Geodesic
Gromov–Witten theory and Noether–Lefschetz theory for holomorphic-symplectic varieties
Forum of Mathematics, Sigma, Tome 10 (2022)

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We use Noether–Lefschetz theory to study the reduced Gromov–Witten invariants of a holomorphic-symplectic variety of $K3^{[n]}$-type. This yields strong evidence for a new conjectural formula that expresses Gromov–Witten invariants of this geometry for arbitrary classes in terms of primitive classes. The formula generalizes an earlier conjecture by Pandharipande and the author for K3 surfaces. Using Gromov–Witten techniques, we also determine the generating series of Noether–Lefschetz numbers of a general pencil of Debarre–Voisin varieties. This reproves and extends a result of Debarre, Han, O’Grady and Voisin on Hassett–Looijenga–Shah (HLS) divisors on the moduli space of Debarre–Voisin fourfolds. 
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     author = {Georg Oberdieck},
     title = {Gromov{\textendash}Witten theory and {Noether{\textendash}Lefschetz} theory for holomorphic-symplectic varieties},
     journal = {Forum of Mathematics, Sigma},
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     doi = {10.1017/fms.2022.10},
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Georg Oberdieck. Gromov–Witten theory and Noether–Lefschetz theory for holomorphic-symplectic varieties. Forum of Mathematics, Sigma, Tome 10 (2022). doi: 10.1017/fms.2022.10

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