Moduli Space Of Vector Bundles Research Articles

Statistics of moduli space of vector bundles

Let X be a smooth projective curve of genus g≥1 over a finite field Fq with q elements, such that the function field Fq(X) is a geometric Galois extension of the rational function field Fq(x) with N=#Gal(Fq(X)/Fq(x)). Let ML(2,1) be the moduli space of rank 2 stable vector bundles over X‾=X×FqFq‾ with fixed determinant L, where L is a line bundle on X‾ of degree 1. Let Nq(ML(2,1)) be the cardinality of the set of Fq-rational points of ML(2,1). We give an estimate of Nq(ML(2,1)) in terms of N,q and g. Further we study the fluctuations of the quantity logNq(ML(2,1))−3(g−1)logq as the curve X as well as L varies over a large family of hyperelliptic curves (N=2) of genus g≥2. We find the limiting distribution of Nq(ML(2,1))−3(g−1)log⁡q as g grows and q is fixed, in terms of its characteristic function. When both g and q grow we see that q3/2(log⁡Nq(ML(2,1))−3(g−1)log⁡q) has a standard Gaussian distribution.

The Picard group of the universal moduli space of vector bundles on stable curves

We construct the moduli stack of properly balanced vector bundles on semistable curves and we determine explicitly its Picard group. As a consequence, we obtain an explicit description of the Picard groups of the universal moduli stack of vector bundles on smooth curves and of the Schmitt's compactification over the stack of stable curves. We prove some results about the gerbe structure of the universal moduli stack over its rigidification by the natural action of the multiplicative group. In particular, we give necessary and sufficient conditions for the existence of Poincaré bundles over the universal curve of an open substack of the rigidification, generalizing a result of Mestrano–Ramanan.

Fourier–Mukai Transform on a Compactified Jacobian

Abstract We use Fourier–Mukai transform to compute the cohomology of the Picard bundles on the compactified Jacobian of an integral nodal curve $Y$. We prove that the transform gives an injective morphism from the moduli space of vector bundles of rank $r \ge 2$ and degree $d$ ($d$ sufficiently large) on $Y$ to the moduli space of vector bundles of a fixed rank and fixed Chern classes on the compactified Jacobian of $Y$. We show that this morphism induces a morphism from the moduli space of vector bundles of rank $r \ge 2$ and a fixed determinant of degree $d$ on $Y$ to the moduli space of vector bundles of a fixed rank with a fixed determinant and fixed Chern classes on the compactified Jacobian of $Y$.

Derived categories of moduli spaces of vector bundles on curves

Let X be a smooth projective curve of genus g≥4 over C and M the moduli space of stable vector bundles of rank 2 and determinant isomorphic to a fixed line bundle of degree 1. We prove in this paper that Db(X) is embedded (via ΦE) in Db(M) as a full subcategory (if g≥4).

Compact moduli spaces for slope-semistable sheaves

We resolve pathological wall-crossing phenomena for moduli spaces of sheaves on higher-dimensional base manifolds. This is achieved by considering slope-semistability with respect to movable curves rather than divisors. Moreover, given a projective n-fold and a curve C that arises as the complete intersection of n-1 very ample divisors, we construct a modular compactification of the moduli space of vector bundles that are slope-stable with respect to C. Our construction generalises the algebro-geometric construction of the Donaldson-Uhlenbeck compactification by Joseph Le Potier and Jun Li. Furthermore, we describe the geometry of the newly construced moduli spaces by relating them to moduli spaces of simple sheaves and to Gieseker-Maruyama moduli spaces.

Finite group actions on moduli spaces of vector bundles

We study actions of finite groups on moduli spaces of stable holomorphic vector bundles and relate the fixed-point sets of those actions to representation varieties of orbifold fundamental groups.

An inequality for Betti numbers of hyper-Kähler manifolds of dimension 6

A compact hyper-Kahler manifold M for which π1(M) = 0 and H2,0(M) = C is said to be simple (irreducible). According to Bogomolov’s celebrated theorem, any compact hyper-Kahler manifold admits a finite covering by a product of the torus and several irreducible hyper-Kahler manifolds [1]. In complex dimensions 4 and higher, two series of hyper-Kahler manifolds are known, the Hilbert schemes of n points over K3 and the generalized Kummer manifolds [2]; in addition, there are two sporadic examples due to O’Grady (see [3], [4]). It has been proved that the moduli spaces of vector bundles with numerical parameters different from those in the examples mentioned above admit no symplectic resolution [5].

