**Dates:** January 30-February 3, 2023

**Location:** 1280 Stanford Dr, Coral Gables, FL 33146 - Lakeside Village

**Live Video Available via Zoom**

In recent years several major developments in Hodge theory and its applications were obtained:

Proof of the period image conjecture and its extension to the mixed case using the new technique of o - minimality from model theory.

Proofs of Hyperbolicity in much greater generality and relatedly introduction of methods from Nevalinna theory.

Studies of harmonic maps from quasiprojective varieties to buildings.

Extension of the proof Shafarevich conjecture to a wide class of quasiprojective varieties.

The goal of this conference is to disseminate new results and discuss new developments and applications.

9:00am |
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. |

10:30am |
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. |

12:00pm |
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. |

2:30pm |
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. |

4:00pm |
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. |

5:00pm | Wine & Cheese Reception |

9:00am |
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. |

10:30am |
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. |

12:00pm |
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. |

2:30pm |
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). |

4:00pm |
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. |

9:00am |
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.) |

10:15am |
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. |

11:30pm |
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. |

2:00pm |
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. |

3:15pm |
Iterated Shimura integrals are iterated integrals of classical modular forms. They are elements of the coordinate ring of the relative unipotent completion of SL_2(Z), which we regard as the fundamental group of the modular curve. Francis Brown has proposed that the coordinate ring of the appropriate relative completions of SL_2(Z) generate the (conjectural) tannakian category of mixed modular motives --- the category of mixed motives generated by the motives of classical modular forms. The goal of this talk is to explain how the classical Hecke operators act on the free abelian group generated by the conjugacy classes of SL_2(Z) and, dually, on those elements of the coordinate ring of the relative completion of SL_2(Z) that are constant on conjugacy classes. This Hecke action commutes with the natural Galois action and each Hecke operator is a morphism of mixed Hodge structure. One surprising fact is that, while the Hecke operators T_N and T_M (acting on conjugacy classes) commute when N and M are relatively prime, T_p and T_{p^2} do not commute when p is a prime. Consequently, the corresponding Hecke algebra is not commutative, in contrast with the classical case. |

4:30pm |
A generalization of the Singer-Hopf conjecture in the Kähler setting predicts that the any perverse sheaf on a compact aspherical Kähler manifold has non-negative Euler characteristic. I will discuss a proof of this conjecture assuming that the fundamental group admits a faithful semi-simple rigid representation. The proof uses ideas from the Shafarevich conjecture. This is a joint work with Donu Arapura. |

9:00am |
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 |

10:30am |
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. |

12:00pm |
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. |

2:30pm |
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. |

4:00pm |
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 . |

9:00am |
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. |

10:30am |
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. |

12:00pm |
There are well-known extension theories for period maps of classical types which carry rich geometric and Hodge-theoretic boundary information, for example, Baily-Borel compactification and toroidal compactification by Ash-Mumford-Rapoport-Tai. However, there is little known about analogs for non-classical period maps. Kato-Usui's theory allows one to obtain extensions for general period maps with image in a logarithmic manifold under the assumption that a specific type of fan exists. The existence of such a fan is only known in classical cases and the resulting Kato-Usui extension agrees with the toroidal compactification. I will briefly review Kato-Usui's theory as well as give an example on which the theory has an application. Some ongoing works and further directions will also be discussed if time permits. |

2:00pm |
The goal of this talk is to introduce orbifold jet differentials (in the sense of Campana) and to discuss the generalization of classical hyperbolicity conjectures. In the classical context, jet differentials allow one to prove algebraic degeneracy of entire curves at the level of jets, for sufficiently large jet orders. Demailly has proven that this result always holds for varieties of general type. We will explain why this does not extend to the orbifold context. This talk is based on joint works with Frédéric Campana, Jean-Pierre Demailly and Erwan Rousseau. |

3:15pm |
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. |

4:30pm |
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. |

Donu Arapura |
Benjamin Bakker |

Ingrid Bauer |
Yohan Brunebarbe |

Benoit Cadorel |
Frédéric Campana |

Fabrizio Catanese |
Benoit Claudon |

Louis Clement |
Lionel Darondeau |

Haohua Deng |
Ya Deng |

Eva Elduque |
Phillipe Eyssidieux |

Manuel Gonzalez Villa |
Mark Green |

R. Hain |
J. H. Keum |

Bruno Klingler |
Daniel Litt |

Laurentiu Maxim |
Phillip Naumann |

Rita Pardini |
Mohan Ramachandran |

Carlos Simpson |
Botong Wang |

**Limited funding is available for young researchers working in the areas of the Conference.**

**Preferences will be given to underrepresented groups.**

**Please contact L. Katzarkov lkatzarkov@gmail.com or R. Aguilar aaguilar.rodolfo@gmail.com**