Periods, Shafarevich Maps and Applications

Carlos Simpson, University of Nice: Twistor Space for Moduli of Local Systems on an Open Curve

We show how to construct a Deligne-Hitchin twistor space for the moduli of framed local systems on an open curve. Harmonic bundles give preferred sections, and the normal bundle of a preferred section supports a natural mixed twistor structure with weights 0, 1, 2. The weight 2 piece corresponds to changes of parabolic structure. The completion along a preferred section then gives a mixed twistor structure for the complete local ring of the moduli space of local systems, at smooth points.  Video

Bruno Klingler, Berlin: On the Algebraicity of the Hodge Locus

Weil (1979) noticed that, according to the Hodge conjecture, the Hodge locus of a polarized variation of Hodge structure of geometric origin should be a countable union of irreducible algebraic varieties. Cattani-Deligne-Kaplan (1995) proved that this is the case, even without the geometricity assumption. In this talk I will discuss the following question: when is the Hodge locus actually algebraic, rather than a countable union of algebraic varieties? Joint work with Baldi and Ullmo.  Video

Mark Green, UCLA: Extension Data of Global Families of Mixed Hodge Structures, with Applications to the Shafarevich Conjecture

Families of mixed Hodge structures have graded pieces which are Hodge structures, and then variable extension data between these Hodge structures.  Extension data between adjacent graded pieces is level 1, and with a degree skip of two are level 2, etc.  For global families, the monodromy on graded pieces plus the extension data of levels 1 and 2 turn out to be enough to capture all but discrete invariants. Using this leads to an alternate proof of the nilpotent case of Shafarevich’s Conjecture.  Video

Philippe Eyssidieux, Université Grenoble Alpes: L2 invariants of Hodge Modules and Shafarevich morphisms

I will outline my work, partially in progress and partially in collaboration with Bastien Jean, on L2 constructible cohomology, Mixed Hodge structures on the reduced L2 cohomology of a Mixed Hodge Module and a (still conjectural) interesting topological consequence of the linear Shafarevich conjecture.  Video

Ya Deng, Université de Lorraine: How Fundamental Groups of Algebraic Varieties Determine their Hyperbolicity

It is natural to ask whether one can characterize the hyperbolicity of algebraic varieties (e.g positivity of canonical bundles and algebraic degeneracy of entire curves) via their topology. In this talk, I will answer this question if their fundamental groups admit a linear representation. Precisely, I will give a sharp condition for the hyperbolicity of complex quasi-projective varieties via representation of fundamental groups. This fits well with the prediction of the strong Green-Griffiths-Lang conjecture. As an application, fundamental groups of special quasi-projective varieties must have nilpotent linear quotient, thus proving a conjecture by Campana in the linear case. This work is based on two joint works with Brotbek-Daskalopoulos-Mese, and Cadorel-Yamanoi.  Video

Donu Arapura, Purdue University: Torelli Action on Representation Varieties

I will report on some work in progress with Dick Hain. The starting point is a paper of Cappell, Lee, and Miller from a couple of decades ago, where they showed that the action of the Torelli group on the cohomology of the SU(2) representation variety of a smooth projective curve is nontrivial. Our goals were to clarify their arguments (for ourselves), and to try to refine them. Currently, we have a proof that the Torelli action on the degree 6 cohomology of the SU(n) representation variety is nontrivial for n > 1. This has some interesting Hodge theoretic consequences for the moduli of semistable bundles of rank n and trivial determinant that I will explain.  Video

