Jungkai Chen (National Taiwan University)
Title: On some moduli spaces of threefolds with small invariants
Abstract: In this talk, we are going to present some recent studies of explicit understanding of threefolds with small invariants.
We are going to demonstrate some different characterizations: as hypersurface in weighted projetive space bundles and as double covering over polarized varieties with small invariants by providing some examples. This is a joint work in progress with Hsin-Ku Chen.
Xun Yu (Center for Applied Mathematics, Tianjin University)
Title: On automorphism groups of smooth hypersurfaces
Abstract: We show that smooth hypersurfaces in complex projective spaces with automorphism groups of maximum size are isomorphic to Fermat hypersurfaces, with a few (explicitly given) exceptions. This is a joint work with Song Yang and Zigang Zhu.
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Yuta Takada (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 are realizable has been considered. We determine the set of the dynamical degrees of automorphisms of projective K3 surfaces with Picard number 2. This extends the result by Hashimoto, Keum, and Lee.
Yohsuke Matsuzawa (Osaka Metropolitan University)
Title: Height growth along orbits and Zariski dense orbit conjecture
Abstract: For a dominant rational self-map on a projective variety defined over ¥overline{Q}, the exponential growth rate of height along orbits is called
arithmetic degree. It is defined as a limit of the form lim_{n ¥to ¥infty} h(f^n(x))^{1/n},where h is a Weil height function.
For any orbit, arithmetic degree is bounded above by the first dynamical degree of the map, and it is conjectured that the equality holds for Zariski dense orbit.
In general, the existence of the limit is not known, but we recently proved that the limit exists for generic orbit, orbit that is infinite and intersects with proper closed subset only finitely many times. Moreover, we also proved that there are many points whose arithmetic degree exists and arbitrary close to the first dynamical degree.
We applied this results to construct points that has Zariski dense orbits, which solves some special cases of Zariski dense orbit conjecture for birational maps. Part of the work is based on joint work with Junyi Xie.
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Wahei Hara (IPMU, University of Tokyo)
Title: Crepant resolutions and tilting objects.
Abstract: The aim of this talk is to report some of my recent works related to the DK conjecture, which predicts that K-equivalence implies D-equivalence.
In the first half, we discuss some examples of simple flops and the derived equivalence for them using tilting bundles.
In the second half, we discuss an obstruction for the existence of a tilting object.
As applications,
(1) examples of threefold crepant resolutions that admit no tilting object, and
(2) examples of higher dimensional flop in which the DK conjecture holds but tilting objects do no exist
will be provided. A part of this talk depends on a joint work with Donovan, Kapustka and Rampazzo, and a joint work with Wemyss.
Tomohiro Karube (IPMU, University of Tokyo)
Title: The Noncommutative MMP for Blowup Surfaces
Abstract: The notion of Bridgeland stability conditions is the derived analogue of the slope stability of coherent sheaves.
Like the ample cone of a variety, the space of Bridgeland stability conditions
reflects properties of a variety.
The noncommutative minimal model program (NCMMP), proposed by Halpern-Leistner, is an analogue of MMP in derived categories.
In this talk, I will show the relation between the NCMMP and quantum cohomology, and I will discuss the blowup of a surface at a point.
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Atsushi Ito (Tsukuba University)
Title: A remark on some punctual Quot schemes on smooth projective curves
Abstract: For a locally free sheaf $E$ on a smooth projective curve, we can define the punctual Quot scheme which parametrizes torsion quotients of $E$ of length $n$ supported at a fixed point. It is known that the punctual Quot scheme is a normal projective variety with canonical Gorenstein singularities. In this talk, I explain that the punctual Quot scheme is a $¥mathbb{Q}$-factorial Fano variety of Picard number one.
Hirotaka Onuki (University of Tokyo)
Title：On the effective generation of direct images of pluricanonical bundles in mixed characteristic
Abstract：In characteristic zero, Popa and Schnell proved an effective global generation theorem for direct images of pluricanonical bundles as a special case of their Fujita-type conjecture. Ejiri showed an analogous result in positive characteristic. These results have been applied to the study of direct images of pluricanonical bundles. Recently, significant progress has been achieved in birational geometry in mixed characteristic. In this talk, we present a mixed characteristic analog of the theorem of Popa and Schnell and that of Ejiri.
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Taro Yoshio (University of Tokyo)
Title: Stable rationality of hypersurfaces in schon affine varieties
Abstract: In recent years, there has been a development in approaching rationality problems through motivic methods. This approach requires the explicit construction of degeneration families over curves with favorable properties. In this talk, I present a combinatorial method for constructing such degeneration families via tropical compactifications. Additionally, I discuss the irrationality of a very general hypersurface in the complex Grassmannian variety Gr(2,n).
Yuto Masamura (University of Tokyo)
Title: Indices of smooth Calabi-Yau varieties
Abstract: The index of a Calabi-Yau variety X, defined as the smallest positive integer m such that mK_X is trivial, plays a central role in its study. A fundamental conjecture in this area, known as the index conjecture, states that the indices of Calabi-Yau varieties of fixed dimension, under suitable assumptions on singularities (e.g., log canonical), are bounded. This conjecture is crucial for understanding the boundedness properties and singularities of varieties. In this talk, I will present recent progress on the indices of smooth Calabi-Yau varieties, focusing on their connection to the indices of lower-dimensional log Calabi-Yau pairs. By using this relationship, we develop an inductive framework that provides new insights into the index conjecture.
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