ATMW Conference on Vector Bundles (2013)

Title:
Stable vector bundles and string theory.
Abstract:
Braun, He, Ovrut and Pantev have proposed a model in string theory which produces an effective theory compatible with the Standard Model of particle physics. Part or the data of this model are two vector bundles on an explicitly given Calabi-Yau manifold. It was conjectured that there was a polarization on this Calabi-Yau such that the given vector bundles are stable with respect to that polarization. In this work we find explicitly a region of the ample cone for which one the vector bundles is stable, and we also proof that the other vector bundle is always unstable. This is joint work with S.Lukic and I. Sols

Title:
Hodge conjecture for moduli space of pairs of rank less than 4 This work is a colaboration between Vicente Mu~noz and Andre G. Oliveira.

Abstract:
Let C be a smooth projective curve of genus g 2 over C. Fix n n and d 2 Z. A pair (E; -) over C consists of an algebraic vector bundle E of rank n and degree d over C and a section - 2 H0 (E). There is a concept of stability for pairs which depends on a real parameter . Let M(n; d) be the moduli space of -polystable pairs of rank n and degree d over C. In this talk we try to prove that for a generic curve C, the moduli space M(n; d) satises the Hodge Conjecture for n 4. To get this result, we prove that M(n; d) is motivated by the motive of C.

Title:
 Gromov h-principle and `special' projective manifolds

Abstract:
A connected complex manifold X is said to satisfy the h-principle if any continuous map cfrom any Stein manifold S to X is homotopic to some holomorphic h from S to X. Grauert showed (in another context: the holomorphic versus topological classification of complex vector bundles on Stein manifolds) results implying that any complex Lie group X satisfies this h-principle. This was later extended, by similar methods, by Gromov to the case of so-called manifolds with a `dominating spray'. This technical conditions implies in particular that X is C-connected (ie: that any two of its points are joint by chains of holomorphic images of C, the complex line).
In the other direction, we show, among other things, that if X is complex projective, and satisfies the h-
principle, then:
1. Any holomorphic map from X to a Brody-hyperbolic projective complex space Y is constant. (Brody- hyperbolic means that any holomorphic map from C to Y is constant).
2. X is `special'. This is an algebro-geometric notion `opposite' to `general type', playing a undamental role in birational classification, and conjecturally meaning `C-connected'. For example, a projective curve is special iff its genus is 0 or 1. A projective surface is special iff it is not of general type, and has an almost abelian fundamental group. In higher dimension, there is no simple characterisation of `special' projective manifolds. Rationally connected manifolds and manifolds with Kodaira dimension zero are special, but there exists n-dimensional `special' manifolds with any Kodaira dimension strictly less than n.
It is unknown whether or not some K3 surface satisfies the h-principle (all are `special') This is a joint work with J. Winkelmann.

Title:
Approximate Hermitian-Einstein structures

Abstract:
Let E be a holomorphic vector bundle over a compact Kähler manifold X. In order for E to carry a Hermitian-Einstein structure, it is necessary and sufficient that E be polystable. If E is not polystable, then by combining the Harder-Narasimhan and socle filtrations, one can still find a filtration of E by coherent subsheaves such that each of the successive quotients is polystable. Using such a filtration, Bradlow described a method for the construction of canonical Hermitian structures on E in the case where X is a compact Riemann surface. We will discuss generalizations of Bradlow's result for the higher-dimensional case and the principal bundle case. We will also mention versions of the result for Higgs bundles on a Kähler manifold and flat vector bundles on an affine manifold. (Joint works with Indranil Biswas, Steven B. Bradlow and Adam Jacob and with Indranil Biswas and John Loftin)

Title:
Quaternions again

Abstract:
We present yet another viewpoint of looking at quaternion bundles over arbitrary base schemes.

Title:
Equivariant principal bundles on nonsingular toric varieties.

Abstract:
 We give a classification of the equivariant holomorphic principal G bundle on nonsingular toric variety when G is an abelian group of $Gl_{k}(\mathbb C)$ or conjugate to some $\mathbb C^{*}^{l}$. This is a partial analogue to Klyachko's classification of equivariant vector bundles on toric varieties. This is a joint work with Mainak Poddar.

Title:
Moduli of equivariant sheaves and representations of Kronecker-McKay quiver

Abstract:
Using King's construction of moduli of representations, Alvarez-Consul and King gave another approach to construction of moduli of sheaves. They realized moduli of sheaves as closed subset of moduli of representations. In this talk we will present the construction of moduli of equivariant sheaves using moduli of representations of Kronecker-McKay quiver. We will also discuss some applications of this approach to theta functions.
This is a joint work with Umesh Dubey.

Title:

A note on Demazure character formula for negative dominant characters.

Abstract:
We describe the character of the top cohomology line bundles on a Schubert variety associated to negative dominant characters in terms of characters of global sections of certain line bundles on Schubert vaerieties.

Title:

Invariant Vector bundles of rank 2 on Hyperelliptic curves

Date

11.00

-

11.30

11.30 - 12.30

12.30

-

2.30

02.30 - 03.30

3.30-4.00

04.00-05.00

25/3/2013

Tea

Break

Lunch
Break

Tea
Break