Bridgeland stability conditions on normal surfaces

We prove a new version of Bogomolov's inequality on normal proper surfaces. This allows to construct Bridgeland's stability condition on such surfaces. In particular, this gives the first known examples of stability conditions on non-projective, proper schemes.


Introduction
Let X be a separated scheme locally of finite type over a fixed base field.Let K 0 (X) be the Grothendieck group of coherent O X -modules and let A * (X) be the Chow group of X (i.e., the quotient of the group of cycles modulo rational equivalence).Then there exists a homomorphism τ X : K 0 (X) → A * (X) Q , which in the case of smooth X coincides with the homomorphism α → ch (α) ∩ Td (X) (see [Fu,Theorem 18.3] for more details).However, in general one cannot define any (additive) homomorphism ch : K 0 (X) → A * (X) Q to the (rational) operational Chow ring A * (X) Q = A * (X → X) ⊗ Q so that τ X (α) = ch (α) ∩ Td (X) for all α ∈ K 0 (X).Such "Chern character" does not exist even on normal surfaces.But in this special case one can use Mumford's intersection theory of Weil divisors on a normal surface S to define a substitute that corresponds to the evaluation ch ∩ [S].More precisely, we define Mumford's Chern character so that for any α ∈ K 0 (S) it satisfies the Riemann-Roch formula τ S (α) = [S] + ch M 1 (α) − THEOREM 0.1.Let S be a normal proper surface over an algebraically closed field.Then there exists an explicit constant C S , depending on the singularities and the birational equivalence class of S, satisfying the following condition.For any numerically non-trivial nef Weil divisor H on S and any slope H-semistable torsion free coherent O S -module E we have Moreover, if S has only rational Gorenstein singularities then C S does not depend on the singularities of S.
The proof of Theorem 0.1 depends on [La1] and [La3] that allow us to reduce to the smooth case.If S is a smooth projective surface and the base field has characteristic zero, the above theorem is classical.If S is smooth, projective and the base field has positive characteristic, the above theorem was proven by N. Koseki in [Ko] building on the methods of [La2].In case X is a normal projective surface with only rational double points, Theorem 0.1 was claimed in [NS].Unfortunately, the notion of a Chern character in [NS] is not correct and the proofs contain various mistakes.
In the above theorem we use only the fact that we can define the number S ∆ M (E).If E is a (non-zero) torsion free coherent O S -module of rank r > 0 and D is a Weil divisor such that detE ≃ O S (D) then this number can be easily recovered from the Riemann-Roch formula In Theorem 0.1 this formula can be taken as a defining property of S ∆ M (E).
Similarly to the smooth projective surface case, the above inequality allows to construct Bridgeland stability conditions on normal proper surfaces: THEOREM 0.2.Let S be a normal proper surface.Then there exist geometric Bridgeland stability conditions on D b (S) satisfying the full support property.
In the above theorem D b (S) denotes the bounded derived category of coherent sheaves on S.This is well-known to be equivalent to the bounded derived category of complexes with coherent cohomology.Theorem 0.2 follows immediately from Lemma 1.1 and Theorem 4.3.
It is well-known that every smooth proper surface is projective.In fact, a normal surface with only Q-factorial singularities is already projective.By Lipman's result (see [Li,Proposition 17.1]) a rational surface singularity is Q-factorial.So Theorem 0.2 for normal surfaces with at most rational singularities is still too weak to provide Bridgeland stability conditions on non-projective varieties.However, there exist many normal proper surfaces that are not projective, or even such that do not contain any non-trivial line bundles (see [Sch]).Using such surfaces and Theorem 0.2, we get the first examples of Bridgeland stability conditions on some proper, non-projective scheme.
Note that Theorems 0.1 and 0.2 are new also in the characteristic zero case.
The structure of the paper is as follows.In Section 1 we gather a few preliminary results.Section 2 contains a study of Mumford's Chern character.In Section 3 we prove Theorem 0.1.The last section contains proof of Theorem 0.2.

Notation.
In the paper we fix a base field k (which unless otherwise stated need not be algebraically closed nor perfect).A variety is an irreducible, reduced scheme X over k such that the structure morphism X → Spec k is separated and of finite type.It is called proper if X → Spec k is proper.A vector bundle on a variety X is a locally free coherent O X -module.A surface is a 2-dimensional variety.

