No abstract uploaded!

No abstract uploaded!

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.

We study the moduli stack of degree 0 semistable G-bundles on an irreducible curve E of arithmetic genus 1, where G is a connected reductive group. Our main result describes a partition of this stack indexed by a certain family of connected reductive subgroups H of G (the E-pseudo-Levi subgroups), where each stratum is computed in terms of bundles on H together with the action of the relative Weyl group. We show that this result is equivalent to a Jordan–Chevalley theorem for such bundles equipped with a framingat a fixed basepoint. In the case where E has a single cusp (respectively, node), this gives a new proof of the Jordan–Chevalley theorem for the Lie algebra g (respectively, group G). We also provide a Tannakian description of these moduli stacks and use it to show that if E is an ordinary elliptic curve, the collection of framed unipotent bundles on E is equivariantly isomorphic to the unipotent cone in G. Finally, we classify the E-pseudo-Levi subgroups using the Borel–de Siebenthal algorithm, and compute some explicit examples.

The theory of virtual fundamental class defines important invariants such as the Gromov–Witten and the Donaldson– Thomas invariants. It has been generalized to the cosection localized virtual cycle which has applications in Seiberg– Witten, Fan–Jarvis–Ruan–Witten and other invariants. In this paper, we prove the formulas of virtual pullback, torus localization and wall crossing for cosection localized virtual cycles.