Higher Dimensional Algebraic Geometry, Holomorphic Dynamics and Their Interactions - IMS

Higher Dimensional Algebraic Geometry, Holomorphic Dynamics and Their Interactions
(3 - 28 Jan 2017)

~ Abstracts ~

The isogeny category of commutative algebraic groups
Michel Brion, Institut Fourier, France

The commutative algebraic groups over a field $k$ form an abelian category, which has nice homological properties when $k$ is algebraically closed (as shown by Serre and Oort), but not over an arbitrary field (as follows from a result of Milne). The talk will discuss the isogeny category of commutative algebraic groups, which turns out to have simpler and more uniform properties. Also, this category is equivalent to the category of all left modules of finite length over some (explicit, very large) non-commutative ring. This yields information on the structure of indecomposable commutative algebraic groups.

Singularities in characteristic two
Paolo Cascini, Imperial College London, UK

We show that many classical results of the three-dimensional minimal model programme do not hold over an algebraically closed field of characteristic two. Joint work with Tanaka.

Rigid manifolds, Hirzebruch-Kummer coverings, projective classifying spaces
Fabrizio Catanese, Universität Bayreuth, Germany

Motivated by recent examples, called BCD surfaces, which recently gave counterexamples to an old question by Fujita concerning Variation of Hodge Structures (joint work with Michael Dettweiler), together with ingrid Bauer, we showed the rigidity the BCD surfaces and of a class of surfaces which includes the Hirzebruch-Kummer coverings of the plane branched over a complete quadrangle.

After this, we become interested about the classification problem of rigid varieties. For curves, the only rigid curve is the projective line, whereas for surfaces the only rigid surfaces not of general type are the projective plane, and some non algebraic Inoue surfaces (Del Pezzo surfaces of degree at least 5 are locally but not globally rigid). So, Kodaira dimensions 0 and 1 do not occur for curves and surfaces. In higher dimension, we show that all Kodaira dimensions occur for rigid varieties, except possibly Kodaira dimension 1.

Until now, excluding the projective plane, the known rigid surfaces were all projective classifying spaces (their universal cover is contractible): they have universal cover which is either the ball or the bidisk (these are the noncompact duals of P^2 and P^1 x P^1), or are the examples of Mostow and Siu, or the Kodaira fibrations of Catanese-Rollenske.

I shall discuss several open problems concerning Kummer coverings of the plane branched over configurations of lines, namely, their rigidity, the existence of metrics of negative curvature, and the question: when are they classifying spaces? In the course of this I shall give criteria for fibred surfaces to be classifying spaces.

Time permitting I shall explain how hypersurfaces in PCS lead to varieties whose moduli spaces can be understood through topology, as the Inoue type varieties, which are etale quotients of hypersurfaces in PCS. Strong rigidity of these depend on strong conditions, and one would like to enlarge the definition in the case of high degree of the classifying map: this leads to the problem of understanding deformation to hyper surface embeddings, which is the object of work in progress together with Yongnam Lee.

On minimal 3-folds of general type with the geometric genus 1,2 or 3
Meng Chen, Fudan University, China

We classify minimal 3-folds of general type with the geometric genus 1,2 or 3 by characterizing the birationality of \Phi_m (where m is small). In this talk, we introduce the main technical part which is the joint work with Yong Hu and Matteo Penegini.

On the boundary of the collection of varieties of general type
Jungkai Chen, National Taiwan University, Taiwan

In the birational classification theory, varieties of general type play an important and fundamental role. Even though varieties of general type does not form a bounded family, their invariants satisfies certain constraints, such as Bogomolov-Miyaoka-Yau inequality, Noether type inequality etc. The purpose of the study of geography is to understand the distribution of fundamental invariants and to explore the interesting geometry of those varieties on the boundary or on some other funny place.

In this talk, we will review some recent progress on threefolds of general type and demonstrate some higher dimensional phenomena. Part of the talk are joint work with Meng Chen.

Dynamical aspects of Hamiltonian systems
Chong-Qing Cheng, Nanjing University, China

We emphasize the topic of invariant Langange sub-manifold and the diffusion in nearly integrable Hamiltonian systems.

Chern classes of automorphic bundles
Hélène Esnault, Freie Universität Berlin, Germany

We prove vanishing of the Chern classes of automorphic bundles in Deligne and continuous l-adic cohomology. We show how it is related to some of Beilinson's motivic conjectures by exhibiting one corollary of them which, it seems, hasn't been discussed in the literature before.
Joint work in progress with Michael Harris, some discussions with Ben Moonen.

