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The Mirror Clemens-Schmid Sequence

Charles F. Doran, Alan Thompson

Abstract

We introduce a four-term long exact sequence that relates the cohomology of a smooth variety admitting a projective morphism onto a projective base to the cohomology of the open set obtained by removing the preimage of a general linear section. We show that this sequence respects the perverse Leray filtration and induces exact sequences of mixed Hodge structures on its graded pieces. We conjecture that this exact sequence should be thought of as mirror to the Clemens-Schmid sequence, which describes the cohomology of degenerations. We exhibit this mirror relationship explicitly for all Type II and many Type III degenerations of K3 surfaces. In three dimensions, we show that for Tyurin degenerations of Calabi-Yau threefolds our conjecture is a consequence of existing mirror conjectures, and we explicitly verify our conjecture for a more complicated degeneration of the quintic threefold.

The Mirror Clemens-Schmid Sequence

Abstract

We introduce a four-term long exact sequence that relates the cohomology of a smooth variety admitting a projective morphism onto a projective base to the cohomology of the open set obtained by removing the preimage of a general linear section. We show that this sequence respects the perverse Leray filtration and induces exact sequences of mixed Hodge structures on its graded pieces. We conjecture that this exact sequence should be thought of as mirror to the Clemens-Schmid sequence, which describes the cohomology of degenerations. We exhibit this mirror relationship explicitly for all Type II and many Type III degenerations of K3 surfaces. In three dimensions, we show that for Tyurin degenerations of Calabi-Yau threefolds our conjecture is a consequence of existing mirror conjectures, and we explicitly verify our conjecture for a more complicated degeneration of the quintic threefold.

Paper Structure

This paper contains 37 sections, 13 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Description of the results
  4. Relationship to other work
  5. Structure of the paper
  6. Acknowledgments
  7. Data availability
  8. Notation and assumptions
  9. The perverse Leray filtration
  10. The perverse Leray filtration and localization
  11. A geometric description of the perverse Leray filtration
  12. Isomorphisms and vanishing
  13. The cup-product filtration and the perverse Leray filtration
  14. The mirror Clemens-Schmid sequence
  15. Mirror symmetry
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $\pi \colon Y \to B$ be a projective, surjective morphism from a smooth complex projective variety $Y$ to a complex projective variety $B$. Fix an embedding $B \subset \mathbb{P}^N$ and let $Z \subset Y$ be the preimage under $\pi$ of a general linear subspace. Let $U := Y - Z$ denote the comple

Theorems & Definitions (39)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Conjecture 1.6
  • Theorem 1.7
  • Conjecture 1.8
  • Theorem 1.9
  • Definition 2.1
  • ...and 29 more