arXiv subjects
Publications and source records attributed to Josiah Foster.
Contingent upon a conjecture of Buchweitz and Flenner, we prove the Lefschetz standard conjectures for 4d-dimensional projective varieties of generalized Kummer deformation type.
In this article, we construct an infinite sequence of irreducible components of Koll\'{a}r--Shepherd-Barron (KSB-) moduli spaces of surfaces of arbitrarily large volumes, and describe the boundary of each component completely. Moreover, we describe the stable reduction steps in finding the KSB-limits in an explicit combinatorial way. Our main approach is to study the moduli spaces of elliptic surfaces with Kodaira dimension one, fibered over rational curves, using the techniques of wall-crossing for KSBA moduli and twisted stable maps.
For a projective $2n$-dimensional irreducible holomorphic symplectic manifold $Y$ of generalized Kummer deformation type and $j$ the smallest prime number dividing $n+1$, we prove the Lefschetz standard conjectures in degrees $<2(n+1)(j-1)/j$. We show that the restriction homomorphism from the cohomology of a projective deformation of a moduli space of Gieseker-stable sheaves on an Abelian surface to the cohomology of $Y$ is surjective in these degrees. A corollary is that the Lefschetz standard conjectures hold for $Y$ when $n+1$ is prime. The proofs rely on Markman's description of the monodromy of generalized Kummer varieties and construction of a universal family of moduli spaces of sheaves, Verbitsky's theory of hyperholomorphic sheaves, and the decomposition theorem.