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Perverse Extensions and Limiting Mixed Hodge Structures for Conifold Degenerations

Abdul Rahman

Abstract

Let $pi:X\toΔ$ 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.

Perverse Extensions and Limiting Mixed Hodge Structures for Conifold Degenerations

Abstract

Let be a one-parameter degeneration whose central fiber has a single ordinary double point. The nearby- and vanishing-cycle formalism determines a canonical perverse sheaf on , 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.

Paper Structure

This paper contains 53 sections, 23 theorems, 162 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Relation to earlier work and Hodge-theoretic framework
  3. Physical and categorical motivation
  4. Main results
  5. Scope and organization
  6. Limiting Mixed Hodge Structures
  7. Monodromy and the limiting mixed Hodge structure
  8. Nearby cycles and vanishing cycles
  9. The ordinary double point case
  10. Mixed Hodge modules and realization
  11. The single-node case
  12. Vanishing cycles and Picard--Lefschetz
  13. The perverse extension
  14. Mixed Hodge modules and nearby cycles
  15. Remarks on the Hodge-theoretic comparison
  16. ...and 38 more sections

Key Result

Theorem 1.1

Let be a one-parameter degeneration whose central fiber $X_0$ has a single ordinary double point $p$, and let Then $\mathcal{P}$ is the unique minimal Verdier self-dual perverse extension of $\mathbb{Q}_U[3]$ across the node, where $U=X_0\setminus\{p\}$. Moreover, the rank-one vanishing contribution in the limiting mixed Hodge structure and the quotient in the exact sequence arise functorially

Theorems & Definitions (50)

  • Theorem 1.1: Single-node case
  • Theorem 1.2: Multiple-node case
  • Theorem 1.3: Stratified singular locus
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • Proposition 4.1
  • ...and 40 more