Moduli Space Of Semistable Bundles Research Articles

On the structure of the algebra generated by the rational equivalence classes of Brill–Noether loci in the Chow ring of the moduli space of semistable bundles on elliptic curve

On the structure of the algebra generated by the rational equivalence classes of Brill–Noether loci in the Chow ring of the moduli space of semistable bundles on elliptic curve

Tautological Algebra of the Moduli Space of Semistable Bundles on an Elliptic Curve

In this paper, our aim is to find the relations amongst the cohomology classes of Brill-Noether subvarieties of the moduli space of semistable bundles over an elliptic curve. We obtain results similar to the Poincare relations on a Jacobian variety.

Spinors, Lagrangians and rank 2 Higgs bundles

The paper considers the Dirac operator on a Riemann surface coupled to a symplectic holomorphic vector bundle W. Each spinor in the null-space generates through the moment map a Higgs bundle, and varying W one obtains a holomorphic Lagrangian subvariety in the moduli space of Higgs bundles. Applying this to the irreducible symplectic representations of SL(2) we obtain Lagrangian submanifolds of the rank 2 moduli space which link up with m-period points on the Prym variety of the spectral curve as well as Brill-Noether loci on the moduli space of semistable bundles. The case of genus 2 is investigated in some detail.

Geometry of Vector Bundle Extensions and Applications to a Generalised Theta Divisor

Let $E$ and $F$ be vector bundles over a complex projective smooth curve $X$, and suppose that $0 \to E \to W \to F \to 0$ is a nontrivial extension. Let $G \subseteq F$ be a subbundle and $D$ an effective divisor on $X$. We give a criterion for the subsheaf $G(-D) \subset F$ to lift to $W$, in terms of the geometry of a scroll in the extension space ${\mathbf{P}} H^{1}(X, \mathrm{Hom}(F, E))$. We use this criterion to describe the tangent cone to the generalised theta divisor on the moduli space of semistable bundles of rank $r$ and slope $g-1$ over $X$, at a stable point. This gives a generalisation of a case of the Riemann-Kempf singularity theorem for line bundles over $X$. In the same vein, we generalise the geometric Riemann-Roch theorem to vector bundles of slope $g-1$ and arbitrary rank.

LOCI OF RATIONAL CURVES OF SMALL DEGREE ON THE MODULI SPACE OF VECTOR BUNDLES

For a smooth algebraic curve C of genus g 4, let (r, d) be the moduli space of semistable bundles of rank r 2 over C with fixed determinant of degree d. When (r,d) = 1, it is known that (r, d) is a smooth Fano variety of Picard number 1, whose rational curves passing through a general point have degree r with respect to the ampl generator of Pic((r, d)). In this paper, we study the locus swept out by the rational curves on (r, d) of degree (r, d).

On the intersection theory of Quot schemes and moduli of bundles with sections

We consider a class of tautological top intersection products on the moduli space of stable pairs consisting of vector bundles together with N sections on a smooth complex curve C. We show that when N is large, these intersection numbers can be viewed as arising on the Grothendieck Quot scheme of coherent sheaf quotients of the rank N trivial sheaf on C. The result has applications to the computation of the intersection theory of the moduli space of semistable bundles on C.

Cohomology of the moduli space of Hecke cycles

Let X be a smooth projective curve of genus g ⩾ 3 and let M 0 be the moduli space of semistable bundles over X of rank 2 with trivial determinant. Three different desingularizations of M 0 have been constructed by Seshadri (Proceedings of the International Symposium on Algebraic Geometry, 1978, 155), Narasimhan–Ramanan (C. P. Ramanujam—A Tribute, 1978, 231), and Kirwan (Proc. London Math. Soc. 65(3) (1992) 474). In this paper, we construct a birational morphism from Kirwan's desingularization to Narasimhan–Ramanan's, and prove that the Narasimhan–Ramanan's desingularization (called the moduli space of Hecke cycles) is the intermediate variety between Kirwan's and Seshadri's as was conjectured recently in (Math. Ann. 330 (2004) 491). As a by-product, we compute the cohomology of the moduli space of Hecke cycles.

Sigma models for bundles on Calabi–Yau: A proposal for Matrix string compactifications

We describe a class of supersymmetric gauged linear sigma-model, whose target space is the infinite-dimensional space of bundles on a Calabi–Yau 3- or 2-fold. This target space can be considered the configuration space of D-branes wrapped around the Calabi–Yau. We propose that this model can be used to define matrix string theory compactifications. In the infrared limit the model flows to a superconformal non-linear sigma-model whose target space is the moduli space of BPS configurations of branes on the compact space, containing the moduli space of semi-stable bundles. We argue that the bulk degrees of freedom decouple in the infrared limit if semi-stability implies stability. We study topological versions of the model on Calabi–Yau 3-folds. The resulting B-model is argued to be equivalent to the holomorphic Chern–Simons theory proposed by Witten. The A-model and half-twisted model define the quantum cohomology ring and the elliptic genus, respectively, of the moduli space of stable bundles on a Calabi–Yau 3-fold.

On four-dimensional compactifications of F-theory

Branches of moduli space of F-theory in four dimensions are investigated. The transition between two branches is described as a 3-brane-instanton transition on a 7-brane. A dual heterotic picture of the transition is presented and the F-theory-heterotic theory map is given. The F-theory data, complex structure of the Calabi-Yau fourfold and the instanton bundle on the 7-brane is mapped to the heterotic bundle on the elliptic Calabi-Yau threefold CY 3. The full moduli space has a web structure which is also found in the moduli space of semi-stable bundles on CY 3. Matter content of the four-dimensional theory is discussed in both F-theory and heterotic theory descriptions.

Local structure of the moduli space of vector bundles over curves

We analyze the local structure of the moduli space of semi-stable bundles on a curve. In particular, a complete description of the local structure is given in the rank 2 case. We obtain as a corollary of this analysis new results about the Kummer variety and the Coble quartic associated to a canonical genus 3 curve.

On the Deformations of Certain Moduli Spaces of Vector Bundles

Introduction. Let X be a smooth projective curve of genus g > 2 and Md(n, L) the moduli space of semi-stable bundles E of rank n and degree d, with det(E) z L. When (n, d) = 1 the variety Md(n, L) is smooth projective. For these varieties a theorem of Narasimhan and Ramanan [NR2] states that the number of moduli of Md(n, L) is the same as that of the curve X. The purpose of this paper is to prove this statement for the smooth compactification N, constructed by Seshadri in [S2], of Mo(2, Ox)s, (the moduli space of stable bundles, over a curve X of genus g > 5). More precisely, we prove the following