Elliptic genera of singular varieties. (English) Zbl 1053.14050

Borisov, Lev; Libgober, Anatoly

Duke Math. J. 116, No. 2, 319-351 (2003).

Let \(X\) be a Kähler manifold and let \(X^{n}/ \Sigma_{n}\) denote the \(n\)-fold symmetric product of \(X\), the quotient of \(X^{n}\) by the action of the symmetric group. In R. Dijkgraaf, G. Moore, E. Verlinde and H. Verlinde [Commun. Math. Phys. 185, No.1, 197-209 (1997; Zbl 0872.32006)] a formula is given, in terms of the elliptic genus of \(X\), for the generating function made from the orbifold elliptic genera of the symmetric products. When \(X\) is an almost complex manifold the generating function is a holomorphic function on the product of the complex numbers and the upper half plane. When \(X\) is Calabi-Yau this function is a weak Jacobi form. This paper sets about justifying the physicists’ conjectural formula mathematically. The authors introduce two notions of elliptic genus for singular varieties. These are (a) the singular elliptic genus, defined for a pair consisting of a variety and a \({\mathbb Q}\)-Cartier divisor on it and (b) the orbifold elliptic genus, defined for a manifold with an action by a finite group \(G\). The authors conjecture that these two elliptic genera coincide for \(X/G\) and the ramification divisor. Using the factorisation of birational maps into a sequence of smooth blow-ups and blow-downs, proved in D. Abramovich, K. Karu, K. Matsuki and J. Wlodarczyk [J. Am. Math. Soc. 15, No. 3, 531–572 (2002; Zbl 1032.14003)] the authors establish their conjecture in a number of cases (and in a footnote apparently promise a complete proof [http://arXiv.org/abs/math.AG/0206241]) as well as deriving a number of general properties such as cobordism invariance.

Reviewer: Victor P. Snaith (Sheffield)

MSC:

14J32	Calabi-Yau manifolds (algebro-geometric aspects)
14J17	Singularities of surfaces or higher-dimensional varieties
32S45	Modifications; resolution of singularities (complex-analytic aspects)
55N34	Elliptic cohomology
11F23	Relations with algebraic geometry and topology

Keywords:

symmetric product; resolution of singularities; generating function

Citations:

Zbl 0872.32006; Zbl 1032.14003

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