On the moduli spaces of vector bundles on algebraic surfaces

Transcription

1 Proceedings of The Thirteenth International Workshop on Diff. Geom. 13(2009) On the moduli spaces of vector bundles on algebraic surfaces Jaeyoo Choy Dept. of Mathematics, Kyungpook National University, Sankyuk-dong, Buk-gu, Daegu , Korea choy@knu.ac.kr Abstract. In this article, we would like to survey the construction of the moduli space of sheaves on projective varieties with Gieseker-Maruyama-Simpson stability and the Donaldson moduli spaces. By these, the Donaldson invariant and K-theory are studied. If the base 4-manifold is sufficiently good, we can generalize the moduli spaces into a new moduli space with a more abundant modular datum. 1 Introduction A vector bundle structured in various Lie algebras is a mathematical counterpart of a heterotic string in Physics. This embodies as a dual of F-theory on an induced fibration, in terms of a spectral curve, so thus a certain duality is computed in detail in Friedman-Morgan-Witten [FMW]. Vector bundles on a 4-dimensional manifold are modularized into a certain space, usually being the moduli spaces of stable sheaves, and the study of the space amounts to 5-dimensional supersymmetric Yang-Mills theory (compactified on a circle) in Physics (for short, 5D gauge theory) [KKV]. As a dual side (or a mirror side), the theory corresponds to Gromov-Witten invariants of a toric Calabi-Yau 3-fold, especially in certain cases via Geometric engineering of Katz, Klemm and Vafa. The gauge theory side are interpreted in accordance with the Chern-Simons theory, rigorously defined by Reshetikhin-Turaev, which is related to 3-dimensional surgery of 3-sphere along embedded knots [RT]. The knots generate some partition function (the Chern-Simons partition function) through the holonomy action, called Wilsonian in Physics literatures. To add a few words on the importance of 5D gauge theory in a mathematical viewpoint, it combines many invariants and predicts some equivalences among them, which usually appear as huge series in multi-variables with coefficients reading the geometry and topology of some moduli and algebras of nice representations. Without the physical predictions, it is never easy to see the series are equal with a small reasonable correction term. Very recent developments have been showing the remarkable predictions by building powerful mathematical theories and the correlations, and motivates the physical insights further in turn. The Verlinde formula in WZW models in 4D gauge theory and the Nekrasov conjecture were good implementations among the numerous successes. The interest here lies in 5D gauge theory. The moduli spaces of vector bundles were constructed in 2 ways; Algebraic Geometry and Differential Geometry. The techniques in Algebraic Geometry side, were initiated by Mumford-Seshadri 175

2 176 Jaeyoo Choy on compact Riemann surfaces and then by Simpson powered by much earlier works of Gieseker-Maruyama, the general moduli is produced in the Algebraic Geometry category. One advantage of this approach is that the moduli space attains a structure of projective scheme of finite type. The other approach, in a parallel vein, was due to Donaldson who succeeded Atiyah s viewpoint. As an advantage of the construction, it can modularize all the vector bundles (in fact, connections) on any 4-dimensional manifolds, which is much wider category than Algebraic Geometry. A useful invariant on the moduli was devised by Donaldson, called the Donaldson polynomial invariant defined the slant product of (the Poincare dual of) cohomology of a 4-dimensional manifold and the universal vector bundle. When a given 4-dimensional manifold is projective, there is a canonical map from the Gieseker- Maruyama moduli to the Donaldson moduli by Morgan-Li; even better, the map extends on the compactifications where the Donaldson moduli is compactified to the Uhlenbeck compactification and the former is to the Gieseker-Maruyama moduli of semistable torsion-free sheaves. Wall-crossing. The base 4-manifold is a K3, which is a good illustration in the mathematical moduli and topology theories as well as in type IIA D-brane duality in Physics. Since K3 is hyperkaehler, moduli are hyperkaehler and the symplectic 2-form is given by the Yoneda product. In a gauge theoretic view not just from the mathematical one as the Bogomolov splitting, hyperkaehlers are source of type IIB string compactification of conformal field theories [Dij] and contain a non-cobordic invariant, called Rozansky-Witten invariant reading a certain Feynman diagram dynamics [RW]. Contents. 2 Moduli of sheaves on projective varieties with Gieseker-Maruyama- Simpson stability and Donaldson moduli spaces 3 the Donaldson invariant and K-theory 4 Desingularization of smooth moduli stacks Note that except 4, all are survey. As the article is submitted to some associated proceeding volume, some parts are not fully explanatory, which are in some details in the presentation. Acknowledgement. This work is based on the author s presentation in 13th International Workshop on Differential Geometry and Related Fields at Kyungpook National University during Nov. 5-7, 2009, organized by Prof. Y. J. Suh. I learned some physical sources here from Prof. Hiraku Nakajima via private communications. 2 Moduli of sheaves on projective varieties with Gieseker-Maruyama- Simpson stability and Donaldson moduli spaces 2.1 Moduli of sheaves on projective varieties with Gieseker-Maruyama- Simpson stability Let X be a projective surface over C. We fix an ample divisor H on X. Denote O X (1) = O X (H). Let E be a coherent sheaf on X. Let P (E) be the Hilbert polynomial of E P (E, n) = χ(e(n)) = i ( 1) i dim H i (X, E(n)).

