Smooth Surfaces in Smooth Fourfolds with Vanishing First Chern Class

Abstract

According to a conjecture attributed to Hartshorne and Lichtenbaum and proven by Ellingsrud and Peskine, the smooth rational surfaces in ${\mathbb{P}}^4$ belong to only finitely many families. We formulate and study a collection of analogous problems in which ${\mathbb{P}}^4$ is replaced by a smooth fourfold $X$ with vanishing first integral Chern class. We embed such $X$ into a smooth ambient variety and count families of smooth surfaces which arise in $X$ from the ambient variety. We obtain various finiteness results in such settings. The central technique is the introduction of a new numerical invariant for smooth surfaces in smooth fourfolds with vanishing first Chern class.

final version, published in the Journal of Pure and Applied Algebra

References in corpus (5)

• All Complete Intersection Calabi-Yau Four-Folds
• On the splitting of the Bloch-Beilinson filtration
• Moduli of Abelian varieties, Vinberg theta-groups, and free resolutions
• Hyper-Kähler fourfolds and Grassmann geometry
• Yet another version of Mumford's theorem