ICTS

An algebraic variety is rational if it is birational to the projective space or affine space of the same dimension. In dimension 1 and 2 and over the complex numbers, smooth projective rational varieties have several characterizations and in particular it is known that they are the same as unirational or rationally connected varieties. Starting from dimension 3, it has been proved in the 70's that there are varieties which are unirational (that is rationally dominated by projective space) but not rational. I will describe in this talk further obstructions to rationality, which have been proved recently to be very effective and powerful, as a consequence of the degeneration argument that I introduced. The key notion is that of decomposition of the diagonal in the spirit of Bloch and Srinivas.

The idea of tame topology was introduced by Grothendieck in "Esquisse d'un programme" and developed by model theorists under the name "o-minimal structures". In the first lecture I will define o-minimal structures, study their basic properties and explain their relevance to both complex algebraic geometry and diophantine geometry. In the second lecture I will explain how tame topology can be applied in Hodge theory to give a new proof of the algebraicity of the components of the Hodge locus , a celebrated result of Cattani-Deligne-Kaplan (joint work with Bakker and Tsimerman); and describe the Zariski-closure of the Hodge locus (joint work with Otwinowska).

Let X be a nonsingular complex projective toric variety. We address the question of semi-stability as well as stability for the tangent bundle TX. A large set of examples of toric varieties is given for which TX is unstable; the dimensions of this collection of varieties are unbounded. When X is a Fano variety of dimension four with Picard number at most two, we give a complete answer. Our method is based on the equivariant approach initiated by Klyachko and developed further by Perling and Kool. This is a joint work with Indranil Biswas, Mainak Poddar and Ozhan Genc.

An algebraic variety is rational if it is birational to the projective space or affine space of the same dimension. In dimension 1 and 2 and over the complex numbers, smooth projective rational varieties have several characterizations and in particular it is known that they are the same as unirational or rationally connected varieties. Starting from dimension 3, it has been proved in the 70's that there are varieties which are unirational (that is rationally dominated by projective space) but not rational. I will describe in this talk further obstructions to rationality, which have been proved recently to be very effective and powerful, as a consequence of the degeneration argument that I introduced. The key notion is that of decomposition of the diagonal in the spirit of Bloch and Srinivas.

This talk deals with holomorphic Cartan geometries on compact complex manifols. The concept of holomorphic Cartan geometry encapsulates many interesting geometric structures including holomorphic Riemannian metrics, holomorphic affine connections or holomorphic projective con- nections. Conjecturaly, a compact complex simply connected manifold bearing a holomorphic Cartan geometry with model the complex homogeneous space G/H, must be biholomorphic to G/H. We present here some results going toward this direction. In particular, we show that compact complex simply connected manifolds do not admit holomorphic Riemannian metrics. We also show that compact complex simply connectd manifolds in Fujiki class C bearing holo- morphic Cartan geometries of algebraic type are projective. Those results were obtain in a joint work with Indranil Biswas (TIFR).

An algebraic variety is rational if it is birational to the projective space or affine space of the same dimension. In dimension 1 and 2 and over the complex numbers, smooth projective rational varieties have several characterizations and in particular it is known that they are the same as unirational or rationally connected varieties. Starting from dimension 3, it has been proved in the 70's that there are varieties which are unirational (that is rationally dominated by projective space) but not rational. I will describe in this talk further obstructions to rationality, which have been proved recently to be very effective and powerful, as a consequence of the degeneration argument that I introduced. The key notion is that of decomposition of the diagonal in the spirit of Bloch and Srinivas.

I will survey my joint work with Thomas Delzant, which describes which aspherical Kähler manifolds have the same homotopy type as a nonpositively curved cubical complex. These manifolds are all finitely covered by holomorphic principal torus bundles over products of Riemann surfaces. Along the way we describe a new criterion to produce fibrations of compact Kähler manifolds over Riemann surfaces.

We discuss estimates of the Bergman kernel associated to the cotangent bundle defined over a hyperbolic Riemann surface, both along the diagonal and away from the diagonal. We then discuss arithmetic applications of these estimates.

An algebraic variety is rational if it is birational to the projective space or affine space of the same dimension. In dimension 1 and 2 and over the complex numbers, smooth projective rational varieties have several characterizations and in particular it is known that they are the same as unirational or rationally connected varieties. Starting from dimension 3, it has been proved in the 70's that there are varieties which are unirational (that is rationally dominated by projective space) but not rational. I will describe in this talk further obstructions to rationality, which have been proved recently to be very effective and powerful, as a consequence of the degeneration argument that I introduced. The key notion is that of decomposition of the diagonal in the spirit of Bloch and Srinivas.

We study the loci of the abelian varieties dominated by hyperelliptic Jacobians. Consider a closed subvariety of A_g of the moduli space of principally polarized varieties of dimension g>3. We prove that if a very general element of Y is dominated by the Jacobian of a curve C and dim Y>2g, then C is not hyperelliptic. Finally we discuss the more intricate problem of the loci of curves such that their Jacobians are dominated by hyperelliptic Jacobians. The results have been obtain is collaboration with J. Carlos Naranjo.

In this talk, we will study the moduli space of genus one curves and and the relation between certain codimension two cycles; this is a result due to Getzler. We will then see how Getzler's relationship, enables one to compute the number of degree d, genus one curves in P^2 (with a variable complex structure), passing through 3d points.