Program

I will talk about various topics on canonical rings and log canonical rings of compact complex manifolds. Now it is well known that the canonical ring of a smooth projective variety is a finitely generated $\mathbb C$-algebra by the great work of Birkar--Cascini--Hacon--McKernan. I have already generalized this result for compact Kähler manifolds. It is conjectured that the log canonical ring is also finitely generated for projective varieties and compact Kähler manifolds. However, it is still widely open. I will explain that the finite generation of the log canonical ring for projective varieties is closely related to several important other conjectures in the theory of minimal models. Some parts of this talk are based on a joint work with Yoshinori Gongyo.

We discuss two particular phenomena where Calabi-Yau metrics or more general Kähler-Einstein metrics play important roles in arithmetic geometry (or, in the study of moduli of varieties).

1. We generalize the Faltings heights for abelian varieties to general arithmetic varieties as a conjectural arithmetic framework of K-stability (Kyoto J.Math 2018). Parts of the logic invoke developments of Arakelov geometry around the 90s, including works of Moriwaki sensei.

2. To classify and determine all compact limits of K3 hyperKähler metrics, we study an explicit compactification of moduli of K3 metrics via tropical geometry and Lie theory (MSJ Memoir vol40, with Y.Oshima, and later subsequels). The boundary parametrizes S^2 with special Kahler metrics, interval with piecewise linear density measures, T^3/± 1 etc. Prof David Morrison kindly told us his expectation that our boundary components should be related to types of some string theories.

The goal of this talk is to introduce the proof of the geometric Bogomolov conjecture in all characteristics by my recent joint work with Junyi Xie. The proof follows the line of the works of Szpiro, Ullmo, Zhang, Gubler and Yamaki, and is especially based on Yamaki's reduction theorem which reduces the conjecture to abelian varieties of good reduction and trivial trace.

Chambert-Loir and Ducros introduced smooth forms and currents on Berkovich spaces using tropicalization maps induced by morphisms to tori. In joint work with Philipp Jell und Joe Rabinoff, we allow more generally harmonic tropicalization maps to define a larger class of weakly smooth forms which has essentially the same properties as the smooth forms, but have a better cohomological behavior.

A family of algebraic varieties gives rise to a family of de Rham cohomology/complex, that is, Gauss-Manin connection equipped with Hodge structure. For a family of noncommutative algebras, the analogues connection was constructed by Getzler. The subject of my talk is about its categorical generalization: we consider a family of stable infinity-categories. I would like to introduce two methods of constructions of Hodge theoretic objects realized as the periodic cyclic complex with a D-module structure and a filtration. Two approaches are conceptual/geometric and interrelated to each another. They are not analogous to procedures in the commutative case. They involve free loop spaces, factorization homology (topological chiral homology), Hochschild pairs and their moduli-theoretic interpretation, duality theorems, and the relation between deformation theory and dg Lie algebras, etc. The talk will start with the perspectives and motivations from examples such as LG models.

We first overview the current status of the minimal model program. I then exhibit several examples which illustrate typical phenomena that occur only in positive characteristic.

Siegel–Jacobi forms are a generalization of Siegel modular forms and can be seen as sections of a line bundle on the universal family of principally polarized abelian varieties. They include important functions as the theta functions that provide projective embeddings of abelian varieties.

The ring of Siegel–Jacobi forms is not finitely generated even if one fixes the ratio between the index and the weight, which makes difficult to estimate the dimension of the spaces of Siegel–Jacobi forms.

The volume of a line bundle measures the asymptotic growth of the space of sections of its powers. Professor Moriwaki has extended the notion of volume to the arithmetic setting, defining the arithmetic volume. One can also extend the notion of volume to b-divisors and to line bundles provided with singular psh metrics. This extension shares many properties with the arithmetic volume.

In a joint work with A. Botero, D, Holmes and R. de Jong we use the theory of b-divisors and volumes to describe the asymptotic growth of the space of Siegel–Jacobi forms.

Freixas and Sankaran (2018) have recently computed arithmetic Chern numbers of some twisted Hilbert modular surfaces, leaving open the question of the relation of their result to a series of conjectures I had previously developped jointly with D. Rossler. I will investigate this irritating problem and (time permitting) I will suggest some generalizations and applications.

Let $K$ be a number field or function field in one variable, and $\iota : C\emb S$ an embedding of a curve into a surface over $K$. Let $X$ and $\Gamma$ denote respectively the product $C\times S$ and the graph of $\iota$. Assume that both $C$ and $S$ have semistable reductions. In this talk, we will describe a modified diagonal cycle $\gamma$ and its Beilinson--Bloch height. When $S=C\times C$ and $\iota$ is the diagonal embedding, the $\gamma$ is defined by Gross and Schoen, and its height have been computed in our previous work.

