The relative Clemens Conjectures for $\frac{1}{2}$-log Calabi-Yau threefolds

TL;DR

This paper formulates a relative version of the Clemens conjectures for 1/2-log Calabi-Yau threefold pairs, establishing a duality framework and verifying finiteness and normal bundle conditions for rational curves, with implications for enumerative geometry.

Contribution

It introduces a new relative Clemens conjecture framework for 1/2-log Calabi-Yau threefolds and verifies it for prime Fano threefolds of index two, simplifying relative Gromov-Witten theory.

Findings

Finiteness of rational curves in generic configurations.

Balanced normal bundle condition for these curves.

Reduction of virtual invariants to classical counts.

Abstract

We formulate a relative analogue of the Clemens conjectures for 1/2-log Calabi-Yau threefold pairs (X,Y) (where K_X+2Y is isomorphic to O_X). This framework rests on the restoration of a perfect deformation/obstruction duality specific to the 1/2-log CY threefold setting. Based on this duality, we conjecture that for a generic intersection configuration on the boundary divisor Y, the number of rational curves anchored to these points is finite, and every such curve possesses the balanced relative normal bundle N_{C/X}(-Y) isomorphic to O_C(-1) + O_C(-1). In a joint appendix with Adrian Zahariuc, we verify this framework for prime Fano threefolds of index two. Using specialization techniques, we demonstrate that the usual virtual complications of relative Gromov-Witten theory are naturally suppressed in this setting. This trivialization of the relative moduli space cleanly reduces the virtual invariants to honest, classical enumerative counts, thereby rigorously proving the geometric conjecture.