TOPOLOGY OF ALGEBRAIC VARIETIES

TBA

The Shafarevich conjecture asserts that the universal cover of a smooth complex projective variety is holomorphically convex. Over the past three decades, this conjecture has been intensively studied in the setting where the fundamental group is a subgroup of a general linear group, a formulation known as the linear Shafarevich conjecture.

Building on the powerful techniques of non-abelian Hodge theory established by Simpson and Gromov–Schoen, and further developed by Katzarkov and Zuo, Eyssidieux first proved the conjecture in the case where the fundamental group is reductive. Later, Eyssidieux, Katzarkov, Pantev, and Ramachandran extended the result to the general linear case. The methods introduced in these works have since found numerous applications in the study of the topology and hyperbolicity of algebraic varieties.

In this minicourse, I will present the main ideas in Eyssidieux’s proof of the reductive Shafarevich conjecture and explain how these techniques can be applied to problems concerning the topology and hyperbolicity of algebraic varieties. If time permits, I will also discuss the proof of the linear Shafarevich conjecture, whose key ingredient is the work of Eyssidieux and Simpson on constructing a variation of mixed Hodge structures associated with suitable tautological representations of the fundamental group.

For a smooth projective complex curve, non-abelian Hodge theory establishes a profound link between its topological and algebro-geometric properties. It specifically relates complex representations of the fundamental group to algebraic vector bundles on the curve augmented by a Higgs field. This creates a transcendental bridge between two distinct moduli spaces: the character variety (which parameterizes representations) and the Hitchin moduli space (which parameterizes Higgs bundles). In 2008, de Cataldo, Hausel, and Migliorini introduced the P=W conjecture, proposing the equality of two filtrations of very different origin on these spaces: this conjecture asserts a precise correspondence between the perverse filtration on the cohomology of Hitchin systems and the weight filtration on the Hodge theory of the character variety, revealing a deep symmetry between their respective topological and algebraic structures. 

Since then, many P=W phenomena have been discovered way beyond the realm of non abelian Hodge Theory and they provide a unified interpretation of the symmetries enjoyed by the cohomology groups of certain moduli spaces and families complex algebraic varieties. In view of the recent proofs of the P=W conjecture, in these three lecture I will present and relate the latest results in the field, with a view on how P=W phenomena appear in multiple areas of algebraic geometry. 

The study of non-abelian cohomology was pioneered by Simpson in the 90s, and is, broadly speaking, the study of representations of the fundamental groups of algebraic varieties; on the other hand, many questions on this subject were already studied in the early 1900s by mathematicians such as Painlevé, Garnier, and Schlesinger. As for usual cohomology, there are many different realizations of non-abelian cohomology, such as Betti, de Rham, étale, and even crystalline, leading to a rich interplay between dynamics, differential equations, and arithmetic. I will give an introduction to this subject and survey some recent developments, in particular the study of an arithmetic invariant of non-abelian cohomology, known as the p-curvature. This minicourse is partly based on joint work with Daniel Litt.