More lectures on Hilbert schemes of points on surfaces (1401.6782v1)

Abstract: This paper is based on author's lectures at Kyoto University in 2010 Summer, and in the 6th MSJ-SI `Development of Moduli Theory' at RIMS in June 2013. The purpose of lectures was to review several results on Hilbert schemes of points which were obtained after author's lecture note was written. Among many results, we choose those which are about equivariant homology groups $H^T_*(X^{[n]})$ of Hilbert schemes of points on the affine plane $X = \mathbb C^2$ with respect to the torus action.

• The paper presents detailed lectures on the equivariant homology of Hilbert schemes, highlighting advanced localization techniques.
• It constructs a Fock representation of the Heisenberg algebra, linking geometric structures with algebraic operations.
• The study elucidates connections to Jack symmetric functions and the Virasoro algebra, offering new pathways for quantum algebra research.

Insights into "More lectures on Hilbert schemes of points on surfaces" by Hiraku Nakajima

This paper, authored by Hiraku Nakajima, explores the intricate field of algebraic geometry, particularly focusing on Hilbert schemes of points on surfaces, with an emphasis on equivariant homology groups concerning the torus action. Building upon Nakajima's previous lecture notes, this work explores advanced topics that have come to prominence in connection with the AGT correspondence and representation theory of $W$-algebras.

Summary and Key Contributions

The paper is structured around a series of lectures that aim to provide a comprehensive understanding of recent developments in the paper of Hilbert schemes. It is particularly focused on the equivariant homology groups $H^T_*(\Hilb{n})$ of Hilbert schemes with respect to a torus $T$ acting on the affine plane $X = \mathbb{C}^2$.

1. Equivariant (Co)homology Basics: The exposition begins with a recap of equivariant cohomology and homology, tailored for algebraic varieties with torus actions. It notably introduces an approach, diverging slightly from traditional methods, by using finite-dimensional approximations of the classifying space $BT$.
2. Localization Theorems: The paper details localization techniques in equivariant Borel-Moore homology, enabling the analysis of fixed point sets $M^T$. This technique is pivotal for understanding the geometry of spaces under group actions.
3. Representation Theoretic Insights: A substantial portion is dedicated to constructing a Fock representation of the Heisenberg algebra on the direct sum of equivariant homology groups. This construction leverages the geometry of Hilbert schemes to illuminate the algebraic structures.
4. Jack Symmetric Functions: A notable discussion revolves around the geometric realization of Jack symmetric functions as fixed point classes in $H^T_*(\Hilb{n})$. This realization connects the algebraic field of symmetric functions with the geometric context of Hilbert schemes, offering a fresh perspective on the norm and Pieri formulas for Jack symmetric functions.
5. Virasoro Algebra: The final sections explore representations of the Virasoro algebra on equivariant homology groups, providing a geometric underpinning for these algebraic constructs. The connection between the first Chern class of the tautological bundle and Jack functions is harnessed to elucidate this algebraic structure.

Implications and Future Directions

The paper's insights into equivariant homology underscore the deep interconnections between geometry and representation theory. The AGT correspondence, a central theme indirectly addressed, is a bridge between these mathematical domains and theoretical physics, particularly in conformal field theory and quantum algebra.

Future work in this area might further refine the connections between geometric properties of moduli spaces and algebraic structures arising in quantum field theories. Moreover, exploring the implications of these findings in the field of $K$-theory or other cohomological invariants could lead to novel theoretical progress.

In essence, this paper not only consolidates various advancements in the paper of Hilbert schemes but also lays the groundwork for ongoing exploration of their interactions with algebraic and physical theories.