镜像 Clemens–Schmid 正合序列
The Mirror Clemens-Schmid Sequence
AI总结：
本文提出联系光滑簇与补开集上同调的四项长正合序列，证明其兼容反常 Leray 滤过和混合 Hodge 结构，并通过 K3、Calabi–Yau 及五次三维簇退化验证其为 Clemens–Schmid 序列的镜像。
摘要：
我们引入一个四项长正合序列，它将允许投射态射到投射基的光滑簇的上同调，与移除一般线性截面的原像后所得开集的上同调联系起来。我们证明该序列尊重反常 Leray 滤过，并在其分次片上诱导混合 Hodge 结构的正合序列。我们猜想，这一正合序列应被视为描述退化上同调的 Clemens–Schmid 序列的镜像。对于 K3 曲面的所有 II 型退化以及许多 III 型退化，我们明确展示了这种镜像关系。在三维情形中，我们证明对于 Calabi–Yau 三维簇的 Tyurin 退化，该猜想是已有镜像猜想的推论；并对五次三维簇的一种更复杂退化明确验证了我们的猜想。
英文摘要：
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.
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