Isotrivial VMRT-structures of complete intersection type
Baohua Fu, Chinese Academy of Sciences, China

The family of varieties of minimal rational tangents on a quasi-homogeneous projective manifold is isotrivial. Conversely, are projective manifolds with isotrivial varieties of minimal rational tangents quasi-homogenous? We will show that this is not true in general, even when the projective manifold has Picard number 1. In fact, an isotrivial family of varie ties of minimal rational tangents needs not be locally flat in differential geometric sense. This leads to the question for which projective variety Z, the Z-isotriviality of varieties of minimal rational tangents implies local flatness. Our main result verifies this for many cases of Z among complete intersections. This is a joint work with Jun-Muk Hwang.

On log CY structure of varieties admitting non-trivial polarized endomorphism
Yoshinori Gongyo, The University of Tokyo, Japan

We mainly discuss the Calabi--Yau type structure of normal projective surfaces and Mori dream spaces admitting a non-trivial polarized endomorphism.

Contraction, deformation and singularity
Zheng Hua, The University of Hong Kong, Hong Kong

Let X->Y be a birational contraction of relative dimension no bigger than one. It is an interesting question whether the infinitesimal deformation of the exceptional curve determine the singularities of Y. When X->Y is a three dimensional flopping contraction, we show that the answer to the above question is yes. This gives a positive answer to a conjecture of Donovan and Wemyss. This talk is based on the a joint work with Yukinobu Toda 1601.04881 and the work 1610.05467.

Infinitesimal automorphisms of cubic hypersurfaces and projective Legendrian manifolds
Jun-Muk Hwang, Korea Institute for Advanced Study, Korea

We study infinitesimal automorphisms of singular cubic hypersurfaces, in particular, their prolongations of special type. This study will be used to investigate the automorphism groups of projective Legendrian manifolds.

Wild group scheme quotient singularities
Hiroyuki Ito, Tokyo University of Science, Japan

Since studying quotient singularities by finite groups is one-sided viewpoint in positive characteristic, we always need to study group scheme quotient singularities in other-side. As usual in positive characteristic, most difficult and essential case is wild case, that is, quotient by a finite group whose order is divisible by the characteristic or a finite group scheme of length divisible by the characteristic. In my talk, we consider such quotients for carious cases with explicit calculations, and try to make unified treatment of these two quotients. Even for the rational double points, we can find very interesting phenomena.

Moduli spaces versus locally symmetric spaces
Lizhen Ji, University of Michigan, USA

There are many similarities between moduli spaces and arithmetic locally symmetric spaces. For example, multiple problems and recent results about the moduli space M_g of compact Riemann surfaces were motivated by locally symmetric spaces. In this talk, I will discuss several results on both classes of spaces by using the interaction and analogy between them.

Explicit birational geometry of Fano and Calabi-Yau 3-folds
Chen Jiang, The University of Tokyo, Japan

According to MMP, it is very important to investigate varieties of general type, Fano varieties, and Calabi-Yau varieties. In dimension three, the explicit geometry was investigated by Professors Jungkai Chen and Meng Chen. I will discuss some recent results on explicit birational geometry of Fano and Calabi-Yau 3-folds. Part of this talk is based on joint work with Professor Meng Chen.

Some results on the eventual paracanonical maps
Zhi Jiang, Fudan University, China

The eventual paracanonical map was introduced by Barja, Pardini, and Stoppino to prove Severi-type inequalities. We will explain that the degrees of the eventual paracanonical maps are bounded in low dimensions.

On derived McKay correspondence for GL(3,C)
Yujiro Kawamata, The University of Tokyo, Japan

We prove that the equivariant derived category for a finite subgroup of GL(3,C) has a semi-orthogonal decomposition into the derived category of a certain partial resolution, called a maximal Q-factorial terminalization, of the corresponding quotient singularity and a relative exceptional collection. This is a generalization of a result of Bridgeland, King and Reid.

Higgs sheaves on singular spaces, the nonabelian Hodge correspondence, and the Miyaoka-Yau Inequality for minimal varieties of general type
Stefan Kebekus, University of Freiburg, Germany

We establish the Miyaoka-Yau inequality for the tangent sheaf of any minimal, complex, projective variety X of general type, with only klt singularities. In the case of equality, we prove that the canonical model of X has only quotient singularities and is uniformized by the unit ball. This is joint with Greb, Peternell, Taji.

