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.

Final version, accepted for publication in the European Journal of Mathematics

References in corpus (6)

• Calabi-Yau manifolds realizing symplectically rigid monodromy tuples
• Families of lattice polarized K3 surfaces with monodromy
• Calabi-Yau Threefolds Fibred by Mirror Quartic K3 Surfaces
• Calabi-Yau threefolds fibred by high rank lattice polarized K3 surfaces
• The Mori fan of the Dolgachev-Nikulin-Voisin family in genus $2$
• Threefolds Fibred by Mirror Sextic Double Planes