Moduli spaces of vector bundles on a singular rational ruled surface

We study moduli spaces M-X (r, c(1), c(2)) parametrizing slope semistable vector bundles of rank r and fixed Chern classes c(1), c(2) on a ruled surface whose base is a rational nodal curve. We showthat under certain conditions, these moduli spaces are irreducible, smooth and rational (when non-empty). We also prove that they are non-empty in some cases. We show that for a rational ruled surface defined over real numbers, the moduli space M-X (r, c(1), c(2)) is rational as a variety defined over R.

Moduli of decorated swamps on a smooth projective curve

In order to unify the construction of the moduli space of vector bundles with different types of global decorations, such as Higgs bundles, framed vector bundles and conic bundles, A. H. W. Schmitt introduced the concept of a swamp. In this work, we consider vector bundles with both a global and a local decoration over a fixed point of the base. This generalizes the notion of parabolic vector bundles, vector bundles with a level structure and parabolic Higgs bundles. We introduce a notion of stability and construct the coarse moduli space for these objects as the GIT-quotient of a parameter space. In the case of parabolic vector bundles and vector bundles with a level structure our stability concept reproduces the known ones. Thus, our work unifies the construction of their moduli spaces.

Flat Rank Two Vector Bundles on Genus Two Curves

We study the moduli space of trace-free irreducible rank 2 connections over a curve of genus 2 and the forgetful map towards the moduli space of under- lying vector bundles (including unstable bundles), for which we compute a natural Lagrangian rational section. As a particularity of the genus 2 case, connections as above are invariant under the hyperelliptic involution : they descend as rank 2 logarithmic connections over the Riemann sphere. We establish explicit links between the well-known moduli space of the underlying parabolic bundles with the classical approaches by Narasimhan-Ramanan, Tyurin and Bertram. This allow us to explain a certain number of geometric phenomena in the considered moduli spaces such as the classical (16, 6)-configuration of the Kummer surface. We also recover a Poincare family due to Bolognesi on a degree 2 cover of the Narasimhan-Ramanan moduli space. We explicitly compute the Hitchin integrable system on the moduli space of Higgs bundles and compare the Hitchin Hamiltonians with those found by vanGeemen-Previato. We explicitly describe the isomonodromic foliation in the moduli space of vector bundles with sl(2,C)-connection over curves of genus 2 and prove the transversality of the induced flow with the locus of unstable bundles.

Degenerating Kähler structures and geometric quantization

We review some recent results on the problem of the choice of polarization in geometric quantization. Specifically, we describe the general philosophy, developed by the author together with his collaborators, of treating real polarizations as limits of degenerating families of holomorphic polarizations. We first review briefly the general framework of geometric quantization, with a particular focus on the problem of the dependence of quantization on the choice of polarization. The problem of quantization in real polarizations is emphasized. We then describe the relation between quantization in real and Kähler polarizations in some families of symplectic manifolds, that can be explicitly quantized and that constitute an important class of examples: cotangent bundles of Lie groups, abelian varieties and toric varieties. Applications to theta functions and moduli spaces of vector bundles on curves are also reviewed.

On rationality of moduli spaces of vector bundles on real Hirzebruch surfaces

Let X be a real form of a Hirzebruch surface. Let M H (r,c 1, c 2) be the moduli space of vector bundles on X. Under some numerical conditions on r, c 1 and c 2, we identify those M H (r,c 1,c 2) that are rational.

On the Hilbert scheme of the moduli space of vector bundles over an algebraic curve

Let M(n, ξ) be the moduli space of stable vector bundles of rank n ≥ 3 and fixed determinant ξ over a complex smooth projective algebraic curve X of genus g ≥ 4. We use the gonality of the curve and r-Hecke morphisms to describe a smooth open set of an irreducible component of the Hilbert scheme of M(n, ξ), and to compute its dimension. We prove similar results for the scheme of morphisms \({M or_P (\mathbb{G}, M(n, \xi))}\) and the moduli space of stable bundles over \({X \times \mathbb{G}}\), where \({\mathbb{G}}\) is the Grassmannian \({\mathbb{G}(n - r, \mathbb{C}^n)}\). Moreover, we give sufficient conditions for \({M or_{2ns}(\mathbb{P}^1, M(n, \xi))}\) to be non-empty, when s ≥ 1.

