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Lectures on Hilbert schemes of points on surfaces. (English) Zbl 0949.14001

University Lecture Series. 18. Providence, RI: American Mathematical Society (AMS). xi, 132 p. (1999).
In this book the author discusses the Hilbert scheme \(X^{[n]}\) of points on a complex surface \(X\). This object is originally studied in algebraic geometry but, as it has been recently realized, it is related to many other branches of mathematics, such as singularities, symplectic geometry, representation theory, and even to theoretical physics. The book reflects this feature on Hilbert schemes and therefore the subjects are analyzed from various points of view. One sees that \(X^{[n]}\) inherits structures of \(X\), e.g., it is a nonsingular complex manifold, it has a holomorphic symplectic form if \(X\) has one, it has a hyper-Kähler metric if \(X= \mathbb{C}^2\), and so on. A new structure is revealed when one studies the homology group of \(X^{[n]}\). The generating function of Poincaré polynomials has a very nice expression. The direct sum \(\bigoplus_n H_* (X^{[n]})\) is a representation of the Heisenberg algebra.
The book, which is nicely written and well-organized, tries to tell the harmony between different fields rather than focusing attention on a particular one. The reader is assumed to have basic knowledge on algebraic geometry and homology groups of manifolds. Some chapters require more background, say spectral sequences, Riemannian geometry, Morse theory, intersection cohomology.

MSC:

14C05 Parametrization (Chow and Hilbert schemes)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14J17 Singularities of surfaces or higher-dimensional varieties
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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