Another Proof for the Presentation of the Quantum Cohomology of the Moduli of Bundles over a Riemann Surface

The presentation of the quantum cohomology of the moduli spaceof stable vector bundles of rank two and odd degree with fixeddeterminant over a Riemann surface of genus g > 2 is obtained.The argument avoids the use of gauge theory, providing an alternativeproof to that given by the author in Duke Math. J. 98 (1999)525–540. 2000 Mathematics Subject Classification 14N35(primary); 14H60, 53D45 (secondary).     

Spin Verlinde Spaces and Prym Theta Functions

It is shown that the theta functions of level n on the principallypolarised Prym varieties of an algebraic curve are dual to thesections of the orthogonal theta line bundle on the moduli spaceof Spin(n)-bundles over the curve. As a by-product of our computations,we also note that when n is odd, the Pfaffian line bundle onmoduli space has a basis of sections labelled by the even thetacharacteristics of the curve. 1991 Mathematics Subject Classification:14D20, 14H42, 14H60, 14K25.     

Jacobi Cohomology, Local Geometry of Moduli Spaces, and Hitchin Connections

We develop some cohomological tools for the study of the localgeometry of moduli and parameter spaces in complex AlgebraicGeometry. Notably, we develop canonical formulae for the differentialoperators of arbitrary order and their natural action on suitable‘natural’ modules (for example, functions); in particular,we obtain a formula, in terms of the moduli problem, for theLie bracket of vector fields on a moduli space. As an application,we obtain another construction and proof of flatness for thefamiliar KZW or Hitchin connection on moduli spaces of curves.2000 Mathematics Subject Classification 14D15, 32G05.     

Complete Sets of Relations in the Cohomology Rings of Moduli Spaces of Holomorphic Bundles and Parabolic Bundles Over a Riemann Surface

The cohomology ring of the moduli space of stable holomorphicvector bundles of rank n and degree d over a Riemann surfaceof genus g > 1 has a standard set of generators when n andd are coprime. When n = 2 the relations between these generatorsare well understood, and in particular a conjecture of Mumford,that a certain set of relations is a complete set, is knownto be true. In this article generalisations are given of Mumford'srelations to the cases when n > 2 and also when the bundlesare parabolic bundles, and these are shown to form completesets of relations. 2000 Mathematics Subject Classification 14H60.     

On the moduli of surfaces admitting genus two fibrations over elliptic curves

In this paper, we study the structure, deformations and the moduli spaces of complex projective surfaces admitting genus two fibrations over elliptic curves. We observe that a surface admitting a smooth fibration as above is elliptic, and we employ results on the moduli of polarized elliptic surfaces to construct moduli spaces of these smooth fibrations. In the case of nonsmooth fibrations, we relate the moduli spaces to the Hurwitz schemes of morphisms of degree n from elliptic curves to the modular curve X(d), d ≥ 3. Ultimately, we show that the moduli spaces in the nonsmooth case are fiber spaces over the affine line with fibers determined by the components of . Received: 30 August 2006     

Murphy’s law in algebraic geometry: Badly-behaved deformation spaces

We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ? (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over $\mathbb{F}_{p}We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ℤ (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over that lifts to ℤ/p⁷ but not ℤ/p⁸. (Of course the results hold in the holomorphic category as well.) It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them to more tractable deformation spaces via smooth morphisms. The essential starting point is Mn?v’s universality theorem. Mathematics Subject Classification (2000) 14B12, 14C05, 14J10, 14H50, 14B07, 14N20, 14D22, 14B05     

A GIT construction of moduli spaces of stable maps in positive characteristic

In a previous paper, the author and Swinarski constructed themoduli spaces of stable maps, _g,n(P^r, d) via geometric invariant theory (GIT). That paper requiredthe base field to be the complex numbers, a restriction thatthis paper removes: here the coarse moduli spaces of stablemaps are constructed via GIT over a more general base.     

Fourier-Mukai transforms for coherent systems on elliptic curves

We determine all the Fourier–Mukai transforms for coherentsystems consisting of a vector bundle over an elliptic curveand a subspace of its global sections, showing that these transformsare indexed by positive integers. We prove that the naturalstability condition for coherent systems, which depends on aparameter, is preserved by these transforms for small and largevalues of the parameter. By means of the Fourier–Mukaitransforms we prove that certain moduli spaces of coherent systemscorresponding to small and large values of the parameter areisomorphic. Using these results we draw some conclusions aboutthe possible birational type of the moduli spaces. We provethat for a given degree d of the vector bundle and a given dimensionof the subspace of its global sections there are at most d differentpossible birational types for the moduli spaces.     

