# A tour of theta dualities on moduli spaces of sheaves

###### Abstract.

The purpose of this paper is twofold. First, we survey known results about theta dualities on moduli spaces of sheaves on curves and surfaces. Secondly, we establish new such dualities in the surface case. Among others, the case of elliptic K3 surfaces is studied in detail; we propose further conjectures which are shown to imply strange duality.

## 1. Introduction

The idea that sections of the determinant line bundles on moduli spaces of sheaves are subject to natural dualities was first formulated, and almost exclusively pursued, in the case of vector bundles on curves. The problem can however be generally stated as follows.

Let be a smooth complex projective curve or surface with polarization , let be a class in the Grothendieck group of coherent sheaves on . Somewhat imprecisely, we denote by the moduli space of Gieseker -semistable sheaves on of class . Consider the bilinear form in -theory

(1) |

and let consist of the -classes orthogonal to relative to this form. There is a group homomorphism

which was extensively considered in [dn] when is a curve, and also in [lp], [jun] in the case when is a surface, as part of the authors’ study of the Picard group of the moduli spaces of sheaves. When admits a universal sheaf , is given by

Here is any sheaf with -type , and and are the two projections from . The line bundle is well defined also when the moduli space is not fine, by descent from the Quot scheme.

Consider now two classes and in , orthogonal with respect to the bilinear form (1) i.e., satisfying Assume that for any points and , the vanishing, vacuous when is a curve,

(2) |

occurs. Suppose further that the locus

(3) |

gives rise to a divisor of the line bundle which splits as

(4) |

We then obtain a morphism, well-defined up to scalars,

(5) |

The main questions of geometric duality in this context are simple to state and fundamentally naive.

Question 1. What are the constraints on , and subject to which one has, possibly with suitable variations in the meaning of ,

(6) |

Question 2. In the cases when the above equality holds, is the map of equation (5) an isomorphism?

This paper has two goals. One is to survey succinctly the existent results addressing Questions and The other one is to study the two questions in new geometric contexts, providing positive answers in some cases and support for further conjectures in other cases. Throughout we refer to the isomorphisms induced by jumping divisors of type (3) as theta dualities, or strange dualities. The latter term is in keeping with the terminology customary in the context of moduli spaces of bundles on curves.

We begin in fact by reviewing briefly the arguments that establish the duality in the case of vector bundles on curves. We point out that in this case the isomorphism can be regarded as a generalization of the classical Wirtinger duality on spaces of level theta functions on abelian varieties. We moreover give a few low-rank/low-level examples. We end the section devoted to curves by touching on Beauville’s proposal [Bea2] concerning a strange duality on moduli spaces of symplectic bundles; we illustrate the symplectic duality by an example.

The rest of the paper deals with strange dualities for moduli spaces of sheaves on surfaces. In this context, Questions and were first posed by Le Potier. Note that in the curve case there are strong representation-theoretic reasons to expect affirmative answers to these questions. By contrast, no analogous reasons are known to us in the case of surfaces.

The first examples of theta dualities on surfaces are given in Section 3. There, we explain the theta isomorphism for pairs of rank 1 moduli spaces i.e., for Hilbert schemes of points. We also give an example of strange duality for certain pairs of rank 0 moduli spaces.

Section 4 takes up the case of moduli spaces of sheaves on surfaces with trivial canonical bundle. The equality (6) of dimensions for dual spaces of sections has been noted [ogr2][gny] in the case of sheaves on surfaces. Moreover, it was recently established in a few different flavors for sheaves on abelian surfaces [mo3]. We review the numerical statements, which lead one to speculate that the duality map is an isomorphism in this context. We give a few known examples on surfaces, involving cases when one of the two moduli spaces is itself a surface or a Hilbert scheme of points. It is likely that more general instances of strange duality can be obtained, starting with the isomorphism on Hilbert schemes presented in Section 3 and applying Fourier-Mukai transformations. We hope to investigate this elsewhere. Finally, the last part of this section leaves the context of trivial canonical bundle, and surveys the known cases of strange duality on the projective plane, due to Dǎnilǎ [D1] [D].

Section 5 is devoted to moduli spaces of sheaves on elliptically fibered K3 surfaces. We look at the case when the first Chern class of the sheaves in the moduli space has intersection number 1 with the class of the elliptic fiber. These moduli spaces have been explicitly shown birational to the Hilbert schemes of points on the same surface [ogr]. We conjecture that strange duality holds for many pairs of such moduli spaces, consisting of sheaves of ranks at least two. We support the conjecture by proving that if it is true for one pair of ranks, then it holds for any other, provided the sum of the ranks stays constant. Since any moduli space of sheaves on a surface is deformation equivalent to a moduli space on an elliptic , the conjecture has implications for generic strange duality statements.

### 1.1. Acknowledgements

We would like to thank Jun Li for numerous conversations related to moduli spaces of sheaves and Mihnea Popa for bringing to our attention the question of strange dualities on surfaces. We are grateful to Kieran O’Grady for clarifying a technical point. Additionally, we acknowledge the financial support of the NSF. A.M. is very grateful to Jun Li and the Stanford Mathematics Department for making possible a great stay at Stanford in the spring of 2007, when this article was started.

## 2. Strange duality on curves

### 2.1. General setup

When is a curve, the topological type of a vector bundle is given by its rank and degree, and its class in the Grothendieck group by its rank and determinant. We let be the moduli space of semistable vector bundles with fixed numerical data given by the rank and degree . Similarly, we denote by the moduli space of semistable bundles with rank and fixed determinant of degree . For a vector bundle which is orthogonal to the bundles in i.e.,

we consider the jumping locus

The additional numerical subscripts of thetas indicate the ranks of the bundles that make up the corresponding moduli space.

