Title and Abstract
Fabrizio Catanese (University of Bayreuth)
Title: Compact Complex Manifolds with some trivial Chern classes or a trivial tangent/cotangent subbundle.
Abstract: Severi and Baldassarri in the 50’s asked for a characterization of projective/Kaehler manifolds with top trivial Chern classes.
I will first survey answers and counterexamples to their conjectures, in particular showing that a free quotient T/G of a torus by a cyclic group G has topologically trivial tangent bundle.
The requirement that some top Chern classes are zero is fulfilled by manifolds (isogenous to one) with a trivial subbundle of the tangent/cotangent bundle. In the tangent case we have the Pseudo-Abelian varieties of Roth, varieties isogenous to a torus product, and suspensions over a torus. Most of the results hold for cKM, or manifolds in the Fujiki class, but we have also suspensions over parallelizable manifolds.
More mysterious, even in the projective case, is the case where the cotangent bundle admits a trivial subbundle. Here the open questions are related to Manifolds admitting a complex submersion onto an Abelian Variety and the associated variation of Hodge structures.
Tien-Cuong Dinh (National University of Singapore)
Title: Exponential mixing of all orders and CLT for automorphisms of compact Kaehler manifolds
Abstract: We consider the unique measure of maximal entropy of an automorphism of a compact Kaehler manifold with simple action on cohomology. We show that it is exponentially mixing of all orders with respect to Hoelder observables. It follows that the Central Limit Theorem (CLT) holds for these observables. In particular, our result applies to all automorphisms of compact Kaehler surfaces with positive entropy. The talk is based on a joint work with Fabrizio Bianchi.
Lawrence Ein (University of Illinois at Chicago)
Title: Some questions on syzygies of algebraic varieties
Abstract: We’ll discuss some of the conjectures on syzygies of algebraic curves and higher dimensional varieties. We’ll also describe some of the recent results on some of these questions.
Yujiro Kawamata (University of Tokyo)
Title: On non-commutative deformations of complex manifolds.
Abstract: We will describe infinitesimal deformations of complex manifolds to the direction of something having possibly non-commutative structure sheaves by using Hochschild cohomology. We will also describe global non-commutative deformations of some surfaces.
JongHae Keum (Korea Institute for Advanced Study)
Yongnam Lee (KAIST & IBS)
Title: Positivity of the tangent bundle and total dual VMRT
Abstract: The total dual VMRT of a family of minimal rational curves carries some information on the positivity of the tangent bundle of rationally connected manifolds. In this talk, we will discuss the total dual VMRT and its application to the pseudo-effective cone of the tangent bundle of weak Fano manifolds of dimension 2 and 3. This talk is based on joint and ongoing works with Hosung Kim and Jeong-Seop Kim.
Qifeng Li (Shandong University)
Title: The geometric structures associated with VMRT-structures
Abstract: The local structures of VMRT's (the varieties of minimal rational tangents) carry much information on global geometry of manifolds. A typical example is the Cartan-Fubini type extension theorem due to Hwang and Mok, which indicates that Fano manifolds of Picard number one can be determined by their local VMRT-structures. We are interested in isotrivial VMRT-structures, the simplest local VMRT-structures. In this talk, we will discuss on the geometric structures associated with the isotrivial VMRT-structures as well as the applications in algebraic geometry. This talk is based on joint works with Jun-Muk Hwang.
Ngaiming Mok (The University of Hong Kong)
Title: Kähler Geometry Motivated by Number Theory: from Abelian Schemes to Uniformization Problems
Abstract: The speaker has long been interested in applications of complex differential geometry to the study of varieties of arithmetico-geometric and algebro-geometric interest, and will trace the trajectory of his involvement revolving around abelian schemes over complex function fields and uniformization problems on quotients of bounded symmetric domains.
We recall first results of Mok (1991) and Mok-To (1993) concerning the finiteness of Mordell-Weil groups of universal abelian varieties AΓ without fixed parts over modular function fields K = ℂ(XΓ). In these early works we introduced in the modular case an invariant semi-Kähler form (currently called the Betti form), and, making use of the classifying map, obtained bounds on ranks of Mordell-Weil groups in geometric terms via the volume of the ramification locus. An important tool was the extendibility of the Betti form as a closed positive current deduced from Bishop's extension theorem for closed positive currents due to Skoda (1982).
Most recently, Mok-Ng (2023) applied the complex differential geometric approach in the above to prove finiteness results on points of Betti multiplicities ≥ 2 of a section σ ∈ E(ℂ(X)) of infinite order in the case of an elliptic scheme over a quasi-projective curve, a result obtained by Corvaja-Demeio-Masser-Zannier, which was rendered effective by Ulmer-Urzúa (2021). Our approach is differential geometric, basing on a fundamental first-order real-linear differential equation satisfied by the "vertical component'' ησ of dσ for σ ∈ E(ℂ(X)), and has the advantage of being applicable in principle to abelian schemes.
Regarding uniformization problems we will discuss applications of complex differential geometry to problems of arithmetic interest, such as Ax-type problems, on quotient of bounded symmetric domains and their algebraic subvarieties. We will discuss the result of Chan-Mok (2022) on projective varieties uniformized by algebraic subsets of bounded symmetric domains (in their Harish-Chandra realizations) deduced from the study of the asymptotic behavior of holomorphic isometries of the Poincaré disk when they exit such domains and Nadel's semisimplicity theorem (1990) on automorphisms of universal covers of projective manifolds with ample canonical line bundle. If time permits, we will also discuss the Ax-Lindemann theorem of the speaker (2019) for arbitrary lattices acting on the complex unit ball 𝔹n and how this combines with uniformization results to prove Ax-Lindemann for arbitrary cocompact lattices acting on bounded symmetric domains.
Shigeru Mukai (RIMS Kyoto University)
Title: Vinberg surface, cubic 4-fold with 10 cusps and Lagrangian fibrations
Abstract: Vinberg(1983) studied "two most algebraic” K3 surfaces and determined the structure of their (infinite) automorphism groups. As a higher dimensional analogue I discuss the birational automorphism groups of holomorphic symplectic 4-, 8- and 10-folds associated with one of discriminant 3. In their study Lagrangian fibrations play a key role. For example the 10-fold is a special case of Beauville’s integrable systems studied by Hwang-Nagai (2008) in connection with O’Grady’s exceptional deformation type.
Katharina Neusser (Masaryk University)
Title: Cone structures and parabolic geometries
Abstract: A cone structure on a complex manifold M is a closed submanifold 𝒞 ⊂ ℙ TM of the projectivized tangent bundle of M which is submersive over M . Such structures arise naturally in differential and algebraic geometry, and when they do, they are typically equipped with a conic connection that specifies a distinguished family of curves on M in directions of 𝒞 . In differential geometry, a classical example is the null cone bundle of a holomorphic conformal structure with the conic connection given by the null-geodesics. In algebraic geometry, one has the cone structures consisting of varieties of minimal rational tangents (VMRT) given by minimal rational curves on uniruled projective manifolds. In this talk we will discuss various examples of cone structures and will introduce two invariants for conic connections. As an application of the study of these invariants and using the theory of parabolic geometries, we will present a local-differential-geometric version of the global algebraic-geometric recognition theorem of Mok and Hong–Hwang, which recognizes certain rational homogeneous spaces from their VMRT-structures. This talk is based on joint work with Jun-Muk Hwang.
Keiji Oguiso (The University of Tokyo)
Title: Smooth projective surfaces of Kodaira dimension zero with non-finitely generated automorphism group
Abstract: We construct very explicitly a surface which is a one point blow-up of a complex K3 surface (resp. Enriques surface) such that the full automorphism group is (necessarily discrete but) not finitely generated. If time allows, we will also discuss the existence of such examples over other algebraically closed fields. This talk is based on my joint works with T.-C. Dinh, C. Gachet, H.-Y. Lin, L. Wang and X. Yu after earlier works with Professors Keum and Professor Dinh.
Thomas Peternell (University of Bayreuth)
Title: Miyaoka-Yau (in)equalities, the topology of klt varieties and degenerations of projective spaces.
Abstract: I will discuss mildly singular (i.e., klt) varieties which are homeomorphic to klt varieties which satisfy the Miyaoka-Yau equality (joint work with D.Greb and S.Kebekus). Further, I will discuss klt degenerations of projective spaces (joint work in progress with A. Höring).
Yum-Tong Siu (Harvard University)
Title: Jet-Differentiation Approach to Global Nondeformability Problem
Abstract: The problem of global nondeformability of irreducible compact Hermtian symmetric manifolds has been open for a long time. It was first proposed by Kodaira and Spencer for the complex projective space in 1958, which was solved around 1990. Jun-Muk Hwang solved the case of the complex hyperquadric in 1995. With the additional assumption that the deformation was assumed to be Kähler, Jun-Muk Hwang and Ngaiming Mok solved the problem of global Kähler nondeformability of irreducible compact Hermitian symmetric spaces in 1998. They introduced the completely new and very powerful method of varieties of tangents of minimal rational curves (each such variety being the set of all tangents to minimal rational curves passing through a point). In this talk we will discuss the approach to solve the global nondeformability problem without the additional Kähler condition, by using jet differentiations of higher order instead of just tangent vectors.
Wing Keung To (National University of Singapore)
Title: Positivstellensätze via geometric and algebraic approaches
Abstract: In this talk, I will discuss some joint works with Colin Tan concerning several positivity conditions on polynomials and Hermitian algebraic functions. We made our studies on this topic via geometric/analytic as well as algebraic approaches.
Hajime Tsuji (Sophia University)
Title: Residue of Kähler-Ricci flows
Abstract: Let f : X → S be a smooth Kähler family. Let ω0 be a Kähler form on X. We shall consider the Kähler flow ωs(t) on each fiber Xs:=f -1(s) starting from ωs(0):= ω0|Xs . We shall study the variation of the family of the Kähler-Ricci flows on the total space X. The key is to relate the KRF on a fiber and a KRF on the total space X.
Sai-Kee Yeung (Purdue University)
Title: Hyperbolicity and the perspective of Carathéodory geometry
Abstract: The goal of the talk is to explain some geometric results on quasi-projective manifolds from the perspective of Carathéodory metrics and distances. We will study some conjectures of Lang on manifolds which satisfy some Carathéodory conditions. The results are also used to study hyperbolicity of suitable compactifications of non-compact manifolds involved, related along the way to earlier results of Nadel and Hwang-To. Most of the results to be presented are joint work with Kwok-Kin Wong.