The arithmetic volume of a pair of an adelic R-Cartier divisor and an R-Cartier divisor is an invariant measuring the asymptotic behavior of the numbers of the strictly small sections of the high multiples of the pair. In this talk, I will explain that the arithmetic volume function defined on the space of pairs is differentiable at a big pair and that its derivative is given by the arithmetic restricted positive intersection number.

In a work in collaboration with Atsushi Moriwaki, we have established a Harder-Narasimhan theory for non-necessarily Hermitian vector bundles on an adelic curve. In other words, any adelic vector bundle admits a unique filtration of semi-stable subquotient with strictly decreasing minimal slopes. In this talk, I will explain a game theory interpretation of this phenomenon. More precisely, we will construct, using the data of the adelic vector bundle, a non-cooperative game of two players. It turns out that the semi-stability of the adelic vector bundle corresponds to the Nash equilibrium condition of the game. Then, in the game theory framework, we discuss conditions which leads to the existence and the uniqueness of Harder-Narasimhan filtration.

In Arakelov geometry, projective arithmetic surfaces play the role of arithmetic analogs of projective surfaces over a field. We introduce a class of formal analytic surfaces, which are the analogs of regular two dimensional formal schemes admitting a (possibly singular) projective curve as scheme of definition. We discuss the Arakelov geometry of these formal analytic arithmetic surfaces and its applications to Diophantine geometry, notably to the study of modular forms. This is joint work with F. Charles.

In this talk, I would like to look back over my research achievements around Arakelov Geometry.

One of the main tasks of theoretical physicists is to calculate the spectra of self-adjoint and bounded-below operators on Hilbert spaces called Hamiltonians. Partition functions are generating functions of the spectra. In this talk, I will discuss a variety of mathematical concepts I have encountered in my work on Hamiltonian spectra. They include modular forms, the elliptic genera, mock-modular forms, finite groups and the moonshine, and the Gromov-Witten invariants and their relation to black-hole microstates.

Bow varieties were introduced by Cherkis and identified with Coulomb branches of quiver gauge theories in my joint work with Takayama. They are moduli spaces of framed sheaves on a certain surface by Cherkis-Hurtubise. In view of Coulomb branches, it is natural to look for similar varieties for moduli spaces of symplectic and orthogonal sheaves. This leads to a definition of symmetric bow varieties, roughly as fixed point sets of involution of bow varieties, whose study was initiated by my student de Campos Affonso. I will explain the definition and properties of symmetric bow varieties.

Tropical toric varieties are partial compactifications of finite dimensional real vector spaces associated with rational polyhedral fans. We introduce pluripotential theory on tropical toric varieties. This theory provides a canonical correspondence between complex and non-archimedean pluripotential theories of invariant plurisubharmonic functions on toric varieties. We apply this correspondence to solve invariant non-archimedean Monge-Ampère equations on toric and abelian varieties over arbitrary non-trivially valued non-archimedean fields. This is joint work with José Ignacio Burgos Gil, Walter Gubler and Philipp Jell.

We shall outline the proof of a conjecture of Esnault and Langer, which says that on an abelian variety defined over a finitely generated field in positive characteristic, a non-torsion line bundle cannot descend along infinitely many relative Frobenii.

It is one of the interesting themes in complex differential geometry to find an equivalence between objects in differential geometry and in algebraic geometry. Rather recently, we obtained equivalences between periodic monopoles on the 3-dimensional Euclidean space and various types of difference modules, as a variant of the Kobayashi-Hitchin correspondences for harmonic bundles, Higgs bundles and flat bundles. After reviewing the equivalences, we would like to discuss a relation between the moduli spaces of monopoles and a class of varieties called Zastava in some easy cases.

The equivariant quantum cohomology (D-module) has the structure of a difference module with respect to shifts of equivariant parameters. In this talk, I will discuss the relation to mirror symmetry and possible applications.

Let $¥varphi: S ¥to C$ be an elliptic surface with a section $O$. For a divisor $D$ on $S$, we say that $D$ is horizontal if no irreducible component is contained in fibers. By a multi-section of $¥varphi$ we mean an effective horizontal divisor. A multi-section is called an $m$-section if $D¥cdot F = m$, where $F$ denotes a fiber of $¥varphi$. Note that a $1$-section means just a section. In this talk, we consider $2$- and $3$-sections on certain rational elliptic surfaces and make use of them in order to construct plane curves with topologically interesting property.