SU X (r, L) is separably unirational

We show that the moduli space of SU _X (r, L) of rank r bundles of fixed determinant L on a smooth projective curve X is separably unirational.      

Chow group of 1-cycles on the moduli space of vector bundles of rank 2 over a curve

Let X be a non-singular complex projective curve of genus ≥3. Choose a point x ∈ X. Let M_x be the moduli space of stable bundles of rank 2 with determinant We prove that the Chow group CH^Q₁(M_x) of 1-cycles on M_x with rational coefficients is isomorphic to CH^Q₀(X). By studying the rational curves on M_x, it is not difficult to see that there exits a natural homomorphism CH₀(J)→CH₁(M_x) where J denotes the Jacobian of X. The crucial point is to show that this homomorphism induces a homomorphism CH₀(X)→CH₁(M_x), namely, to go from the infinite dimensional object CH₀(J) to the finite dimensional object CH₀(X). This is proved by relating the degeneration of Hecke curves on M_x to the second term I^*2 of Bloch's filtration on CH₀(J). Insong Choe was supported by KOSEF (R01-2003-000-11634-0).     

Monads and rank 3 vector bundles on quadrics

In this paper we give the classification of rank 3 vector bundles without “inner” cohomology on a quadric hypersurface (n > 3) by studying the associated monads.      

Rational curves on moduli spaces of vector bundles

We identify the spaces Hom_i(ℙ¹,M) fori = 1, 2, whereM is the moduli space of vector bundles of rank 2 and determinant isomorphic to ,x ₀ ∈X, on a compact Riemann surface of genusg ≥ 2.     

Automorphic pseudodifferential operators and vector bundles

Given a discrete subgroup Г of SL(2, ?), we consider its action on pseudodifferential operators whose coefficients are holomorphic functions on the Poincaré upper half plane H and construct a vector bundle over the quotient space Г\H whose sections can be identified with pseudodifferential operators invariant under such Г-action.     

Nef Divisors in Codimension One on the Moduli Space of Stable Curves

Let M _g be the moduli space of smooth curves of genus g 3, and M¯ _g the Deligne-Mumford compactification in terms of stable curves. Let M¯ _g ^[1] be an open set of M¯ _g consisting of stable curves of genus g with one node at most. In this paper, we determine the necessary and sufficient condition to guarantee that a -divisor D on M¯ _g is nef over M¯ _g ^[1], that is, (D · C) 0 for all irreducible curves C on M¯ _g with C M¯ _g ^[1] .     

A few splitting criteria for vector bundles

We prove a few splitting criteria for vector bundles on a quadric hypersurface and Grassmannians. We give also some cohomological splitting conditions for rank 2 bundles on multiprojective spaces. The tools are monads and a Beilinson’s type spectral sequence generalized by Costa and Miró-Roig.      

Density of vector bundles periodic under the action of Frobenius

Let X be a proper and smooth curve of genus g?2 over an algebraically closed field k of positive characteristic. If , it follows from Hrushovski's work on the geometry of difference schemes that the set of rank r vector bundles with trivial determinant over X that are periodic under the action of Frobenius is dense in the corresponding moduli space. Using the equivalence between Frobenius periodicity of a stable vector bundle and its triviality after pull-back by some finite étale cover of X (due to Lange and Stuhler) on the one hand, and specialization of the fundamental group on the other hand, we prove that the same result holds for any algebraically closed field of positive characteristic.     

Torelli Theorem for the Moduli Spaces of Rank 2 Quadratic Pairs

Let X be a smooth projective complex curve. We prove that a Torelli type theorem holds, under certain conditions, for the moduli space of α-polystable quadratic pairs on X of rank 2.     

The universal theta divisor over the moduli space of curves

By computing the class of the universal antiramification locus of the Gauss map, we obtain a complete birational classification by Kodaira dimension of the universal theta divisor over the moduli space of curves.     

Betti Numbers of parabolic U(2,1)-Higgs bundles moduli spaces

Let X be a compact Riemann surface together with a finite set of marked points. We use Morse theoretic techniques to compute the Betti numbers of the parabolic U(2,1)-Higgs bundles moduli spaces over X. We give examples for one marked point showing that the Poincaré polynomials depend on the system of weights of the parabolic bundle.      

Codes associated to the zero-schemes of sections of vector bundles

We consider the algebraic geometric codes associated to the zero-schemes of sections of vector bundles on a smooth projective variety. We give lower bounds for the minimum distances of the codes exploiting the Cayley–Bacharach property of zero-dimensional subschemes.     

