Perverse Schober Structures for Conifold Degenerations

TL;DR

This paper constructs a canonical perverse sheaf capturing the vanishing cycle contribution in conifold degenerations of Calabi-Yau threefolds, linking sheaf theory with monodromy phenomena.

Contribution

It introduces a sheaf-theoretic object that isolates the vanishing cycle in conifold degenerations, connecting it to categorical monodromy effects.

Findings

Constructed a canonical perverse sheaf on the singular fiber.

Showed the sheaf differs from the intersection complex by a rank-one contribution.

Linked the sheaf construction to spherical monodromy phenomena.

Abstract

We study a one parameter degeneration of Calabi Yau threefolds whose central fiber contains a single ordinary double point. Using the nearby and vanishing cycle formalism, we construct a canonical perverse object on the singular fiber from the variation morphism between vanishing and nearby cycles. We show that this object restricts to the constant perverse sheaf on the smooth locus and differs from the intersection complex by a single rank one contribution supported at the node. Thus the object isolates the vanishing cycle contribution associated with the conifold degeneration in a canonical sheaf theoretic form. We also explain how this construction aligns with the rank-one Picard Lefschetz phenomenon that appears categorically through spherical monodromy, making it a natural comparison object for the decategorified effect of spherical twists in the ordinary double point case.