2024 KIAS HCMC
Algebraic Geometry Seminar

Previous Talks

December 2 (Mon) 2024, 16:00~17:30 - HCMC Seminar Room

Speaker: Changho Han (Korea University)
Title: Compact moduli of K3 surfaces via trigonal maps

K3 surfaces, as a generalization of elliptic curves, have a rich amount of geometric properties. Recalling that elliptic curves are double covers of rational curves branched over 4 distinct points, there are K3 surfaces that are cyclic triple covers of rational surfaces; Artebani and Sarti classified such generic K3 surfaces depending on lattice invariants. Such K3 surfaces admit Kulikov and KSBA degenerations, each leading to toroidal and KSBA compactifications of the moduli spaces of such K3 surfaces. As joint works in progress with Valery Alexeev, Anand Deopurkar, and Philip Engel, I will explain how to use trigonal curves (triple covers of rational curves) to obtain aforementioned degenerations, leading to more explicit understandings of boundaries of those compactifications: such as classifications of generic members and the dimensions.

November 19 (Tue) 2024, 11:00~12:30 - HCMC Seminar Room

Speaker: Shin-young Kim (Yonsei University)
Title: Global deformation rigidity of complex projective manifold

A complex projective manifold X is said to be globally rigid under projective deformations if for any smooth projective morphism over a disc with one fiber biholomorphic to X, then all fibers are biholomorphic to X. For example, the projective space of dimension n is rigid by a classical result of Hirzebruch and Kodaira; the same holds for quadrics of dimension n at least 3 (Brieskorn and Hwang). More generally, projective homogeneous varieties of Picard rank 1 were shown to be rigid by Hwang and Mok, with one (remarkable) exception. We will survey the results of Hwang and Mok, and further recent developments about Fano quasi-homogeneous varieties with Picard rank one.

November 15 (Fri) 2024, 11:00~12:00 - HCMC Seminar Room

Speaker: Han-Bom Moon (Fordham University)
Title: Derived category of moduli space of vector bundles on a curve

The derived category of moduli spaces of vector bundles on a curve is expected to be decomposed into the derived categories of symmetric products of the base curve. I will briefly explain the expectation and known results, and some consequences. This is joint work in progress with Kyoung-Seog Lee.

November 11 (Mon) 2024, 11:15~12:15 - HCMC Seminar Room

Speaker: Livia Campo (University of Vienna)
Title: K-stablity of Fano threefold hypersurfaces of index 1

The existence of Kaehler-Einstein metrics on Fano 3-folds can be determined by studying lower bounds of stability thresholds. An effective way to verify such bounds is to construct flags of point-curve-surface inside the Fano 3-folds. This approach was initiated by Abban-Zhuang, and allows us to restrict the computation of bounds for stability thresholds only on flags. We employ this machinery to prove K-stability of terminal quasi-smooth Fano 3-fold hypersurfaces. This is deeply intertwined with the geometry of the hypersurfaces: in fact, birational rigidity and superrigidity play a crucial role. The superrigid case had been attacked by Kim-Okada-Won. In this talk I will discuss the K-stability of strictly rigid Fano hypersurfaces via Abban-Zhuang Theory. This is a joint work with Takuzo Okada.

July 31 (Wed) 2024, 16:00~17:00 - HCMC Seminar Room

Speaker: Jae Hwang Lee (Colorado State University)
Title: A Quantum H*(T)-module via Quasimap Invariants

For X a smooth projective variety, the quantum product of the quantum cohomology ring QH*(X) is a deformation of the product of the usual cohomology ring H*(X), where the product structure is modified to incorporate quantum corrections. These correction terms are defined using Gromov-Witten invariants. When X is toric with its geometric quotient description V//T, the cohomology ring H*(V// T) also has the structure of a H*(T)-module. In this paper, we introduce a new deformation of this module structure using quasimap invariants with a light point. This defines a quantum H*(T)-module structure on H*(X) through a modified version of the WDVV equations. We explicitly compute this structure for the Hirzebruch surface of type 2. We conjecture that this new quantum module structure is isomorphic to the natural module structure of the Batyrev ring for a smoother projective semipositive toric variety.

June 26 (Wed) 2024, 16:00~17:00 - HCMC Seminar Room

Speaker: Sungwook Jang (Yonsei University)
Title: Anticanonical minimal models and Zariski decomposition

Let (X,Δ) be an lc pair. The minimal model program is a sequence of birational maps, which makes the divisor Kx+Δ closer to a nef divisor. Now, we know that we can run a minimal model program. However, we do not know whether the sequence terminates in finite steps or not. If the sequence stops, then we obtain either a minimal model or a mori fiber space. Besides the termination, Birkar and Hu showed that if a pair (X,Δ) is lc and Kx+Δ admits a birational Zariski decomposition, then (X,Δ) has a minimal model. Analogously, we prove that if a pair (X,Δ) is pklt and -(Kx+Δ) admits a birational Zariski decomposition, then (X,Δ) has an anticanonical minimal model.

June 19 (Wed) 2024, 16:00~17:00 - KIAS 1424

Speaker: Kisun Lee (Clemson University)
Title: Symmetric Tropical Rank 2 Matrices

Tropical geometry replaces usual addition and multiplication with tropical addition (the min) and tropical multiplication (the sum), which offers a polyhedral interpretation of algebraic variety. This talk aims to pitch the usefulness of tropical geometry in understanding classical algebraic geometry. As an example, we introduce the tropicalization of the variety of symmetric rank 2 matrices. We discuss that this tropicalization has a simplicial complex structure as the space of symmetric bicolored trees. As a result, we show that this space is shellable and delve into its matroidal structure. It is based on the joint work with May Cai and Josephine Yu.

June 4 (Tue) 2024, 16:00~18:00 - HCMC Seminar Room

Speaker: Donggun Lee (IBS Center for Complex Geometry)
Title: Automorphisms and deformations of regular semisimple Hessenberg varieties

Regular semisimple Hessenberg varieties are smooth subvarieties in flag varieties which have interesting nontrivial symmetric group actions on their cohomology. In this talk, we discuss their automorphisms and deformations, when they are divisors in the flag varieties. Especially in type A, we provide a complete classification, along with an interpretation in terms of pointed rational curves, and describe the moduli stack of such Hessenberg varieties as a quotient stack of the moduli space of pointed rational curves. This is a joint work with P. Brosnan, L. Escobar, J. Hong, E. Lee, A. Mellit and E. Sommers.

May 29 (Wed) 2024, 10:30~12:30 - KIAS 1424

Speaker: Lorenzo Barban (IBS Center for Complex Geometry)
Title: Geometric realization of birational maps

In this talk we aim to describe the rich relation between 1-dimensional algebraic torus actions on projective varieties and birational geometry. This seminar is divided in two parts. In the first one we give a provide a general introduction to the subject: we describe for instance the Bialynicki-Birula decomposition, GIT quotients of a polarized pair by a C*-action and the recent results obtained by G. Occhetta, E. Romano, L. E. Solà Conde and J. Wisniewski. The second part of the talk is based on joint works with E. Romano, G. Occhetta, L. Sola Conde and S. Urbinati. Given a birational map among projective varieties, we introduce a projective algebraic version of the cobordism notion coming from Morse theory, which we call geometric realization. We show how to construct geometric realizations of maps which are small modifications of dream type, that is birational maps which are isomorphism in codimension 1 associated to a finitely generated multisection ring. 

April 29 (Mon) 2024, 13:30~15:00 - KIAS 1424

Speaker: Yat-Hin Suen (KIAS)
Title: Tropical Lagrangian multi-sections and locally free sheaves

The Gross-Siebert program is usually referred to as the algebraic version of the famous SYZ mirror symmetry. The fundamental tool in their program is tropical geometry. A natural question that we want to address is how can one understand homological mirror symmetry under the Gross-Siebert framework. In this talk, I am going to introduce the notion of tropical Lagrangian multi-sections, which is a combinatorial replacement of Lagrangian multi-sections in the SYZ proposal. Such tropical objects can be used to construct locally free sheaves on log Calabi-Yau varieties. I will discuss the construction and smoothability of these locally free sheaves that arise from tropical geometry.

April 9 (Tue) 2024, 10:30~11:30 - HCMC Seminar Room

Speaker: Guolei Zhong (IBS Center for Complex Geometry)
Title: Structure of projective varieties with almost nef or positively curved tangent sheaves

Based on the first talk, we shall sketch the proof of our main result on the structure of projective varieties with almost nef or positively curved tangent sheaves. As applications, we first study the Fujita decomposition of reflexive sheaves on those varieties. We shall also prove that the almost nefness of tangent sheaves will impose rather restrictive conditions on the singularities. If the time is allowed, I will ask some open questions on the equivariant minimal model program concerning the positivity of tangent sheaves. This talk is based on the joint work with Masataka Iwai and Shin-ichi Matsumura.

April 8 (Mon) 2024, 16:00~17:00 - KIAS 1424

Speaker: Guolei Zhong (IBS Center for Complex Geometry)
Title: Introduction to the positivity of tangent sheaves of projective varieties

After Mori's solution to the Hartshorne conjecture, it has become evident that a certain positivity condition of the tangent bundle would produce rich geometry on the underlying variety. Considering the minimal model program for the classification of algebraic varieties, the appearance of singularities is indispensable. In this talk, I will first briefly recall various notions on the positivity of tangent sheaves in the singular setting. Then we compare them with the classical notions in the smooth case and some pathological examples will be given. Finally, after a review of previous results in this direction, we state our main theorem on the structure of projective varieties with certain positive tangent sheaves. This talk is based on the joint work with Masataka Iwai and Shin-ichi Matsumura.

March 6 (Wed) 2024, 16:00~18:00 - KIAS 1424

Speaker: Sanghyeon Lee (Shanghai Center for Mathematical Sciences)
Title: Computing Vafa-Witten invariants of surface with elliptic fibrations

By applying Nesterov's theory on surface with elliptic fibrations, we compute Vafa-Witten invariants via some assumption on surface invariant, in following two cases:
1) S = C x E / G, G is a finite group acting on curve C and elliptic curve E freely.
2) S -> P^1 is an elliptic surface, whose fibers are irreducible elliptic curves, at most have nodal singularity.
We will also give conjectures on VW invariants for elliptic surfaces whose fibers are elliptic curves(not necessarily irreducible), at most have nodal singularity. The contents of the talk is based on a joint work with Yaoxiong Wen.

Past Algebraic Geometry Seminars ...... 2023

Organizers
Junho Choe, In-Kyun Kim, Jeong-Seop Kim, Hyunsuk Moon, Hyeonjun Park
supported by June E Huh Center for Mathematical Challenges at Korea Institute for Advanced Study