Title and Abstract (workshop home)
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Yu-Shen Lin (slides)
Title: SYZ Mirror Symmetry of Del Pezzo Surfaces
Abstract: Strominger-Yau-Zaslow has been a long time the guiding principle for mirror symmetry. In this talk, I will discuss the recent progress of the existence of SYZ fibrations in del Pezzo surfaces and rational elliptic surfaces. Not only these will be the dual fibration of each other, the SYZ fibrations also help to detect new Ricci-flat metrics on some log Calabi-Yau surfaces compactified to rational elliptic surfaces and different from those constructed by Hein. Except from the analytic aspects, I will also explain how the true SYZ fibration may benefit the enumerative geometry by developing the tropicalization of the Lagrangian Floer theory. In particular, The talk is partly based on the joint work with Collins, Adam and Tsung-Ju, Siu-Cheong.
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Ryota Mikami (slides)
Title: A tropical analog of the Hodge conjecture for smooth algebraic varieties over trivially valued fields
Abstract: We propose a tropical approach to problems cycle class maps such as the Hodge conjecture. In this talk, I will explain a proof a tropical analog of the Hodge conjecture for smooth algebraic varieties over trivially valued fields. The main ingredients are a algebro-geometric theorem on cohomology theories (the existence of the Gersten resolutions), developed by many mathematicians (e.g., Quillen), and explicit calculations of tropical cohomology of the trivial line bundles by non-archimedean geometry.
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Yuji Odaka (slides)
Title: Towards canonical compactifications of moduli of K3 via hyperKahler metrics, and some mirror symmetric phenomena
Abstract: This is a joint work with Yoshiki Oshima (Osaka univ). To moduli of K3s, we apply some Satake’s compactifications (symmetric space theory)/Morgan-Shalen type ("tropical") compactification, while confirming the equivalence. Then we put a "geometric meaning" by letting the boundary parametrize explicit metric spaces with additional structures (e.g. "tropical K3" spheres and intervals...), which we conjecture to all the (compact) limits of K3 metrics, and partially prove it.
Some byproducts so far: LCSL (type III degeneration) of K3 surfaces is confirmed to (give rise to the SYZ fibration and) satisfy the Kontsevich-Soibelman conjecture (arXiv:1810.07685). Also type II degeneration has some PL convex function invariants (arXiv:2010.00416) whose combinatorial types are classified by some Dynkin diagrams. For Tjurin degeneration case, this seems to be related to the talk of Y-S.Lin.
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Takumi Otani (slides)
Title: Gamma integral structure for an invertible polynomial of chain type
Abstract: The Gamma integral structure for the quantum cohomology of an algebraic variety was introduced by Iritani, Katzarkov-Kontsevich-Pantev. In this talk, we will consider the Gamma integral structure for an invertible polynomial of chain type. Based on the Gamma conjecture II by Galkin-Golyshev-Iritani, we will explain the conjecture that the Gamma integral structure should be identified with the natural integral structure for the Berglund-Hübsch transposed polynomial by the mirror isomorphism.
This is a joint work with Atsushi Takahashi.
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Tokio Sasaki (slides,
with_explanation)
Title: Limits of geometric higher normal functions and Apéry constants
Abstract: The irrationality of $\zeta (3)$ was historically proven by R. Apéry via the approximation by the ratio of two sequences of integers. For each of five Mukai Fano threefolds with Picard rank 1, V.Golyshev obtained a special value of $L$-function as the ratio of similar two sequences which arise from the quantum recursion. In terms of the mirror symmetry, this construction in the A-model side can be generalized to Fano threefolds with Picard rank 1. The Arithmetic Mirror Symmetry Conjecture states that a corresponding construction in the B-model side will be obtained from the limits of geometric higher normal functions. In this talk, we show that this conjecture holds for five Golyshev’s examples by constructing specific higher Chow cycles. This is joint work with V. Golyshev and M. Kerr.
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Hsian-Hua Tseng (slides)
Title: Relative Gromov-Witten theory without log geometry
Abstract:
We describe a new Gromov-Witten theory of a space relative to a simple normal-crossing divisor constructed using multi-root stacks.
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Junxiao Wang (slides,
examples)
Title: The Gamma Conjecture for the Tropical 1-cycles in Local Mirror Symmetry
Abstract: The Gamma Conjecture in mirror symmetry relates the central charges of dual objects. Mathematically, periods of a Lagrangian submanifold are related to characteristic classes of the mirror coherent sheaf. In this talk, I will test the Gamma Conjecture in the setting of local mirror symmetry. For a given coherent sheaf on the canonical bundle of a smooth toric surface, I will identify a 3-cycle in the mirror using tropical geometry by comparing its period with the central charge of the coherent sheaf through the Gamma Conjecture. If time permits, I will also discuss about some higher dimensional cases. This work is based on Ruddat and Siebert's work on the period computation and is inspired by Abouzaid, Ganatra, Iritani and Sheridan's work on the Gamma Conjecture.
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Fenglong You (slides)
Title: Structures of relative Gromov-Witten theory
Abstract: Absolute Gromov-Witten theory is known to have many nice structural properties, such as quantum cohomology, WDVV equation,
Givental's formalism, mirror theorem, CohFT etc.. In this talk, I will explain how to obtain parallel structures for relative Gromov-Witten
theory via the relationship between relative and orbifold Gromov-Witten invariants.
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Jeng-Daw Yu (slides)
Title: Irregular Hodge structures in Landau-Ginzburg models and in arithmetic
Abstract: From the viewpoint of irregular Hodge theory, we discuss the cohomological structures for some varieties equipped with regular functions, as the mirrors of certain Fano varieties and as ingredients toward arithmetic applications.
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Ilia Zharkov (slides)
Title: Topological SYZ fibrations with discriminant in codimension two
Abstract: To date only for K3 surfaces (trivial) and the quintic threefold (due to M. Gross) the discriminant can be made to be in codimension two. I will outline the source of the problem and how to resolve it in much more general situations using phase and over-tropical pairs-of-pants. Joint project with Helge Ruddat. If time permits, I'll explain an application to lifting tropical cycles from the SYZ base to "holomorphic" and "Lagrangian" type objects in the torus fibrations.