Hodge theory, tropical geometry and o-minimality  
Berlin, October 30th-31th 2023




Monday October 30th


9:15 - 10:30 Gregorio Baldi
The tame geometry of the Hodge locus
11 - 12:15 Yohan Brunebarbe
Uniformization of complex algebraic varieties
14 - 15:15 Martin Ulirsch
Archimedean and non-Archimedean P=W phenomena on abelian varieties
15:30 - 16:45 Mirko Mauri
Remarks on the topology of hyperkähler varieties
17 - 18:15 Vasily Rogov
Definable structures on universal covers



Tuesday October 31st


9:15 - 10:30 Javier Fresan
Exponential period functions
11 - 12:15 Helge Ruddat
Explicit period integrals over tropically constructed cycles
14 - 15:15 Beatrice Pozzetti
The real spectrum compactification of character varieties
15:30 - 16:45 Benjamin Schröter
Chow rings and matroids



Abstracts

  • Karim Adiprasito (cancelled)

    The Chern-Hopf-Thurston conjecture states that the Euler characteristic of an aspherical closed and compact manifold should be of a sign dependent only on its dimension, and it remains mysterious to this day. Even when asphericity is more manageable, and the manifold nonpositively curved (which implies that it is aspherical) the conjecture, in that form emphasized by Chern and Thurston, is elusive.
    And even more puzzling is a strengthening of this particular case of the conjecture due to Gal, who, based on work of the combinatorialists Foata, Schützenberger and Strehl, introduced several other numbers, called γ-numbers, which generalize the signed Euler characteristic and are equally conjectured to be nonnegative. A topological or algebraic interpretation has eluded us so far.
    I will provide a gentle introduction for everyone, provide such an interpretation that implies nonnegativity, and relate it to work of Gromov on hyperkähler manifolds.

  • Gregorio Baldi

    I will give an overview of various forms of the Zilber-Pink conjecture and stress the role o-minimality plays both in the geometric and arithmetic parts of the conjecture. After a historical overview, I will focus on the case of V a variation of Hodge structures and explain the ‘’typical and atypical’’ dichotomy, which governs the Hodge locus of V. If time permits, and depending on the interest of the audience, I will present some recent applications of the theory in some ‘special’ cases.

  • Yohan Brunebarbe

    In an attempt to understand which complex analytic spaces can be realised as the universal covering of a complex algebraic variety, Shafarevich asked whether the universal covering of any smooth projective variety X is holomorphically convex. In other words, does there exists a proper holomorphic map from the universal covering of X to a Stein analytic space? Although still open, Shafarevich question has been answered positively e.g. when the fundamental group of X admits a faithful complex linear representation (Eyssidieux-Kaztarkov-Pantev-Ramachandran). In my talk, I will discuss the generalization of Shafarevich question to non-compact algebraic varieties.

  • Javier Fresan

    I will explain why every exponential period function of the form $\int_\sigma e^{-zf}\omega$, where $f$ is a regular function on an algebraic variety $X$ defined over the field of algebraic numbers, $\omega$ is an algebraic differential form on $X$, and $\sigma$ is a rapid decay cycle on $X(\mathbb{C})$, is a linear combination of E-functions ”with monodromy” with coefficients in the field generated by usual periods, special values of the gamma function and Euler’s constant. This is how E-functions arise from geometry and gives some intuition of why a positive answer to Siegel’s question about every E-function being a polynomial expression in hypergeometric E-functions was extremely unlikely. (Joint work with Peter Jossen).

  • Mirko Mauri

    The Nagai conjecture and the SYZ conjecture concern respectively the geometry of degenerations and fibrations of hyperkähler varieties.
    In this talk I will explore some topological consequences of these conjectures.
    In particular, I will show that the fundamental group of the regular locus of a hyperkähler variety endowed with a Lagrangian fibration is finite,
    and that the Hodge diamond of a hyperkähler manifold admits a three-dimensional refinement called perverse–Hodge octahedron (modulo Nagai conjecture).

