Abstract
AbstractWe prove a conjecture of Maulik, Pandharipande and Thomas expressing the Gromov–Witten invariants of K3 surfaces for divisibility 2 curve classes in all genera in terms of weakly holomorphic quasi-modular forms of level 2. Then we establish the holomorphic anomaly equation in divisibility 2 in all genera. Our approach involves a refined boundary induction, relying on the top tautological group of the moduli space of smooth curves, together with a degeneration formula for the reduced virtual fundamental class with imprimitive curve classes. We use double ramification relations with target variety as a new tool to prove the initial condition. The relationship between the holomorphic anomaly equation for higher divisibility and the conjectural multiple cover formula of Oberdieck and Pandharipande is discussed in detail and illustrated with several examples.
Highlights
Let be a complex nonsingular projective K3 surface and ∈ 2(, Z) be an effective curve class
Gromov–Witten invariants of are defined via intersection theory on the moduli space, (, ) of stable maps from -pointed genus curves to. This moduli space comes with a virtual fundamental class
Gromov–Witten invariants of K3 surfaces for primitive curve classes have been well understood since the seminal paper by Maulik, Pandharipande and Thomas [29]
Summary
Let be a complex nonsingular projective K3 surface and ∈ 2( , Z) be an effective curve class. Gromov–Witten invariants of are defined via intersection theory on the moduli space , ( , ) of stable maps from -pointed genus curves to. This moduli space comes with a virtual fundamental class. The virtual class vanishes for ≠ 0, so instead we use the reduced class. ∪ ev∗ ( ) , where ev : , ( , ) → is the evaluation at the th marking and is the cotangent class at the th marking. By the deformation invariance of the reduced class, the invariant depends only on the norm. And the divisibility of the curve class
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