3 On the moduli spaces of vector bundles on algebraic surfaces 177 Let d(e) = dim SuppE. We say E has pure dimension d if d(f ) = d(e) for every subsheaf F E. Let us define rank r(e) = p d(e) d(e)! where p d(e) is the coefficient of P (E) of degree d(e). A coherent sheaf E of pure dimension d > 0 is semistable (stable, resp.) if P (F, n) r(f ) P (E, n) r(e) (<, resp.) for all subsheaf F E of pure dimension d and n 0. Let µ(e) = p d 1 (d 1)! r(e), which is called slope of E. Then E is µ-semistable (µ-stable, resp.) if µ(f ) µ(e) (<, resp.) for all subsheaf F E of pure dimension d. The relation between the stability and the µ-stability is as follows: µ stable stable semistable µ semistable. By the Harder-Narasimhan filtration of E, we understand a (unique) filtration of E 0 = E 0 E 1... E k = E such that each E i /E i 1 is semistable P (E i /E i 1 )/r(e i /E i 1 ) strictly decreases (as polynomials). Suppose E is semistable. By the Jordan-Hölder filtration of E, we understand a filtration 0 = E 0 E 1... E k = E such that each E i /E i 1 is stable. The Jordan-Hölder filtration is not unique but it is true that for any other Jordan-Hölder filtration (E j ) j J, there exists a bijective correspondence a IJ : I = {1,.., i,..., k} J satisfying E i /E i 1 = E aij (i) /E a IJ (i 1). Equivalently gr((e i ) i I ) = gr((e j ) j J) where gr((e i ) i I ) = i I E i/e i 1 (similarly for (E j )), hence gr(e) is well-defined. Boundedness. As observed in Introduction, in order to get sheaves together in one scheme, we need boundedness. Maruyama obtains the following type of boundedness which is strong enough to construct a moduli space. The stability in usual is called the Simpson stability. The Harder-Narasimhan filtration is defined with respect to the Simpson s stability, but can be defined with respect to the µ-stability. However the filtration with respect to the µ-stability needs not be unique. Also the Jordan-Hölder filtration with respect to the µ-stability can be defined and the uniqueness of gr holds.

4 178 Jaeyoo Choy Theorem 2.1. (Maruyama [Mar81]) Fix a polynomial P. The set of torsion-free sheaves E on P n with P (E) = P such that there exists b Z satisfying for every subsheaf F E, µ(f ) < b, is bounded. By Maruyama s theorem Simpson could obtain the general boundedness directly. Theorem 2.2. Fix a polynomial P with d = deg P. The set of sheaves E of pure dimension d on X with P (E) = P such that there exists b Z satisfying for every subsheaf F E with µ(f ) < b, is bounded. sheaves is bounded. Specifically the set of µ-semistable Proof. ([Si94]) The idea is simple. Because X is a projective scheme with a polarization H, embed X by H into some projective space P n. Denote this embedding by i : X P n. Then i E is a sheaf of pure dimension d. It is immediate that the boundedness of i E implies the boundedness of E, so we replace i E by E. Consider a linear projection π : P n P d whose projection center avoids Supp(E). Hence π E is of pure dimension d on P d i.e. it is a torsion-free sheaf. It is easy to check by computation that every subsheaf of π E satisfies the slope equality in Maruyama s theorem. Thus π E form a bounded family by Maruyama s theorem. Because π is a finite map, there is a vector space V of finite dimension such that π π E = V π E and V π E φ E Since π E and the choice of φ are bounded, E form a bounded family as far as π avoids SuppE. Now by choosing another linear projections π finitely many, we can conclude that E form a bounded family. We introduce k-boundedness. Let F be a family of sheaves of pure dimension d on P n. Then F is k-bounded (0 k < d) if there exists a bounded family F A set A of sheaves is bounded if there is a scheme of finite type S such that there exists a universal sheaf U over X S satisfying that any sheaf of A is isomorphic to U s for some s S. In other expression, the sheaves with the bounded µ max is bounded.

5 On the moduli spaces of vector bundles on algebraic surfaces 179 such that F P n k is a element of F for each generic P n k P n. A predecessor of Maruyama s theorem is a lemma below called Maruyama s lemma, which we translate in a serviceable fashion. Lemma 2.3. The set of sheaves E on P n of pure dimension d with r(e) = r and 0 µ(e) 1, such that for any subsheaf F E we have µ(f ) b for some b Z, is (d 1)-bounded. Proof. See [Si94] Lemma 1.4. The k-boundedness gives a bound of h 0 (X, E(m)) as follows. Lemma 2.4. Let F be a k-bounded family of sheaves of pure dimension d and rank r on P n with k d 1. Then there exists an integer B such that for all subsheaves E in F, for all m. h 0 (P n, E(m)) 0 if m + B 0 r(m+b) d d! if m + B 0 Proof. The proof is done by inductive method by restring E along codimension 1 subspace. For the detail see [Si94] Lemma 1.5. Because semistable implies µ-semistable, if we fix a polynomial P then we have a bounded family of the semistable coherent sheaves of pure dimension on X with Hilbert polynomial P. Now by Grothendieck [EGA], we can take a sufficiently large M 0 such that for all M M 0 and E in the family, H i (X, E(M)) = 0 and thus H 0 (X, E(M)) is constant. Let l = h 0 (X, E(M)). By Grothendieck [Gr95] there is a projective scheme Quot(O( M) l, P ) which is a universal object in the sense that for any scheme S such that S parametrizes morphisms O( M) l E with P (E) = P, there exists a unique morphism S Quot(O( M) l, P ) such that the family over S is pulled back from the family over Quot(O( M) l, P ). Such a Quot(O( M) l, P ) is called Quot-scheme. Note that any two morphism It follows from the noetherian property of the family.

