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The mirror Clemens-Schmid sequence.

Charles F Doran1,2,3, Alan Thompson4

  • 1Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, AB T6G 2G1 Canada.

European Journal of Mathematics
|October 28, 2024
PubMed
Summary
This summary is machine-generated.

Researchers developed a new cohomology exact sequence for smooth varieties, revealing connections to degeneration and mixed Hodge structures. This sequence offers insights into K3 surfaces and Calabi-Yau threefolds, supporting existing mirror conjectures.

Keywords:
Calabi–Yau manifoldDegenerationsFibrationsK3 surfaceMirror symmetry

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Area of Science:

  • Algebraic Geometry
  • Hodge Theory
  • Enumerative Geometry

Background:

  • Smooth projective varieties and their cohomology are fundamental in algebraic geometry.
  • Understanding degenerations and their impact on cohomology is crucial for studying moduli spaces.
  • Mixed Hodge structures provide a powerful framework for analyzing the cohomology of singular varieties.

Purpose of the Study:

  • Introduce a novel four-term long exact sequence relating cohomology of varieties to open sets.
  • Investigate the sequence's interaction with the perverse Leray filtration and mixed Hodge structures.
  • Explore the conjectured mirror relationship between this sequence and the Clemens-Schmid sequence.

Main Methods:

  • Construction of a four-term long exact sequence in cohomology.
  • Demonstration of the sequence's compatibility with the perverse Leray filtration.
  • Explicit verification of the sequence for specific degenerations of K3 surfaces and Calabi-Yau threefolds.

Main Results:

  • A new long exact sequence is established for smooth varieties under projective morphisms.
  • The sequence preserves the perverse Leray filtration and induces exact sequences of mixed Hodge structures.
  • Mirror symmetry is explicitly demonstrated for Type II and Type III K3 surface degenerations and certain Calabi-Yau threefold degenerations.

Conclusions:

  • The introduced exact sequence provides a new tool for studying the cohomology of smooth varieties.
  • The conjectured mirror relationship to the Clemens-Schmid sequence is supported by explicit examples.
  • The findings contribute to the understanding of degenerations and mirror symmetry in algebraic geometry.