The degeneration formula for stable log maps

We give a direct proof for the degeneration formula of Gromov–Witten invariants including its cycle version for degenerations with smooth singular locus in the setting of stable log maps of Abramovich-Chen, Chen, Gross–Siebert.

Refined floor diagrams from higher genera and lambda classes

We show that, after the change of variables $$q=e^{iu}$$ q = e iu , refined floor diagrams for $${\mathbb {P}}^2$$ P 2 and Hirzebruch surfaces compute generating series of higher genus relative Gromov–Witten invariants with insertion of a lambda class. The proof uses an inductive application of the degeneration formula in relative Gromov–Witten theory and an explicit result in relative Gromov–Witten theory of $${\mathbb {P}}^1$$ P 1 . Combining this result with the similar looking refined tropical correspondence theorem for log Gromov–Witten invariants, we obtain a non-trivial relation between relative and log Gromov–Witten invariants for $${\mathbb {P}}^2$$ P 2 and Hirzebruch surfaces. We also prove that the Block–Göttsche invariants of $${\mathbb {F}}_0$$ F 0 and $${\mathbb {F}}_2$$ F 2 are related by the Abramovich–Bertram formula.

Curves on K3 surfaces in divisibility 2

We 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.