6 180 Jaeyoo Choy φ, ψ Quot(O( M) l, P ) are an identical element if and only if as twisted morphisms φ, ψ : O l F (M) have the same kernel. Let us denote the kernels of φ : O l F (M) by G. Then Grothendieck observed that H 0 (X, G) can viewed a g- dimensional subspace of a l-dimensional vector space, thus that Quot(O( M) l, P ) is naturally embedded in Gr(g, l). As we soon realize, there can be redundant elements in Quot(O( M) l, P ) i.e. some elements in Quot(O( M) l, P ) are not semistable. It was Mumford for dim X = 1, Gieseker for dim X = 2 and Simpson for any dimension ([MFK94], [Gi77], [Si94] Lemma 1.19) that ( ) the semistability (stability, resp.) of E implies the semistability (stability, resp.) of P GL(l)-action on Gr(g, l). We will introduce the most general proof by Simpson. the construction of moduli spaces completely. Before it, we want to work out The converse of ( ) is not true, thus we take a component Q of Quot(O( M) l, P ) which is the Zariski closure of the set of [O( M) l E] where E is semistable of pure dimension d. By the Nagata-Mumford theorem [Le97, Theorem 6.3.1], we obtain a quotient map q : Q ss Q /P GL(l), where the superscript ss denotes the locus of semistable points. The quotient map gives a coarse moduli property in the sense that for any scheme Z with a P GL(l)-invariant morphism z : Q Z there exists a morphism γ : Q /P GL(l) Z such that the diagram commutes Q z q Q /P GL(l) γ Z We summarize with a few more addenda. Here twisted morphisms simply stands for morphisms after twisting by O(M). Semistability and stability of a linear group action K on a finite dimensional vector space V is defined as follows. A nonzero vector v V is semistable with respect to K if k.v 0 for k K. In other expression, K.v is a closed subset in V. A nonzero vector v V is stable with respect to K if it is semistable and its stabilizer is finite. The locus is Zariski open in Quot(O( M) l, P ), because the semistable condition is always an open condition on any parametrizing family. And the pure dimension condition is again an open condition.

7 On the moduli spaces of vector bundles on algebraic surfaces 181 Theorem 2.5. Let M(P ) = Q /P GL(l). Then M(P ) is projective. Moreover we have (1) If we denote the corresponding moduli functor by M(P ) then there exists a natural transform M(P ) M(P ) such that M(P ) corepresents M(P ). (2) Every point of M(P ) represents the equivalence class (S-equivalence) of semistable sheaves where E 1 S E 2 if and only gr(e 1 ) = gr(e 2 ). (3) If we denote the stable locus by M s (P ) then it is a fine moduli space in the étale sense i.e. there exists an étale universal sheaf U on X Q s. Proof. For the verification of (2) and (3), see [Si94] Theorem We now finish this section by the lemma characterizing subsheaves of a semistable sheaf by Hilbert polynomials, as promised. Lemma 2.6. Let E be a semistable sheaf of pure dimension d with Hilbert polynomial P. Then there exists N 0 such that for all N N 0 and all subsheaves F E we have h 0 (F (N)) r(f ) P (N) r(e) and moreover if the equality for some N N 0 then for all m. P (F, m) r(f ) = P (m) r(e) Proof. Let µ = µ(e) and r = r(f ). Let (F i ) be the Harder-Narasimhan filtration of F. Let Q i = F i /F i 1 and r i = r(q i ), µ i = µ(q i ). By induction we have h 0 (F (N)) i h0 (Q i (N)). Also µ i µ and i r i = r. By Lemma 2.4 we have h 0 0 if µ i + N + B 0 (Q i (N)) r i (µ i + N + B) d /d! if µ i + N + B 0 Consider 0 F 1 F F/F 1 0 and compute h 0 (F (N)), h 0 (F 1(N)), h 0 (F/F 1(N)).

8 182 Jaeyoo Choy Let ν = min µ i. First we assume that there exists a sufficiently large A Z, uniform with respect to F, such that h 0 0 if µ + N + B 0 (F (N)) (r 1)(µ + N + B) d /d! if µ + N + B 0 0 if ν + N + B 0 + (ν + N + B) d /d! if ν + N + B 0 r(n A) d /d! for all N A with ν + A µ. We can impose a further condition on A such that (N A) d /d! < P (E, N)/r(E) for all N A. Let N 0 = A. Therefore if ν µ A then we obtain for all N N 0. h 0 (F (N))/r(F ) < P (E, N)/r(E) Now we want to consider the region where A satisfies ν µ A. Let F sat be the saturation of F defined by the maximal sheaf F E with r(f ) = r(f ) containing F as a subsheaf. We can pick N 0 such that h 0 (F sat (N)) = P (F sat (N)) and F sat (N) is globally generated for all N N 0 by the observation of Grothendieck and Lemma Now by the global generation of F sat, the set {P (F sat )} is finite. Hence there exists N 0 such that for all N N 0 P (F sat, N)/r(F ) P (E, N)/r(E). Replacing N 0 by N 0 + d + 1 if necessary, we can observe that if P (F sat, N)/r(F ) = P (E, N)/r(E) for some N N 0 then it holds for all N. Since h 0 (F (N)) h 0 (F sat (N)) and F sat (N) is globally generated for all N N 0, h 0 (F (N)) = h 0 (F sat (N)) implies that F = F sat. Hence the lemma is proven. 2.2 Donaldson moduli spaces We construct the differential geometry counterpart of the moduli spaces of Gieseker-Maruyama-Simple semistable sheaves on a projective manifold. Since there