Yang-Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics

Atiyah and Bott used equivariant Morse theory applied to theYang–Mills functional to calculate the Betti numbers ofmoduli spaces of vector bundles over a Riemann surface, rederivinginductive formulae obtained from an arithmetic approach whichinvolved the Tamagawa number of SL_n. This article attempts tosurvey and extend our understanding of this link between Yang–Millstheory and Tamagawa numbers, and to explain how methods usedover the last three decades to study the singular cohomologyof moduli spaces of bundles on a smooth projective curve over can be adapted to the setting of ¹-homotopy theory to studythe motivic cohomology of these moduli spaces over an algebraicallyclosed field.     

A Duality for Spin Verlinde Spaces and Prym Theta Functions

The paper proves canonical isomorphisms between Spin Verlindespaces, that is, spaces of global sections of a determinantline bundle over the moduli space of semistable Spin_n-bundlesover a smooth projective curve C, and the dual spaces of thetafunctions over Prym varieties of unramified double covers ofC.     

Conformal Versus Topological Conjugacy of Automorphisms on Compact Riemann Surfaces

We produce a family of algebraic curves (closed Riemann surfaces)S admitting two cyclic groups H₁ and H₂ of conformal automorphisms,which are topologically (but not conformally) conjugate andsuch that S/H_i is the Riemann sphere . The relevance of thisexample is that it shows that the subvarieties of moduli spaceconsisting of points parametrizing curves which occur as cycliccoverings (of a fixed topological type) of need not be normal.1991 Mathematics Subject Classification 14H10, 30F10.     

Moduli of sheaves,Fourier–Mukai transform,and partial desingularization

We study birational maps among (1) the moduli space of semistable sheaves of Hilbert polynomial \(4m+2\) on a smooth quadric surface, (2) the moduli space of semistable sheaves of Hilbert polynomial \(m^{2}+3m+2\) on \(\mathbb {P}^{3}\), (3) Kontsevich’s moduli space of genus-zero stable maps of degree 2 to the Grassmannian Gr(2, 4). A regular birational morphism from (1) to (2) is described in terms of Fourier–Mukai transforms. The map from (3) to (2) is Kirwan’s partial desingularization. We also investigate several geometric properties of 1) by using the variation of moduli spaces of stable pairs.     

Twisted stability and Fourier-Mukai transform II

 In this note, we define the twisted stability for a purely 1-dimensional sheaf and study the problem of the preservation of the stability condition under the relative Fourier-Mukai transform on an elliptic surface. As an application, we compute the Hodge polynomials of some moduli spaces of sheaves on an elliptic surface. We also construct the moduli space of twisted semi-stable sheaves. Received: 29 January 2002 / Revised version: 16 October 2002 Published online: 24 January 2003 Mathematics Subject Classification (2000): 14D20     

Exact Sequences of Semistable Vector Bundles on Algebraic Curves

Let X be a smooth complex projective curve of genus g 1. Ifg 2, then assume further that X is either bielliptic or withgeneral moduli. Fix integers r, s, a, b with r > 1, s >1 and as br. Here we prove the existence of an exact sequence [formula] of semistable vector bundles on X with rk(H) = r, rk(Q) = s,deg(H) = a and deg(Q) = b. 1991 Mathematics Subject Classification14H60.     

On the curvature of vortex moduli spaces

We use algebraic topology to investigate local curvature properties of the moduli spaces of gauged vortices on a closed Riemann surface. After computing the homotopy type of the universal cover of the moduli spaces (which are symmetric products of the surface), we prove that, for genus $g>1$ , the holomorphic bisectional curvature of the vortex metrics cannot always be nonnegative in the multivortex case, and this property extends to all Kähler metrics on certain symmetric products. Our result rules out an established and natural conjecture on the geometry of the moduli spaces.     

Moduli of rank 4 symplectic vector bundles over a curve of genus 2

Let X be a complex projective curve which is smooth and irreducibleof genus 2. The moduli space ₂ of semistable symplectic vectorbundles of rank 4 over X is a variety of dimension 10. Afterassembling some results on vector bundles of rank 2 and odddegree over X, we construct a generically finite cover of ₂by a family of 5-dimensional projective spaces, and outlinesome applications.     

Moduli for pairs of elliptic curves¶with isomorphic N-torsion

We study the moduli surface for pairs of elliptic curves together with an isomorphism between their N-torsion groups. The Weil pairing gives a “determinant” map from this moduli surface to (Z/N Z)^*; its fibers are the components of the surface. We define spaces of modular forms on these components and Hecke correspondences between them, and study how those spaces of modular forms behave as modules for the Hecke algebra. We discover that the component with determinant −1 is somehow the “dominant” one; we characterize the difference between its spaces of modular forms and the spaces of modular forms on the other components using forms with complex multiplication. In addition, we prove Atkin–Lehner-style results about these spaces of modular forms. Finally, we show some simplifications that arise when N is prime, including a complete determination of such CM-forms, and give numerical examples. Received: 20 September 2000 / Revised version: 7 February 2001