Extension of the Topological Abel-Jacobi Map for Cubic Threefolds

TL;DR

This paper extends the topological Abel-Jacobi map for cubic threefolds by constructing a compactification of a covering space related to vanishing cycles, analyzing boundary points, and proving surjectivity on fundamental groups.

Contribution

It introduces a compactification of the covering space associated with vanishing cycles on cubic threefolds and extends the Abel-Jacobi map to this setting, analyzing boundary behavior.

Findings

Constructed a 72-to-1 covering space $T_v$ for cubic threefolds.

Extended the topological Abel-Jacobi map to a compactification of $T_v$.

Proved the map on fundamental groups is surjective.

Abstract

The difference $[L_1]-[L_2]$ of a pair of skew lines on a cubic threefold defines a vanishing cycle on the cubic surface as the hyperplane section spanned by the two lines. By deforming the hyperplane, the flat translation of such vanishing cycle forms a 72-to-1 covering space $T_v$ of a Zariski open subspace of $(\mathbb P^4)^*$. Based on a lemma of Stein on the compactification of finite analytic covers, we found a compactification of $T_v$ to which the topological Abel-Jacobi map extends. Moreover, the boundary points of the compactification can be interpreted in terms of local monodromy and the singularities on cubic surfaces. We prove the associated map on fundamental groups of topological Abel-Jacobi map is surjective.