Programme

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Cours 1 : Local Systems in Arithmetic Geometry
Hélène Esnault

Cours 2 : Resolution of Singularities
Dan Abramovich

Cours 3 : Birational complexity of algebraic varieties
Robert Lazarsfeld

I will talk about a circle of ideas loosely centered around the theme of studying "how complicated" a given variety is from a birational viewpoint. For example, given a projective variety X whose irrationality is known, how can one quantify and control “how irrational” it is? Given two varieties of the same dimension, how far are they from being birationally isomorphic? The work in this direction is still at an experimental stage, so I will emphasize examples and especially open questions. They involve a lot of concrete down-to-earth geometry.

Cours 4 : Non Abelian Hodge Theory and Applications to the Shafarevich Conjecture
Philippe Eyssidieux

We will give an introduction to Corlette-Simpson’s non abelian Hodge theory and some if its applications.

Exposés des postdocs et des étudiants

Generic vanishing of Dolbeault cohomology of mixed Hodge modules on abelian varieties
Ze Yun, Stony Brook
Horaire : lundi

We study generic vanishing for the Dolbeault cohomology of mixed Hodge modules, where the differentials in the Dolbeault complex are modified by a one-form. We prove that, on the Dolbeault moduli space of Higgs line bundles, the cohomology support loci of mixed Hodge modules agree with the images of the cohomology support loci of their underlying \(D\)-modules on the de Rham moduli space under the map induced by the nonabelian Hodge correspondence. As an application, this gives an algebraic proof, without using harmonic metrics, of an equality in the same spirit as Simpson between the dimensions of Dolbeault cohomology and de Rham cohomology for mixed Hodge modules tensored with rank one flat connections. We further apply this result to study Fourier-Mukai transforms of mixed Hodge modules on abelian varieties. We show that the extended Fourier-Mukai transform for bounded complexes of mixed Hodge modules has cohomology sheaves that are flat over the twistor line of \(\lambda\)-connections.

Holomorphic Maps from \(\mathbb{C}^p\) into Semi-Abelian Varieties
Zhe Wang, Houston University
Horaire : lundi

In 1974, W. Stoll proposed a method of studying holomorphic functions of several complex variables by reducing them to one variable through fiber integration. In this talk, we use this method to extend two important results from holomorphic curves to holomorphic maps from \(\mathbb{C}^p\) into semi-abelian varieties: Bloch's theorem and the Second Main Theorem of Noguchi--Winkelmann--Yamanoi.

https://link.springer.com/content/pdf/10.1007/s12220-026-02508-8.pdf et https://arxiv.org/abs/2512.09805

Noether-Lefschetz Theory for Degree 5 and 6 Surfaces
Vidhu Adhihetty, Columbia University
Horaire : mercredi

We show that there are only finitely many Noether-Lefschetz loci in the moduli of degree 5 surfaces in \(\mathbb{P}^3_{\mathbb{C}}\) on which the generic Picard rank is larger than 2. We also show that the same is true for degree 6 surfaces, and moreover that there are only finitely many Noether-Lefschetz components of codimension less than \(h^{2,0}.\) The key inputs are recent developments in the field of atypical intersections developed by Baldi, Klingler, and Ullmo, as well as a key technical result on Mumford-Tate groups of Hodge structures which have indecomposable IVHS.

The Tate conjecture for abelian fourfolds over finite fields
Matt Broe, Boston University
Horaire : mercredi

We prove the Tate conjecture for abelian fourfolds over finite fields. The proof relies on techniques from Ancona’s proof of the Hodge standard conjecture for abelian fourfolds, and ultimately reduces to Markman’s results on the algebraicity of Weil classes on complex abelian varieties.

Reduction of Singularities in Poisson Geometry via Weighted Blowups
Boris Zupancic, McGill University
Horaire : mercredi

Hironaka's theorem on resolution of singularities via blowups fails in Poisson geometry, since blowups generally destroy Poisson structures. However, there exists a more general operation called a weighted blowup, which may provide a viable alternative. Weighted blowups have proven useful in the recent literature on resolution of singularities; for instance, Abramovich-Temkin-Wlodarczyk have shown that fast functorial resolution of singularities can always be achieved by sequentially blowing up the worst singularities via a canonical choice of weighted blowup. We take a similar approach to show that appropriate choices of weighted blowups produce reduction of singularities in low-dimensional Poisson geometry. In joint work with Simon Lapointe, Mykola Matviichuk and Brent Pym, we obtain: (1) a criterion for lifting a Poisson structure to a weighted blowup, (2) a classification of normal forms of Poisson varieties not admitting any weighted blowup, in low-dimension, and (3) a theorem on the extent to which weighted blowups can reduce the singularities of Poisson varieties, in low-dimension.

https://arxiv.org/pdf/2604.16698

Computing Jet Differentials and the Green-Griffiths-Lang Conjecture for Complements of Smooth Plane Curves
Jaziel Torres, University of Notre Dame
Horaire : mercredi pendant le buffet

We study the Green-Griffiths-Lang Conjecture for complements of smooth curves in \(\mathbb{P}^2\).

In particular, we develop a technique for explicitly computing jet differentials that obstruct entire curves in the complement.

For many curves, this allows us to give a computational certification that the complement of a given smooth curve satisfies the Green-Griffiths-Lang Conjecture, and in some examples, it allows us to explicitly compute the exceptional locus.

On Resultant and q-bic Polynomial
Runchi Tan, EPFL
Horaire : mercredi pendant le buffet

We are interested in understanding when two polynomials on \(\mathbb P^1_k\) share a common root. Although the question of whether two polynomials share a common root is classical, it admits a rich geometric interpretation. Rather than studying individual polynomials, one may consider the parameter space of all pairs of polynomials of prescribed degrees and ask for the locus consisting of those pairs that possess a common zero.