Fabrizio Catanese, Bayreuth University: Nodal surfaces and Coding Theory

One of the most famous applications of the theory of period maps is the Torelli theorem for K3 surfaces. I will use it do determine in a simple way all the irreducible (connected) components of the variety of polarized  nodal K3 ’s, and their incidence relation. The main idea is to show that these components are in bijection with the isomorphism classes of some associated binary code \sK’, which in the geometric situation  is the quotient L’/L where L’ is the saturation of the lattice L generated by the hyperplane class H and by the classes of the (-2)-curves in the cohomology of the resolution (the K3 lattice). We classify the potentially ocurring  K3 codes \sK’, and then determine when does the associated lattice L’ admit a primitive embedding in the K3 lattice. A key concept is  of the shortening of a code: the codes of quartic surfaces are for instance all the shortenings of the Kummer code (and shortening corresponds to partial smoothing). Time permitting, I will also try to present the situation for surfaces of degree 5 and 6 in 3-space.  Video

Frédéric Campana, University of Nancy: On Nilpotent Quotients of Quasi-K\"ahler Groups

We show that if $X$ is a smooth complex quasi-projective manifold the quasi-Albanese map of which is proper, then the torsionfree nilpotent quotients of $\pi_1(X)$ are the same ones as those of the Stein factorisation of its quasi-Albanese image. This implies that the \'etale Galois cover of $X$ associated to the nilpotent completion of $\pi_1(X)$ is holomorphically convex. This last result is proved in the quasi-projective case by $3$ other methods in \cite{GGK}, which motivated the present work. When $X$ is `special' in the sense of \cite{Ca11}, we deduce that the torsion free nilpotent quotients of $\pi_1(X)$ are abelian. This property fails (as first observed in \cite{CDY}) when the quasi-Albanese map is not proper.  Joint work with R. Aguilar.  Video

Yohan Brunebarbe, Université de Bordeaux: Hyperbolicity in Presence of a Large Local System

Green-Griffiths and Lang have proposed several influential conjectures relating different notions of hyperbolicity for projective complex algebraic varieties. For example, they conjectured that the locus swept out by entire curves coincides with the locus swept out by subvarieties not of general type, at least after taking Zariski closures. I will explain that some of these conjectures (including the one above) are true for varieties admitting a large complex local system (e.g. any variety admitting a variation of mixed Hodge structures with a finite period map).  Video

Benoit Cadorel, Institut Élie Cartan de Lorraine: Complex Hyperbolicity and Representations of the Fundamental Group in the Non-Compact Case

Complex varieties which admit "big" representations of their fundamental group tend to satisfy many algebraic or transcendental hyperbolicity properties. We will present here a joint work with Y. Deng and K. Yamanoi, where we prove such hyperbolicity results for complex quasi-projective varieties admitting a semi-simple, Zariski dense, big representation; this extends to the quasi-projective setting some important earlier work of Campana-Claudon-Eyssidieux, Yamanoi, Brunebarbe... As is now quite usual in this theory, the proof mixes tools from the archimedean and non-archimedean setting: this talk will mostly focus on the archimedean part; the non-archimedean constructions based on building theory and the Katzarkov-Zuo reduction will be introduced in an earlier talk of Y. Deng.  Video

Laurentiu Maxim, University of Wisconsin-Madison: A Homological Interpretation of Higher Du Bois and Higher Rational Singularities

The notions of higher Du Bois and higher rational singularities of hypersurfaces were recently introduced and studied by Jung-Kim-Saito-Yoon, Mustata-Olano-Popa-Witaszek and Friedman-Laza, as natural generalizations of Du Bois and rational singularities, respectively. In this talk I will present a homological characterization of these notions, in terms of characteristic classes introduced in prior work with M. Saito and J. Schuermann. As a preliminary step, we describe such singularities using the Hodge filtration on the vanishing cycle complex. (Joint work with R. Yang.)  Video

Louis-Clément Lefèvre, Lycée Hoche, Versailles: Mixed Hodge Structures on Cohomology Jump Ideals

The formal local ring in the representation variety of Kähler manifolds was first described by Goldman and Millson. Reviewing their arguments, we constructed mixed Hodge structures on these local rings at representations given by the monodromy of variations of mixed Hodge structures and extended this to non-compact varieties. Inside the formal local rings are the cohomology jump ideals, defining locally the locus of representations with dimensions constraints on their cohomology groups. Our goal is now to show that these ideals are sub-mixed Hodge structures.  Video