Vanishing thetanull and hyperelliptic curves

Let ??_g,2 be the moduli space of curves of genus g with a level‐2 structure. We prove here that there is always a non hyperelliptic element in the intersection of four thetanull divisors in ??_6,2. We prove also that for all g ≥ 3, each component of the hyperelliptic locus in ??_g,2 is a connected component of the intersection of g – 2 thetanull divisors. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Mumford relations and the moduli of rank three stable bundles

The cohomology ring of the moduli space M(n,d) of semistable bundles of coprime rank n and degree d over a Riemann surface M of genus g 2 has again proven a rich source of interest in recent years. The rank two, odd degree case is now largely understood. In 1991 Kirwan [8] proved two long standing conjectures due to Mumford and to Newstead and Ramanan. Mumford conjectured that a certain set of relations form a complete set; the Newstead-Ramanan conjecture involved the vanishing of the Pontryagin ring. The Newstead–Ramanan conjecture was independently proven by Thaddeus [15] as a corollary to determining the intersection pairings. As yet though, little work has been done on the cohomology ring in higher rank cases. A simple numerical calculation shows that the Mumford relations themselves are not generally complete when n>2. However by generalising the methods of [8] and by introducing new relations, in a sense dual to the original relations conjectured by Mumford, we prove results corresponding to the Mumford and Newstead-Ramanan conjectures in the rank three case. Namely we show (Sect. 4) that the Mumford relations and these dual Mumford relations form a complete set for the rational cohomology ring of M(3,d) and show (Sect. 5) that the Pontryagin ring vanishes in degree 12g-8 and above.     

Vector bundles on a product of real cubic curves

In this paper we give a characterization of then-tuples (C ₁,...,C _n ) of nonsingular projective real cubic curves such that every topological complex vector bundle onC ₁×...×C _n admits an algebraic structure. The results are very explicit and can be expressed in an especially simple form for cubies defined over the rationals.The second author was supported by an NSF grant.     

A criterion for the strongly semistable principal bundles over a curve in positive characteristic

Let X be an irreducible smooth projective curve over an algebraically closed field k of positive characteristic and G a simple linear algebraic group over k. Fix a proper parabolic subgroup P of G and a nontrivial anti-dominant character λ of P. Given a principal G-bundle E_G over X, let E_G(λ) be the line bundle over E_G/P associated to the principal P-bundle E_G→E_G/P for the character λ. We prove that E_G is strongly semistable if and only if the line bundle E_G(λ) is numerically effective. For any connected reductive algebraic group H over k, a similar criterion is proved for strongly semistable H-bundles.     

Cohomology of the moduli of parabolic vector bundles

The purpose of this paper is to compute the Betti numbers of the moduli space ofparabolic vector bundles on a curve (see Seshadri [7], [8] and Mehta & Seshadri [4]), in the case where every semi-stable parabolic bundle is necessarily stable. We do this by generalizing the method of Atiyah and Bott [1] in the case of moduli of ordinary vector bundles. Recall that (see Seshadri [7]) the underlying topological space of the moduli of parabolic vector bundles is the space of equivalence classes of certain unitary representations of a discrete subgroup Γ which is a lattice in PSL (2,R). (The lattice Γ need not necessarily be co-compact). While the structure of the proof is essentially the same as that of Atiyah and Bott, there are some difficulties of a technical nature in the parabolic case. For instance the Harder-Narasimhan stratification has to be further refined in order to get the connected strata. These connected strata turn out to have different codimensions even when they are part of the same Harder-Narasimhan strata. If in addition to ‘stable = semistable’ the rank and degree are coprime, then the moduli space turns out to be torsion-free in its cohomology. The arrangement of the paper is as follows. In § 1 we prove the necessary basic results about algebraic families of parabolic bundles. These are generalizations of the corresponding results proved by Shatz [9]. Following this, in § 2 we generalize the analytical part of the argument of Atiyah and Bott (§ 14 of [1]). Finally in § 3 we show how to obtain an inductive formula for the Betti numbers of the moduli space. We illustrate our method by computing explicitly the Betti numbers in the special case of rank = 2, and one parabolic point.     

On a period map for SU(2) conformal vacua

We study some properties of a natural period map associated to SU(2) theta functions or, in other words, to SU(2) conformal vacua. In particular we will consider for such a map an analogue of the classical Torelli’s problem. The main result of the paper is a sufficient conditions for our period map to have the infinitesimal Torelli’s property.      

Nefness of adjoint bundles for ample vector bundles of corank 3

Let be an ample vector bundle of rank on a smooth complex projective variety X of dimension n. The aim of this paper is to describe the structure of pairs as above whose adjoint bundles are not nef for . Furthermore, we give some immediate consequences of this result in adjunction theory.     

Characteristic classes for complex bundles with trivial real reduction

This note concerns itself with a theory of characteristic classes for those complex bundles whose real reductions are trivial.