Gromov–Witten theory of complete intersections via nodal invariants

Abstract

We provide an inductive algorithm computing Gromov-Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov-Witten classes of all smooth complete intersections in projective space belong to the tautological ring of the moduli space of stable curves. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov-Witten invariants, we introduce the new notion of nodal relative Gromov-Witten invariants. We then prove a nodal degeneration formula and a relative splitting formula. These results for nodal relative Gromov-Witten theory are stated in complete generality and are of independent interest.

A Lie algebra action on the Chow ring of the Hilbert scheme of points of a K3 surface

“…Note that D(Σ) 2 | W = 0 since D(Σ) 2 can be presented as the locus of subschemes incident to two different sections Σ 1 , Σ 2 , and choosing these generic, this locus is disjoint from W . Hence(e c1(S) D(Σ), D(Σ)[W ]) S [2] = S [2] D(c 1 (S))D(Σ)D(Σ)[W ] = 0.Hence we find that the left hand side is equal to Q PT D(Σ), D(Σ)[W ] and can be easily computed from Theorem 8.9.Similar, with δ = c 1 (O [n] S ) we have δ[W ] = −2q 2 (Σ)v ∅ , and hence k>0 δ, δ[W ] S [2] 0,kA (−p) k = (q 2 (1), Q Hilb q 2 (Σ)) = 8(log(1 − p) − log(1 − p 2 )).…”

“…Hence arguing as in the proof of Proposition 7.19 (see [2] for the use of the symplectic instead of the orthogonal group) it suffices to prove the GW/PT correspondence where (parallel to (30)) the insertions which are vanishing can be grouped into pairs given by a product of ∆ Σ times classes from H * (E ×E). These can then be degenerated by the methods of [2,45]. After resolving Σ 0 , we are reduced to proving Theorem 7.16 for (E × P 1 , E x1,...,x2g ).…”

“…We have the following basic computation, which exactly matches [46,Theorem 39]. 8 8 The series T in [46,Thm 39] computes the genus 0 invariants of the subspace W ⊂ S [2] in classes that push forward to df + kA. The variable y is related to our variable p by y = −p, and the q variables are the same.…”

“…This problem has received considerable attention over the years and is of interest both from a geometric and representation-theoretic viewpoint. Graber [9] computed the quantum cohomology of (P 2 ) [2] and used this to determine counts of hyperelliptic plane curves passing through an appropriate number of points. Li-Li [21] computed for Hilbert schemes of n points, for any n, on a simply connected surface the 2-pointed Gromov-Witten invariants for extremal curve classes, i.e.…”

“…For surfaces S with p g (S) > 0, Hu-Li-Qin [12] showed that all Gromov-Witten invariants of S [n] vanish unless they are a linear combination of (K S ) n and extremal curve classes (see Section 2 for notation). For K3 surfaces S the reduced 1 quantum cohomology of S [2] was computed in [46], and a conjectural formula for quantum multiplication with divisor classes for any S [n] was given in [46,13]. In [47] the structure constants of the reduced quantum product of K3 [n] were shown to be quasi-Jacobi forms.…”