Perverse Extensions and Limiting Mixed Hodge Structures for Conifold Degenerations

TL;DR

This paper investigates the structure of perverse sheaves and limiting mixed Hodge structures in conifold degenerations, revealing their unique minimal extensions and the role of Saito's divisor-gluing formalism.

Contribution

It characterizes the canonical perverse sheaf as the minimal self-dual extension in conifold degenerations and connects it with limiting mixed Hodge structures and Saito's theory.

Findings

The corrected perverse object is the unique minimal Verdier self-dual extension.

Rank-one contributions in perverse sheaves and Hodge structures originate from the same nearby-cycle formalism.

Saito's divisor-gluing formalism provides a natural framework for these constructions.

Abstract

Let $pi:X\to\Delta$ be a one-parameter degeneration whose central fiber $X_0$ has a single ordinary double point. The nearby- and vanishing-cycle formalism determines a canonical perverse sheaf on $X_0$, obtained from the variation morphism and fitting into an extension of the intersection complex by a point-supported rank-one contribution. We study this object from the perspective of limiting mixed Hodge theory and Saito's theory of mixed Hodge modules. In the ordinary double point case, we show that the corrected perverse object is the unique minimal Verdier self-dual perverse extension of the shifted constant sheaf across the node, and that its rank-one singular contribution and the corresponding rank-one vanishing contribution in the limiting mixed Hodge structure arise from the same nearby-cycle formalism. We also formulate the analogous structural statements for multi-node degenerations and for more general stratified singular loci. Finally, we explain how Saito's divisor-gluing formalism provides the natural framework for a fuller mixed-Hodge-module refinement of these constructions.