Smooth Surfaces in Smooth Fourfolds with Vanishing First Chern Class

TL;DR

This paper investigates finiteness properties of smooth surfaces in special fourfolds with zero first Chern class, introducing a new numerical invariant to analyze their families and generalize classical results.

Contribution

It formulates and studies finiteness problems for smooth surfaces in fourfolds with vanishing first Chern class, introducing a novel numerical invariant for these surfaces.

Findings

Finiteness results for families of smooth surfaces in such fourfolds.

Introduction of a new numerical invariant for smooth surfaces in fourfolds.

Generalization of classical finiteness conjectures to higher-dimensional settings.

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.