9 On the moduli spaces of vector bundles on algebraic surfaces 183 is no exact corresponding notion to sheaves which is not vector bundles in differential geometry, we have to construct the moduli space of vector bundles. It is a collection of metric connections on a fixed differential type of a vector bundles which minimizes the curvature volume, as we will see. Since for a complex surface, the metric connection is the Chern connection and it determines the complex structure of the vector bundle by its (0, 1)-part; thus the moduli of vector bundles in algebraic geometry and the moduli of vector bundles in differential geometry coincide. Furthermore, if one deals with a Kaehler manifold, one has the notion of stability of Mumford-Takemoto. Therefore, as shown in the previous subsection, we have a canonical map from the moduli of vector bundles in DG to the moduli in AG. Let us scroll down the details above now. Let (X, ω) be a compact complex manifold of dim C X = 2 with a Kähler form ω on X. Assume that there is a hermitian vector bundle E on X which admits a metric connection d A with the curvature 2-form F A := d A d A. Because a metric connection d A with the vanishing (0, 2)-part of F A = d A d A determines a holomorphic structure A := d 0,1 A on E, we have a holomorphic vector bundle E = (E, A ). By Kobayashi-Hitchin s theorem ([Ko82]), if a metric connection d A is irreducible and minimizes the Yang-Mills equation (2.1) X F A 2 vol g then E is stable in the following sense of Mumford-Takemoto (µ-stable in the previous subsection): for every nonzero proper (holomorphic) subsheaf F of E, (2.2) [ω] c 1 (F) rank F < [ω] c 1(E) rank E. Conversely, if a holomorphic structure on E with the prescribed Chern classes is given as (E, ) is stable, then there exists an irreducible connection d A such that d 0,1 A = up to complexified gauge group action and d A minimizes the Yang-Mills equation (Donaldson-Uhlenbeck-Yau [Do85][UY86]). We consider the set of isomorphism classes of holomorphic vector bundles E = (E, A ) where E is a vector bundle of rank r with the vanishing Chern classes, satisfying the stability condition. This moduli set, in order to be a moduli space,

10 184 Jaeyoo Choy needs to obtain a complex analytic structure. It is constructed as the quotient space of integrable and stable partial connections on E by the complexified gauge group, denoted by M st g (E) where g stands for the Kähler metric associated to ω (ref. [LT95] p.3 4). It is easy to prove M st g (E) does not depend on the choice of a hermitian metric on E as the hermitian metric is not involved in the notation ([LT95] (4.3.7)). Let F be the set of diffeomorphic classes of vector bundles of rank r on X with prescribed Chern classes in Q-coefficient. Then, we define M := E F M st g (E), which we call the moduli space of stable bundles of rank r (with the fixed rational topological type). By the construction of the moduli, we have embedding M M(P ) for the Hilbert polynomial determined by the rational Chern classes, and its inverse is also defined on the stable vector bundle locus of M(P ). We will denote the map by (2.3) φ : M(P ) M. 3 the Donaldson invariant and K-theory Let M be the Donaldson moduli on a 4-manifold constructed in the previous section. We define the Donaldson invariant of M. Let us mention a word on the motivation from Physics. The coupling of the Ramond-Ramond background gauge fields (given by some action functional on a 6-manifold ref.[dij]) to the D5-brane gauge theory inherits canonical cohomology classes on the instanton moduli. This produces (2 kinds of) 2-forms on M and they are B-fields in the topological sigma model. The 2-forms correspond mathematically to the Donaldson invariant [Do85], and physically to the topological field theory by Witten [Wit]. Definition 3.1. ([GNY, pp.5 7]) Let X, S be Noetherian schemes over C. Let E be a S-flat sheaf over X S. Let p : X S X and q : X S S be the projections. Let K(X) be the K-groups for locally free sheaves on X (i.e. Grothendieck s group). Let us define λ E : K(X) P ic(s)

11 On the moduli spaces of vector bundles on algebraic surfaces 185 by the composition K(X) q K(X S) E K(X S) p! K(S) det K(S). Here, p! denotes the following operator: for F K(X S), the alternating sum p! F = i ( 1)i R i f F in the K-group. Now let S = M H (r, c 1, c 2 ) be the Gieseker-Maruyama-Simpson moduli space of semistale pure sheaves of rank r and the indicated Chern classes on a smooth projective surface/c X (i.e. the topological type (r, c 1, c 2 ), and E be a universal sheaf. Here we choose the ample polarization H as a general element in the ample cone such that the strict H-semistability equals to the strict H -semistability for all nearby polarization H. In the case, even in the locus of strictly semistable sheaves, λ is well-defined on the span of double duals of (1, 0, 0), (0, H, 0) and (0, 0, H 2 ). (here, the duals are defined with respect to the holomorphic Euler characteristic 2-form χ on K(X).) We define µ(l) = λ(l) P ic(s) for L P ic(x) assuming < c 1 (L), c 1 > 2Z. The (K-theory) Donaldson invariant of X is the holomorphic Euler characteristic χ(s, µ(l)). The generating function is χ(l; Λ) = d 0 Λ d χ((m H (2, c 1, c 2 )), µ(l)). The construction of µ is parallel to the Donaldson µ-map[do85]. Viewing χ(s, µ(nl)) as a polynomial in n, its highest order coefficient is Φ H c 1 (c 1 (L) d /d!) the differential geometry Donaldson invariants, if the moduli space is of expected dimension (otherwise, it is defined by the metric-perturbation). A universal sheaf is not unique, nor always existing. Thus, λ is not defined But, there is a canonical replacement in the virtual sense; conveniently here we pretend it exists. The non-uniqueness does not affect here so the canonical polarization of S is induced by λ after all.

