Lectures on Birational Geometry (17-19 Jan)

• Guodu Chen (Shanghai Jiao Tong University)  
  Title: On the Minimal Model Program for generalized pairs and foliations
  Foliations and generalized pairs are two structures that play important roles in modern birational geometry, specifically in the minimal model program. The purpose of these lectures is to is to introduce these two new structures and then present some recent development in the minimal model program for both structures. The lectures are planned as follows:

1.  We first briefly review the classical minimal model program and then we will introduce the notion of generalized pairs as well as its history.

2. We will discuss recent progress on the minimal model program for generalized pairs, such as the cone theorem, the contraction theorem, and the existence of flips, etc.

3.  We will also provide some applications. We will discuss the minimal model program for foliations, focusing in particular on algebraically integrable foliations. Presenting some results parallel to those of the classical MMP.
   
   

• Zhan  Li  (Southern University of Science and Technology)
  Title: On relative Morrison-Kawamata cone conjecture for Calabi-Yau fiber spaces
  Abstract: A fibration with a relatively trivial canonical divisor is called a Calabi-Yau fiber space. The Morrison-Kawamata cone conjecture relates the birational geometry of a Calabi-Yau fiber space to the convex geometry of its movable cone. It has many interesting implications for the geometry of algebraic varieties not only restricted to Calabi-Yau varieties. This lecture series focuses on establishing the cone conjecture under the assumptions of its validity for the generic fiber and the abundance conjecture. The lectures are planned as follows:

1. Explaining background material on cones and birational automorphism groups. Establishing the structure theorem of Shoukurov polytopes. Reviewing the history of the cone conjecture.

2. Discussing the convex geometry of cones. Studying the cone conjecture by Shoukurov polytopes.

3. Explaining the structure of the Albanese morphisms for Calabi-Yau varieties. Establishing the  cone conjecture under the assumptions of its validity for the generic fiber and the abundance conjecture.  Discussing consequences of the cone conjecture and related problems.

• BANQUET 1 on JAN 18 is at  ASHLEY QUEENS at Hongdae (애쉴리 퀸즈 홍대점). 

It is within a 5-minute walking distance from Holiday inn Express Hongdea.

JAN 22 MONDAY

• Laurentiu Maxim (University of Wisconsin, Madison) - I
  Title: Singularities through the lens of characteristic classes
  - Lecture 1. Introduction to characteristic classes for singular varieties
  Abstract: I will start by introducing characteristic numbers and characteristic classes for singular complex algebraic varieties, with an emphasis on Hirzebruch classes. I will then exploit the Hodge theoretic and motivic nature of the Hirzebruch classes to compute them in several instances, e.g., in the toric context.
  
  

• Claude Sabbah (CNRS, École polytechnique, Institut Polytechnique de Paris) - I
  Title: A global point of view on singularities of functions
  Abstract: We will illustrate the notion of irregular Hodge theory on the example of pairs $(U,f)$ formed by a smooth quasi-projective variety with a regular function on it. Such a datum can be seen as a mirror image of a projective Fano manifold (or orbifold).
  - Lecture 1. After indicating Deligne's (1984) initial motivation for such a theory, we will introduce the notion of exponential mixed Hodge structure (Kontsevich-Soibelman).
  
  

• Nero Budur (KU Leuven)
  Title: Contact loci of arcs
  Abstract: Contact loci are sets of arcs on a smooth variety with prescribed contact order along a fixed hypersurface. They appear in motivic integration, where motivic zeta

functions are generating series for classes of contact loci in appropriate Grothendieck groups. We give an overview of recent results on the topology of contact loci, relating their cohomology with the Floer cohomology of monodromy iterates, and their irreducible components with dlt models. 

• Cheol-Hyun Cho (Seoul National University)
  Title: Floer theory for the variation operator of an isolated singularity

Abstract: The variation operator in singularity theory maps relative homology cycles to compact cycles in the Milnor fiber using the monodromy. We construct its symplectic analogue for an isolated singularity. We define a new Floer cohomology, called monodromy Lagrangian Floer cohomology, which provides categorifications of the standard theorems on the variation operator and the Seifert form. The key ingredients are a special class in the symplectic cohomology of the inverse of the monodromy and its closed-open images. For isolated plane curve singularities whose A'Campo divide has depth zero, we find an exceptional collection consisting of non-compact Lagrangians in the Milnor fiber corresponding to a distinguished collection of vanishing cycles under the variation operator. This is a joint work with Hanwool Bae, Dongwook Choa and Wonbo Jeong.

• Jinhyung Park (KAIST)
  Title: K-polystability of the first secant varieties of rational normal curves
  Abstract: The first secant variety of a rational normal curve is a Fano threefold with log terminal singularities. In this talk, we show that this Fano threefold is K-polystable and admits an anticanonical polar cylinder. This is joint work with In-Kyun Kim and Joonyeong Won.

JAN 23 TUESDAY

• Claude Sabbah (CNRS, École polytechnique, Institut Polytechnique de Paris) - II
  Title: A global point of view on singularities of functions
  Abstract: We will illustrate the notion of irregular Hodge theory on the example of pairs $(U,f)$ formed by a smooth quasi-projective variety with a regular function on it. Such a datum can be seen as a mirror image of a projective Fano manifold (or orbifold).

- Lecture 2. We will illustrate the theory for pairs $(U,f)$ by introducing the notion of space of global vanishing cycles and its irregular Hodge filtration, together with its Brieskorn lattice.

• Laurentiu Maxim (University of Wisconsin, Madison) - II
  Title: Singularities through the lens of characteristic classes
  - Lecture 2. Characteristic classes of hypersurfaces via specialization
  Abstract: I will discuss characteristic class theories of Milnor type (e.g., Hirzebruch-Milnor classes) associated with complex hypersurfaces. Such classes measure the complexity of hypersurface singularities, providing vast generalizations of the notion of Milnor number, and they can be described via specialization. Several geometric and topological applications will be also discussed.
  
  

• Shunsuke Takagi (University of Tokyo)
  Title: On the behavior of adjoint ideals under pure morphisms

Abstract: Recently, Z. Zhuang proved that the property of being of plt type descends under pure morphisms. In this talk, I will discuss a generalization of Zhuang's result. To do so, I will give a characterization of adjoint ideal sheaves using ultraproducts. This talk is based on joint work with Tatsuki Yamaguchi.

• Osamu Fujino (Kyoto University)
  Title: Minimal model program for projective morphisms between complex analytic spaces

Abstract: We discuss the minimal model program for projective morphisms of complex analytic spaces. Roughly speaking, we show that the results obtained by Birkar—Cascini—Hacon—McKernan hold true for projective morphisms between complex analytic spaces. Our formulation will play an important role in the study of complex analytic singularities.

• Caucher Birkar (Cambridge University/Tsinghua University)   *ONLINE*  
  The ZOOM link.  passcode is  today's  YYYYMMDD
  Title: Singularities of algebraic varieties
  Abstract: Singularities play an important role in algebraic geometry and other areas of mathematics. In this talk we will discuss some generalities and then some recent progress on singularities in birational geometry. 

• The BANQUET 2 on January 23 will be held at Chio Young Hall. It is located beneath the campus eagle statue. (6:30 pm)

JAN 24 WEDNESDAY

• Laurentiu Maxim (University of Wisconsin, Madison) - III
  Title: Singularities through the lens of characteristic classes
  - Lecture 3. Spectral classes and Applications to rational and du Bois singularities
  Abstract: I will make use of the monodromy action to introduce spectral versions of the Hirzebruch-Milnor characteristic classes of hypersurfaces, generalizing the notion of Steenbrink spectrum. I will then explain how these enhanced classes can be used to detect jumping numbers of (higher) multiplier ideals, as well as (higher) du Bois and (higher) rational hypersurface singularities.
  
  

• Claude Sabbah (CNRS, École polytechnique, Institut Polytechnique de Paris) - III
  Title: A global point of view on singularities of functions
  Abstract: We will illustrate the notion of irregular Hodge theory on the example of pairs $(U,f)$ formed by a smooth quasi-projective variety with a regular function on it. Such a datum can be seen as a mirror image of a projective Fano manifold (or orbifold).
  - Lecture 3. We will conclude with some general results on the theory of irregular mixed Hodge modules, including a Kodaira-Saito-type vanishing theorem for irregular Hodge bundles.

JAN 25 THURSDAY

• Mihai Tibar (Université de Lille)
  Title:  Enumerative geometry of the gradient
  Abstract.  We will present, under a unifying viewpoint, although using different techniques, some recent results involving the degree of the gradient, namely results on the the polar degree, on the Euclidean distance degree (ED-degree), and on the linear Morsification of complex polynomials.
  
  

• Javier Fernandez de Bobadilla (BCAM - Basque Center for Applied Mathematics)
  Title: Symplectic degenerations at radius 0 and their applications
  Abstract: Given a normal crossings degeneration $f:(X,\Omega_X)\to\Delta$ of complex Kahler manifolds, in recent work together with T. Pelka, we have shown how to associate a smooth locally trivial fibration $f_A:A\to \Delta_{log}$ over the real oriented blow up of the disc $\Delta$. It is moreover endowed with a closed $2$-form $\omega_A$ giving it the structure of a symplectic fibration. The restriction of $\omega_A$ to every fibre of $f_A$ "at positive radius" (that is over a point of $\Delta\setminus\{0\}$ is the modification by a potential of the restriction of $\omega_X$ to the same fibre. The construction is so that:
  (1) We can produce symplectic representatives of the monodromy with very special dynamics, and based on this and on a spectral sequence due to McLean prove the family version of Zariski's multiplicity conjecture.
  (2) If $f$ is a maximal Calabi-Yau degeneration we can produce Lagrangian torus fibrations over a the complement of a codimension 2 set over the (expanded) essential skeleton of the degeneration, satisfying many of the properties conjectured by Kontsevich and Soibelman.
   

• Guodu Chen (Shanghai Jiao Tong University)
  Title: Minimal Model Program for Q-factorial foliated dlt algebraically integrable foliations

Abstract: Foliated dlt foliations play the same role as dlt pairs in the classical minimal model program, making it a natural class of singularities to study in the theory of foliations. We show that we can run an MMP on a Q-factorial foliated dlt foliation by proving the cone theorem, contraction theorem, and the existence of flips. Numerous applications also will be presented. Joint work with Jingjun Han, Jihao Liu, and Lingyao Xie.

• Kiwamu Watanabe (Chuo University)
  Title: Extremal contractions and non-free rational curves on Fano varieties

Abstract: Let X be a complex smooth Fano variety whose minimal anticanonical degree of non-free rational curves on X is at least dimX−2. We give a classification of extremal contractions of such varieties. As applications, we obtain a classification of Fano fourfolds whose pseudoindex and Picard number are greater than one and study the structure of Fano varieties with nef third exterior power of the tangent bundle. 

• Joaquín Moraga (UCLA)
  Title: Birational complexity of Fano varieties.

Abstract: The complexity is an invariant that measures how far is a log pair from being toric. In this talk, we will introduce a birational variant of the complexity; the birational complexity. The birational complexity measures how far is a Fano variety from admitting a  crepant birational toric structure. This invariant has several interesting implications on the study of higher-dimensional Fano varieties,  for instance, rationality, singularities, and the topology of dual complexes.

JAN 26 FRIDAY

• Yulia Gorginyan (IMPA - Instituto de Matemática Pura e Aplicada)
  TItle: Complex curves in nilmanifolds
  Abstract: A nilmanifold is a compact manifold obtained as a quotient of a nilpotent Lie group by a cocompact lattice. When the Lie group is equipped with a left-invariant complex structure, the nilmanifold is called a complex nilmanifold. It is known that almost all complex nilmanifolds, except a torus, are non-Kahler. Among all complex non-Kahler surfaces only three types of Inoue surfaces have no curves. It is natural to study the existence of complex curves in a non-Kahler manifold. For example, any deformation of the Iwasawa manifold, which is a 3-dimensional complex nilmanifold, possesses a complex curve. We will prove that a general twistor deformation of a hypercomplex nilmanifold admits no complex curves.  
  
  

• Takahiro Saito (Chuo University)
  Title: Mixed Hodge modules of normal crossing type on smooth toric varieties
  Abstract: In general, mixed Hodge modules (MHMs) are complicated and difficult to deal with, while the general theory is well established. However, if the underlying D-module of a MHM on C^n satisfies the condition: "of normal crossing type", it can be expressed in a linear algebraic way. This is a generalization to MHM of the well-known "can-var description" of perverse sheaves on C. As an application, we can give a natural definition of "the Fourier-Laplace transform of a MHM of normal crossing type". In the first half, I will introduce these facts. In the second half, I will talk on the recent progress on the generalization of them to MHMs on smooth toric varieties.

• Christopher Chiu (KU Leuven)
  Title: Schematic structure of arc spaces and invariants of singularities
  Abstract: Starting with Nash's work in the 1960s, the topology of the arc space has been investigated in relationship with the singularities of the underlying variety. For example, codimensions of contact loci compute both discrepancies and log canonical thresholds. However, recent work has supported the idea that the singularities of the arc space itself should also be useful to control such invariants. In this talk, we will introduce a structural result on arc fibers of quasi-finite morphisms. As consequences, we will obtain both finiteness results on stable points of the arc space, as well as connections between their embedding (co)dimensions and Mather(-Jacobian) discrepancies. This is joint work with T. de Fernex and R. Docampo.