2024/2025

Hodge Structure and A-discriminant of Affine Hypersurface

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

Where: Faculty of Mathematics

When: 4 module

Open to: students of all HSE University campuses

Language: English

ECTS credits: 2

Contact hours: 20

Course Syllabus

Abstract

"We aim at an introduction to fundamental theory on affine hypersurfaces in algebraic toric variety and on their moduli spaces. This kind of knowledge is necessary for further studies on mirror symmetry, Gromov-Witten invariants and Gamma classes of Galkin-Golyshev-Iritani etc. The course consists of two parts. In the first part, we recall basic facts from the toric geometry that are necessary to describe the mixed Hodge structure of an affine hypersurface. Two filtrations – Hodge and weight filtrations – defined on the cohomology carry fundamental information about its monodromy. These topological data are reduced to combinatorics of the Newton polyhedron and the related fan. At the end of the first part, we shall take a look of Stanley-Reisner ring that describes the cohomology with the aid of generating class cycles. In the second part, we shall study moduli space of affine hypersurfaces in making use of A-discriminant and A-discriminantal loci introduced by Gel’fand-Kapranov-Zelevinsky. In order to get A-discriminant, we have recourse to the construction of secondary polytope that is obtained from regular triangulations of the Newton polyhedron. As an application, we will analyze the convergent domains of A-hypergeometric series. We know utility of this kind of approach to the moduli space of affine hypersurfaces in studies of global monodromy of homological cycles. It is widely applied in the homological mirror symmetry. At the end of the second part, we shall recall several fundamental properties of the amoeba of A-discriminantal loci. In the last years, the amoeba notion attracts more attention as it serves a bridge between toric and tropical geometry."