# Hodge Theory and Related Topics

Dates: March 29, 2022 - April 2, 2022
Physical Location: 1365 Memorial Drive, Ungar Building, Room 528-B
Online via Zoom

#### Tuesday, March 29, 2022

 Opening Phillip Griffiths, Institute for Advanced Study, University of MiamiOpening Remarks 9:30am Claire Voisin, French National Centre for Scientific ResearchHyper-Kähler Manifolds I   (Video) 11:00am András Némethi, Rényi Institute of MathematicsLattice Cohomology I   (Video) 1:30pm Bhargav Bhatt, University of MichiganConstructible Sheaves and Filtered D-modules   (Video) 3:00pm Chenyang Xu, Princeton UniversityK-stability of Fano Varieties   (Video)

#### Wednesday, March 30, 2022

 9:30am Claire Voisin, French National Centre for Scientific ResearchHyper-Kähler Manifolds II   (Video) 11:00am András Némethi, Rényi Institute of MathematicsLattice Cohomology II   (Video) 1:30pm Morgan Brown, University of MiamiBirational Geometry of Moduli Spaces via the Essential Skeleton   (Video) 3:00pm Ziquan Zhuang, Massachusetts Institute of TechnologyFinite Generation and K-stability   (Video) 4:15pm Sebastián Torres, University of MiamiWindows and the BGMN Conjecture   (Video)

#### Thursday, March 31, 2022

 9:30am Ivan Cheltsov, University of EdinburghEquivariant Birational Geometry of Three-dimensional Projective Space   (Video)  (Slides) 11:00am Greg Pearlstein, Texas A&M UniversityExtensions in MHM and Hain's Biextension Line Bundle I   (Video) 1:30pm Rodolfo Aguilar, University of MiamiHomology Planes and Fundamental Groups   (Video) 3:00pm Boris Tsygan, Northwestern UniversityOperations on Cyclic Complexes, the Getzler-Gauss-Manin Connection, and Noncommutative Crystalline Cohomology   (Video)  (Slides) 4:15pm Christian Schnell, SUNY, Stony BrookThe Nilpotent Orbit Theorem, New and Old   (Video)   (Slides)

#### Friday, April 1, 2022

 9:30am Erik Paemurru, University of MiamiCounting Divisorial Contractions with Centre a $cA_n$-singularity   (Video) 11:00am Patrick Brosnan, University of MarylandExtensions in MHM and Hain's Biextension Line Bundle II   (Video) 1:30pm Giulia Saccà, Columbia UniversityModuli Spaces on K3 Categories Are Irreducible Symplectic Varieties   (Video) 3:00pm Bernardo Uribe, Universidad del NorteOriented and Unitary Equivariant Bordism of Surfaces   (Video)

#### Saturday, April 2, 2022

 9:30am Alberto Verjovsky, Universidad Nacional Autónoma de MéxicoAdelic Toric Varieties and Loop Groups   (Video) 11:00am José Seade, Instituto de Matemáticas, UNAMTopological Invariants of Compact Singular Varieties   (Video) 2:30pm René Mboro, University of MiamiBasic Remarks on Lagrangian Submanifolds of Hyper-Kähler Manifolds   (Video) 3:45pm Miguel Xicoténcatl, Centro de Investigación en MatemáticasOn the Nielsen Realization Problem and the Cohomology of Mapping Class Groups of Non-orientable Surfaces   (Video)  (Slides)

#### Abstracts

Rodolfo Aguilar: Homology Planes and Fundamental Groups

Abstract: We will present some results around the fundamental group of homology planes: smooth, affine complex surfaces with trivial reduced homology. The main tool is an algorithm to compute a presentation of it when the homology planes arise from an arrangement of lines. As an application, we obtained what may be the first examples of homology planes of log-general type with an infinite fundamental group.

Bhargav Bhatt: Constructible Sheaves and Filtered D-modules

Abstract: The Riemann-Hilbert correspondence relates constructible sheaves to D-modules on complex algebraic varieties. The theory of mixed Hodge modules equips the D-module attached to a constructible sheaf of geometric origin with a natural "Hodge" filtration; the associated graded of this filtration plays a central role in many applications of this theory in algebraic geometry.

In this talk, I will explain why the Hodge filtration is in fact automatic in the p-adic world: one can functorially attach a filtered D-module to any constructible sheaf on an algebraic variety over a p-adic field. I will also illustrate how one might use this functor by deducing Kollár's vanishing theorem for higher direct images of the canonical bundle from the BBDG decomposition theorem. (Joint work in progress with Jacob Lurie.)

Patrick Brosnan: Extensions in MHM and Hain's Biextension Line Bundle

Abstract: The so-called biextension line bundle L is an analytic line bundle over a complex manifold U associated to a pair of normal functions on U. In the case that U is a complement of a normal crossing divisor in a projective variety S, Pearlstein and I proved that (a) L can be extended to a line bundle on S and (b) L has a canonical extension to S as a Q-line bundle. (a) proves that L has an algebraic structure, but (b) begs the question of describing what the extension is. The first Chern class of L in the cohomology of U is a topological invariant of the normal functions which is classical to compute. (I.e., it goes back to the foundational papers of Hain and Hain-Reed.) In this talk, I will give a formula for the Q-line bundle.

