Elliptic genera of singular varieties

Abstract

Abstract. Orbifold elliptic genus and elliptic genus of singular varieties are introduced and relation between them is studied. Elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of Calabi-Yau hypersurfaces in Fano Gorenstein toric varieties introduced earlier. Orbifold elliptic genus is given in terms of the fixed point sets of the action. We show that the generating function for this orbifold elliptic genus Ell orb (X n , Σn)p n for symmetric groups Σn acting on n-fold products coincides with the one proposed by Dijkgraaf, Moore, Verlinde and Verlinde. Two notions of elliptic genera are conjectured to coincide.

Sheaves on surfaces and virtual invariants

“…Just like (1) describes the Euler characteristics of S [n] in terms of e(S), one can express the elliptic genera of S [n] in terms of Ell(S). This is achieved by a famous formula originating from string theory in work of Dijkgraaf, G. Moore, E. Verlinde, and H. Verlinde [DMVV] and proved by L. Borisov and A. Libgober [BL1,BL2]. In order to describe the formula, we need the notion of a Borcherds lift.…”

Refined Verlinde formulas for Hilbert schemes of points and moduli spaces of sheaves on K3 surfaces

“…In [EGL] such a formula was proven in the cases r = −1, 0, 1 or K 2 S = K S L = 0. On the other hand the celebrated Dijkgraaf-Moore-Verlinde-Verlinde formula [DMVV], shown in [BL1,BL2,BL3], relates the generating function of the elliptic genera Ell(S [n] ) of Hilbert schemes of points to Siegel modular forms.…”

Twisted motivic Chern class and stable envelopes

“…The elliptic class. The elliptic class of a singular pair was defined by Borisov and Libgober [BL03]. The torus equivariant version, when there is only a finite number of fixed points was considered in [RW19, §2.4, formula 8], [MW19].…”

“…The formulas below define a class in equivariant K-theory (extended by a formal parameters q and h). Applying the Riemann-Roch transformation we obtain an element in equivariant cohomology, which was studied in [BL03,BL05].…”