Hilbert Schemes of Points and Infinite Dimensional Lie Algebras

Zhenbo Qin (Autor)

Buch | Hardcover

336 Seiten

2018
American Mathematical Society (Verlag)
978-1-4704-4188-3 (ISBN)

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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 (zero-dimensional 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 Gromov-Witten 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.

Zhenbo Qin, University of Missouri, Columbia, MO.

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/Gromov-Witten correspondence
Gromov-Witten theory of the Hilbert schemes of points: Cosection localization for the Hilbert schemes of points
Equivariant quantum operator of Okounkov-Pandharipande
The genus-0 extremal Gromov-Witten invariants
Ruan's Cohomological Crepant Resolution Conjecture
Bibliography
Index

Erscheinungsdatum	07.01.2018
Reihe/Serie	Mathematical Surveys and Monographs
Verlagsort	Providence
Sprache	englisch
Maße	178 x 254 mm
Gewicht	765 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Algebra
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
ISBN-10	1-4704-4188-8 / 1470441888
ISBN-13	978-1-4704-4188-3 / 9781470441883
Zustand	Neuware