Gromov–Witten theory and Noether–Lefschetz theory for holomorphic-symplectic varieties

Abstract

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.

Tropical curves in abelian surfaces III: pearl diagrams and multiple cover formulas

“…The above multiple cover formulas (ii) is a particular case of the more general Oberdieck's multiple cover formula conjectured by G. Oberdieck in [29]. The formula is as follows.…”

“…Conjecture. 5.1 [29] Let α be a tautological class in the cohomology ring H * (M g,n ) and γ i ∈ H * (CA, R) be arbitrary insertions, where CA is an abelian surface. Then, one has…”

“…Complex multiple cover formulas. The multiple cover formulas proven in Theorem 5.13 are a particular case of the general multiple cover formula which is conjectured by G. Oberdieck in [29]. Oberdieck's formula has the following form.…”

“…In the cover formula, we now take n = g 0 , g 0 point insertions, and α = λ g−g 0 . The multiple cover formula from [29] makes the following prediction:…”

“…We now insert a λ-class, and take g 0 − 2 point insertions, and four 1-cycle insertions forming a basis of the first homology group. The multiple cover formula from [29] predicts…”