Preliminaries 1.Intersection theory on normal surfaces
Let S be a normal surface.Let f : S → S be any resolution of singularities, i.e., a proper birational morphism from a regular surface S. Then one can define the Mumford pullback of Weil divisors f * : Z 1 (S) → Z 1 ( S) Q (see [Fu,Example 7.1.16]).This is defined by the property that for every Weil divisor D on S the difference of f * D and the proper transform of D is supported on the exceptional locus and for any irreducible comonent C of the exceptional locus of f the intersection number f * D.C vanishes.
Note that there exists a positive integer N such that the image of We can take as N, e.g., the product over all x ∈ S of the determinants of intersection matrices of irreducible components of f −1 (x).
The above pullback descends to the pullback homomorphism f * : A 1 (S) → A 1 ( S) Q .This allows us to define Mumford's intersection product [Fu,Example 8.3.11]).Here f * α • f * β denotes the intersection product of the class of the Cartier Q-divisor f * α with the class f * β of a 1-cycle (see [Fu,2.3]).This product is symmetric (by [Fu,Theorem 2.4]), bilinear and independent of the choice of resolution f .By definition, the image of Mumford's intersection product is contained in 1 N A 0 (S) ⊂ A 0 (S) Q for N as above.Note that since S is only regular, we cannot use the intersection product on non-singular varieties as defined in [Fu,Chapter 8].However, instead of the above intersection product on S one can equivalently use the rational intersection product provided by [Kl,Proposition 3.9] (see also [Vi,Proposition 2.3]) or the intersection product provided by Quillen's higher K-theory by using [Fu,20.5,Theorem].
In the following we also write α.β to denote S α • β .By abuse of notation we also write α 2 to denote α.α.By construction the image of the corresponding intersection product

Ampleness and nefness on normal proper surfaces
Let S be a normal proper surface over an algebraically closed field k.We say that an R-Weil divisor H on S is nef if for every closed curve C ⊂ S we have H.C ≥ 0. We say that an R-Weil divisor H is numerically ample if for every closed curve C ⊂ S we have H.C > 0 and H 2 > 0.
The following lemma, showing that such divisors always exist, seems to be well-known but we recall its proof for convenience of the reader.
LEMMA 1.1.Every proper normal surface S admits a numerically ample Weil divisor.
Proof.Let f : S → S be a resolution of singularities with projective S (to obtain such a resolution one can use Chow's lemma and then a resolution of singularities).Let A be an effective ample Cartier divisor on S. If C is an effective Weil divisor on S then f * C is also effective.Indeed, if we write f * C = f −1 * C + ∑ a i E i , where a i ∈ Q and E i are the irreducible components of the exceptional locus of f , then for all j which implies that a i ≥ 0 (see, e.g., [Gi,1.1]).This shows that H = f * A satisfies inequality Since by construction H is effective, this also shows that H 2 > 0.
Let N(S) denote the group of Weil divisors on S modulo the radical of Mumford's intersection pairing.One can check that N(S) is isomorphic to the quotient of the group B 1 (S) of Weil divisors modulo algebraic equivalence by torsion.If H is a numerically ample R-Weil divisor on S then passing to a resolution of singularities one can easily see that Mumford's intersection form is negative definite on H ⊥ ⊂ N(S) R , i.e., Hodge's index theorem holds on normal proper (possibly non-projective!) surfaces.If H is an R-Cartier divisor then by the Nakai-Moishezon criterion H is numerically ample if and only if it is ample (see [FM,Theorem 1.3]).In particular, a numerically ample R-Weil divisor on a non-projective surface is not numerically equivalent to a (numerically) ample R-Cartier divisor.Note also that by [Sch] there exist normal proper surfaces S with the trivial intersection form on Pic S (or even with trivial Pic S, i.e., with no non-trivial line bundles).

