In this talk I will give an overview of the main concepts and results in D-module theory, and then switch to the notion of relative D-module. I will explain the main motivation for the study of holonomic relative modules given by Mochizuki's notion of mixed twistor D-module. We will explain the Riemann-Hilbert correspondence in this case as a joint work with Claude Sabbah.
A smooth cubic hypersurface X of dimension >1 is unirational. The variety of lines F(X) on these hypersurfaces is an essential tool to understand the geometry of X. In dimension 3, the study of F(X) enables to prove that X is always irrational.
In this talk we study the zeta function of F(X) and we obtain a simplified proof of the irrationality of a dense set of smooth cubic threefold. This is a joint work with D. Markouchevitch.
The geodesics for the Mabuchi metric on the space of Kaehler metrics on a manifold correspond to solutions of the homogeneous complex Monge-Ampere (HCMA) equation. We will describe a method for reducing the Cauchy problem for the HCMA equation with analytic initial data to finding a related Hamiltonian flow followed by a "complexification". Examples and applications will be discussed.
Given a reductive group G and an affine G-scheme X, constellations are G-equivariant sheaves over X such that their module of global sections has finite multiplicities. Prescribing these multiplicities by a function h, and imposing a stability condition $\theta$ there is a moduli space for $\theta$-stable constellations constructed by Becker and Terpereau, using Geometric Invariant Theory. This construction depends on a finite subset D of the set of irreducible representations of G.
Y. Yang observed 20 years ago that the 2-sphere is the only compact orientable surface admitting solutions of the Einstein-Bogomol'nyi equations, coupling vortices with gravity, and obtained sufficient conditions for the existence of cosmic strings in this situation. In this talk, we will give an algebro-geometric interpretation of Yang's conditions, and explain why they are in fact necessary for the existence of solutions.
The concept of curvature ellipse at a point of a surface immersed in 4-space has been known since a long time ago  and it has proven to be a useful tool in the study of the geometrical properties from both, the local and global viewpoint [1, 3, 4]. Its natural generalization to higher dimensional manifolds is given by the image of a convenient linear projection of a Veronese submanifold of order 2 in the normal space of the manifold at each point . We call it the curvature locus or curvature veronese.
Holomorphic chains on a Riemann surface are sequences of holomorphic bundles, connected by holomorphic maps. Triples are chains of length one. Chains arise naturally as fixed points of the C^*-action on the moduli space of Higgs bundles.
I will talk about Nijenhuis deformations of Lie-infinity algebras, a notion that unifies several Nijenhuis deformations, namely those of Lie algebras, Lie algebroids and Poisson structures. This is a Joint work with M. Azimi and C. Laurent-Gengoux.