Morgan Brown: Birational Geometry of Moduli Spaces via the Essential Skeleton

Abstract: Given $X$, a variety over a valued field $K$, we can define the essential skeleton of $X$, a polyhedral complex contained in the Berkovich analytification of $X$. I will present the essential skeleton and discuss how it can be used to link birational geometry and tropical geometry. Our main case study will be the moduli space $\mathcal{M}_{0,n}$. I will also present a few speculative remarks about the higher dimensional case of moduli spaces of hyperplane arrangements in $\mathbb{P}^2$.

Ivan Cheltsov: Equivariant Birational Geometry of Three-dimensional Projective Space

Abstract: In this talk, we will describe G-equivariant birational geometry of the three-dimensional projective space for a finite group G acting faithfully on the projective space. In particular, we will describe all possibilities for the group G such that the projective space is G-solid, i.e. it is not G-birational to a conic bundle and it is not G-birational to a del Pezzo fibration. For these groups, we will explicitly describe all G-Mori fibre spaces that are G-birational to the projective space. This is a joint project with Arman Sarikyan.

René Mboro: Basic Remarks on Lagrangian Submanifolds of Hyper-Kähler Manifolds

Abstract: We will present some properties of Lagrangian subvarieties which can be seen to some extent as higher dimensional analogue of what happens for curves on K3 surfaces. These concern their interaction with Lagrangian fibrations of the hyper-Kähler and the albanese dimension of Lagrangian surfaces.

András Némethi: Lattice Cohomology

Abstract: The topological lattice cohomology is associated with negative definite plumbed 3-manifolds, that is, with links of (analytic) normal surface singularities. The theory for these manifolds is equivalent with the Heegaard Floer theory, its Euler characteristic is the Seiberg-Witten invariant. Its construction was motivated by its connection with analytic invariants of the singularities. This object will be the starting point of our presentation. Then we introduce the analytic lattice cohomology associated with the analytic types of isolated singularities. It is a categorification of the geometric genus. Then we discuss how the variation of the analytic lattice cohomology measures the family of different analytic structures supported on a fixed topological type. Some deformation theoretical connections will also be discussed.

Greg Pearlstein: Extensions in MHM and Hain's Biextension Line Bundle

Abstract: The so-called biextension line bundle L is an analytic line bundle over a complex manifold U associated to a pair of normal functions on U. In the case that U is a complement of a normal crossing divisor in a projective variety S, Pearlstein and I proved that (a) L can be extended to a line bundle on S and (b) L has a canonical extension to S as a Q-line bundle. (a) proves that L has an algebraic structure, but (b) begs the question of describing what the extension is. The first Chern class of L in the cohomology of U is a topological invariant of the normal functions which is classical to compute. (I.e., it goes back to the foundational papers of Hain and Hain-Reed.) In this talk, I will give a formula for the Q-line bundle.

Erik Paemurru: Counting Divisorial Contractions with Centre a $cA_n$-singularity

Abstract: First, we simplify the existing classification due to Kawakita and Yamamoto of 3-dimensional divisorial contractions with centre a $cA_n$-singularity. Next we consider divisorial contractions of discrepancy at least~2 to a fixed variety with centre a $cA_n$-singularity. We show that if there exists one such divisorial contraction, then there exist uncountably many such divisorial contractions.

Giulia Saccà: Moduli Spaces on K3 Categories Are Irreducible Symplectic Varieties

Abstract: Recent developments by Druel, Greb-Guenancia-Kebekus, Horing-Peternell have led to the formulation of a decomposition theorem for singular (klt) projective varieties with numerical trivial canonical class. Irreducible symplectic varieties are one the building blocks provided by this theorem, and the singular analogue of irreducible hyper-Kahler manifolds.

In this talk I will show that moduli spaces of Bridgeland stable objects on the Kuznetsov component of a cubic fourfold (or of a Gushel Mukai four or six fold) with respect to a generic stability condition are always projective irreducible symplectic varieties. I will rely on the recent work of Bayer-Lahoz-Macri-Neuer-Perry-Stellari, which, ending a long series of results by several authors, proved the analogue statement in the smooth case.

José Seade: Topological Invariants of Compact Singular Varieties

Christian Schnell: The Nilpotent Orbit Theorem, New and Old

Abstract: I will quickly sketch Schmid's proof of the nilpotent orbit theorem, and then explain a completely different argument that works for polarized complex variations of Hodge structure (where the local monodromy is no longer quasi-unipotent).

