Borel-Weil-Bott theory on the moduli stack of G-bundles over a curve

Let G be a semi-simple group and M the moduli stack of G-bundles over a smooth, complex, projective curve. Using representation-theoretic methods, I prove the pure-dimensionality of sheaf cohomology for certain “evaluation vector bundles” over M, twisted by powers of the fundamental line bundle. This result is used to prove a Borel-Weil-Bott theorem, conjectured by G. Segal, for certain generalized flag varieties of loop groups. Along the way, the homotopy type of the group of algebraic maps from an affine curve to G, and the homotopy type, the Hodge theory and the Picard group of M are described. One auxiliary result, in Appendix A, is the Alexander cohomology theorem conjectured in [Gro2]. A self-contained account of the “uniformization theorem” of [LS] for the stack M is given, including a proof of a key result of Drinfeld and Simpson (in characteristic 0). A basic survey of the simplicial theory of stacks is outlined in Appendix B. Oblatum 17-XII-1996 & 26 VI-1997     

The motive of the moduli stack of <Emphasis Type="Italic">G</Emphasis>-bundles over the universal curve

We define relative motives in the sense of André. After associating a complex in the derived category of motives to an algebraic stack we study this complex in the case of the moduli of G-bundles varying over the moduli of curves.     

The embedding dimension of the formal moduli space of certain curve singularities

For curve singularities which can be deformed into complete intersections the vector space dimension of T¹ is estimated. Thus in the case that T² is trivial we prove a formula of Deligne on the dimension of smoothing components with purely local methods.The author was supported by a grant of the DAAD (Deutscher Akademischer Austauschdienst)     

Desingularizations of the moduli space of rank 2 bundles over a curve

Let X be a smooth projective curve of genus g3 and M₀ be the moduli space of rank 2 semistable bundles over X with trivial determinant. There are three desingularizations of this singular moduli space constructed by Narasimhan-Ramanan [NR78], Seshadri [Ses77] and Kirwan [Kir86b] respectively. The relationship between them has not been understood so far. The purpose of this paper is to show that there is a morphism from Kirwans desingularization to Seshadris, which turns out to be the composition of two blow-downs. In doing so, we will show that the singularities of M₀ are terminal and the plurigenera are all trivial. As an application, we compute the Betti numbers of the cohomology of Seshadris desingularization in all degrees. This generalizes the result of [BS90] which computes the Betti numbers in low degrees. Another application is the computation of the stringy E-function (see [Bat98] for definition) of M₀ for any genus g3 which generalizes the result of [Kie03].Young-Hoon Kiem was partially supported by KOSEF R01-2003-000-11634-0 and SNU; Jun Li was partially supported by NSF grants.Mathematics Subject Classification (2000): 14H60, 14F25, 14F42     

The determinant bundle on the moduli space of stable triples over a curve

We construct a holomorphic Hermitian line bundle over the moduli space of stable triples of the form (E₁, E₂,?), where E₁ and E₂ are holomorphic vector bundles over a fixed compact Riemann surfaceX, and?: E₂ → E₁ is a holomorphic vector bundle homomorphism. The curvature of the Chern connection of this holomorphic Hermitian line bundle is computed. The curvature is shown to coincide with a constant scalar multiple of the natural Kähler form on the moduli space. The construction is based on a result of Quillen on the determinant line bundle over the space of Dolbeault operators on a fixed C^∞ Hermitian vector bundle over a compact Riemann surface.     

Compactified Picard stacks over the moduli stack of stable curves with marked points

In this paper we give a construction of algebraic (Artin) stacks endowed with a modular map onto the moduli stack of stable curves of genus g with n marked points. The stacks we construct are smooth, irreducible and have dimension 4g−3+n, yielding a geometrically meaningful compactification of the universal Picard stack parametrizing n-pointed smooth curves together with a line bundle.     

The moduli space of stable vector bundles over a real algebraic curve

We study the spaces of stable real and quaternionic vector bundles on a real algebraic curve. The basic relationship is established with unitary representations of an extension of \mathbbZ/2{\mathbb{Z}/2} by the fundamental group. By comparison with the space of real or quaternionic connections, some of the basic topological invariants of these spaces are calculated.     

Upper bounds for the spread of a matrix

In this paper, we exhibit new and sharper upper bounds of the spread of a matrix.     

Orthogonality of natural sheaves on moduli stacks of SL(2)-bundles with connections on \Bbb P^1 minus 4 points

A special kind of SL(2)-bundles with connections on \Bbb P¹\{x₁,...,x₄}\Bbb P^1\setminus\{x_1,\dots,x_4\} is considered. We construct an equivalence between the derived category of quasicoherent sheaves on the moduli stack of such bundles and the derived category of modules over a TDO ring on some (non-separated) curve.     

Stability of Picard bundle over moduli space of stable vector bundles of rank two over a curve

Answering a question of [BV] it is proved that the Picard bundle on the moduli space of stable vector bundles of rank two, on a Riemann surface of genus at least three, with fixed determinant of odd degree is stable.     

The Brauer group of desingularization of moduli spaces of vector bundles over a curve

Let C be an irreducible smooth projective curve, of genus at least two, defined over an algebraically closed field of characteristic zero. For a fixed line bundle L on C, let M _C (r; L) be the coarse moduli space of semistable vector bundles E over C of rank r with ∧ ^r E = L. We show that the Brauer group of any desingularization of M _C (r; L) is trivial.     

A note on 1-cycles on the moduli space of rank-2 bundles over a curve

Over a smooth complex projective curve of genus ≥3, we study 1-cycles on the moduli space of rank-2 stable vector bundles with fixed determinant of degree 1. We show the first Chow group of the moduli space is isomorphic to the zeroth Chow group of the curve.     

A remark on the moduli field of a curve

We give a purely algebro-geometric proof of the fact that every nonsingular projective curve can be defined over a finite extension of its moduli field. This extends a result byWolfart [7] to curves over fields of arbitrary characteristic. Received: 30 November 2001     

Classification of modules of dimension n and multiplicity one over the weyl algebraA n

We give a classification of modules with Gel’fand-Kirillov dimensionn and multiplicity one over the Weyl algebran A _n.