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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {P}^2$$\end{document} has a principal connected component Stab†(P2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Stab}^\dag (\mathbf{P }^2)$$\end{document}. We show that Stab†(P2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Stab}^\dag (\mathbf{P }^2)$$\end{document} is the union of geometric and algebraic stability conditions. As a consequence, we give a cell decomposition for Stab†(P2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Stab}^\dag (\mathbf{P }^2)$$\end{document} and show that Stab†(P2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {Stab}^\dag (\mathbf{P }^2)$$\end{document} is contractible.


Introduction
Motivated by the concept of -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 [8]. 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 [10], 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 its central charge, which is a linear subspace in K R (X ) of real codimension two. Conversely, 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 in 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 [14] 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 [6] that D b (P 2 ) can be generated by an exceptional collection {O 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 [20], 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 E with parameters m j and φ j . The space of algebraic stability conditions Stab Alg (P 2 ) is the union of all E . Note that theGL + (2, R)-action is not free on Stab Alg (P 2 ). Each E can be divided into three parts: the head Geo E ; the legs + E,E 1 , − E,E 3 ; and the tail Pure E (see Definition 2.3). The head part is the overlap part with the geometric stability conditions, this is the only part of 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 E and E are either the same, or separated from each other (see Proposition 3.4). Each tail part Pure E is a private area for E , which is separated from any other 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 the legs, and then contracting all the legs to their boundaries with heads. The union of all heads Geo E is aGL + (2, R)-fiber space 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 [19] and [20], 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 [7,11,13,16,23,24]. 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 to 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 [24], in which the authors proved the contractibility for many interesting examples, does not apply to the case of 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 [24].

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 in 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, [11,16], 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 E on (H) 3 E independent of the triples E such that all the R E 's form a disjoint cover of Stab Alg (P 2 ).

Notations
The Picard group of P 2 is of rank one with generator H = [O(1)], and we will, by the abuse of notation, identify the ith 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, 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 [2,15,22].
An ordered collection of exceptional objects E = {E 0 , . . . , E m } is called an exceptional collection 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 [14,15,17]. There is a one-to-one correspondence between the dyadic integers p 2 m and exceptional bundles E p . We use Chern characters [ch 0 , ch 1 , 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. 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 L P . We call the cone in K R (P 2 ) spanned by the origin and C L P as the Le Potier cone. We also make a notation for the following open region above C L P .
is above C L P and not on any segment l ee + }. 1. it is proportional to an exceptional character e p 2 m ; 2. The point 1, ch 1 ch 0 , ch 2 ch 0 is on or below C L P in the {1, ch 1 ch 0 , ch 2 ch 0 }−plane.

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 [15]. 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 conditions on a triangulated category from [8]. Let T be the bounded derived category of coherent sheaves on a smooth projective 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: 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 [9] and [4] and conclude that the space of geometric stability condition on P 2 is aGL + (2, R) fiber space over Geo L P .
is called geometric if all skyscraper sheaves k(x) are σ -stable with the same phase. We denote the subset of all geometric stability conditions 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.
The ray along L E F from v(F) to infinity and does not contain v(E) This definition of the central charge Z s,q is slightly different from the usual case as that in the [1]. 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 + P E and l P− (for definition, see Table 1) at P on the H P half plane.

Proposition 1.10
For any (s, q) ∈ Geo L P , σ s,q = (Z s,q , P s,q ) is a geometric stability condition.
For the proof, readers are referred to the arguments in [9] and [4] 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 L P , 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: Knowing the classification result Theorem 1.5 on stable characters, the property is proved in the same way as that in the local P 2 and K3 surfaces cases.

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 the set of all nondegenerate 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 ).

Lemma 1.14GL
+ (2, R) acts freely on Stab nd (P 2 ) with closed orbits, and is a local homeomorphism.
Proof By Theorem 1.2 in [8], Stab nd (P 2 ) → Hom Z (K(P 2 ), C) is a local homeomorphism. The image is in the non-degenerate part of Hom Z (K(P 2 ), C). Hom nd Z (K(P 2 ), C) /GL + (2, R) is just the quotient Grassmannian Gr 2 (3) as a topological space. if and only if the ray l + P E is above l + P F .
Proof By the definition of l + P E , l P− , Z P and Remark 1.9, the angle spanned by the rays l + P E 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 L P and the origin. 2. There exists a slope semistable sheaf F such that the point P is in the region bounded by l r E F and l F− . In either case, the line l E P is not inside Geo L P . In particular, at least one of v(E) and P is outside <0 .
Proof Supposeṽ(E) is in the Geo L P -cone, in particular, ch 0 is not 0.
When ch 0 (E) > 0, H 0 (E) is non-zero. Let F = H 0 (E) min be the quotient sheaf of H 0 (E) with the minimum slope. Let D be H −1 (E) and G be the kernel of H 0 (E) → F. We have μ(D) < μ(F) < μ(G), when D and G are non-zero. We have the relation The equality only holds when D and G are both zero, but this is not possible as else v(E) is equal to v(F), which is inside Geo L P 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 E F . Therefore, P is in the region bounded by l r E F 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 E F . As l F− does not intersect Geo L P , P is not on L F(0,0,1) . Therefore, P is in the region bounded by l r E F 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 E P is also in the region which is contained in Geo L P .