Sebastián Torres: Windows and the BGMN Conjecture

Abstract: Let C be a smooth projective curve of genus at least 2, and let N be the moduli space of semistable rank-two vector bundles of odd degree on C. We construct a semi-orthogonal decomposition in the derived category of N conjectured by Belmans, Galkin and Mukhopadhyay and by Narasimhan. It has blocks of the form D(Cd) where Cd are d-th symmetric powers of C, and the semi-orthogonal complement to these blocks is conjecturally trivial. In order to prove our result, we use the moduli spaces of stable pairs over C. Such spaces are related to each other via GIT wall crossing, and the method of windows allows us to understand the relationship between the derived categories on either side of a given wall. This is a joint work with J. Tevelev.

Boris Tsygan: Operations on Cyclic Complexes, the Getzler-Gauss-Manin Connection, and Noncommutative Crystalline Cohomology

Abstract: I will review recent work of Petrov-Vologodsky and of my own. In particular, I will discuss the algebra of operations on the negative cyclic complex of an algebra and explain how it provides the constructions of Getzler's Gauss-Manin superconnection and of a noncommutative crystalline complex of an associative algebra.

Bernardo Uribe: Oriented and Unitary Equivariant Bordism of Surfaces

Abstract: Together with collaborators Angel, Segovia and Samperton we have been able to determine explicitly the torsion part of both the equivariant oriented and unitary bordism groups. The key result is the calculation of the obstruction class for a surface with free action to bound equivarianltly. I will present this obstruction class as well as an explicit group where the invariant is non trivial.

Alberto Verjovsky: Adelic Toric Varieties and Loop Groups

Abstract: We describe the proalgebraic space which is the inverse limit of all finite branched covers over a normal toric variety with branching set the invariant divisor under the algebraic torus action. These are completions (compactifications) of the adelic abelian algebraic group which is the profinite completion of the algebraic torus. We prove that the vector bundle category of the proalgebraic toric completion of a toric variety is the direct limit of the respective categories of the finite toric varieties coverings defining the completion. In the case of the complex projective line, we obtain as proalgebraic completion the adelic projective line P. We define holomorphic vector bundles over P. We also introduce the smooth, Sobolev, and Wiener adelic loop groups and the corresponding Grassmannans; we describe their properties and prove Birkhoff's factorization for these groups. We prove that the adelic Picard group of holomorphic line bundles is isomorphic to the rationals and prove the Birkhoff-Grothendieck splitting theorem for holomorphic bundles of higher rank over P.

The paper on the subject appears in arXiv: arXiv:2001.07997v2 [math.AG]

Claire Voisin: Hyper-Kähler Manifolds

Abstract: Hyper-Kähler manifolds form a special class of compact Kähler manifolds with trivial canonical bundle. They are higher-dimensional generalizations of K3 surfaces, and a number of deformation classes of hyper-Kähler manifolds can be constructed starting from either a K3 or an abelian surface. In the first lecture, I will describe the main general classical properties of hyper-Kähler manifolds that will be needed in the second lecture, where I will focus to dimension 4 and sketch the proof of a simple topological characterization of hyper-Kähler manifolds of Hilb2(K3) deformation type (joint work with Debarre, Huybrechts, and Macrì).

Miguel Xicoténcatl: On the Nielsen Realization Problem and the Cohomology of Mapping Class Groups of Non-orientable Surfaces

Abstract: The Nielsen realization problem is the question about whether finite subgroups of the mapping class group Mod(Sg) can act on the surface Sg. A complete proof of this fact was given by Kerckhoff in the '80s in the case of orientable surfaces. In this talk we will show that Nielsen realization holds in the case of non-orientable surfaces Ng with marked points, that is every finite subgroup of the mapping class group Mod(Ng, k) lifts isomorphically to a subgroup of the Diff(Ng, k). In contrast, we also show the natural projection from the group of diffeomorphisms Diff(Ng) to the mapping class group Mod(Ng) does not admit a section for large g. We introduce the notion of fixed point data for a diffeomorphism of Ng and use these to classify the normalizers of cyclic subgroups of prime order p of Mod(Ng , k) and to study the p-periodicity of this group. As an application we determine the p-primary component of the Farrell cohomology of Mod(Ng , k) in some cases. This is joint work with N. Colin.

Chenyang Xu: K-stability of Fano Varieties

Abstract: (This is the first part of the joint talk with Ziquan Zhuang.) K-stability of Fano varieties was initiated as a central topic in complex geometry, for its relation with the Kahler-Einstein problem. It turns out that the machinery of higher dimensional geometry, centered around the minimal model program, provides a fundamental tool to study it, and therefore makes it an active algebraic subject. This meeting of two well-studied fields has made a number of major conjectures solved, including the Yau-Tian-Donaldson Conjecture, and the construction of a moduli space for Fano varieties. In this talk, I will survey the recent development. More details will be discussed in the second part of the talk by Ziquan Zhuang.

Ziquan Zhuang: Finite Generation and K-stability

Abstract: (This is the second part of the joint talk with Chenyang Xu.) K-stability is an algebraic condition that characterizes the solution of the Kähler-Einstein problem on Fano varieties. In this talk, I will discuss some recent progress in birational geometry on the finite generation property of valuations, and explain its connection to the solutions of some major questions in K-stability such as the Yau-Tian-Donaldson conjecture and the K-moduli conjecture.