12 186 Jaeyoo Choy 4 Desingularization of smooth moduli stacks We suppose M be a smooth moduli stacks of rank 2 objects. The method presented here basically extends to arbitrary rank, but unless rank 2 it must involve a certain non-commutative procedure, i.e. quantized moduli. We will suppress the latter notion. Rank 2 case is a lucky case since the descent to the moduli space from the moduli stack is possible in the classical commutative geometry. Our viewpoint in some aspects, resembles Bagger-Lambert s SU(2) k SU(2) k -gauge theory for N = 8 as a modification of U(2)-gauge theory from the physics which is not mathematically rigorously established. And the non-commutative generalization looks related (but more generalized) to ABJM theory in the physics. In mathematics, the method appeared in 70 s due to Seshadri who aimed to resolve the singularities of the moduli spaces of vector bundles on a compact reimann surfaces. His name for the desingularization is parabolic moduli space, so we follow it. The moduli stack M here can parameterize vector bundles of rank 2 with additional structures (say, framed modules, decorated bundles, so on; Higgs bundles are a kind of them). But vector bundles themselves are most basic, so we assume: our smooth stack M parameterize the vector bundles E of rank 2 on a projective surface X with a fixed Hilbert polynomial P. In M, there is a substack M s of stable vector bundles. This is an open substack. The substack M ss of Gieseker-Maruyama-Simpson stable vector bundles are also open substack and we have a canonical classifying map M ss M(P ). Since a stable vector bundle has only trivial automorphism and M is smooth, the stable locus M s (P ) is smooth. But, along the strictly semistable locus, it may have quotient singularity. 4.1 Parabolic moduli space The material of the subsection is a review from [Ch]. In this section, we review some results in [HL94] and [Se82] about moduli spaces In fact, we can analyze the quotient singularity further using the transverse slice. It appears as quiver singularities.

13 On the moduli spaces of vector bundles on algebraic surfaces 187 of parabolic structures (=framed modules in [HL94]). Also the space of algebras is explained in order to introduce a Seshadri-type moduli functor. Let us start with the general assumptions: X is a projective manifold of any dimension and the rank can be any positive integer. Let (X, O X (1)) be a smooth projective variety with an ample line bundle O X (1). Let p X. Let δ p be a positive rational number. Let (E, α) be a pair such that E is a coherent sheaf and α : E C p. Let P (E,α) (n) = χ(e(n)) ε(α)δ p n where ε(α) = 1 if α 0 and ε(α) = 0 otherwise. The chosen point p is called a parabolic point. The morphism α of (E, α) is called a framing of E. The pair (E, α) is called a quasi-parabolic structure. If E is a coherent subsheaf of E, then α = α E defines an induced quasiparabolic structure (E, α ). Definition 4.1. ([HL94, Definition 1.1]) A quasi-parabolic structure (E, α) where E is of rank r is called δ p -parabolic (semi)stable if for n 0 (4.1) rp (E,α )(n)( ) < r P (E,α) (n) (respectively) for all induced pairs (E, α ) where 0 E E and r = rank E. We call a (δ p -)parabolic semistable quasi-parabolic structure as a (δ p -)parabolic structure. Throughout this paper we consider the parabolic semistability and the standard semistability only for torsion-free sheaves. Let (a 1, a 2 ) be a pair of positive rational numbers such that δ = a 2 a 1 > 0. Then (4.1) can be written as follows: (4.2) P E (n) r + ε(α )a 1 + (r ε(α ))a 2 r ( ) < P E(n) r + ε(α)a 1 + (r ε(α))a 2 r When dim X = 1, the definition coincides with Seshadri s one ([Se82] 3). We call (a 1, a 2 ) a parabolic weight. Remark 4.2. Let rank r be fixed. If δ p > 0 is taken sufficiently small, then δ p - parabolic semistability and δ p -parabolic stability coincide for any p X. Moreover,

14 188 Jaeyoo Choy if a pair (E, α) is δ p -parabolic semistable for a sufficiently small δ p > 0 then E is (Gieseker-Maruyama-Simpson) semistable. Definition 4.3. Let (E, α) and (F, β) be parabolic structures. A homomorphism between parabolic structures Φ : (E, α) (F, β) is a commutative diagram (4.3) E φ F α C p c C p β for some c C. A homomorphism Φ is an isomorphism if E φ = F and c C. Remark 4.4. Any stable parabolic structure (E, α) has only trivial automorphisms. I.e. Aut(E, α) = C or equivalently End(E, α) = C. Indeed by definition every nonzero endomorphism Φ End(E, α) should be an isomorphism, hence End(E, α) is a field. Since End(E, α) is a finite field extension of C, it is a trivial extension by Schur s lemma. We also use the term parabolic sheaf E for abbreviation if E is equipped with a framing α so that (E, α) is a parabolic structure with respect to some δ p (p X). Definition 4.5. Let T be a scheme of finite type over C. A flat family of parabolic sheaves parametrized by T is a T -flat coherent O X T -sheaf E with a homomorphism α : E p X C p where p X : X T X is the projection to X such that α t 0 for all t T and (E t, α t ) is a parabolic sheaf. The main theorem of [HL94] is modified to our case. The original statement in [HL94], rather, covers the broader definition of parabolic structure. Let (Sch) be the category of locally Noetherian schemes over C and (Sets) be the category of sets. Theorem 4.6. ([HL94, Theorem 0.1]) Let δ be a sufficiently small positive rational number. Let P be a polynomial of the rational coefficients. Let M δp (P ) : (Sch)