Corollary 1.19 Let E be an exceptional bundle, and P = (1, s, q) be a point in Geo L P , then E is σ P -stable if s < μ(E) and l E P is contained in Geo L P . On the shifted side, E[1] is σ P -stable, if μ(E) ≤ s and l E P is contained in Geo L P .
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: we get H −1 (F) ⊂ H −1 (E) = 0, and v(F) is 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 P E , and l E+ . As l E P is in Geo L P , the whole open region bounded by these three segments is also in Geo L P . The whole line segment l F P is contained in Geo L P (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 E P is contained in Geo L P ' 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 E P intersects some ee + segments, we may choose the one (denoted by F) with minimum ch 1 ch 0 coordinate. The segment l F P is contained in Geo L P , and the φ s,q (F) > φ s,q (E). By [15], Hom(F, E) = 0 when μ(F) < μ(E). This leads to a contradiction if E is σ s,q -stable.

Definition 2.1 We call an ordered set
if E is a full strong exceptional collection of coherent sheaves on D b (P 2 ).
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 ([20] 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 E .

Fig. 1 The Le Potier curve C L P
We denote Stab Alg as the union of E for all exceptional triples on D b (P 2 ), and call the element of it the algebraic stability conditions.

Lemma 2.4 Let
We may therefore assume a = 2 and b = 1, then F is in the form of E ⊕n 1

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 the algebraic part E 'glues' on to Stab Geo . We denote TR E the triangle region on  Fig. 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 L P , by Corollary 1.19, E 2 is σ s,q -stable 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 E P is contained in Geo L P . 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 Coh #s . By Lemma 1.17, 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 φ s, 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 E , this finishes the second statement of the proposition. For the first statement, since φ 3 − φ 1 is not an integer, E ∈ Stab nd . The image of Ker 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 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 (Definition of ± E ) Given exceptional triples E and
. We denote this subspace by + E . In a similar way, we can define a subspace − E . Proof Let the three objects in E (E ) be E, E 2 , E 3 (E, E 2 , E 3 ). By [15], 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 [20] Proposition 3.17, at a point ( − → m , − → φ ) in E , when φ 3 < φ 2 + 1, L E 2 E 3 is stable at the point and its phase satisfies When φ 2 − φ 1 > 1, this is a stability condition in E . Under the coordinate of E , is stable at the point and E is a chamber that the skyscraper sheaf k(x) is destabilized by E. + E is a chamber in which the skyscraper sheaf k(x) is co-destabilized by E [1]. We may also use the notation + E,E 1 in some situations, since it has the chart induced from E .

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 + F and F is not stable under any stability condition in
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.
We may consider the image W of Ker 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 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 F E is in the angle spanned by l Ee + 1 and l E− . Let Q be the intersection of L E F 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 E F ∩ W = l E Q 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 Q E ; and we have φ(E) > φ(F), when Ker Z is in TR E Qe + . As l E Q is the only wall, in , by definition, we must have φ(F)−φ(F 2 ) < 1, and this implies the stability is in F ⊂ Stab Geo (P 2 ), which is a contradiction.
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.
As a conclusion, − F ∩ + E is empty when E = F.

Corollary 3.5
The union of geometric and algebraic stability conditions has the following decomposition: We are now ready to show Stab Geo (P 2 )∪ Stab Alg (P 2 ) forms 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 [19]: the boundary of finitely many E is contained in Stab Alg . Theorem 3.6 (Theorem 4.7 in [19]) 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 ± E .
, in which the third part is also contained in ∂ F and hence ∂ Stab Geo (P 2 ). Suppose σ is in F (φ 2 − φ 1 > φ 3 − φ 2 = 1), by a similar argument as that in Lemma 2.4, the only σ -semistable objects are F 1 [n] or in ⊥ F 1 . By Proposition 3.1, for any exceptional bundle E in ⊥ E, E is σ -semistable. The object F 1 must be E [n]. Due to the same argument, σ is not in F (φ 3 − φ 2 > φ 2 − φ 1 = 1). We may choose F such that F 1 = E. Proof Let σ = (Z , P) be a stability condition on ∂ Stab Geo .
I. σ is non-degenerate. By Lemma 1.14, Ker Z is on the boundary of Geo L P .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 Ker Z is not at the infinity line, by Proposition 2.5, σ is either on the boundary of 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 passing through 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 [14], M MG (v n ) is non-empty. Adopting the notation in [18], 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 [12] or [18], 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 [8], Stab † (P 2 ) → Hom Z (K(P 2 ), C) is a local homeomorphism, the degenerate locus has codimension 2 in Stab † (P 2 ). By [4] 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) + E as the open neighborhood. We may contract Stab Geo (P 2 ) ( E exc sheaves ± E ) to Stab Geo which is a contractible space.