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
Instructors:
Танабэ Сусуму
Language:
English
ECTS credits:
2
Course Syllabus
Abstract
"We aim at an introduction to fundamental theory on affine hypersurfaces in algebraic toric variety andon 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 arenecessary to describe the mixed Hodge structure of an affine hypersurface. Two filtrations – Hodge andweight 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. Atthe end of the first part, we shall take a look of Stanley-Reisner ring that describes the cohomology withthe aid of generating class cycles. In the second part, we shall study moduli space of affine hypersurfacesin 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 isobtained from regular triangulations of the Newton polyhedron. As an application, we will analyze theconvergent domains of A-hypergeometric series. We know utility of this kind of approach to the modulispace of affine hypersurfaces in studies of global monodromy of homological cycles. It is widely appliedin the homological mirror symmetry. At the end of the second part, we shall recall several fundamentalproperties of the amoeba of A-discriminantal loci. In the last years, the amoeba notion attracts moreattention as it serves a bridge between toric and tropical geometry."