Plücker forms and the theta map

Let ${\rm SU}_X(r,0)$ be the moduli space of semistable vector bundles of rank r and trivial determinant over a smooth, irreducible, complex projective curve $X$. The theta map $\theta_r: {\rm SU}_X(r,0) \rightarrow \Bbb{P}^N$ is the rational map defined by the ample generator of ${\rm Pic}{\rm SU}_X(r,0)$. The main result of the paper is that $\theta_r$ is generically injective if $g \gg r$ and $X$ is general. This partially answers the following conjecture proposed by Beauville: $\theta_r$ is generically injective if $X$ is not hyperelliptic. The proof relies on the study of the injectivity of the determinant map $d_E: {\wedge}^r H^0(E) \rightarrow H^0(\det E)$, for a vector bundle $E$ on $X$, and on the reconstruction of the Grassmannian $G(r,rm)$ from a natural multilinear form associated to it, defined in the paper as the Pl\"ucker form. The method applies to other moduli spaces of vector bundles on a projective variety $X$.

Automorphisms of moduli spaces of vector bundles over a curve

Let X be an irreducible smooth complex projective curve of genus g≥4. Let M(r,Λ) be the moduli space of stable vector bundles E⟶X or rank r and fixed determinant Λ. We show that the automorphism group of M(r,Λ) is generated by automorphisms of the curve X, tensorization with suitable line bundles, and, in case r divides 2deg(Λ), also dualization of vector bundles.

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 MC (r; L) be the coarse moduli space of semistable vector bundles E over C of rank r with ∧rE = L. We show that the Brauer group of any desingularization of MC(r; L) is trivial.

Bubble tree compactification of moduli spaces of vector bundles on surfaces

In this article we announce some results on compactifying moduli spaces of rank-2 vector bundles on surfaces by spaces of vector bundles on trees of surfaces. This is thought as an algebraic counterpart of the so called bubbling of vector bundles and connections in differential geometry. The new moduli spaces are algebraic spaces arising as quotients by group actions according to a result of Koll\'ar. As an example the compactification of the space of stable rank-2 vector bundles with Chern classes $c_1=0, c_1=2$ on the projective plane is studied in more detail. Proofs are only indicated and will appear in separate papers.

On the Brill-Noether theory for K3 surfaces

Let (S, H) be a polarized K3 surface. We define Brill-Noether filtration on moduli spaces of vector bundles on S. Assume that (c 1(E), H) > 0 for a sheaf E in the moduli space. We give a formula for the expected dimension of the Brill-Noether subschemes. Following the classical theory for curves, we give a notion of Brill-Noether generic K3 surfaces. Studying correspondences between moduli spaces of coherent sheaves of different ranks on S, we prove our main theorem: polarized K3 surface which is generic in the sense of moduli is also generic in the sense of Brill-Noether theory (here H is the positive generator of the Picard group of S). The harder part of the proof is proving the non-emptiness of the Brill-Noether loci. In the case of algebraic curves such a theorem, proved by Griffiths and Harris and, independently, by Lazarsfeld, is sometimes called the strong theorem of the Brill-Noether theory. We finish by considering a number of projective examples. In particular, we construct explicitly Brill-Noether special K3 surfaces of genus 5 and 6 and show the relation with the theory of Brill-Noether special curves.

The Picard group of a coarse moduli space of vector bundles in positive characteristic

Let C be a smooth projective curve over an algebraically closed field of arbitrary characteristic. Let M ss r,L denote the projective coarse moduli scheme of semistable rank r vector bundles over C with fixed determinant L. We prove Pic(M ss r,L ) = ℤ, identify the ample generator, and deduce that M ss r,L is locally factorial. In characteristic zero, this has already been proved by Drézet and Narasimhan. The main point of the present note is to circumvent the usual problems with Geometric Invariant Theory in positive characteristic.

COHERENT SYSTEMS AND MODULAR SUBAVRIETIES OF $\mathcal{SU}_C(r)$

Let C be an algebraic smooth complex curve of genus g > 1. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on C and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients (ℙr-1)rg// PGL (r), where r is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, α-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when C has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces.