2024 KIAS HCMC
Algebraic Geometry Seminar
Upcoming Talks
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.
Previous Talks
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