Integral transforms for coherent sheaves
Integral transforms for coherent sheaves
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相干滑轮的积分变换
DOI：
10.4171/jems/753
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
Anatoly Preygel
中科院分区：
文献类型：
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作者：
David Ben;D. Nadler;Anatoly Preygel
积分理论或傅里叶-向井理论（Fourier-Mukai）是非交换代数几何中一个成熟的工具。一般的“表示定理”确定了在框架上的完美复形（或其“大”版本，拟凝聚层）的范畴之间的所有合理的线性函子，以及在纤维积上具有层（相同性质）形式的积分核的某些固定基上的栈。然而，对于镜像对称和几何表示论中的许多应用，人们感兴趣的是相干层的有界导出范畴（或其“大”版本，ind-coherent层），一旦底层的种类是奇异的，它就不同于完美复形（和准相干层）。本文利用纤维积上的积分核给出了基上凝聚层范畴间线性函子的一般表示定理。也就是说，我们确定相干核函子采取完美的复相干层，和内核是相干的相对于源函子采取所有相干层相干层。证明依赖于派生类别的“功能分析”的关键方面，即小类别和大类别之间的区别及其使用$t$-结构的测量。这些特别用于校正独立相干层上的积分变换的故障，以对应于纤维产品上的这种层。结果适用于在同伴纸的仿射Hecke类别的表示理论，确定仿射字符层的光谱几何朗兰兹类别在属1。
The theory of integral, or Fourier-Mukai, transforms between derived categories of sheaves is a well established tool in noncommutative algebraic geometry. General "representation theorems" identify all reasonable linear functors between categories of perfect complexes (or their "large" version, quasi-coherent sheaves) on schemes and stacks over some fixed base with integral kernels in the form of sheaves (of the same nature) on the fiber product. However, for many applications in mirror symmetry and geometric representation theory one is interested instead in the bounded derived category of coherent sheaves (or its "large" version, ind-coherent sheaves), which differs from perfect complexes (and quasi-coherent sheaves) once the underlying variety is singular. In this paper, we give general representation theorems for linear functors between categories of coherent sheaves over a base in terms of integral kernels on the fiber product. Namely, we identify coherent kernels with functors taking perfect complexes to coherent sheaves, and kernels which are coherent relative to the source with functors taking all coherent sheaves to coherent sheaves. The proofs rely on key aspects of the "functional analysis" of derived categories, namely the distinction between small and large categories and its measurement using $t$-structures. These are used in particular to correct the failure of integral transforms on Ind-coherent sheaves to correspond to such sheaves on a fiber product. The results are applied in a companion paper to the representation theory of the affine Hecke category, identifying affine character sheaves with the spectral geometric Langlands category in genus one.