Let S be a K3 surface and let E be an elliptic curve. We solve the reduced Gromov-Witten theory of the Calabi-Yau threefold S × E for all curve classes which are primitive in the K3 factor. In particular, we deduce the Igusa cusp form conjecture.The proof relies on new results in the Gromov-Witten theory of elliptic curves and K3 surfaces. We show the generating series of Gromov-Witten classes of an elliptic curve are cycle-valued quasimodular forms and satisfy a holomorphic anomaly equation. The quasimodularity generalizes a result by Okounkov and Pandharipande, and the holomorphic anomaly equation proves a conjecture of Milanov, Ruan and Shen. We further conjecture quasimodularity and holomorphic anomaly equations for the cycle-valued Gromov-Witten theory of every elliptic fibration with section. The conjecture generalizes the holomorphic anomaly equations for elliptic Calabi-Yau threefolds predicted by Bershadsky, Cecotti, Ooguri, and Vafa. We show a modified conjecture holds numerically for the reduced Gromov-Witten theory of K3 surfaces in primitive classes.

Let S be a nonsingular projective K3 surface. Motivated by the study of the Gromov-Witten theory of the Hilbert scheme of points of S, we conjecture a formula for the Gromov-Witten theory (in all curve classes) of the Calabi-Yau 3-fold S × E where E is an elliptic curve. In the primitive case, our conjecture is expressed in terms of the Igusa cusp form χ 10 and matches a prediction via heterotic duality by Katz, Klemm, and Vafa. In imprimitive cases, our conjecture suggests a new structure for the complete theory of descendent integration for K3 surfaces. Via the Gromov-Witten/Pairs correspondence, a conjecture for the reduced stable pairs theory of S × E is also presented. Speculations about the motivic stable pairs theory of S × E are made.The reduced Gromov-Witten theory of the Hilbert scheme of points of S is much richer than S × E. The 2-point function of Hilb d (S) determines a matrix with trace equal to the partition function of S × E. A conjectural form for the full matrix is given.

We conjecture that the relative Gromov-Witten potentials of elliptic fibrations are (cycle-valued) lattice quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture for the rational elliptic surface in all genera and curve classes numerically. The generating series are quasi-Jacobi forms for the lattice E8. We also show the compatibility of the conjecture with the degeneration formula. As Corollary we deduce that the Gromov-Witten potentials of the Schoen Calabi-Yau threefold (relative to P 1 ) are E8 × E8 quasi-bi-Jacobi forms and satisfy a holomorphic anomaly equation. This yields a partial verification of the BCOV holomorphic anomaly equation for Calabi-Yau threefolds. For abelian surfaces the holomorphic anomaly equation is proven numerically in primitive classes. The theory of lattice quasi-Jacobi forms is reviewed.In the Appendix the conjectural holomorphic anomaly equation is expressed as a matrix action on the space of (generalized) cohomological field theories. The compatibility of the matrix action with the Jacobi Lie algebra is proven. Holomorphic anomaly equations for K3 fibrations are discussed in an example.Recently it became clear that we should expect properties (i, ii) not only for Calabi-Yau threefolds but also for varieties X (of arbitrary dimension) which are Calabi-Yau relative to a base B, i.e. those which admit a fibration π : X → B whose generic fiber has trivial canonical class. The potential F g (q) is replaced here by a π-relative Gromov-Witten potential which takes values in cycles on M g,n (B, k), the moduli space of stable maps to the base. In this paper we conjecture and develop such a theory for elliptic fibrations with section. Our main theoretical result is a conjectural link between the Gromov-Witten theory of elliptic fibrations and the theory of lattice quasi-Jacobi forms. This framework allows us to conjecture a holomorphic anomaly equation. 4 The elliptic curve (or more generally, trivial elliptic fibrations) is the simplest case of our conjecture and was proven in [33]. In this paper we prove the following new cases (see Section 5.3):(a) The P 1 -relative Gromov-Witten potentials of the rational elliptic surface are E 8 -quasi-Jacobi forms numerically 5 .(b) The holomorphic anomaly equation holds for the rational elliptic surface numerically.In particular, (a) solves the complete descendent Gromov-Witten theory of the rational elliptic surface in terms of E 8 -quasi-Jacobi forms. We also show:(c) The quasi-Jacobi form property and the holomorphic anomaly equation are compatible with the degeneration formula (Section 4.6).These results directly lead to a proof of Theorem 1 and 2 as follows. The Schoen Calabi-Yau X admits a degenerationwhere E i ⊂ R i are smooth elliptic fibers. By the degeneration formula [27] we are reduced to studying the case R i × E j . By the product formula [25] the claim then follows from the holomorphic anomaly equation for the rational elliptic surface and the elliptic curve [33]. For completeness we also prove the follo...

We study the enumerative geometry of rational curves on the Hilbert schemes of points of a K3 surface.Let S be a K3 surface and let Hilb d (S) be the Hilbert scheme of d points of S. In case of elliptically fibered K3 surfaces S → P 1 , we calculate genus 0 Gromov-Witten invariants of Hilb d (S), which count rational curves incident to two generic fibers of the induced Lagrangian fibration Hilb d (S) → P d . The generating series of these invariants is the Fourier expansion of a power of the Jacobi theta function times a modular form, hence of a Jacobi form.We also prove results for genus 0 Gromov-Witten invariants of Hilb d (S) for several other natural incidence conditions. In each case, the generating series is again a Jacobi form. For the proof we evaluate Gromov-Witten invariants of the Hilbert scheme of 2 points of P 1 × E, where E is an elliptic curve.Inspired by our results, we conjecture a formula for the quantum multiplication with divisor classes on Hilb d (S) with respect to primitive curve classes. The conjecture is presented in terms of natural operators acting on the Fock space of S. We prove the conjecture in the first non-trivial case Hilb 2 (S). As a corollary, we find that the full genus 0 Gromov-Witten theory of Hilb 2 (S) in primitive classes is governed by Jacobi forms.We present two applications. A conjecture relating genus 1 invariants of Hilb d (S) to the Igusa cusp form was proposed in joint work with R. Pandharipande in [OP14]. Our results prove the conjecture in case d = 2. Finally, we present a conjectural formula for the number of hyperelliptic curves on a K3 surface passing through 2 general points.

Abstract. Let X = S × E be the product of a K3 surface S and an elliptic curve E. Reduced stable pair invariants of X can be defined via (1) cutting down the reduced virtual class with incidence conditions or (2) the Behrend function weighted Euler characteristic of the quotient of the moduli space by the translation action of E. We show that (2) arises naturally as the degree of a virtual class, and that the invariants (1) and (2) agree. This has applications to deformation invariance, rationality and a DT/PT correspondence for reduced invariants of S × E.