Analysis and Geometry of Period Integrals

2024/2025

Academic Year

ENG

Instruction in English

2

ECTS credits

Delivered at:

Faculty of Mathematics

Course type:

Optional course (faculty)

When:

1 module

Course Syllabus

Abstract

"On the very fundamental level of transcendental algebraic geometry, we encounter the notion ofso-called «periods» of an algebraic variety. We define period as a coupling betweenhomological 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 properdimension. With the aid of such period-integrals, we can investigate monodromy of homologyor cohomology of the variety. For a special class of varieties, the global monodromy group mayturn 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 theutility and importance of period-integrals. Analysis of this example will give us the followinglesson: periods can be represented in terms of special hypergeometric functions (Gausshypergeometric function… A-hypergeometric functions of Gel’fand-Kapranov-Zelevinsky), fromthe periods we obtain such global objects like Picard-Fuchs equation or Gauss-Maninconnection (Ph.Griffiths), special value of a period integral calculates cardinality of p-adic pointson an algebraic curve (Yu.I.Manin)."