We show that the product in the quantum K-ring of a generalized flag manifold G/P involves only finitely many powers of the Novikov variables. In contrast to previous approaches to this finiteness question, we exploit the finite difference module structure of quantum K-theory. At the core of the proof is a bound on the asymptotic growth of the J-function, which in turn comes from an analysis of the singularities of the zastava spaces studied in geometric representation theory. An appendix by H. Iritani establishes the equivalence between finiteness and a quadratic growth condition on certain shift operators.

We study here the crepant resolution correspondence for the torus equivariant descendent Gromov-Witten theories of Hilb(C2) and Sym(C2).The descendent correspondence is obtained from our previous matching of the associated CohFTs by applying Givental's quantization formula to a specific symplectic transformation K. The first result of the paper is an explicit computation of K. Our main result then establishes a fundamental relationship between the Fourier-Mukai equivalence of the associated derived categories (by Bridgeland, King, and Reid) and the symplectic transformation K via Iritani's integral structure. The results use Haiman's Fourier-Mukai calculations and are exactly aligned with Iritani's point of view on crepant resolution.

Given a smooth projective variety X and a smooth divisor D\subset X. We study relative Gromov-Witten invariants of (X,D) and the corresponding orbifold Gromov-Witten invariants of the r-th root stack X_{D,r}. For sufficiently large r, we prove that orbifold Gromov- Witten invariants of X_{D,r} are polynomials in r. Moreover, higher genus relative Gromov-Witten invariants of (X,D) are exactly the constant terms of the corresponding higher genus orbifold Gromov-Witten invari- ants of X_{D,r}. We also provide a new proof for the equality between genus zero relative and orbifold Gromov-Witten invariants, originally proved by Abramovich-Cadman-Wise. When r is sufficiently large and X=C is a curve, we prove that stationary relative invariants of C are equal to the stationary orbifold invariants in all genera.

Roughly speaking, to any space M with perfect obstruc- tion theory we associate a space N with symmetric perfect obstruction theory. It is a cone over M given by the dual of the obstruction sheaf of M, and contains M as its zero section. It is locally the critical locus of a function. More precisely, in the language of derived algebraic geometry, to any quasi-smooth space M we associate its (−1)-shifted cotangent bundle N. By localising from N to its C∗-fixed locus M this gives five notions of virtual signed Euler characteristic of M: (1) TheCiocan-Fontanine-Kapranov/Fantechi-G ̈ottschesignedvirtual Euler characteristic of M defined using its own obstruction theory, (2) Graber-Pandharipande’s virtual Atiyah-Bott localisation of the virtual cycle of N to M, (3) Behrend’s Kai-weighted Euler characteristic localisation of the vir- tual cycle of N to M, (4) Kiem-Li’s cosection localisation of the virtual cycle of N to M, (5) (−1)vd times by the topological Euler characteristic of M. Our main result is that (1)=(2) and (3)=(4)=(5). The first two are deformation invariant while the last three are not.

Let X and Y be K-equivalent toric Deligne–Mumford stacks related by a single toric wall-crossing. We prove the Crepant Transformation Conjecture in this case, fully-equivariantly and in genus zero. That is, we show that the equivariant quan- tum connections for X and Y become gauge-equivalent after analytic continuation in quantum parameters. Furthermore we identify the gauge transformation involved, which can be thought of as a linear symplectomorphism between the Given- tal spaces for X and Y , with a Fourier–Mukai transformation between the K-groups of X and Y , via an equivariant version of the Gamma-integral structure on quantum cohomology. We prove similar results for toric complete intersections. We im- pose only very weak geometric hypotheses on X and Y : they can be non-compact, for example, and need not be weak Fano or have Gorenstein coarse moduli space. Our main tools are the Mirror Theorems for toric Deligne–Mumford stacks and toric complete intersections, and the Mellin–Barnes method for analytic continuation of hypergeometric functions.