A Character variety X_FG is a space of representations of a finitely generated group F into a Lie group G. The most interesting cases are when F is the fundamental group of a Kähler manifold M, and G is a reductive group, since then X_FG is homeomorphic to a space of so-called G-Higgs bundles over M. Typically, X_FG are singular algebraic varieties, defined over the integers, and many of its topological and arithmetic properties can be encoded in a polynomial generalization of the Euler-Poincaré characteristic: the E-polynomial.

# Seminars

The modular class is a generalization of the divergence of vector fields to other geometric structures such as Poisson manifolds and Lie algebroids. The modular class of a regular foliation involves a volume form on the conormal bundle and the associated Bott connection. It is a closed one form along the leaves. The vanishing of this modular class implies that there exists a volume form which is invariant along the leaves. In the singular case, the above definition can not be summoned since the conormal bundle may not be even well defined.

Links of Gorenstein toric isolated singularities are good toric contact

manifolds with zero first Chern class. In this talk I will present some

results relating contact and singularity invariants in this particular

toric context. Namely,

(i) I will explain why the contact mean Euler characteristic is equal

to the Euler characteristic of any crepant toric smooth resolution of

the singularity (joint work with Leonardo Macarini).

(ii) I will discuss applications of contact invariants of Lens spaces

Given a complex reductive group G, the moduli space M(G) of G-Higgs bundles on a curve has a natural hyperkähler structure and it comes equipped with an algebraically completely integrable system through the Hitchin fibration. These moduli spaces have played an important role in mirror symmetry and in the geometric Langlands program and thus it has become of particular interest the study of certain decorated special subvarieties (branes) of M(G).

Kiem proved that the moduli space of SL(2,C)-Higgs bundles over a smooth projective curve X admits a symplectic desingularization if and only if g(X) = 2. Based on work of Schedler and Bellamy, Tirelli reproduced this result to the case of GL(2, C)-Higgs bundles, proving as well that no symplectic desingularization of SL(n, C) and GL(n, C)-Higgs bundles exists besides these two cases where (n, g) = (2, 2).

Plücker formulas for complex plane algebraic curves, relate the genus, number of double points, cusps, inflections and bitangencies. These formulas are almost two hundred years old, but their real analogues for plane curves or surfaces in R4 are from the second half of the last century and later. I will briefly describe the theorem of Fabricius-Bjerre for plane curves and present some of the results and problems in generalising them for surfaces.

In this talk, we will cover some results on the nature of mixed Hodge structures on the cohomology of complex quasi-projective varieties. Specifically, we will be interested in varieties whose cohomology behaves as an exterior algebra. For those, we will manage to deduce the mixed Hodge structure on their symmetric products. These results will allows us to obtain a general formula for the mixed Hodge structure of free abelian character varieties for some reductive groups. We will study this formula in some detail for *G=Sp(n, C)*, a case handled by me and C.

We will describe a method for constructing geodesics on the space of Kähler metrics on a compact Kähler manifold. Applications to geometric quantization will also be discussed.

V. Alexeev proved in 1994 that the set *S* of self-intersections of the canonical class of stable surfaces satisfies the descending chain condition, this is, any monotone sequence is increasing. (This set *S* is a subset of the positive rational numbers.) In particular *S* has a minimum, and it may have accumulation points. I will discuss what is known about *S*, certain new theorems on accumulation points, and open questions. This is a joint work with José Ignacio Yáñez.

In this talk, we will review the famous Hopf problem, which dates back to 1948, of whether there is a complex structure on the round 6-sphere. Although some attempted answers have been advanced, both in the negative and in the positive directions, this problem is still open. We will review some of the work and partial results that appear in the literature and take a closer look at almost complex structures compatible with the metrics of constant sectional curvature.