwise by line segments l e 1 e 2 , l e 2 e 3 and l e 3 e 1 (the edges l e 1 e 2 , l e 2 e 3 are defined to be in the TR E , the three vertices are not).We denote MZ E as the open region on {1, ch 1 ch 0 , ch 2 ch 0 }−plane bounded anti clockwise by line segments l e 1 e + 1 , l e + 1 e 2 , l e 2 e + 3 , l e + 3 e 3 and l e 3 e 1 . Proposition 2.5. Let E be an exceptional triple, then we have: Proof. We first prove the second statement. As MZ E is contained in Geo LP , by Corollary 1.19, E 2 is σ s,qstable for any point (1, s, q) in MZ E . As e + 1 , e r 1 , e 2 , e 3 are collinear on the line of χ(−, E 1 ) = 0, for any point P in MZ E , l EP is contained in Geo LP . By Corollary 1.19, E 3 is stable in MZ E . For the same reason, E 1 is stable MZ E .
For any (1, s, q) in MZ E , E 3 and E 1 [1] are in the heart of Coh #s . By Lemma 1.17, φ s, When s ≥ µ(E 2 ), E 3 and E 2 [1] are in the heart Coh #s , we have As (1, s, q) is above l e 1 e 2 , by Lemma 1.17, we also have When s < µ(E 2 ), by a similar argument we also have the same inequalities for φ s,q (E i )'s. By Proposition 2.2, we get the embedding For (1, s, q) outside the area MZ E , we have either Because either E 1 and E 3 are in the same heart (when s > µ(E 3 ) or s ≤ µ(E 1 )), or the slope φ s,q (E 1 [1]) is greater than φ s,q (E 3 ); or both E 1 [1] and E 2 [1] are in Coh #s but (1, s, q) is below l e 1 e 2 ; or both E 2 and E 3 are in Coh #s but (1, s, q) is below l e 2 e 3 . Hence σ s,q is not contained in Θ E , this finishes the second statement of the proposition. For the first statement, since φ 3 − φ 1 is not an integer, Θ ▽ E ∈ Stab nd . The image of Ker Θ ▽ E is in Tr E . By the previous argument, we also have the embedding The map Ker is local homeomorphism and the composition is an isomorphism. Since Θ ▽ E is path connected, the two maps are both isomorphism. We get the first statement of the proposition.

Neighbor cells of geometric stability conditions
Proposition and Definition 3.1 (Definition of Θ ± E ). Given exceptional triples E and E ′ on D b (P 2 ) with the same . We denote this subspace by Θ + E . In a similar way, we have the subspace Θ − E .
Proof. Let the three objects in E (E ′ ) be E, E 2 , E 3 (E, E ′ 2 , E ′ 3 ). By [GR87], E ′ 2 , E ′ 3 is constructed from E 2 , E 3 by consecutive left or right mutations. Without loss of generality, we may assume By [Ma07] Proposition 3.17, at a point ( − → m, − → φ ) in Θ E , when φ 3 < φ 2 + 1, L E 2 E 3 is stable at the point and its phase satisfies On the other hand, at a point ( Therefore, the left and right mutation identify the following two subsets in Θ E and Θ E ′ .
Now by the first statement of Proposition 2.5, By taking off Stab Geo on both sides of (1), we get Θ + E,E 1 = Θ + E ′ ,E 1 . The Θ − E case is proved in the same way.

Remark 3.2. Θ − E is a chamber that the skyscraper sheaf k(x) is destabilized by E. Θ + E is a chamber that the skyscraper sheaf k(x) is co-destabilized by E[1]
. We may also use the notation Θ + E,E 1 in some situations, since it has the chart induced from Θ E .

