Procesi bundle

In algebraic geometry, Procesi bundles are vector bundles of rank ${\displaystyle n!}$ on certain symplectic resolutions of quotient singularities, particularly on the Hilbert scheme of ${\displaystyle n}$ points in the complex plane. ^[1] They play a fundamental role in geometric representation theory and were crucial in Mark Haiman's proof of the n! theorem and Macdonald positivity conjecture, and were named after Italian mathematician Claudio Procesi.

Definition

Let ${\displaystyle H_{n}={\text{Hilb}}^{n}(\mathbb {C} ^{2})}$ denote the Hilbert scheme of n points in the complex plane ${\displaystyle \mathbb {C} ^{2}}$, which provides a resolution of singularities of the quotient ${\displaystyle (\mathbb {C} ^{2})^{n}/S_{n}}$, where ${\displaystyle S_{n}}$ is the symmetric group of degree ${\displaystyle n}$. A Procesi bundle ${\displaystyle {\mathcal {P}}}$ on ${\displaystyle H_{n}}$ is a ${\displaystyle \mathbb {C} ^{\times }}$-equivariant vector bundle of rank ${\displaystyle n!}$ together with an isomorphism ${\displaystyle {\text{End}}({\mathcal {P}})\cong \mathbb {C} [x,y]\#S_{n}}$ (where ${\displaystyle \mathbb {C} [x,y]\#S_{n}}$ is the smash product algebra) of ${\displaystyle \mathbb {C} [V]^{S_{n}}}$-algebras, such that ${\displaystyle {\text{Ext}}^{i}({\mathcal {P}},{\mathcal {P}})=0}$ for all ${\displaystyle i>0}$. The isomorphism ensures that each fiber of ${\displaystyle {\mathcal {P}}}$ is naturally the regular representation of ${\displaystyle S_{n}}$. ^[1]

More generally, for a finite subgroup ${\displaystyle \Gamma \subset {\text{SL}}_{2}(\mathbb {C} )}$ and its wreath product ${\displaystyle \Gamma _{n}=S_{n}\ltimes \Gamma ^{n}}$, Procesi bundles can be defined on symplectic resolutions of ${\displaystyle \mathbb {C} ^{2n}/\Gamma _{n}}$. ^[1]

Properties

Procesi bundles provide a derived McKay equivalence between the derived category of coherent sheaves on ${\displaystyle H_{n}}$ and the derived category of ${\displaystyle S_{n}}$-equivariant modules over ${\displaystyle \mathbb {C} [x,y]}$. For one distinguished Procesi bundle ${\displaystyle {\mathcal {P}}}$, the ${\displaystyle \Gamma _{n-1}}$-invariants ${\displaystyle {\mathcal {P}}^{S_{n-1}}}$ coincide with the tautological bundle on ${\displaystyle H_{n}}$. On any symplectic resolution of ${\displaystyle \mathbb {C} ^{2n}/\Gamma _{n}}$, there are exactly two normalized (meaning ${\displaystyle {\mathcal {P}}^{S_{n}}\cong {\mathcal {O}}}$) Procesi bundles which are dual to each other. ^[2]

History and constructions

The first construction of a Procesi bundle was given by American mathematician Mark Haiman in his proof of the n! theorem, using intricate combinatorial methods. ^[3] Alternative constructions were later developed by Roman Bezrukavnikov and Dmitry Kaledin using quantization in positive characteristic, ^[4] and by Victor Ginzburg using D-modules and the Hotta-Kashiwara construction. ^[5]

Belarusian-American mathematician Ivan Losev provided significant further developments in the theory of Procesi bundles, including a complete classification of Procesi bundles on Hamiltonian reductions, ^[1] and an inductive construction showing how Procesi bundles relate to nested Hilbert schemes. ^[2] His work established that there are exactly two normalized Procesi bundles on any given symplectic resolution obtained by Hamiltonian reduction.

As a result of their use in the proof of the n! theorem, Procesi bundles have also found important applications in the proof of the Macdonald positivity conjecture, ^[3] the study of rational Cherednik algebras and their representations, and understanding derived equivalences for symplectic quotient singularities. ^{[4] [6]}

See also

• Derived category
• Geometric invariant theory
• Hilbert scheme
• McKay correspondence

References

1. Losev, Ivan (2014). "On Procesi bundles". Mathematische Annalen. 359 (3): 729–744. doi:10.1007/s00208-014-1017-7.
2. Losev, Ivan (2019). "On inductive construction of Procesi bundles". arXiv. 1901.05862. arXiv: 1901.05862 .
3. Haiman, Mark (2001). "Hilbert schemes, polygraphs and the Macdonald positivity conjecture". Journal of the American Mathematical Society. 14 (4): 941–1006. doi:10.1090/S0894-0347-01-00373-3.
4. Bezrukavnikov, Roman; Kaledin, Dmitry (2004). "McKay equivalence for symplectic quotient singularities". Proceedings of the Steklov Institute of Mathematics. 246: 13–33.
5. Ginzburg, Victor (2012). "Isospectral commuting variety, the Harish-Chandra D-module, and principal nilpotent pairs". Duke Mathematical Journal. 161 (11): 2023–2111. arXiv: 1108.5367 . doi:10.1215/00127094-1699392.
6. Gordon, Iain (2012). "Macdonald positivity via the Harish-Chandra D-module" (PDF). Inventiones Mathematicae. 187 (3): 637–643. Bibcode:2012InMat.187..637G. doi:10.1007/s00222-011-0339-2.