2025/2026

Интегралы-периоды алгебраических многообразий и ГЗК А-гипергеометрические функции

Статус: Дисциплина общефакультетского пула

Кто читает: Факультет математики

Когда читается: 1, 2 модуль

Охват аудитории: для всех кампусов НИУ ВШЭ

Язык: английский

Контактные часы: 30

Course Syllabus

Abstract

On the very fundamental level of the transcendental algebraic geometry, we encounter the notion of so-called «periods» of an algebraic variety. We define period as a coupling between homological cycle and cohomology element represented by a differential form on the variety i.e. it is defined as an integral of a differential form along some homological cycle of proper dimension. With the aid of such period-integrals, we can investigate monodromy of homology or cohomology of the variety. For a special class of varieties, the global monodromy group may turn out to be highly non-trivial discrete group, embedded into some algebraic group (G.D. Mostow). Local monodromy of period-integrals describesHodge structre of the cohomology (P. Deligne, А.N.Varchenko). In the first part of the course, we start from the example of a family of elliptic curves to furnish a survey on the utility and importance of period-integrals. Analysis of this example will give us the following lesson : periods can be represented in terms of special hypergeometric functions (Gauss hypergeometric function… A-hypergeometric functions of Gel’fand-Kapranov-Zelevinsky), from the periods we obtain such global objects like Picard-Fuchs equation or Gauss-Manin connection (Ph.Griffiths), special value of a period integral calculates cardinality of p-adic points on an algebraic curve (Yu.I.Manin). In the second part, we begin by recalling 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 – put into the cohomology carry fundamental information on its monodromy. These topological data are reduced to combinatorics of the Newton polyhedron and the related fan. Further we shall take a look of Stanley-Reisner ring that describes the cohomology with the aid of generating class cycles. After that 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 built 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.