第12次 아시아ㆍ太平洋 矯政局長會議 參加記

/pdf/classical-algebraic-geometry-3wzv0dreho.pdf

Classical Algebraic Geometry

We propose Gamma conjectures for Fano manifolds which can be thought of as a square root of the index theorem. Studying the exponential asymptotics of solutions to the quantum differential equation, we associate a principal asymptotic class $A_{F}$ to a Fano manifold $F$. We say that $F$ satisfies Gamma conjecture I if $A_{F}$ equals the Gamma class $\widehat{\Gamma}_{F}$. When the quantum cohomology of $F$ is semisimple, we say that $F$ satisfies Gamma conjecture II if the columns of the central connection matrix of the quantum cohomology are formed by $\widehat{\Gamma}_{F}\operatorname{Ch}(E_{i})$ for an exceptional collection $\{E_{i}\}$ in the derived category of coherent sheaves $\mathcal{D}^{b}_{\mathrm{coh}}(F)$. Gamma conjecture II refines a part of a conjecture by Dubrovin. We prove Gamma conjectures for projective spaces and Grassmannians.

Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures

We study Fano threefolds with terminal Gorenstein singularities admitting a `minimal' action of a finite group. We prove that under certain additional assumptions such a variety contains no planes. We also obtain upper bounds for the number of singular points of certain Fano threefolds with terminal factorial singularities.

http://iopscience.iop.org/article/10.1070/IM2015v079n04ABEH002761/pdf

On G-Fano threefolds

Introduction.- Birational Geometry of Surfaces.- Logarithmic Category.- Overview of Mori's Program.- Singularities.- Vanishing Theorems.- Base Point Freeness of Linear Systems.- Cone Theorem.- Contraction Theorem.- Flip.- Cone Theorem Revisited.- Logarithmic Mori's Program.- Birational Relations among Minimal Models.- Birational Relations among Mori Fiber Spaces.- Birational Geometry of Toric Varieties.

Introduction to the Mori Program

We observe a method for finding weak Landau-Ginzburg models for Fano varieties and find them for smooth Fano threefolds of genera 9, 10, and 12.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cntp/2007/0001/0004/CNTP-2007-0001-0004-a004.pdf

On Landau–Ginzburg models for Fano varieties

https://pure.mpg.de/pubman/item/item_3121492_1/component/file_3121493/Ilten_toric_oa_2013.pdf

Toric degenerations of Fano threefolds giving weak Landau–Ginzburg models

The paper is joined with arXiv:0911.5428 and improved. 
We prove that Landau-Ginzburg models for all 17 smooth Fano threefolds with Picard rank 1 can be represented as Laurent polynomials in 3 variables exhibiting them case by case. We check that these Landau-Ginzburg models can be compactified to open Calabi-Yau varieties. In the spirit of L. Katzarkov's program we prove that numbers of irreducible components of the central fibers of compactifications of these pencils are dimensions of intermediate Jacobians of Fano varieties plus 1. In particular these numbers do not depend on compactifications. We state most of known methods of finding Landau-Ginzburg models in terms of Laurent polynomials. We discuss Laurent polynomial representation of Landau-Ginzburg models of Fano varieties and state some problems related to it.

Weak Landau-Ginzburg models for smooth Fano threefolds

We prove that Hori-Vafa mirror models for smooth Fano complete intersections in weighted projective spaces admit an interpretation as Laurent polynomials.

Victor Przyjalkowski

Papers

On Landau–Ginzburg models for Fano varieties

Toric degenerations of Fano threefolds giving weak Landau–Ginzburg models

Weak Landau-Ginzburg models for smooth Fano threefolds

Hori-Vafa mirror models for complete intersections in weighted projective spaces and weak Landau-Ginzburg models

On Landau--Ginzburg models for Fano varieties