The Mirror Clemens-Schmid Sequence

TL;DR

This paper introduces a new long exact sequence relating cohomology of smooth varieties with degenerations, demonstrating its compatibility with mixed Hodge structures and proposing it as a mirror to the Clemens-Schmid sequence, with explicit examples in K3 surfaces and Calabi-Yau threefolds.

Contribution

It proposes a novel mirror sequence to the Clemens-Schmid sequence, extending understanding of degenerations and cohomology in algebraic geometry.

Findings

Sequence respects perverse Leray filtration

Sequence induces exact sequences of mixed Hodge structures

Verified conjecture for specific degenerations of K3 surfaces and Calabi-Yau threefolds

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.