Boundary of geometric stability conditions Lemma 3.3. Let E and F be two exceptional bundles such that µ(E) < µ(F), then E is not stable under any stability condition in Θ + F and F is not stable under any stability condition in
Proof. Let E = {E 1 , E 2 , E 3 = E} be an exceptional triple extended from E. We may choose E 2 such that µ(E 2 ) < µ(F)−3. This can be done because of the correspondence between dyadic triples (♣) and exceptional triples. In particular, we may choose dyadic triples (♣) of the second type for some q large enough. By [GR87], since µ(E) < µ(F), Hom(E, F) 0. Therefore, we have φ(F) > φ(E). On the other hand, We may consider the image W of Ker Θ E (φ 3 − φ 1 < 2, φ 3 − φ 2 > 1) on P K R (P 2 ) and the wall µ(E) = µ(F). By similar arguments in Proposition 2.5 and the result of Lemma 1.16, W is connected and is a 'triangle' on the projective {1, ch 1 ch 0 , ch 2 ch 0 }−plane. On the {1, ch 1 ch 0 , ch 2 ch 0 }−plane, W is the union of two regions bounded by {l r E 1 E 2 , l E 2 E , l r E 1 E } and {l r EE 1 , l r E 2 E 1 } respectively.
As µ(E 1 ) < µ(F(−3)), F is above L e + 1 e r 1 E 2 E , on which χ(−, E 1 ) = 0. The ray l r FE is in the angle spanned by l Ee + 1 and l E− . Let Q be the intersection of L EF and L e + 3 e l 3 E 2 E 1 , then it is on the segment l E 2 e + . By the position of the lines, L EF ∩ W = l EQ and it is the only wall on which φ(F) = φ(E). By Proposition 2.5 and Lemma 1.17, we have φ(E) < φ(F), when Ker Z is in the triangle area TR E 2 QE ; and we have φ(E) > φ(F), when Ker Z is in TR EQe + . As l EQ is the only wall, in We get the contradiction that F cannot be stable at any point in Θ − E . The Θ + F part is proved in the same way. Proof. For any stability condition in Θ ± E , E is stable. Assuming µ(E) < µ(F), by Lemma 3.3, we only need prove that Θ − F ∩ Θ + E is empty. When µ(E) + 3 > µ(F), we may choose an exceptional triple F = {F 1 , F 2 , F 3 = F} being an extension of F such that µ(F 1 ) < µ(E). Such triple exists due to the correspondence of triples of dyadic numbers and exceptional triples. By Lemma 3.3, F 1 is not stable in Θ + E , the intersection Θ − F ∩ Θ + E is empty. When µ(E)+3 = µ(F), in other words, E = F(−3), we may choose an exceptional triple F = {F 1 , F 2 , F 3 = F} being an extension of F. There is another exceptional triple In particular, F 3 is stable, we may assume that it is in a heart E[a 1 ], E 2 [a 2 ], E 3 [a 3 ] for some integers a 1 , a 2 , a 3 such that a 2 − a 1 ≥ 1 and a 3 − a 2 ≥ 1. Since Under the chart of Θ F , this stability condition is in Θ F (φ 3 − φ 2 < 1). By Proposition 2.5, is empty. The last case is when µ(E) + 3 < µ(F). We may choose an exceptional triple F = {F 1 , F 2 , F 3 = F} being an extension of F such that µ(F 1 ) > µ(E) + 3. Again, such triple exists due to the correspondence of triples of dyadic numbers and exceptional triples.
By the same argument, F i 's are all in the same heart E 1 [2], E 2 [1], E 3 . As φ(F 3 ) − φ(F 1 ) < 1, this stability condition is not in Θ F . As a conclusion, Θ − F ∩ Θ + E is empty when E F. Corollary 3.5. The union of geometric and algebraic stability conditions has the following decompositions: We are now ready to show Stab Geo (P 2 )∪ Stab Alg (P 2 ) form the whole connected component. To do this, we need to prove that Stab Geo (P 2 )∪ Stab Alg (P 2 ) has no boundary point. The following important result is from [Ma04]: the boundary of finitely many Θ E is contained in Stab Alg .
Theorem 3.6 (Theorem 4.7 in [Ma04]). Let E be an exceptional triple, we have To prove the main result, we also need the following description for details of the boundary of Θ ± E . Lemma 3.7. Let E be an exceptional bundle, the boundary of Θ + E (as well as Θ − E ) is contained in the union of the boundary of Stab Geo and the boundary of Θ pure E for exceptional triples E that contain E: Proof. Let σ ∈ Stab † be a point on the boundary of Θ ± E . By Theorem 3.6, σ belongs to Θ F for some exceptional triple F = {F 1 , F 2 , F 3 }. The point σ is not in Θ Geo Proof. Let σ = (Z, P) be a stability condition on ∂ Stab Geo , by the principal of chambers ([BM11] Proposition 3.3) we may assume that the skyscraper sheaf k(x) is a semistable object with phase 1 and destabilized by F x with the same phase. I. σ is non-degenerate. By Lemma 1.14, Ker Z is on the boundary of Geo LP .When Ker Z is at the infinity line of P K R (P 2 ) , its locus is (0, 0, 1) as this is the only asymptotic line of the parabola. However, σ cannot be a stability condition, since Z(k(x)) = Z([0, 0, 1]) = 0, contradicting the fact that k(x) is semistable on the boundary. When KerZ is not at the infinity line, by Proposition 2.5, σ is either on the boundary of Θ E for an exceptional triple E or on the∆ 1 2 but not between any e r and e l . The first case is due to Theorem 3.6. The second case is more complicated, we will show that σ cannot satisfy the support condition. Let Ker Z be (1, s, q), then q = 1 2 s 2 − 1 2 . Let L 1 be the line on the {1, ch 1 ch 0 , ch 2 ch 0 }−plane across the points (1, s, q) and 1, s − 3, 1 2 (s − 3) 2 − 1 2 . Let L 2 be the line ch 1 ch 0 = s. Let D r be the set of characters defined as D r := {v ∈ K(P 2 ) | v is strictly below L 1 and to the right of L 2 , ||v − (1, s, q)|| < r}.
Let v n be a character in D 1 n , as v n is below L 1 , it is below the Le Potier curve. By the classification result of [DP85], M MG (v n ) is non-empty. Adopting the notation in [LiZ16], as v n is below L 1 , it is in R E for exceptional E with ch 1 ch 0 (E) < s. By the criterion for the last wall in [CHW14] or [LiZ16], the stability condition σ is above the last wall of v n , in another word, there are σ-stable objects with character v n . On the other hand, as ||v − (1, s, q)|| < 1 n and the Ker Z is (1, s, q), we have |Z(v n )| 1 n ||v n ||.
The stability condition σ does not satisfy the support condition. We get the contradiction. II. σ is degenerate. By [Br07], Stab † (P 2 ) → Hom Z (K(P 2 ), C) is a local homeomorphism, the degenerate locus has codimension 2 in Stab † (P 2 ). By [BM11] Proposition 3.3, the destabilizing wall W k(x) P for the skyscraper sheaf is of codimension 1. As the destabilizing walls are locally finite, we may assume σ is on the boundary of W k(x) P for a character P in K R (P 2 ). By Lemma 1.16, the kernel of the central charge of any stability condition on W k(x) P ∩Stab nd is on the line L Pk(x) , which is a line parallel to the ch 2 ch 0 -axis on the {1, ch 1 ch 0 , ch 2 ch 0 }−plane. As the kernel of W k(x) P ∩Stab nd has codimension one and is on the boundary of Geo LP , it is the segment of l ee + for some exceptional bundle E. W k(x) P ∩Stab nd is contained in the closure of Θ Geo E ∪ Θ Geo E ′ for any E = {E 1 , E 2 , E} and E ′ = {E, E ′ 2 , E ′ 3 }. Therefore, σ is contained in the closure of Θ Geo E ∪ Θ Geo E ′ . By Theorem 3.6, σ ∈ Stab Alg (P 2 ).