English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We prove a conjecture of Maulik, Pandharipande, and Thomas expressing theGromov--Witten invariants of K3 surfaces for divisibility two curve classes inall genus in terms of weakly holomorphic quasimodular forms of level two. Then,we establish the holomorphic anomaly equation in divisibility two in all genus.Our approach involves a refined boundary induction, relying on the toptautological group of the moduli space of smooth curves, together with adegeneration formula for reduced virtual fundamental class with imprimitivecurve classes. We use the double ramification relations with target variety asa new tool to prove the initial condition. The relationship between theholomorphic anomaly equation for higher divisibility and the conjecturalmultiple cover formula of Oberdieck and Pandharipande is discussed in detailand illustrated with several examples.

Curves on K3 surfaces in divisibility two