15 On the moduli spaces of vector bundles on algebraic surfaces 189 (Sets) be a (contravariant) moduli functor such that M δp (P )(T ) = { the equivalence classes of flat families of δ p -parabolic semistable sheaves of Hilbert polynomial P parametrized by T } Then there is a projective scheme M δp (P ) which is a fine moduli space for M δp (P ). Combined with Remark 4.2, the above theorem bears a corollary that there is a forgetful map π δp : M δp (P ) M(P ) where M(P ) is a coarse moduli space of M(P ) the moduli functor of (Simpson) semistable sheaves with Hilbert polynomial P. Since M(P ) consists of only torsion-free sheaves, E is torsion-free for every (E, α) in M δp (P ). Theorem 4.7. ([HL94, Theorem 4.1] and [HL94, p.319 iv]) Under the same assumption as in Theorem 4.6, there is an isomorphism (4.4) T [(E,α)] M δp (P ) = Ext 1 (E, E α C) where Ext denotes the hyper-ext. Moreover, for any tangent vector v Ext 1 (E, E α C), there is a canonical obstruction map (4.5) o : Ext 1 (E, E α C) Ext 2 (E, E α C) Corollary 4.8. With the same assumption as above, we further assume that dim End(E) = rank E = rank E p. Then, there are isomorphisms (4.6) T [(E,α)] M δp (P ) = Ext 1 (E, E α C) = Ext 1 (E, E) Moreover for any tangent vector v Ext 1 (E, E obstruction map α C) there is a corresponding (4.7) o : Ext 1 (E, E α C) Ext 2 (E, E α C)

16 190 Jaeyoo Choy which fits into the following commutative diagram: (4.8) Ext 1 (E, E α C) o Ext 2 (E, E α C) Ext 1 (E, E) Ext 2 (E, E) where is the obstruction map corresponding to the image of v in Ext 1 (E, E). A lemma is required for the proof. Lemma 4.9. For any δ p -parabolic stable structure (E, α), α E p has no nontrivial AutE-stabilizer. Proof. Otherwise, AutE has a nontrivial stabilizer. But this implies that (E, α) has a nontrivial automorphism. Contradiction to Remark 4.4. Proof of Corollary 4.8. Because α is a surjective map, we have [E α C] = ker α in the bounded derived category D b (X) of coherent sheaves. Therefore Ext i (E, E α C p ) = Ext i (E, ker α) for any i 0. Take the derived functor of Hom(E, ) to the exact sequence 0 ker α E C p 0. Then, we have the long exact sequence (4.9) 0 Hom(E, ker α) Hom(E, E) Hom(E, C p ). By Lemma 4.9, Hom(E, E) Hom(E, C p ) is surjective. Since E is locally free at p, Ext i (E, C p ) = 0 for i 1. Therefore, we obtain (4.10) Ext i (E, E α C p ) = Ext i (E, ker α) = Ext i (E, E) for i 1. Now the isomorphism for the tangent space in (4.6) follows from [HL94, Theorem 4.1]. Moreover, the commutativity of the obstruction map (4.8) follows from the functoriality of the isomorphisms (4.10) and [HL94, p.319 iv].

17 On the moduli spaces of vector bundles on algebraic surfaces 191 Corollary The canonical obstruction map (4.7) in Theorem 4.7 o : Ext 1 (E, E α C) Ext 2 (E, E α C) = Ext 2 (E, E) has the image in the traceless part Ext 2 0(E, E) of Ext 2 (E, E). Proof. The isomorphism Ext 2 (E, E α C) = Ext 2 (E, E) is shown in the proof of Corollary 4.8. By the functoriality, we have the commutative diagram Ext 1 (E, E α C) o Ext 2 (E, E α C) Ext 1 (E, E) Ext 2 (E, E) (compared to (4.8), the left vertical map needs not be an isomorphism). Since it is the standard fact that has the image in the traceless part, the proof is done. For simplicity sometimes we omit p and α in δ p and (E, α) respectively as long as no confusion arises. For a sufficiently small positive rational number δ, a pair (E, α) M δp (P ) has only trivial endomorphisms i.e. End(E, α) = C by Remark 4.2 and Remark 4.4. Therefore, when we say a δ-parabolic sheaf E has endomorphism group isomorphic to A for some algebra A, we mean End(E) = A. Some background materials for the space of algebras. Let A(r) be the subset of Hom(C r2 C r2, C r2 ) whose element is a unitary algebra structure of rank r 2 with a fixed identity element e 0. Any element in A(r) defines a unitary subalgebra of M(r 2 ) the (r 2 r 2 )-matrix algebra over C with e 0 = I r 2 r2, since the elements of an algebra A A(r) act on A by the left multiplication which are linear transformations of A. Endowing A(r) with the affine substructure of Hom(C r2 C r2, C r2 ), we let A r be the subscheme of A(r) whose element is isomorphic to M(r). Let A r = A r be the Zariski closure of A r in A(r). We say that an algebra A is (isomorphic to) an M(r)-specialization if A is isomorphic to an element in A r. Let W be a sheaf of algebras on A r such that for each a A r, W C a is an algebra represented by a ([Se82, I Chapter 5]).

18 192 Jaeyoo Choy Notation. In the rest of the section, the polynomial P is assumed to be a Hilbert polynomial for some sheaf of rank r over X. Let F δp (rp ) : (Sch) (Sets) be a moduli functor such that F δp (rp )(T ) = { the equivalence classes of T -flat families (E, α) of δ p -parabolic semistable sheaves of Hilbert polynomial rp such that for each t T, there exists an open subvariety (t )T 1 of T and a morphism f : T 1 A r satisfying that f W = (p T ) (End E) T1 } where p T : X T T is the projection to T. Let S δp (rp ) : (Sch) (Sets) be a moduli functor which is a closure of F δp (rp ) in M δp (rp ). This is motivated by Seshadri s moduli functor in [Se82]. Let S δp (rp ) be the closure of the locus {(F r, α) F M s (P ) & rank F p = r} in M δp (rp ). The verification of the reason why the sheaves of the above form F r admit the δ p -parabolic structure will be done in Proposition Let M δp, (P ) be the open subvariety of M δp (P ) such that for (E, α) M δp (P ), (E, α) M δp, (P ) if and only if E is locally free at p (ref. [HL97] Lemma 2.1.8). Let S δp, (rp ) = S δp (rp ) M δp, (rp ). Let S δp, (rp ) be a subfunctor of S δp (rp ) which parameterizes the sheaves locally free at p. Let M p (P ) be the open subvariety of M(P ) parameterizing the sheaves E locally free at p. This definition is well-defined because if we write the associated grading gr(e) = F i for the Jordan-Holder filtration, all F i are locally free at p. Note the equivalent definition was given in??. We draw a diagram with the entries of the moduli spaces. Let i M : M(P ) M(rP ) be a natural closed embedding given by i M (E) = E r ([HL97, Lemma 4.B.6]). Let i S δp, : S δp, (rp ) M δp, (rp ) be the natural inclusion. Let π δp :