    This is an account on a joint project with Daniel Huybrechts and an on-going project with Stefano Filipazzi and Roberto Svaldi.

  • Beatrice Pozzetti

    I will discuss joint work with Burger, Iozzi and Parreau in which we investigate properties of a natural (real) algebraic compactification of character varieties of finitely generated groups in semisimple Lie groups. After describing how points at infinity in such compactification can be interpreted as equivalence classes of actions on buildings, I will explain the relation with the Thurston-Parreau marked lenght spectrum compactification and mention applications to Hitchin and maximal character varieties.

  • Vasily Rogov

    The universal cover of a smooth complex algebraic variety is merely a complex manifold and does not a priori possess any algebraic or real semi-algebraic structure. A conjecture by Kollár and Pardon says that if the universal cover of a smooth projective variety is biholomorphic to a real semi-algebraic set (or, more generally, to a complex manifold definable in an o-minimal structure) it splits as a product of a simply connected projective variety and a Kähler homogeneous space. The action of the fundamental group by deck transformations extends in this case to a Lie group action by biholomorphic automorphisms.
    I am going to discuss the Kollár-Pardon conjecture and its variations. I am also going to address a counterpart of this conjecture in the context of Riemannian geometry. Surprisingly, the Riemannian version of Kollár-Pardon conjecture turns out to be much simpler, than its complex algebraic sibling.

  • Helge Ruddat

    The intrinsic mirror construction introduced by Mark Gross and Bernd Siebert associates to a log Calabi-Yau pair (Y,D) an integral affine pseudomanifold with a canonical wall structure from which one constructs a degenerating family of mirror dual varieties X. An explicit computation of period integrals of this dual family X exhibits the period as the logarithm of a product wall function in the wall structure. These wall functions on the other hand are generating functions of Gromov-Witten invariants of (Y,D). This relationship gives a new proof of Takahashi's conjecture and has the potential to yield a fundamental understanding and proof of enumerative mirror symmetry, a phenomenon that was first observed by mathematical physicists in the early 90s.

  • Benjamin Schröter

    nitiated by the work of Feichtner and Yuzvinsky and successfully extended by Adiprasito, Huh and Katz Chow rings and Hodge theory found their way into combinatorics. From a polyhedral and combinatroical point of view the stable intersection of tropical linear spaces is a matroid intersection and the chow relations reflect the geometry of valuations.

    In my talk I will explain this connection, show how one can obtain explicit expressions for many matroid invariants and take a brief look at other matroid rings and their homology. Based on work with Austin Alderete and Luis Ferroni.

  • Martin Ulirsch

    Let $X$ be a smooth projective complex variety. Simpson's non-abelian Hodge correspondence provides us with a real analytic isomorphism between the Betti moduli space of characters of $\pi_1(X)$ and the moduli space of topologically trivial semistable Higgs bundles on $X$. The P=W conjecture, recently proved by Maulik--Shen and Hausel--Mellit--Minets--Schiffmann, predicts that the perverse filtration on the cohomology of the Dolbeault moduli space agrees (up to index shift) with the weight filtration on the cohomology of the Betti moduli space, when $X$ is a compact Riemann surface. In this talk I will report on a project, in which we extend this $P=W$ phenomenon to $X$ being a complex abelian variety, where it takes a particularly simple form. The insights gained in this situation lead us to a non-Archimedean incarnation of the $P=W$ phenomenon on the $\ell$-adic cohomology of the Betti/Dolbeault moduli space of an abelian variety $X$ with maximally degenerate reduction over an algebraically closed non-Archimedean field $K$ of characteristic zero. A central new insight is that the reduced cohomology of the tropicalization of the moduli space of topologically trivial vector bundles on $X$ plays a role as a correction term.

    This talk is based on joint work with B. Bolognese and A. Küronya and joint work in progress with A. Gross, I. Kaur, and A. Werner.