Coimbra Salamanca

Algebraic Geometry Seminar

III Meeting - Coimbra, 9th and 10th of September 2011

Abstracts:

Mario García Fernández: In two well known cases the existence of a canonical metric in Kähler geometry is related to a stability condition in algebraic geometry -- first, the Hitchin-Kobayashi correspondence for Hermitian-Einstein metrics on vector bundles and second the Yau-Tian-Donaldson conjecture for constant scalar curvature Kähler (cscK) metrics on projective manifolds. In each of these theories balanced metrics play a crucial role. On the one hand the existence of a balanced metric can be shown to be equivalent to a stability condition in the sense of finite dimensional Geometric Invariant Theory. On the other hand, the asymptotic behaviour of a sequence of balanced metrics is governed by a ``density of states'' expansion, from which the Hermitian-Einstein or cscK equations can be extracted. In this talk we combine these ideas by considering simultaneously stability of a vector bundle and its underling manifold. This is joint work with Julius Ross.

Ana Cristina López Martín: Since its introduction by Mukai, the theory of integral functors and Fourier-Mukai transforms has been an important tool in the study of the geometry of varieties and moduli spaces. Working with a fibered scheme over a base T it is quite natural to look at the group of T-linear autoequivalences. The description of this group seems a hard problem. We will restrict ourselves to the subgroup given by relative Fourier-Mukai transforms. In this talk, I will explain how for a projective fibration the knowledge of the structure of the group of autoequivalences of its fibres and the properties of relative integral functors provide a machinery to study that subgroup. I will work out the case of a Weierstrass fibration and report about the results for abelian schemes and Fano or anti-Fano fibrations. This is a joint work with D. Sánchez Gómez and C. Tejero Prieto

Peter Gothen: We study the C*-action on Hausel and Schmitt's compactification of the moduli space of Hitchin pairs and use it to prove a Torelli type theorem. The talk is based on joint work with Biswas and Logares: arxiv:0912.4615.

Tomas L. Gómez: We compute the automorphism group of the moduli space of symplectic bundles over an algebraic curve. Joint work with I. Biswas and V. Muñoz.

André G. Oliveira: We consider holomorphic quadratic pairs of rank 2 over a smooth projective curve. The stability condition for these objects depends on a real parameter α, and we denote by N_α the moduli space of α-semistable quadratic pairs. As α varies, the difference between the moduli spaces are confined to certain subvarieties, which are called the flip loci, and it is through the study of the flip loci that we show that N_α is connected for every α. We shall also mention the relation between these spaces and the spaces of surface group representations. This is joint work with Peter Gothen.

Rita Pardini: The irregularity of a smooth complex projective surface is the number q of independent global 1-forms of S; there exist a complex torus of dimension q, the Albanese variety Alb(S), and a map S-->Alb(S), the Albanese map, through which any map S-->T, T a complex torus, factorizes. The Albanese dimension of a surface is the dimension of the image of the Albanese map. Little is known on surfaces of general type with Albanese dimension 2. I will propose an approach to the study of these surfaces via the analysis of the curves of small genus on them. This leads naturally to considering the Brill-Noether locus W(C) of a curve C of S, namely the set of line bundles P in Pic^0(S) such that the divisor C+P is effective. I will give a structure result for W(C) and show that it gives numerical restrictions on the curves of small genus on S. This is joint work with Margarida Mendes Lopes and Gian Pietro Pirola.