Benoit Claudon, Rennes: The Miyaoka--Yau Equality for Pairs

In this talk, I will report on a result that provides us with a full characterization of ball quotients in terms of Chern classes for klt pairs. This is a joint work with P. Graf and H. Guenancia.  Video

Ingrid Bauer, Bayreuth University: On Rigid Compact Complex Manifolds

Roughly speaking a compact complex manifold is rigid, if it has no (non trivial) deformations. In this talk I will introduce several notions of rigidity of compact complex manifolds, discuss their relations and present open questions and conjectures. Then I will concentrate on product - quotient varieties, i.e., (not necessarily smooth) quotients of a product of compact Riemann surfaces by the action of a finite group, showing how to obtain in this way rigid manifolds with interesting properties. I will report on examples of rigid, but not infinitesimally rigid surfaces of general type, giving an answer to a question of Morrow-Kodaira, which was open for more than 40 years.  Finally, I will discuss  recent results on rigid product-quotient varieties of dimension at least three with Kodaira dimension 0 and 1.  Video

B. Bakker, UIC: The Geometric Andre--Grothendieck Period Conjecture

A period integral of a complex algebraic variety is the integral of an algebraic differential form along a topological cycle.  These numbers are at the heart of Hodge theory.  In this talk I will explain how to prove a version of the Ax--Schanuel conjecture for these period integrals in families, and how it provides a capstone to the advances
in the transcendence theory of period maps made over the past decade. I will also discuss the relationship with the functional version of the Andre--Grothendieck period conjecture, which predicts that all algebraic relations between such periods integrals arise from geometry. This is joint work with J. Tsimerman.  Video

Rita Pardini, Università di Pisa: Exploring the Boundary of the Moduli Space of Stable Surfaces: I-Surfaces, a Test Case

An I-surface is a minimal complex surface of general type with K^2=1 and p_g=2. I will review recent joint work with Coughlan, Franciosi, Rana, Rollenske (in various combinations) giving a partial description of the boundary points of the KSBA compactification of the moduli space of I-surfaces.  Video

JongHae Keum, Korea Institute for Advanced Study: Fake Projective Planes

A smooth compact complex surface with the same Betti numbers as the complex projective plane P^2 is either P^2 or is called a fake projective plane(FPP). Indeed, a smooth compact complex surface with Betti numbers b_0=b_2=b_4=1, b_1=b_3=0 has Chern numbers c_2 =3,  c_1^2 = K^2 = 9, Picard number= 2nd Betti number = 1; thus, its canonical class K is either ample or anti-ample, and in the latter case it is isomorphic to P^2. In other words, a FPP is a surface of general type with geometric genus p_g = 0 and K^2 = 9. Furthermore, it can be uniformized by it universal cover, the unit complex 2-ball, by Aubin and Yau, and its fundamental group is a co-compact arithmetic subgroup of PU(2, 1) by Klingler. The existence of such a surface was first proved by Mumford in 1979, via 2-adic uniformization method. Algebraic varieties are not always described via polynomial equations: sometimes they are constructed via uniformization: this means, as quotients of bounded symmetric domains, via the action of discontinuous groups. General theorems (as Kodaira's) imply the algebraicity of these quotient complex manifolds. The problem concerning the algebro-geometrical properties of such varieties constructed via uniformization and especially the description of their projective embeddings (and the corresponding polynomial equations) lies at the crossroads of several allied fields: the theory of arithmetic groups and division algebras, complex algebraic and differential geometry, use of group symmetries, and topological and homological tools in the study of quotient spaces. Of particular importance are the so-called ball quotients, especially in dimension 2, since they yield the surfaces with the maximal possible canonical volume K^2 for a fixed value of the geometric genus p_g. In this talk I will report recent progress on FPPs, such as their derived categories, bicanonical maps and their equations.  Video