Q-homology projective planes
Jonghae Keum, Korea Institute for Advanced Study, Korea

A normal projective surface with at worst quotient singularities is called a Q-homology projective plane if it has the same Betti numbers as the complex projective plane.
In this lecture, I will explain many examples, then discuss the current state of the art on the classification of such surfaces. This problem contains the so called Montgomery-Yang problem from differential topology, originated from the study of circle actions on the 5-sphere.

Grothendieck Riemann-Roch for Deligne-Mumford stacks
Amalendu Krishna, Tata Institute of Fundamental Research, India

Grothendieck Riemann-Roch theorem is one of the most important and deep results in algebraic geometry. But this theorem is restricted only to schemes. In recent years, there has been numerous need for such a result for stacks in order to compute cohomology groups of moduli spaces and stacks. In this talk, I shall provide a generalization of Grothendieck Riamann-Roch theorem to quotient stacks.

On the Eisenbud-Goto conjecture
Sijong Kwak, Korea Advanced Institute of Science and Technology, Korea

The Eisenbud-Goto regularity conjecture has been long open and proved for integral projective curves due to Castelnuovo, Gruson-Lazarsfeld-Peskine(1986). Recently, J. Mccullough and I. Peeva found interesting counter examples for singular varieties and many experts in this area are interested in this topics again. I will introduce this topic and report the recent progress for smooth cases.

Birational geometry of compactifications of Drinfeld half-spaces over a finite field
Adrian Langer, University of Warsaw, Poland

I will talk about compactifications of Drinfeld half-spaces over a finite field and new interesting phenomena occuring in their study. In particular, I will study endomorphisms of Drinfeld half-spaces, leading to a new type of inseparable Cremona transformation. I will also show some various interesting singularities and unexpected rational surfaces and 3-folds.

On Q-Gorenstein morphisms
Yongnam Lee, Korea Advanced Institute of Science and Technology, South Korea

The notions of Q-Gorenstein scheme and of Q-Gorenstein morphism will be introduced for locally Noetherian schemes. These cover all the previously known notions of Q-Gorenstein algebraic variety and of Q-Gorenstein deformation satisfying Koll\'ar condition, over a field. In this talk, we inspect relations between Q Gorenstein morphism and its two weak forms: naively Q-Gorenstein morphism and virtually Q-Gorenstein morphism, and discuss several interesting examples related with these notions. This is a joint work with Noboru Nakayama.

Mixed spin fields on the quintic
Weiping Li, Hong Kong University of Science and Technology, Hong Kong

The Gromov-Witten invariants of the quintic is one of the most famous research topics from the very first day of mirror symmetry to today. There is another physical theory for the quintic polynomial, called Landau-Ginzburg theory. Physicists conjectured that these two theories can be identified via some mysterious transformations. The analytical construction of enumerative invariants in the affine LG-space was given by Fan-Jarvis-Ruan (FJRW invariants). Mixed Spin Field theory is a mathematical attempt to unlock the mysterious link between GW-invariants and FJRW-invariants. It uses the newly developed technologies such as the cosection localization by Kiem and J.Li and the P-fields theory of Chang and J. Li. It is an on-going joint work with H.L. Chang, J. Li and Melissa Liu.

Geometric substructures on uniruled projective manifolds
Ngaiming Mok, The University of Hong Kong, Hong Kong

Let X be a uniruled projective manifold equipped with a minimal rational component. Let (S;p) be a germ of complex submanifold at some general point p of S. We say that S inherits a geometric substructure essentially to mean that the projectivized tangent space at each point of S intersects nontrivially the VMRT of X at the point. We will study the question whether S is uniruled by germs of minimal rational curves of X lying on S and the existence problem of a compactification of S as a uniruled projective subvariety. Applications will be made to transcendental problems such as the study of holomorphic isometries of the complex unit ball into an irreducible bounded symmetric domain of rank at least 2.

Rational points on log Fano threefolds over a finite field
Yusuke Nakamura, The University of Tokyo, Japan

Esnault proved that smooth Fano varieties defined over a finite field have a rational point. In this talk, I will discuss the singular case. This is joint work with Yoshinori Gongyo and Hiromu Tanaka. We proved that log Fano threefolds with klt singularities defined over a finite field of characteristic larger than 5 have a rational point.

On compactifying moduli of Kähler-Einstein varieties
Yuji Odaka, Kyoto University, Japan

The classical moduli space of compact Riemann surfaces of genus g>1, exists as an algebraic variety of dimension 3g-3. The (somewhat implicit) idea to think of such space and the dimension counts essentially goes back to B.Riemann in mid 19c. A century later, in 1960s, this moduli space was compactified in a "geometrically meaningful" way as known as the Deligne-Mumford compactification (Mayer-Mumford, Deligne-Mumford, Knudsen-Mumford, Gieseker...). This nice moduli theory has inseparable but often hidden relation to the existence of the hyperbolic metrics on the Riemann surfaces and this survey talk will focus on how to generalize such story and uncover its nature.

Since 1980s, along with the development of the Minimal Model theory, "Kollár-Shepherd-Barron-Alexeev" theory has generalized to compactified moduli of higher dimensional varieties with ample canonical classes, and around 2009-2010 the speaker found that generalization can be regarded as a special case of "moduli of K-stable varieties", through birational geometric study of K-stability. For the case of Fano manifolds, which extends the Riemann sphere ("g=0"), benefitting from the recent wonderful well-known breakthrough of the Kahler-Einstein metric geometry, we have observed the existence of analogous compact moduli algebraic spaces in cooperation with many people. If time permits, we may also try to survey some still-related developments of different flavors.

Differential forms and the abundance conjecture
Thomas Peternell, University of Bayreuth, Germany

I will discuss various results towards the abundance conjecture on projective varieties with mild singularities. This is joint work with V. Lazic. Also, I will mention the connection to the abundance conjecture for line bundles on Calabi-Yau manifolds (joint work with V.Lazic and K.Oguiso).

On the parametric differential Galois group
Ho Hai Phung, Institute of Mathematics, Vietnam

Let $R$ be a DVR and $X/R$ a smooth scheme. The Tannakian duality applied to the category of stratified sheaves on $X/R$ yields the (relative) differential fundamental group scheme of $X/R$, which is the inverse limit of the one determined (through Tannakian duality) from single stratified bundles. Some algebraic properties of these group schemes reflex the degeneration of the bundles as a one-parametric family of stratified bundles (or connections, in the case of characteristic zero). Our perspectives is to understand the structure of the differential Galois groups as well as the differential fundamental groups. In this talk I shall report our recent progress. This is a joint work with J. P. dos Santos (Paris).

Monotonicity of entropy for one-parameter families of unimodal interval maps
Weixiao Shen, Fudan University, China

We modify Thurston's pull back map so that it applies to maps which are only locally holomorphic. We obtain 'positively-oriented'transversality and monotonicity of entropy for certain families of unimodal interval maps with 'essential singularities'. This is a joint work with Genadi Levin and Sebastian van Strien.

Higgs de-Rham flous and applications
Mao Sheng, University of Science and Technology of China, China

Deligne-Illusie's Lemma in the smooth case is the key to the algebraic proof of the $E_1$-degeneration of Hodge-to-de Rham spectral sequence. In this talk, I will report a Deligne-Illusie's Lemma in the semi-stable case. It has applications to arithmetics: we construct a smooth lifting of the nodal rational curve over a perfect field of positive characteristic to the Witt ring, which is analogous to the Serre-Tate canonical lifting for an ordinary elliptic curve. This is a joint work in progress with Professor Kang Zuo.

Holomorphic dynamics and currents
Nessim Sibony, Universite Paris-Sud, France

I will survey some recent developments in holomorphic dynamics in connection with the theory of positive closed or ddc -closed currents on compact Kähler manifolds. Many natural questions can be read as intersection problems in the theory of currents, the theory of densities gives the appropriate calculus. The talk is based on joint work with T.C Dinh.

Globally F-regular type of moduli spaces of parabolic sheaves
Xiaotao Sun, Tianjin University, China

A variety over a perfect field of characteristic p>0 is called globally F-regular if it is stably Frobenius split along every effective divisor. A variety over a field of characteristic zero is called globally F-regular type if its modulo p reduction is globally F-regular for almost p. In this talk, I will report a joint work with Mingshuo Zhou that moduli spaces of semi-stable parabolic bundles and generalized parabolic sheaves with fixed determinant on curves are globally F-regular type. In particular, they have vanishing cohomologies for any nef line bundles.

