Elliptic classes, McKay correspondence and theta identities
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Elliptic classes, McKay correspondence and theta identities Małgorzata Mikosz1 · Andrzej Weber2 Received: 17 September 2019 / Accepted: 4 January 2020 © The Author(s) 2020
Abstract We revisit the construction of elliptic class given by Borisov and Libgober for singular algebraic varieties. Assuming torus action we adjust the theory to the equivariant local situation. We study theta function identities having a geometric origin. In the case of quotient singularities Cn /G, where G is a finite group the theta identities arise from McKay correspondence. The symplectic singularities are of special interest. The Du Val surface singularity An leads to a remarkable formula. Keywords Theta function · McKay correspondence · Elliptic class of singular varieties · Quotient singularities
Contents 1 Basic functions . . . . . . . . . . . . . . . . . . 2 The An−1 identity . . . . . . . . . . . . . . . . 3 Elliptic class . . . . . . . . . . . . . . . . . . . 4 Resolution of singularities and theta identities . 5 The orbifold elliptic class . . . . . . . . . . . . 6 Orbifold elliptic class of symplectic singularities 7 Examples . . . . . . . . . . . . . . . . . . . . . 8 Diagonal quotient . . . . . . . . . . . . . . . . 9 Proof of Theorem 1 . . . . . . . . . . . . . . . 10 Self-duality of An−1 singularity . . . . . . . . . 11 Hirzebruch class—the limit with q → 0 . . . . . References . . . . . . . . . . . . . . . . . . . . . .
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Andrzej Weber is supported by the Polish National Science Center project Algebraic Geometry: Varieties and Structures 2013/08/A/ST1/00804 and 2016/23/G/ST1/04282 (Beethoven 2).
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Andrzej Weber [email protected] Małgorzata Mikosz [email protected]
1
Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland
2
Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
123
Journal of Algebraic Combinatorics
The theory of theta functions is a classical subject of analysis and algebra. It had a prominent role in 19th century mathematics, as one can see reading the monograph about Algebra [28]. Nowadays, it seems that the intriguing combinatorics related to theta functions has been put aside. Nevertheless, there are modern sources treating the subject of theta functions in a wider context. For example, the Mumford three-volume book [17] is devoted to the theta function. Th
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