The zeroth and first Chern characters on normal varieties
Let X be a normal variety of dimension n.The aim of this subsection is to define the Chern characters Embedding j : X reg ֒→ X of the regular locus induces the restriction homomorphism j * : A * (X) → A * (X reg ).By normality of X the closed subset Y = X\X reg ⊂ X has codimension ≥ 2 and hence the exact sequence shows that j * is an isomorphism on the Chow groups of (n − 1) and n-cycles.So for any class α in the Grothendieck group K 0 (X) of coherent sheaves on X, we can define ch M 0 (α) and ch M 1 (α) by setting for i = 0, 1.For any class α ∈ K 0 (X) we consider ch M 0 (α) as an integer (using the canonical isomorphism A n (X) → Z).If α is the class of a coherent O S -module E then ch M 0 (E) is equal to the the rank of E at the generic point of X.If E is a reflexive coherent O X -module of rank 1 then there exists a Weil divisor D with In particular, the anticanonical divisor −K X = c M 1 (T X ) is the unique class of Weil divisors extending −K X reg (note that unlike the cotangent sheaf Ω X , the tangent sheaf T X is torsion free and even reflexive).

Chern character on normal surfaces
Let S be a normal surface.By Subsection 1.3 to define a homomorphism

Let us recall that by definition Td
The homomorphism ch M : K 0 (S) → A * (S) Q is called Mumford's Chern character.Let K 0 (S) be the Grothendieck group of vector bundles on S. Composing ch M with the canonical homomorphism ϕ : K 0 (S) → K 0 (S) one immediately sees that for any α ∈ K 0 (S) where ch : K 0 (S) → A * (S) Q is the usual Chern character of vector bundles.In particular, if S is smooth then ϕ is an isomorphism and ch By the covariance property for proper morphisms (see [Fu,Theorem 18.3,(1)]), if S is a proper normal surface then χ(S, E) = S τ S (E).In this case the above definition implies that for any coherent O S -module E we have the Riemann-Roch formula (1)

Computation of ch M
In this subsection we show how to compute ch M using a resolution of singularities.Let Ref (S) denotes the category of reflexive coherent O S -modules.The following lemma says that it is sufficient to compute ch M on Ref (S): LEMMA 2.1.Let S be a normal surface.Then the canonical homomorphism K(Ref (S)) → K 0 (S), from the Grothendieck group of Ref (S) to K 0 (S), is an isomorphism.
Proof.The proof is analogous to the one showing that on a smooth variety X the canonical homomorphism K 0 (X) → K 0 (X) is an isomorphism.So it is sufficient to show that any coherent O S -module F has a finite resolution 0 in which all E i are reflexive coherent O S -modules.By [SV,Theorem 2.1] S has sufficiently many vector bundles and hence there exists an exact sequence of the form in which both E 1 and E 0 are vector bundles (in particular, they are reflexive).Then im ϕ is torsion free (as it is contained in E 0 ) and hence ker ϕ is reflexive by [SP,Lemma 31.12.7].
Let f : S → S be any resolution of singularities of S (existence of such resolutions has been established by J. Lipman; see [SP,Theorem 54.14.5]).By [Fu,Theorem 18.3,(1)] for any coherent O S-module F we have τ S (R f * F) = f * (τ S(F )).
In particular, if F is a vector bundle on S and E = ( f * F) * * then we have For a sheaf G on S supported in dimension 0 and a closed point x ∈ X we denote by l x (G) the length of the stalk G x over the local ring O X,x .Since both R 1 f * F and E/ f * F are supported on a finite set of points (the points over which f is not an isomorphism), [Fu,Example 18.3.11]implies that where χ(x, F) recovering the formula from [Fu,Example 18.3.4].
For every point x ∈ S there exists a unique Weil Q-divisor c 1 (x, F) supported on the set f −1 (x) such that for every irreducible component C of f −1 (x) we have where Using the definition of ch M 2 and the above formula, we get In particular, the above formula for ch M 2 (E) does not depend on the choice of F such that E = ( f * F) * * .To finish the computation it is sufficient to note that for every E ∈ Ref (S) there exists a vector bundle F on S such that E = ( f * F) * * (e.g., we can take F = ( f * E) * * ).
Remark 2.2.Note that Remark 2.3.In [La1] and [La3] the author introduced and studied a completely different notion of the Chern character on normal surfaces, that is defined only for reflexive sheaves on normal surfaces over an algebraically closed field.The relation between this character and the one considered here is provided by the formula c M that holds in A 0 (S) R (for proper normal surfaces this follows from [La3,Theorem 4.4] and a proof in the non-proper case is similar).