Manuel Gonzalez Villa, CIMAT: On a Quadratic Form Associated with a Surface Automorphism and its Applications to Singularity Theory

We study the nilpotent part N of a pseudo-periodic automorphism h of a real oriented surface with boundary Σ. We associate a quadratic form Q defined on the first homology group (relative to the boundary) of the surface Σ. Using the twist formula and techniques from mapping class group theory, we prove that after killing ker N the form becomes positive definite if all the screw numbers associated with certain orbits of annuli are positive. The case of monodromy automorphisms of Milnor fibers Σ = F of germs of curves on normal surface singularities will be discussed in detail.  Video

Mohan Ramachandran, University at Buffalo: Filtered Ends and Bochner Hartogs Dichotomy on Complete Kahler Manifolds 

We will discuss the relationship between a generalization of the notion of ends and the  first compactly supported cohomology with coefficients in the structure sheaf of a complete Kahler manifold. This is joint work with T Napier.  Video

Philipp Naumann, Bayreuth University: Curvature Formula for Direct Images of Relative Canonical Bundles with a Poincaré Type Twist

We give a curvature formula of the L^2 metric on the direct image of the relative canonical bundle twisted by a holomorphic line bundle endowed with a positive singular metric whose inverse has Poincaré type singularities along a relative snc divisor. The result applies to families of log canonically polarized pairs. Moreover, we show that it improves the general positivity result of Berndtsson-Paun in a special situation of a big line bundle. A generalisation to higher direct images would allow to prove the Kobayashi hyperbolicity of the moduli space of log canonically polarised manifolds.  Video

Daniel Litt, University of Toronto: Non-Abelian Big Monodromy

Let $X\to S$ be a family of smooth proper curves, and let $s\in S$ be a point. Then $\pi_1(S,s)$ has a natural outer action on $\pi_1(X_s)$. In joint work with Aaron Landesman, we study manifestations of the slogan that "monodromy groups should be as big as possible" in this non-abelian setting, by studying the action of $\pi_1(S, s)$ on the character variety $r$-dimensional representations of $\pi_1(X_s)$. Our main result, which relies on non-abelian Hodge theory and input from the Langlands program is that if $r<\sqrt{g(X_s)+1},$ then the finite orbits are precisely the representations with finite image. This resolves conjectures of Esnault-Kerz, Budur-Wang, Kisin, and Whang.  Video

Haohua Deng, Duke University: A Kato-Usui Type Extension of General Period Map

Eva Elduque, Universidad Autónoma de Madrid: Hodge Theory of Abelian Covers of Algebraic Varieties

Let f : U -> C^* be an algebraic map from a smooth complex connected algebraic variety U to the punctured complex line C^*. Using f to pull back the exponential map C -> C^*, one obtains an infinite cyclic cover U^f of the variety U. The homology groups of this infinite cyclic cover, which are endowed with Z-actions by deck transformations, determine the family of Alexander modules associated to the map f. In previous work jointly with Geske, Herradón Cueto, Maxim and Wang, we constructed a mixed Hodge structure (MHS) on the torsion part of Alexander modules. In this talk, we will talk about work in progress aimed at generalizing this theory to abelian covering spaces of algebraic varieties which arise in an algebraic way, i.e. from maps f:U->G, where G is a semiabelian variety. Joint work with Moisés Herradón Cueto.  Video

Marco Franciosi, University of Pisa: I-surfaces: Hodge Structures and Canonical Rings

An I-surface is a minimal complex surface of general type with K^2=1 and p_g=2. In this talk firstly we exhibit some examples of I-surfaces, and show how the geometric constructions correspond to  degenerations of the Hodge type; secondly we show how to realize the canonical ring of 2-Gorenstein stable I-surfaces by analyzing a canonical curve.  Video