Mini-Workshop:
Young Perspectives in Algebraic Geometry
Nov. 25-26, 2024
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Dates and Location
Dates and Location
Mini-Workshop: November 25 -26, 2024
Venue: R440, Astronomy Mathematics Bldg., National Taiwan University, Taipei
Registration
Registration
Registration form: [Link] Deadline: November 15, 2024
Invited Speakers
Invited Speakers
Jheng-Jie Chen (National Centeral University)
Keita Goto (National Taiwan University)
Yuto Masamura (The University of Tokyo)
Flora Poon (NCTS)
Yuya Sasaki (The University of Tokyo)
Yutaro Sugimoto (The University of Tokyo)
Yuta Takada (The University of Tokyo)
Tomoki Yoshida (Waseda University)
Agenda
11/25 Mon.
10:30-11:30 Keita Goto
12:50-13:50 Yutaro Sugimoto
14:10-15:10 Yuya Sasaki
11/26Tue.
09:30-10:30 Flora Poon
10:50-11:50 Yuta Takada
13:30-14:30 Tomoki Yoshida
14:50-15:50 Yuto Masamura
16:10-17:10 Jheng-Jie Chen
Title & Abstract
Jheng-Jie Chen (National Centeral University)
Title: Computations on the set of threefold canonical thresholds
Abstract:In this talk, we will briefly introduce the minimal model program and the Sarkisov program. Then, we will recall some classifications of divisorial contractions that contract divisors to points in dimension 3 due to the works of Professors Mori, Cutkosky, Kawamata, Hayakawa, Kawakita, and Yamamoto. These help us to classify the set T^can_3 of threefold canonical thresholds. An interesting result is that the set T^can_{3,sm} of smooth threefold canonical threshold coincides with the set HT_2 of 2-dimensional hypersurface log canonical thresholds completely classified by Professor Takayasu Kuwata in 1999. If time permits, I will explain the classification of the set T^can_3 . This is a recent joint work with Professor Jiun-Cheng Chen and my current student Hung-Yi Wu.
Keita Goto (National Taiwan University)
Title: On a non-Archimedean analog of SYZ Fibration for toric degenerations of Calabi--Yau complete intersections
Abstract: When we consider a degenerating family of algebraic varieties, non-Archimedean geometry often provides a useful perspective in understanding what’s going on. As such an example, we would like to discuss SYZ fibration from the perspective of non-Archimedean geometry. After Kontsevich and Soibelman proposed this non-Archimedean method to understand mirror symmetry, Yang Li developed a more sophisticated method that takes pluripotential theory into account. Since then, many researchers have studied degenerating families of Calabi--Yau hypersurfaces in a toric Fano manifold to apply Li's method. In this talk, I’ll explain our result on toric degenerations of Calabi--Yau complete intersections as a generalization of such studies. This talk is based on a joint work with Yuto Yamamoto.
Yuto Masamura (The University of Tokyo)
Title: Indices of smooth Calabi--Yau varieties
Abstract: A Calabi--Yau variety is a normal projective variety with numerically trivial canonical divisor $K_X$. The index of a Calabi--Yau variety $X$ is defined as the smallest positive integer $m$ such that $mK_X$ is trivial. A fundamental open problem in this area, known as the index conjecture, suggests that for Calabi--Yau varieties of fixed dimension and with appropriate assumptions on singularities (such as log canonical), their indices are bounded. In this talk, I will discuss recent developments in understanding the relationship between the indices of smooth Calabi--Yau varieties and the indices of lower-dimensional log Calabi--Yau pairs. This approach gives an inductive framework that may lead to a proof of the index conjecture.
Flora Poon (NCTS)
Title: Kuga-Satake construction on families of K3 surfaces of Picard rank 14.
Abstract: Classically, it is known that the period domain D of a moduli space of K3 surfaces of Picard rank r > 13 is diffeomorphic to the period domain of a different moduli space of some polarized varieties. The lowest Picard rank known for such a coincidence happens is r = 14: there is a diffeomorphism from D to the period domain of a moduli of polarized abelian 8-folds with totally definite quaternion multiplication, which descends to a map between the corresponding moduli spaces. We will describe the latter map explicitly by considering the Kuga-Satake construction on lattice polarized K3 surfaces. Using lattice theoretical arguments, we will also show that the map of moduli exhibits exceptional behaviour when specialised to families of K3 surfaces of Picard rank 18 admitting a Shioda-Inose or a Kummer structure.
Yuya Sasaki (The University of Tokyo)
Title: On automorphisms of the Hilbert scheme of $n$ points of some simple abelian varieties
Abstract: For a smooth projective variety $X$, an automorphism of $X^{[ n ]}$ is said to be natural if it is induced by an automorphism of $X$. I found simple abelian surfaces $X$ having nonnatural automorphisms. In this talk, I about automorphisms of such varieties.
Yutaro Sugimoto (The University of Tokyo)
Title: On controlling the dynamical degrees of automorphisms of complex simple abelian varieties
Abstract: Dynamical degrees are the feature value for birational self-maps of non-singular projective varieties, and it takes the value at least 1. Recently, I have been studying about dynamical degrees and its related topics on abelian varieties and on rational varieties.
In the talk, I will speak about the results on complex simple abelian varieties. The dynamical degrees of an endomorphism of a complex simple abelian variety can be calculated by the minimal polynomial of the endomorphism in the endomorphism algebra of the abelian variety. By using this, problems about the dynamical degrees on the complex simple abelian varieties are transformed to algebraic problems.
I would like to explain how we use this fact for resolving the problems on dynamical degrees: (1) realization of Salem numbers as the first dynamical degree, and (2) the smallest first dynamical degree greater than 1, with (or without) fixing the dimension of abelian varieties.
Yuta Takada (The University of Tokyo)
Title: Dynamical degrees of automorphisms of K3 surfaces with Picard number 2
Abstract: It is known that the dynamical degree of an automorphism of a K3 surface is 1 or a Salem number, and the question of which Salem numbers can be realized has been considered. We determine the set of dynamical degrees of automorphisms of projective K3 surfaces with Picard number 2. This extends the result by Hashimoto, Keum, and Lee.
Tomoki Yoshida (Waseda University)
Title: A simple derived categorical generalization of Ulrich bundles
Abstract: In this talk, we define Ulrich objects as a derived categorical generalization of Ulrich bundles, using the cohomological characterization of Ulrich bundles. We then provide a characterization of these Ulrich objects. As an application of this characterization, we demonstrate that considering the generalization to Ulrich objects offers a new approach to the existence problem for Ulrich bundles.
Organizing Committee
Organizing Committee
Hsueh-Yung Lin (National Taiwan University)
Keiji Oguiso (The University of Tokyo)
Sponsor:
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