Poincaré problem and birational invariants of a differential equation
Sheng-Li Tan, East China Normal University, China

In the 19th century, Darboux, Painlevéand Poincaré studied differential equations of the first order by using complex algebraic geometry. More precisely, the theory of integrable curves defined by complex differential equations is similar to the theory of families of algebraic curves, i.e., fibrations on an algebraic surface. Poincaré proposed the following research program.

1) Study the (topological) properties of families of algebraic curves on a complex algebraic surface, and check if they are the properties of differential equations.
2) Find numerical invariants of complex differential equations.
3) Classify complex differential equations according to their invariants.
4) Characterize those complex differential equations which are algebraically integrable.
5) Apply to some problems on real differential equations.

I will talk about some recent progress in Poincaré Program.

K-stability and Kähler metrics
Gang Tian, Princeton University, USA and Peking University, China

In these expository lectures, I will discuss the K-stability and its connections to existence of canonical metrics in Kähler geometry. I will show how to establish the existence of Kähler-Einstein metrics on Fano manifolds. Some open problems will be also discussed near the end.

Flag varieties, a geometric characterization and rigidity
Jaroslaw Wisniewski, University of Warsaw, Poland

Flag varieties can be characterized as these Fano manifolds whose all extremal contractions are smooth P^1 fibrations. Subsequently they are (globally) rigid in families of Fano manifolds. I will report on these results obtained in collaboration with Gianluca Occhetta, Luis E. Solá Conde, Kiwamu Watanabe, and Andrzej Weber.

For a conjecture of Silverman for the endomorphisms of the affine plan
Junyi Xie, IRMAR - Université de Rennes 1, France

Let f be an endomorphism of the affine plan defined over a number field K. For any K-point x in the plan, Silverman have defined the arithmetic degree of x, which measures the growth of the heights of the orbits of x and showed that it is bouned by the dynamical degree of f. He conjecured that the arithmetic degree is equal to the dynamical degree unless the orbit of x is not Zariski dense. With Jonsson and Wulcan, we proved this conjecture for the endomorphisms of the affine plan when the dynamical degree is not less than the topological degree.

Lecture 1: Torsion points and preperiodic points: the Manin-Mumford conjecture and its dynamical analogue
Shou-Wu Zhang, Princeton University, USA

I will talk about techniques used in different proofs of the Manin - Mumford conjecture and its analogue in dynamical systems: o-minimality geometry (Pila-Zannier), Arakelov geometry (Ullmo-Zhang), and perfectoid geometry (Xie).

Iitaka conjecture and abundance for 3-folds in char p
Lei Zhang, Shaanxi Normal University, China

Recently, Birkar, Hacon and Xu have proved existence of minimal model of 3-folds in char p >5. It remains to prove abundance. In this talk, we will discuss some related topics, and introduce the recent progresses in Iitaka conjecture and abundance for 3-folds in positive characteristic p >5. If time permitting, we will explain the proof of abundance for 3-folds with non-trivial Albanese maps.

Rational points on curves: the ABC conjecture and BSD conjecture
Shou-Wu Zhang, Princeton University, USA

I will survey basic conjecture and recent results about rational on curves:
- the ABC conjecture and its relation to effective Mordell conjecture, Szpiro's conjecture, and the Milnor-Wood inequality;
- the BSD conjecture and its relation to congruent number problem.

Lecture 2: CM points and derivatives of L-functions: the Andre-Oort conjecture and Colmez' conjecture
Shou-Wu Zhang, Princeton University, USA

I will talk about recent work about the Andre-Oort conjecture (Pila-Tsimmerman, et al), Colmez' conjecture (Yuan-S. Zhang), and some related work on derivatives of L-functions and Drinfeld's moduli of Shtukas over function fields(Yun-W. Zhang).

On smooth projective curves with given Hodge type
Kang Zuo, Mainz University, Germany

Motived by Coleman conjecture on finiteness of CM jacobians of smooth curves of genus g>7 and Ekedahl-Serre conjecture on the existence or the finiteness of completely decomposble jacobians of smooth curves of higher genus, we introduce the notion of smooth projective curves with given Hodge type via Mumford Tate group. We show some finiteness (non-existence) theorem of curves with very special Hodge types, which answers Coleman conjecture and Ekedahl-Serre conjecture partially.

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