October 30 - November 3 KIAS 1503 |

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**Florin Ambro (Simion Stoilow)**

**Title: **On toric Fano fibrations

**Abstract:** We discuss the classification of germs of toric Fano fibrations, extending work of A. Borisov in the case of Q-factorial toric singularities. As an application, we verify in the toric case a conjecture of V. Shokurov on the existence of complements with bounded index and prescribed singularities.

**Kenny Ascher (Irvine)**

**Title: **Moduli of low degree K3 surfaces

**Abstract: **The explicit descriptions of low degree K3 surfaces lead to natural compactifications coming from geometric invariant theory (GIT) and Hodge theory. The relationship between these compactifications for degree two K3 surfaces was studied by Shah and Looijenga, and revisited by Laza and O’Grady, who also provided a conjectural description for the case of degree four K3 surfaces. I will discuss these results, as well as a verification of this conjectural picture using tools from K-moduli. This is joint work with Kristin DeVleming and Yuchen Liu.

**Paolo Cascini (Imperial College London)**

**Title: **On the Chern numbers of a complex threefold

**Abstract: **We study the Chern numbers of a complex terminal threefold through a flip. Joint work with Hsin-Ku Chen.

**Ivan Cheltsov (University of Edinburgh)**

**Title: **Equivariant geometry of singular cubic threefolds

**Abstract:** I will report on a joint work with Yuri Tschinkel (Simons Foundation) and Zhijia Zhang (New York University) on linearizability of actions of finite groups on singular cubic threefolds.

**Kristin DeVleming (University of Massachusetts Amherst)**

**Title: **K-moduli of a family of conic bundle threefolds

Abstract: Recently, there has been significant progress in understanding the K-moduli spaces of Fano varieties and log Fano pairs (X,cD). When D is a rational multiple of the anticanonical divisor of X, the K-moduli spaces of log Fano pairs (X,cD) admit a wall crossing framework as c varies and there is a finite collection of rational values of c where the K-moduli spaces change. With Lena Ji, Patrick Kennedy-Hunt, and Ming Hao Quek, we explore the K-moduli spaces in an example where D is not proportional to the anticanonical divisor. We study the K-moduli space of pairs (P1xP2, cD) where D is a (2,2) divisor and prove that there is exactly one irrational value of c where the moduli spaces change. We further relate these moduli spaces to several related spaces: the GIT of (2,2) divisors in P1xP2, K-moduli of the conic bundle threefold that is the double cover of P1xP2 branched over D, and various moduli spaces of quartic plane curves arising as the discriminant of these conic bundles.

**Maksym Fedorchuk (Boston College)**

**Title: **Good limits of smooth quartic hypersurfaces

**Abstract: **Consider a family of smooth quartic 3-folds over a punctured disk, or more generally of Fano quartic hypersurfaces in a projective space. How can we complete it without a base change? If we think of a quartic 3-fold as a quartic in P^4, then Koll'ar's stability for hypersurfaces gives one way to complete any such family. If we think of it as a smooth Fano 3-fold with Picard rank 1 and anticanonical volume 4, then it is a (2,4)-complete intersection in P(1^5,2) and the generalization of Koll'ar's stability for such complete intersections another (and slightly better) completion to the original family.

However, neither of these two approaches gives a completely satisfactory answer. In this talk, I will describe the work in progress on finding a correct Koll'ar stability setup for this question that leaves the world of complete intersections. If time allows, I will draw analogies with the recently studied K-moduli space of quartic 3-folds. (This is joint work with Abban and Krylov.)

**Kento Fujita (Osaka University)**

**Title:** K-stability of Casagrande--Druel varieties

Abstract: We focus on a certain subclass of Fano varieties named Casagrande—Druel varieties. Especially, we see the K-polystability of several Casagrande—Druel threefolds whose general members are in Mori—Mukai's list Nos. 3.9 and 4.2, and see the K-moduli spaces parametrizing those varieties. This is a joint work with Ivan Cheltsov, Tiago Duarte Guerreiro, Igor Krylov and Jesus Martinez Garcia.

**Masafumi Hattori (Kyoto Univ.)**

**Title: **On boundedness and moduli spaces of K-stable Calabi-Yau fibrations over curves

**Abstract: **The characterization of K-stable varieties is well-studied when $K_X$ is ample or X is a Calabi-Yau or Fano variety. However, K-stability of Calabi-Yau fibrations (i.e., $K_X$ is relatively trivial) is not known much in algebraic geometry. We introduce uniform adiabatic K-stability (if $fcolon (X,H)to (B,L)$ is a fibration of polarized varieties, which means that K-stability of $(X,aH+L)$ for sufficiently small $a>0$).

In this talk, I would like to explain that uniform adiabatic K-stability of a Calabi-Yau fibration over a curve is equivalent to K-stability of the base curve in some sense. Furthermore, we construct separated moduli spaces of polarized uniformly adiabatically K-stable Calabi-Yau fibrations over curves. This talk is based on a joint work with Kenta Hashizume.

**Jun-Muk Hwang (Institute for Basic Science - Center for Complex Geometry)**

**Title: **Minimal rational curves whose VMRT at a general point is an adjoint variety

**Abstract: **For a family of minimal rational curves on a uniruled projective manifold, its VMRT at a point is the projective subvariety consisting of tangent directions of minimal rational curves through that point. In a joint work with Qifeng Li, we study families of minimal rational curves whose VMRT at a general point is the adjoint variety of a simple Lie algebra. Nontrivial examples arise from wonderful group compactifications and hyperplane sections of certain Grassmannians. We show that when the simple Lie algebra is not sl_n, n >3, these are the only nontrivial examples, modulo the equivalence of germs of minimal rational curves.

**In-Kyun Kim (KIAS)**

**Title: **Classification of singular del Pezzo surfaces

**Abstract: **In the field of Kähler geometry, the classification of Fano varieties equipped with a Kähler-Einstein metric is an important problem. Although proving the existence of such a metric on a Fano variety is challenging, recent progress has provided strong tools to attack this problem. In this talk, we will study how to prove the existence of Kähler-Einstein metrics on quasi-smooth del Pezzo hypersurfaces with higher index.

**Igor Krylov (IBS-CGP)**

**Title:** 2n^2-inequality for cA_1-singularities and applications to birational rigidity

**Abstract:** I will discuss the idea of proof of birational rigidity of threefolds and the importance of local inequalities for their proof. Then I will discuss the birational rigidity results that follow from 2n^2-inequality for cA_1 points, in particular I will talk about birational rigidity of sextic double solids with cA_n-singularities. At the end of the talk I will talk a bit about the idea of the proof of local inequalities and why our approach did not work for cA_2 points.

**Yongnam Lee (Institute for Basic Science - Center for Complex Geometry)**

**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, I will discuss the total dual VMRT and its application to the pseudo-effective cone of the projectivized tangent bundle on Fano manifolds.

**Takuzo Okada (Saga University)**

**Title:** Birationally solid Fano 3-fold hypersurfaces

**Abstract:** Fano 3-folds that are embedded as (quasismooth) hypersurfaces in weighted projective 4-spaces are classified and they form 130 families. Among them 95 families consist of Fano 3-fold weighted hypersurfaces of Fano index 1, and Cheltsov-Park proved that they are all birationally rigid. Recently, Abban-Cheltsov-Park showed that none of Fano 3-fold weighted hypersurfaces of Fano index at least 2 is birationally rigid. The aim of this talk is to explain birational properties of these Fano 3-fold weighted hypersurfaces of Fano index at least 2, and explain the classification of those with the property of "birational solidity" which is a notion weaker than birational rigidity.

**Karol Palka (IMPAN)**

**Title: **Singular del Pezzo surfaces of rank one in arbitrary characteristic

**Abstract: **We discuss the geometry of normal del Pezzo surfaces of Picard rank one over an algebraically closed field of arbitrary characteristic. We introduce a new invariant, which guides our uniform approach to the classification, the height. By definition, it is the minimal intersection number of a fiber of any P^1-fibration of the minimal resolution of singularities with the exceptional divisor. The geometry of del Pezzo surfaces gets more constrained as the height grows. Moreover, it turns out that the height is at most 4, with minor exceptions in case the characteristic of the field is positive and small, which allows for a complete classification. This is a joint project with Tomasz Pełka.

**Yuri Prokhorov (Steklov Mathematical Institute)**

**Title:** Tetragonal conic bundles

**Abstract:** I outline the classification of type IV Sarkisov links between tetragonal threefold conic bundles. In particular, the behaviour of the invariant $2K_S+C$ under these transformations will be discussed. This is a part of a joint work in progress with V. Shokurov.

**Miles Reid (University of Warwick)**

**Title:** Partial results on some cases of Fano 3-folds in the Graded Ring Database

**Abstract:** The Graded Ring Database (work of Gavin Brown and Al Kasprzyk) contains lists of candidate constructions of Fano 3-folds. A few hundred case can be settled by constructions or impossibility proofs, but it is hard to reach convincing conclusions in the majority of cases. I will discuss the overall picture, and a few special cases (largely based on ideas of Prokhorov).

**Evgeny Shinder (Univ. of Sheffield)**

**Title: **Nodal Fano threefolds: from defect to derived categories

**Abstract:** I will report on the recent progress in decomposing derived categories of non-factorial nodal Fano threefolds, following joint work with Pavic and Kalck, and more recent joint work with Kuznetsov.

**Konstantin Shramov (HSE/Steklov Mathematical Institute)**

**Title:** Automorphism groups of compact complex manifolds

**Abstract: **Automorphism groups of compact complex manifolds may have complicated structure. However, sometimes they satisfy certain nice properties on the level of their finite subgroups. I will survey results and expectations concerning the boundedness of finite subgroups in automorphism groups of compact complex manifolds and some of their quotients.

**Sho Tanimoto (Nagoya)**

**Title: **Non-free sections of Fano fibrations

**Abstract: **Manin’s Conjecture predicts the asymptotic formula for the counting function of rational points over number fields or global function fields. In the late 80’s, Batyrev developed a heuristic argument for Manin’s Conjecture over global function fields, and the assumptions underlying Batyrev’s heuristics are refined and formulated as Geometric Manin’s Conjecture. Geometric Manin’s Conjecture is a set of conjectures regarding properties of the space of sections of Fano fibrations, and it consists of three conjectures: (i) Pathological components are controlled by Fujita invariants; (ii) For each nef algebraic class, a non-pathological component which should be counted in Manin’s Conjecture is unique (This component is called as Manin component); (iii) Manin components exhibit homological or motivic stability. In this talk we discuss our proofs of GMC (i) over complex numbers using theory of foliations and the minimal model program. Using this result, we prove that these pathological components are coming from a bounded family of accumulating maps. This is joint work with Brian Lehmann and Eric Riedl.

**Andrei Trepalin (HSE / Steklov Mathematical Institute)**

**Title: **Quotients of pointless del Pezzo surfaces of degree 8

**Abstract: **In the talk we will consider del Pezzo surfaces of degree 8 over algebraically nonclosed fields of characteristic 0. Any quadric surface in three-dimensional projective space is a del Pezzo surface of degree 8, and it is well known that such surface can be pointless. We want to study birational classification of quotients of pointless del Pezzo surfaces of degree 8 by finite automorphism groups. In particular, we want to find conditions on the surface and the group for which the quotient can be not rational over the main field. We will show that the quotient by any group of odd order is birationally equivalent to the original surface, and the quotient by any group of even order is birationally equivalent to a quadric surface. Also we give a list of groups, for which the quotient can be not rational, and show, that the quotient is rational for other groups.

**Antonio Trusiani (Chalmers University of Technology)**

**Title: **A relative Yau-Tian-Donaldson conjecture and stability thresholds

**Abstract:** In the first part of the talk, partly motivated by the study of Kähler-Einstein metrics with prescribed singularities, a new relative K-stability notion will be introduced for a fixed smooth Fano variety. A particular focus will be given to motivations and intuitions, making a comparison with the log K-stability/log Kähler-Einstein metrics. The relative K-stability and the Kähler-Einstein metrics with prescribed singularities will then be related to each other through a Yau-Tian-Donaldson correspondence, which will be the core of the talk. An important role will be played by relative versions of the Fujita-Odaka δ-invariant, which will be also used to link the relative K-stability to the genuine K-stability.

**Chuyu Zhou (Yonsei Univ.)**

**Title:** K-semistable domain and wall crossing for K-stability

**Abstract:** In this talk, we will define K-semistable domain of a pair consisting of a Fano variety and multiple boundaries. We will present many important properties of the K-semistable domain. Based on this, we will see a wall crossing theory for K-stability. Time permits, we will also talk about some interesting examples.