Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes
Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes
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DOI：
10.1515/crelle-2013-0037
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发表时间：
2011-02
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影响因子：
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通讯作者：
Timo Schürg;B. Toën;G. Vezzosi
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文献类型：
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作者：
Timo Schürg;B. Toën;G. Vezzosi
我们展示了Deligne-Mumford堆栈X的准光滑导出增强如何自然地赋予X在Behrend-Fantechi意义下的函数式完全阻塞理论。然后将这一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，对于X=S的一个代数K3曲面，g∈N，且β=0在H2(S，Z)中的一个曲线类，我们构造了一个导出的叠层Rmred g，n(S；β)，它截断为从亏格g到S的曲线到达类β的点稳定映射的一般叠层Mg，n(S；β)，且使得包含Mg(S；β)→Rmred g(S；β))诱导到Mg(S；β)的一个完全阻塞理论，它的切线和阻塞空间与Okounkov-Maulik-Pandharipande-Thomas[O-P2，M-P，M-P-T]的相应约化空间重合。我们在这里提出的方法使用了导出的代数几何，不仅给出了一个完全严格的证明--不依赖于任何半正则映射的结果--而且给出了一种新的全局几何解释。我们给出了复数模的两个进一步的应用。对于K3-曲面S，我们证明了S上的简单完美复形的堆叠是光滑的。对于相应的粗模空间，Inaba([In])用不同的方法证明了这一结果。最后，我们构造了从曲线的稳定嵌入的派生堆栈(到光滑复射影簇X)到X上具有零负Ext的简单完美复形的派生堆栈的映射，并证明了当X是Calabi-Yau三重映射时，该映射如何诱导相应的障碍理论的态射。我们构造的一个重要组成部分是从完美复形的派生堆叠到线丛的派生堆叠的完美行列式映射，其切线态射逐点是完美复形的Illusie迹映射。我们希望这个决定因素图在其他情况下也会有用。
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, for X = S an algebraic K3-surface, g ∈ N, and β = 0 in H 2 (S, Z) a curve class, we construct a derived stack RM red g,n (S; β) whose truncation is the usual stack M g,n (S; β) of pointed stable maps from curves of genus g to S hitting the class β, and such that the inclusion M g (S; β) → RM red g (S; β) induces on M g (S; β) a perfect obstruction theory whose tangent and obstruction spaces coincide with the corresponding reduced spaces of Okounkov-Maulik-Pandharipande-Thomas [O-P2, M-P, M-P-T]. The approach we present here uses derived algebraic geometry and yields not only a full rigorous proof of the existence of a reduced obstruction theory-not relying on any result on semiregularity maps-but also a new global geometric interpretation. 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 ([In]) 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.