Virtual Residue and an Integral Formalism

Abstract

We generalize Grothendieck's residues Res ψ s to virtual cases, namely cases when the zero loci of the section s has dimension larger than the expected dimension(zero). We also provide an exponential type integral formalism for the virtual residue, which can be viewed as an analogue of the Mathai-Quillen formalism for localized Euler classes. * Partially supported by Hong Kong GRF grant 16301515 and 16301717. 1 A Landau Ginzburg space is a pair (X, W ) of a complex manifold X and a holomorphic function W : X → C with compact critical locus;

Aspects of (2,2) and (0,2) hybrid models

“…The formula for the S 2 correlators we derived in the previous section enjoys a series of properties that we will argue for in this section. We remark that some of these properties have already been shown in [28] and [38].…”

“…As a final comment, we were not able to reduce our formula to an integration over a cycle in Y \B (and neither are the authors of [38]). More importantly perhaps, it seems challenging to implement (4.61) for explicit calculations.…”

“…In this section we are going to analyze more closely the integral arising in the S 2 correlators. Our goal is to show that such integral reduces to an integral over B, and we will compare it with the residue formula given in [38]. Given the factor exp(−v dW 2 /4) in (4.30) and the assumption of polynomial growth along X, the integration over the fiber coordinates is absolutely convergent.…”

Virtual residue and a generalized Cayley-Bacharach theorem

“…using the Koszul complex of (V, s), the authors [2] constructed a closed form η ψ ∈ Ω n,n−1 (M \ Z) via Griffiths-Harris's construction [5,Chapter 5]. Then they define the virtual residue as…”