19 On the moduli spaces of vector bundles on algebraic surfaces 193 M δp, (rp ) M(rP ) be the morphism forgetting framings i.e. π δp (E, α) = E. (4.11) S δp, (rp ) M(P ) i S δp, M δp, (rp ) i M π δp M(rP ) In summary, the moduli functors and the corresponding moduli spaces are listed in Table 1. For the fine moduli property of F δp, (rp ), one has to confer Lemma 4.16 and Proposition Moduli What are Moduli functor parametrized scheme M(P ) semistable sheaves M(P ): projective of Hilbert polynomial P (coarse) M δp (P ) δ p -parabolic semistable sheaves M δp (P ): projective of Hilbert polynomial P (fine) δ p -parabolic semistable sheaves M δp, (P ) locally free at p X M δp, (P ): quasi-projective of Hilbert polynomial P (fine) δ p -parabolic semistable sheaves F δp (rp ) of Hilbert polynomial rp some quasi-projective scheme with endomorphism = M(r) (coarse) δ p -parabolic semistable sheaves F δp, (rp ) locally free at p X M s (P ) M p (P ): quasi-projective of Hilbert polynomial rp (fine under the condition with endomorphism = M(r) of Proposition 4.18) S δp, (rp ) the closure S δp, (rp ): quasi-projective of F δp, (rp ) in M δp, (rp ) (fine under the condition of Main Theorem) Table 1. list of moduli functors and moduli spaces of sheaves of Hilbert polynomial P

20 194 Jaeyoo Choy 4.2 Basic properties of parabolic structures Let X be any projective surface over C with any polarization H. Let P be a Hilbert polynomial of some sheaf of rank r over X. The Riemann-Roch formula computes that the coefficient of m 2 of P (m) is r 2 H2. In this section we extract basic properties of parabolic structures on X from [Se82, 5.3]. Still δ p is taken a sufficiently small positive rational number for a fixed parabolic point p X. Lemma Let (E 1, α 1 ) and (E 2, α 2 ) be parabolic structures over X in M δp, (rp ) with Hilbert polynomials P Ei = rp. Assume dim EndE i = r 2. Suppose that E i is locally free at p. Then (E 1, α 1 ) = (E 2, α 2 ) if and only if E 1 = E2. Proof. The proof is identical with the case over a Riemann surface ([Se82] Chapter 5 Proposition 9). One implication is obvious and the other way implication follows from Lemma 4.9 that EndE i acts freely on E i p. Corollary Let (E, α) be a parabolic structure over X in M δp, (rp ) with Hilbert polynomial P E = rp. Then, dim EndE r 2. Remark Let (4.12) F : M δp, (rp ) M(rP ) by (E, α) E be the forgetful natural transform. By Remark 4.2, F factors through M ss (rp ) where M ss (rp ) is the subfunctor of M(rP ) parameterizing the semistable sheaves. Restricted to S δp, (rp ), F S δp, (rp ) : Sδp, (rp ) M ss (rp ) is injective by Lemma Note that it is obvious that passing to coarse moduli spaces, F induces the diagram (4.11). We introduce some elementary algebraic operation on torsion-free sheaves. For a torsion-free sheaf F on an algebraic surface Z, a torsion-free sheaf F is called a local double dual of F with respect to z Z if F is locally free at z and there is an injective morphism F F which is an isomorphism outside z. The existence of a

21 On the moduli spaces of vector bundles on algebraic surfaces 195 local double dual of F with respect to z is always guaranteed by the construction of a double dual of F on an affine open set U such that F is locally free on U {z} and pasting. Details are left as an easy exercise. Also it is easy to show a local double dual with respect to a given point is unique. Proposition Let E be a semistable sheaf with Hilbert polynomial P E = rp. Suppose E is locally free at p. Then the following two are equivalent: (1) (E, α) M δp, (rp ) for some α : E C p ; (2) for every nonzero stable sheaf F of rank s such that P F s = P r, there does not exist any injective homomorphism F (s+1) E. Proof. The key idea of the proof is identical with the case over a Riemann surface ([Se82] Chapter 5 Proposition 7). We prove the implication: (2) implies (1). Observe that by the definition of parabolic semistability, it is enough to show that: ( ) there exists a codimension 1 subspace H E E p such that for all nonzero stable subsheaves F E of rank r > 0 with P F r = P r, we have F p H E, because the subsheaves E E of rank r > 0 with P E r < P E r automatically fulfill (4.1) for a sufficiently small δ p > 0. Note that the implication (1) (2) easily follows from the observation( ), which we omit the proof. Notice that F of rank r > 0 with P F r = P E r must be locally free at p because otherwise we can consider the local double dual F with respect to p so that we have F F E and P F (m) < P F (m) for m 0 which contradicts to the semistability of E. Let gr(e) = i E ri i where E i are all stable and non-isomorphic each other and r i are positive numbers. It suffices to check ( ) for F = E i0 for some i 0. We may assume that the number of the direct summands of gr(e) is larger than 1, to avoid the triviality. Hom(F, E) = C r. Then r r i0. Since F, E i are locally free at p, the subspaces F p E p are parametrized by P r 1. Let I i0 be the incidence variety in P(E p) Let

22 196 Jaeyoo Choy P r 1 defined by (H E, F p) such that F p H E. Because the condition H E F p imposes dim F p linear constraints on P(E p), we have The assumption of (2) means rank E i0 cod(i i0, P(E p) P r 1 ) = rank E i0. r i0. Consequently, the image p 1 (I i0 ) P(E p) for the projection p 1, has the codimension rank E i0 < dim P(E p) i.e. p 1 (I i0 ) is a proper subvariety of P(E p). Because h p 1 (I i0 ) if and only if F p h for some F, any element H E P(E p) p 1 (I i0 ) satisfies ( ). We have an immediate corollary of the above proposition. Corollary Let (E, α) M δp, (rp ). Let E = 1 i m F si i Hölder grading, i.e. the direct sum of stable subsheaves F i of F i F j for i j. Then, s i rank (F i ) for all 1 i m. be the Jordan- P (F i) rank F i = P r with Lemma Let (E, α) M δp, (rp ). If EndE = M(r), then E = F r for some F M s (P ). Proof. The key idea of the proof is identical with the case over a Riemann surface ([Se82] Chapter 5 Proposition 8). Let E = be the direct sum of stable subsheaves F i with F i F j 1 i m F si i for i j. Let gr(e) be the associated grading to the Jordan-Holder grading of E. Then, we have dim End(E) 1 i m Since s i rank F i by Corollary 4.15, we obtain dim End(E) 1 i m s i dim Hom(F i, gr(e)). rank F i dim Hom(F i, gr(e)) = rank E. Because the equality dim End(E) = rank E holds, we have rank F i = s i for 1 i m and (4.13) dim End(E) = 1 i m s i dim Hom(F i, gr(e)).

23 On the moduli spaces of vector bundles on algebraic surfaces 197 Note that any endomorphism in End(E) induces an endomorphism in End(E ) by restriction. Hence we have a canonical exact sequence: 0 Hom(E/E, E) End(E) q End(E ). By direct computation, the following (in)equalities hold: dim End(E ) = dim Hom(E/E, E) 1 i m 1 i m s 2 i s i dim Hom(F i, gr(e/e )). By the equality dim Hom(F i, gr(e/e )) = dim Hom(F i, gr(e)) s i and (4.13), we have dim End(E) = dim End(E ) + dim Hom(E/E, E) and thus the map q should be surjective. By our assumption EndE = M(r), we have the surjective algebra homomorphism q : M(r) M(s i ). 1 i m Thus we induce a surjective algebra homomorphism q : M(r) M(s 1 ). Suppose s 1 < r. Since ker q is a two-sided ideal of M(r), for any nonzero matrix B in ker q, ker q contains all the matrices A with rank A rank B. Since such A generate M(r), we obtain q is 0. This is contradiction. Thus s 1 = r. This forces s i = 0 for all i 2. Remark The proof of triviality of q in the above proof follows Seshadri s argument. However, a quick proof is possible by the observation that End(E) acts transitively on E p. Lemma 4.16 applies to finding the fine moduli space of F δp, (rp ). Recall that the moduli functor F δp, (rp ) parameterizes parabolic structures (E, α) with EndE = M(r).

24 198 Jaeyoo Choy Proposition If M(P ) is a smooth stack, then the moduli functor F δp, (rp ) is representable by M s (P ) M p (P ). Remark It is easy to see that S δp, (rp ) is a coarse moduli space of S δp, (rp ). It is unknown that S δp, (rp ) is a fine moduli space in general. In contrast, when X is a smooth projective curve, due to the vanishing of the obstruction space, Seshadri shows that S δp (rp ) is a fine moduli space of the parabolic moduli functor S δp (rp ) and a closed subvariety of M δp (rp ) ([Se82, Theorem 15, 17 Chapter 5]). Unfortunately, his method cannot apply to the higher dimensional cases. However, the restriction of the domain (Sch) of S δp, (rp ) to the category (Red) of locally Noetherian reduced schemes, is representable as follows. Corollary Let S δp, (rp ) be restricted to the function over (Red). Then S δp, (rp ) is a fine moduli space of the restricted functor of S δp, (rp ). Moreover, S δp, (rp ) is the Zariski closure in M δp, (rp ) of the locally closed subvariety whose element is given as (E, α) M δp, (rp ) with E = F r for some stable sheaf F. Proof. The result comes from the universality of M δp, (rp ). Let T be an irreducible reduced scheme parametrizing parabolic sheaves in M δp, (rp ) with T T parameterizes (E, α) having EndE = M(r) for some proper closed subvariety T T. By the universal property of M δp, (rp ), the morphism T M δp, (rp ) exists. By the universal property of F δp, (rp ) (Proposition 4.18), the morphism T T M s (P ) M p (P ) M δp, (rp ) exists. Therefore T M δp, (rp ) factors through the embedding S δp, (rp ) M δp, (rp ). Theorem If r = 2 and M(P ) is a smooth stack (of vector bundles of rank r with the Hilbert polynomial P on a projective surface X), then S δp, (2P ) is a smooth variety. Furthermore, as p moves, S δp, (2P ) patches to a variety S(2P ) which is a resolution of M(P ). References [Ch] J. Choy. A Moduli-theoretic desingularization of moduli spaces of rank 2 sheaves over a K3. Preprint (2005)

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