Equivariant elliptic genera and local McKay correspondences. (English) Zbl 1165.14017

Waelder, Robert

Asian J. Math. 12, No. 2, 251-284 (2008).

Let \(G\) be a finite group which acts on a smooth complex variety \(X\) via holomorphic maps. Generally speaking, a McKay correspondence establishes an equivalence between invariants of the orbifold \((X,G)\) and invariants of crepant resolutions of the quotient \(X/G\).
In the present writing, the author establishes such an equivalence between equivariant elliptic genera for open varieties with torus actions, building on work of L. Borisov and A. Libgober [Ann. Math. (2) 161, No. 3, 1521–1569 (2005; Zbl 1153.58301)] in the non-equivariant closed case. The localization techniques of M. F. Atiyah and R. Bott [Topology 23, 1–28 (1984; Zbl 0521.58025)] are used to extend the earlier results to the new setting.
As corollaries, the author proves an equivariant analogue of the Dijkgraaf-Moore-Verlinde-Verlinde formula and reproves the equivariant elliptic genus analogue of the classical McKay correspondence from his earlier paper [Pac. J. Math. 235, No. 2, 345–377 (2008; Zbl 1203.58007)].

Reviewer: Markus Szymik (Bochum)

MSC:

14E99	Birational geometry
55N91	Equivariant homology and cohomology in algebraic topology

Keywords:

elliptic genus; equivariant cohomology; McKay; localization; Hilbert scheme

Citations:

Zbl 1153.58301; Zbl 0521.58025; Zbl 1203.58007