Let X be a smooth projective curve of genus at least two. Let G be an almost simple affine algebraic group. The universal principal G-bundle is called the Poincaré bundle. It is a principal G-bundle on the product of the curve and the moduli stack of principal G-bundles. We prove that it is stable with respect to any polarization (joint work with I. Biswas and N. Hoffmann).
In this introduction for geometers and topologists, we explain the role that central extensions of L-infinity algebras, a concept we will define, plays in physics. This connection first appeared with the work of physicists D’Auria and Fré in 1982, but is beautifully captured by the “brane bouquet” of Fiorenza, Sati and Schreiber which shows how physical objects such as “strings”, “D-branes” and “M- branes” can be classified by taking successive central extensions of an especially simple L-infinity algebra called the “supertranslation algebra”.
Mumford introduced in the 1960ies an algebraic approach to the construction of (almost) canonical bases of sections of ample line bundles on abelian varieties that permitted him to construct quasi-projective moduli spaces. His construction was later re-interpreted by Welters as a flat projective connection before being generalized by Hitchin to the non-abelian setting.
Let S be a smooth projective surface over C and B a smooth projective curve. A fibration f : S → B is a surjective morphism such that the general fibre is a smooth connected curve.
This talk will focus on some properties of fibrations with general fibre of genus ≥ 2, discussing in particular the existence and number of singular fibres on a fibration.
Let Λ be a D-algebra in the sense of Bernstein and Beilinson. Higgs bundles, vector bundles with flat connections, co-Higgs bundles... are examples of Λ-modules for particular choices of Λ. Simpson studied the moduli problem for the classification of Λ-modules over Kähler varieties, proving the existence of a moduli space of Λ-modules. Using the Polishchuck-Rothstein transform for modules of D-algebras over abelian varieties, we obtain a description of the moduli spaces of Λ-modules of rank 1. We also proof that polystable Λ-modules decompose as a direct sum of rank 1 Λ-modules.
Non-abelian gerbes are a generalization of principal G-bundles, involving the replacement of the Lie group G by a Lie 2-group, or crossed module of groups, not necessarily Abelian. Apart from providing a nice example of categorification in geometry, they have found a number of applications in physics, e.g. in higher gauge theory and topological states of matter.
We generalize the classical MacDonald formula for smooth curves to reduced curves with planar singularities. More precisely, we show that the cohomologies of the Hilbert schemes of points on a such a curve are encoded in the cohomologies of the fine compactified Jacobians of its connected subcurves, via the perverse Leray filtration. A crucial step in the proof is the case of nodal curves, where the weight polynomials of the spaces involved can be computed in terms of the underlying dual graph. This is a joint work with Luca Migliorini and Vivek Schende.
Among abelian varieties, Jacobians of smooth projective curves C have the important property of being autodual, i.e., they are canonically isomorphic to their dual abelian varieties. This is equivalent to the existence of a Poincaré line bundle P on J(C)×J(C) which is universal as a family of algebraically trivial line bundles on J(C). A yet other instance of this fact was discovered by S. Mukai, who proved that the Fourier-Mukai transform with kernel P is an auto-equivalence of the bounded derived category of J(C).
We present a recent result about the Riemannian metric structure of the tangent manifold TM, the total space of the tangent bundle T M → M of any given Riemannian manifold M. We recall how such space is endowed with a metric, due to S. Sasaki, and which are its main properties. Following this, we show the construction of a fully original Hermitian structure, called ciconia, which leads to interesting Kähler-Einstein and, in particular, non-compact Calabi-Yau manifolds.
The group of holomorphic diffeomorphisms, Aut(M), of a compact complex manifold M is a Lie group of
finite dimension. To provide bounds for the dimension of these groups is a classical problem in complex
analysis. It is well known that the dimension of Aut(M) cannot be bounded in terms of the dimension of M
solely. However several important problems arise once specific constraints are imposed on the manifold M.
For example, the case of homogeneous manifolds has been intensively studied in connection with which is