Triangle varieties and surface decomposition of hyper-Kähler manifolds

Abstract

We introduce and study the notion of "surface decomposable" variety, and discuss the possibility that any projective hyper-Kähler manifold is surface decomposable, which would produce new evidence for Beauville's weak splitting conjecture. We show that surface decomposability relates to the Beauville-Fujiki relation, a constraint on the cohomology ring of the variety, and that general varieties with $h^{2,0}\ne 0$ are not surface decomposable. We also formalize the notion of triangle variety that is useful to produce surface decomposition. We show the existence of these geometric structures on most explicitly constructed classes of projective hyper-Kähler manifolds of Picard number $1$.

References in corpus (5)

• Families of rational curves on holomorphic symplectic varieties and applications to zero-cycles
• The Fourier transform for certain hyperKaehler fourfolds
• Lagrangian constant cycle subvarieties in Lagrangian fibrations
• Characteristic foliation on non-uniruled smooth divisors on hyperkaehler manifolds
• On the variety of triangles for a hyper-Kaehler fourfold constructed by Debarre and Voisin