2024/2025

Analysis and Geometry of Period Integrals

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

Where: Faculty of Mathematics

When: 1 module

Open to: students of all HSE University campuses

Language: English

ECTS credits: 2

Contact hours: 16

Course Syllabus

Abstract

"On the very fundamental level of 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 describes Hodge structre of the cohomology (P. Deligne, А.N.Varchenko, Morihiko Saito). In this 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). "