Markman–Markushevich principle for O’Grady six-manifolds

Establish a dominant, generically injective map from the moduli space of marked cubic surfaces to the Markman–Markushevich space of six-dimensional O’Grady manifolds that identifies the essential Hodge structure of each O’Grady manifold with the transcendental lattice of the Shioda–Inose partner of the corresponding marked cubic surface, and prove that degeneration of the cubic surface to a nodal cubic with its six marked lines passing through the node yields O’Grady’s original example associated with a Jacobian of a genus-two curve.

Background

The paper seeks an analogue for six-dimensional O’Grady manifolds of the Markman–Markushevich descriptions already available for manifolds of K3^[n], generalized Kummer, and O’Grady ten-dimensional types. The proposed source of the relevant four-dimensional moduli space is the moduli space of marked cubic surfaces, while the associated K3 surface is obtained as a Shioda–Inose partner constructed from the cubic surface and its marked double-six.

The conjecture asserts both the existence and generic injectivity of the period-type map into the Markman–Markushevich space and a precise Hodge-theoretic identification. It also predicts compatibility with a boundary degeneration: the nodal cubic-surface case should recover the original O’Grady six-dimensional construction from a genus-two Jacobian. The paper presents evidence but does not establish these claims.