2026/2027
Структура Ходжа и А-дискриминант аффинной гиперповерхности
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
1, 2 модуль
Охват аудитории:
для всех кампусов НИУ ВШЭ
Язык:
английский
Кредиты:
3
Контактные часы:
30
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."
Learning Objectives
- To master the fundamental theory of affine hypersurfaces in algebraic toric varieties and their moduli spaces via Newton polyhedra, Stanley-Reisner rings, A-discriminants, secondary polytopes, and amoebas as a necessary foundation for advanced studies in mirror symmetry, Gromov-Witten invariants, Gamma classes, and homological mirror symmetry.
Expected Learning Outcomes
- The ability to compute the mixed Hodge structure and monodromy of affine hypersurfaces based on the combinatorics of the Newton polyhedron using Stanley-Reisner rings; to construct their moduli spaces via A-discriminants and secondary polytopes; to analyze convergence domains of A-hypergeometric series; as well as to apply these methods for calculating Gromov-Witten invariants, Gamma classes, and investigating global monodromy in homological mirror symmetry.
Course Contents
- Graded ring of polynomials. Ehrhart polynomial. Euler characteristic of an affine hypersurface in alge- braic tori.
- Laurent polynomial nondegenerate with respec to its Newton polyhedron. Jacobian ring of a Laurent polynomial. Koszul complex defined by the Jacobian ideal of a Laurent polynomial
- The mixed Hodge structure of the cohomology of an affine hypersurface. Hodge and weight filtrations of the cohomology. Primitive part of the chomology.
- Stanley – Reisner ring and the cohomology of an affine hypersurface.
- Regular triangulations of a polyhedron. Secondary polytope. A-discriminant of an affine hypersurface. Secondary fan.
- A-hypergeometric functions of Gel’fand – Kapranov – Zelevinsky. Domains of convergence of A- hyperge- ometric series. A-discriminantal loci of an affine algebraic variety. Horn – Kapranov uniformisation of A-discriminantal loci. Principal A-discriminant and amoebas.
Interim Assessment
- 2026/2027 2nd moduleEvaluation will be made according to the rule: 60% -answers to homework exercises, 40% - the final examination. In both cases, answers shall be submitted in a written form. During the course, more than 14 exercise questions will be spread in the classroom. Each question corresponds to 5 – 15 points. One who gains more than 100 points for solution of exercises will get full 60% mark attributed to homework exercises
Bibliography
Recommended Core Bibliography
- Introduction to toric varieties, Fulton, W., 1993
Recommended Additional Bibliography
- Fulton, W. (1993). Introduction to Toric Varieties. (AM-131), Volume 131. Princeton: Princeton University Press. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1432979