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 {mathbb {C}}^n/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 A_n leads to a remarkable formula.
Highlights
The theory of theta functions is a classical subject of analysis and algebra
In the beginning of 2000s, theta function was applied by Borisov and Libgober [2] to construct an elliptic genus of singular complex algebraic varieties
We explain the normalization constants, which allow to place the elliptic class in the common formalism used in algebraic topology
Summary
The function δ is well defined on C∗ × C∗ since the fractional powers cancel out It has the following quasi-periodic properties δ(q A, B) = B−1δ( A, B), δ( A, q B) = A−1δ( A, B). Resolutions of quotient singularities give rise to certain identities for theta function. We will explain this mechanism in the subsequent sections, but first we would like to give an example of the cyclic symplectic quotient C2/Zn, i.e., the singularity An−1. It leads to the following interesting identity: Theorem 1 Let n > 0 be a natural number.
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