导出的algebraic geometry，完全复形的行列式及在映射和复形的阻塞理论中的应用
Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes
摘要：
我们展示了如何一个准光滑的Deligne-Mumford堆栈X的增强自然赋予X的函子完美的障碍理论的意义上的Bewend-Fantechi。然后将该结果应用于光滑复射影簇上的映射和完全复形的模。对于映射的模，我们认为X=S是一个代数K ~ 3-曲面，$g\geq 0$，和$β$是一个曲线类，并且我们构造了一个导出的堆栈，其截断是从亏格g到S的曲线的点稳定映射的通常堆栈，这些曲线击中类$β$，并且使得截断的包含导致完全阻塞理论，其切线和阻塞空间与对应的约化空间一致，奥库科夫-莫里克-潘达里潘德-托马斯。我们给出复体模的两个进一步的应用。对于K3-曲面S，我们证明了S上的简单完全复形的堆叠是光滑的。Inaba对相应的粗模空间用不同的方法证明了这一结果。最后，我们构建了一个映射，从导出堆栈的稳定嵌入的曲线（到一个光滑的复杂的射影品种X）的导出堆栈的简单完美的复杂的X与消失的负Ext的，并显示这个映射如何诱导相应的障碍理论的态射时，X是一个Calabi-Yau三倍。我们的建设的一个重要组成部分是一个完美的行列式映射从派生堆栈的完美复形派生堆栈的线丛的切线态射是，逐点，Illusie的完美复形的跟踪地图。我们希望这个决定性映射在其他情况下也是有用的。
英文摘要：
We show how a quasi-smooth derived enhancement of a Deligne-Mumford stack X naturally endows X with a functorial perfect obstruction theory in the sense of Behrend-Fantechi. This result is then applied to moduli of maps and perfect complexes on a smooth complex projective variety. For moduli of maps, we consider X=S an algebraic K3-surface, $g\geq 0$, and $β$ a curve class, and we construct a derived stack whose truncation is the usual stack of pointed stable maps from curves of genus g to S hitting the class $β$, and such that the inclusion of the trunaction induces on a perfect obstruction theory whose tangent and obstruction spaces coincide with the corresponding reduced spaces of Okounkov-Maulik-Pandharipande-Thomas. We give two further applications to moduli of complexes. For a K3-surface S we show that the stack of simple perfect complexes on S is smooth. This result was proved with different methods by Inaba for the corresponding coarse moduli space. Finally, we construct a map from the derived stack of stable embeddings of curves (into a smooth complex projective variety X) to the derived stack of simple perfect complexes on X with vanishing negative Ext's, and show how this map induces a morphism of the corresponding obstruction theories when X is a Calabi-Yau threefold. An important ingredient of our construction is a perfect determinant map from the derived stack of perfect complexes to the derived stack of line bundles whose tangent morphism is, pointwise, Illusie's trace map for perfect complexes. We expect that this determinant map might be useful in other contexts as well.
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