The well-understood structure of the Picard group of , given in [dn], reveals immediately that the line bundle associated to depends only on the rank and determinant of . Moreover, on the moduli space , depends only on the rank and degree of . In fact, the Picard group of has a unique ample generator , and therefore

for some integer .

For numerical choices and orthogonal to each other, the construction outlined in the Introduction gives a duality map

(7) |

The strange duality conjecture of Beauville [bsurvey] and Donagi-Tu [dt] predicted that the morphism is an isomorphism. This was recently proved for a generic smooth curve in [bel1], which inspired a subsequent argument of [mo1] for all smooth curves. The statement for all curves also follows from the generic-curve case in conjunction with the recent results of [bel2]. We briefly review the arguments in the subsections below.

### 2.2. Degree zero

The duality is most simply formulated when , on the moduli space of rank bundles with trivial determinant. There is a canonical line bundle on the moduli space associated with the divisor

The map (7) becomes

(8) |

Note that this interchanges the rank of the bundles that make up the moduli space and the level (tensor power) of the determinant line bundle on the moduli space.

To begin our outline of the arguments, let

(9) |

be the Verlinde number of rank and level . As the theta bundles have no higher cohomology, the Verlinde number computes in fact the dimension of the space of sections. The most elementary formula for reads

(10) |

This expression for was established through a concerted effort and variety of approaches that spanned almost a decade of work in the moduli theory of bundles on a curve. It is beyond the purpose of this note to give an overview of the Verlinde formula. Nonetheless, let us mention here that (10) implies the symmetry

required by Question , setting the stage for the strange duality conjecture.

###### Example 1.

Level 1 duality. Equation (10) simplifies dramatically in level yielding

This coincides with the dimension of the space of level classical theta functions on the Jacobian. The corresponding isomorphism

(11) |

was originally established in [bnr], at that time in the absence of the Verlinde formula. Nonetheless, once the Verlinde formula is known, the isomorphism (11) may be proved by an easy argument which we learned from Mihnea Popa; see also [bel2]. Let denote the group of -torsion points on the Jacobian, and let be the Heisenberg group of the line bundle , exhibited as a central extension

Now is the Schrödinger representation of the Heisenberg group i.e., the unique irreducible representation on which the center acts by homotheties. We argue that the left hand side of (11) also carries such a representation of . Indeed, the tensor product morphism

is invariant under the action of on the source, given by

The pullback

carries an action of , while the theta bundle on the right carries an action of . This gives an induced action of the Heisenberg group on covering the tensoring action of on . The non-zero morphism is clearly -equivariant, hence it is an isomorphism.

It is more difficult to establish the isomorphism (8) for arbitrary levels. The argument in [bel1] starts by noticing that since the dimensions of the two spaces of sections involved are the same, and since the map is induced by the divisor (3), is an isomorphism if one can generate pairs

(12) |

such that

Equivalently, one requires

(13) |

Finding the right number of such pairs relies on giving a suitable enumerative interpretation to the Verlinde formula (10), and this is the next step in [bel1]. The requisite vector bundles (12) satisfying (13) are assembled first on a rational nodal curve of genus . They are then deformed along with the curve, maintaining the same feature on neighboring smooth curves, and therefore establishing the duality for generic curves. To obtain the right count of bundles on the nodal curve, the author draws benefit from the fusion rules [ueno] of the Wess-Zumino-Witten theory, which express the Verlinde numbers in genus in terms of Verlinde numbers in lower genera, and with point insertions. The latter are Euler characteristics of line bundles on moduli spaces of bundles with parabolic structures at the given points. Belkale realizes the count of vector bundles with property (13) on the nodal curve as an enumerative intersection in Grassmannians with recursive traits precisely matching those of the fusion rules.

Recently, the author extended the result from the generic curve case to that of arbitrary curves [bel2]. He considers the spaces of sections relatively over families of smooth curves. These spaces of sections give rise to vector bundles which come equipped with Hitchin’s projectively flat connection [hitchin]. Belkale proves, importantly, that

###### Theorem 1.

[bel2] The relative strange duality map is projectively flat with respect to the Hitchin connection. In particular, its rank is locally constant.

This result moreover raises the question of extending the strange duality isomorphism to the boundary of the moduli space of curves.

The alternate proof of the duality in [mo1] is inspired by [bel1], in particular by the interpretation of the isomorphism as solving a counting problem for bundles satisfying (12), (13). This count is carried out in [mo1] using the close intersection-theoretic rapport between the moduli space of bundles and the Grothendieck Quot scheme parametrizing rank , degree coherent sheaf quotients of the trivial sheaf on . The latter is irreducible for large degree , and compactifies the space of maps from to the Grassmannian of rank planes in . The -asymptotics of tautological intersections on the Quot scheme encode the tautological intersection numbers on [mo2]. This fact prompts one to take the viewpoint that the space of maps from the curve to the classifying space of is the right place to carry out the intersection theory of the moduli space of -bundles on the curve.

As a Riemann-Roch intersection on , the Verlinde number is also expressible on the Quot scheme, most simply as a tautological intersection on which parametrizes rank quotients of of suitably large degree . Precisely, let denote the rank universal subsheaf of on and set

Then

(14) |

This formula was essentially written down by Witten [witten] using physical arguments; it reflects the relationship between the WZW model and the sigma model of the Grassmannian, which he explores in [witten]. Therefore, the modified Verlinde number

becomes a count of Quot scheme points which obey incidence constraints imposed by the self intersections of the tautological class . Geometrically interpreted, these constraints single out the finitely many subsheaves

which factor through a subsheaf of of the same rank but of lower degree. Therefore, one obtains diagrams

(15) |