论文标题
派生的$θ$ - stratifications和$ d $ - 等价猜想
Derived $Θ$-stratifications and the $D$-equivalence conjecture
论文作者
论文摘要
$θ$ stratifications的理论将矢量束模量的经典分层概括为平滑曲线的模量，即较难的narasimhan-shatz分层，以通过代数堆栈来表示的任何模量问题。使用派生的代数几何形状，我们开发了一种结构理论，是对局部共同体理论的完善，对于配备$θ$ stratatification的代数堆栈上的准共晶复合物的派生类别。然后，我们将其应用于$ d $等价的猜想，该猜想预测Biration等同于Calabi-yau歧管具有相同的相干滑轮类别。我们证明，在$ k3 $表面上，吉赛克半固定的连贯滑轮的平滑模量在$ k3 $的表面上的平滑模量在同等的模量上等同于同等的类别。这确立了第一个已知的$ d $等值猜想的构想，即在三个以上的基础等价类别中。
The theory of $Θ$-stratifications generalizes a classical stratification of the moduli of vector bundles on a smooth curve, the Harder-Narasimhan-Shatz stratification, to any moduli problem that can be represented by an algebraic stack. Using derived algebraic geometry, we develop a structure theory, which is a refinement of the theory of local cohomology, for the derived category of quasi-coherent complexes on an algebraic stack equipped with a $Θ$-stratification. We then apply this to the $D$-equivalence conjecture, which predicts that birationally equivalent Calabi-Yau manifolds have equivalent derived categories of coherent sheaves. We prove that any two projective Calabi-Yau manifolds that are birationally equivalent to a smooth moduli space of Gieseker semistable coherent sheaves on a $K3$ surface have equivalent derived categories. This establishes the first known case of the $D$-equivalence conjecture for a birational equivalence class in dimension greater than three.
