Hilbert schemes of points and infinite dimensional lie algebras
 Responsibility
 Zhenbo Qin.
 Language
 English. Text in English.
 Publication
 Providence, Rhode Island : American Mathematical Society, [2018]
 Copyright notice
 ©2018
 Physical description
 xi, 336 pages ; 26 cm.
 Series
 Mathematical surveys and monographs ; no. 228.
Access
Available online
Science Library (Li and Ma)
Stacks
Call number  Status 

QA3 .A4 V.228  Unknown 
More options
Creators/Contributors
 Author/Creator
 Qin, Zhenbo, author.
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 325334) and index.
 Contents

 Hilbert schemes of points on surfaces: Basic results on Hilbert schemes of points The nef cone and flip structure of $(\mathbb{P}^2)^{[n]}$ Hilbert schemes and infinite dimensional Lie algebras: Hilbert schemes and infinite dimensional Lie algebras Chern character operators Multiple $q$zeta values and Hilbert schemes Lie algebras and incidence Hilbert schemes Cohomology rings of Hilbert schemes of points: The cohomology rings of Hilbert schemes of points on surfaces Ideals of the cohomology rings of Hilbert schemes Integral cohomology of Hilbert schemes The ring structure of $H^*_{\textrm{orb}}(X^{(n)})$ Equivariant cohomology of the Hilbert schemes of points: Equivariant cohomology of Hilbert schemes Hilbert/GromovWitten correspondence GromovWitten theory of the Hilbert schemes of points: Cosection localization for the Hilbert schemes of points Equivariant quantum operator of OkounkovPandharipande The genus0 extremal GromovWitten invariants Ruan's Cohomological Crepant Resolution Conjecture Bibliography Index.
 (source: Nielsen Book Data)9781470441883 20180403
 Publisher's Summary
 Hilbert schemes, which parametrize subschemes in algebraic varieties, have been extensively studied in algebraic geometry for the last 50 years. The most interesting class of Hilbert schemes are schemes $X^{[n]}$ of collections of $n$ points (zerodimensional subschemes) in a smooth algebraic surface $X$. Schemes $X^{[n]}$ turn out to be closely related to many areas of mathematics, such as algebraic combinatorics, integrable systems, representation theory, and mathematical physics, among others. This book surveys recent developments of the theory of Hilbert schemes of points on complex surfaces and its interplay with infinite dimensional Lie algebras. It starts with the basics of Hilbert schemes of points and presents in detail an example of Hilbert schemes of points on the projective plane. Then the author turns to the study of cohomology of $X^{[n]}$, including the construction of the action of infinite dimensional Lie algebras on this cohomology, the ring structure of cohomology, equivariant cohomology of $X^{[n]}$ and the GromovWitten correspondence. The last part of the book presents results about quantum cohomology of $X^{[n]}$ and related questions. The book is of interest to graduate students and researchers in algebraic geometry, representation theory, combinatorics, topology, number theory, and theoretical physics.
(source: Nielsen Book Data)9781470441883 20180403
Subjects
 Subject
 Hilbert schemes.
 Schemes (Algebraic geometry)
 Lie algebras.
 Hilbert schemes.
 Lie algebras.
 Schemes (Algebraic geometry)
 Algebraic geometry > Cycles and subschemes > Parametrization (Chow and Hilbert schemes)
 Nonassociative rings and algebras > Lie algebras and Lie superalgebras > Infinitedimensional Lie (super)algebras.
 Algebraic geometry > (Colo.)homology theory > Other algebrogeometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
 Algebraic geometry > Surfaces and higherdimensional varieties > Vector bundles on surfaces and higherdimensional varieties, and their moduli.
 Algebraic geometry > Projective and enumerative geometry > GromovWitten invariants, quantum cohomology, GopakumarVafa invariants, DonaldsonThomas invariants.
 Nonassociative rings and algebras > Lie algebras and Lie superalgebras > Vertex operators; vertex operator algebras and related structures.
Bibliographic information
 Publication date
 2018
 Copyright date
 2018
 Series
 Mathematical surveys and monographs ; volume 228
 ISBN
 9781470441883 paperback alkaline paper
 1470441888 paperback alkaline paper