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We show how a quasi-smooth derived enhancement of a Deligne-Mumford stack Xnaturally endows X with a functorial perfect obstruction theory in the sense ofBehrend-Fantechi. This result is then applied to moduli of maps and perfectcomplexes on a smooth complex projective variety. For moduli of maps, weconsider X=S an algebraic K3-surface, $g\geq 0$, and $\beta$ a curve class, andwe construct a derived stack whose truncation is the usual stack of pointedstable maps from curves of genus g to S hitting the class $\beta$, and suchthat the inclusion of the trunaction induces on a perfect obstruction theorywhose tangent and obstruction spaces coincide with the corresponding reducedspaces of Okounkov-Maulik-Pandharipande-Thomas. We give two furtherapplications to moduli of complexes. For a K3-surface S we show that the stackof simple perfect complexes on S is smooth. This result was proved withdifferent methods by Inaba for the corresponding coarse moduli space. Finally,we construct a map from the derived stack of stable embeddings of curves (intoa smooth complex projective variety X) to the derived stack of simple perfectcomplexes on X with vanishing negative Ext's, and show how this map induces amorphism of the corresponding obstruction theories when X is a Calabi-Yauthreefold. An important ingredient of our construction is a perfect determinantmap from the derived stack of perfect complexes to the derived stack of linebundles whose tangent morphism is, pointwise, Illusie's trace map for perfectcomplexes. We expect that this determinant map might be useful in othercontexts as well.

Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes