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Perverse Schober Structures for Conifold Degenerations

Abdul Rahman

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.

Perverse Schober Structures for Conifold Degenerations

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.

Paper Structure

This paper contains 31 sections, 12 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Related work and context
  4. Perverse models in conifold settings
  5. Categorical Picard--Lefschetz theory and spherical monodromy
  6. Perverse schobers
  7. Position of the present work
  8. Conifold degenerations, nearby and vanishing cycles
  9. Geometric setup
  10. Milnor fiber and vanishing cohomology
  11. Nearby and vanishing cycles
  12. Canonical morphisms
  13. Vanishing cycles for an ordinary double point
  14. Monodromy
  15. Canonical perverse models
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $\pi:\mathcal{X}\to\Delta$ be a one-parameter degeneration of Calabi--Yau threefolds whose central fiber $X_0$ has a single ordinary double point $p$. Let Assume that $\mathrm{var}_F:\phi_\pi(F)\to\psi_\pi(F)$ is a monomorphism in $\mathrm{Perv}(X_0)$. Then the object lies in $\mathrm{Perv}(X_0)$ and satisfies: $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (31)

  • Theorem 1.1: Canonical perverse model
  • Remark 1.1
  • Proposition 1.2: Categorical comparison, conditional form
  • Remark 2.1: Canonical vs. rank-prescribed perverse models
  • Remark 2.2: Relation with earlier constructions
  • Remark 3.1: Dimension of the total space
  • Lemma 3.1: Vanishing cohomology for an ODP
  • proof
  • Proposition 3.2
  • proof
  • ...and 21 more