科研之友
微信
新浪微博
Facebook
分享链接
摘要
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. We apply this result 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 is an element of N, and beta not equal 0 in H-2 (S, Z) a curve class, we construct a derived stack R (M) over bar (red)(g,n) (S; beta) whose truncation is the usual stack (M) over bar (g,n) (S; beta) of pointed stable maps from curves of genus g to S hitting the class beta, and such that the inclusion (M) over bar (g) (S; beta) hooked right arrow R (M) over bar (red)(g) (S; beta) induces on (M) over bar (g) (S; beta) a perfect obstruction theory whose tangent and obstruction spaces coincide with the corresponding reduced spaces of Okounkov-Maulik-Pandharipande-Thomas. 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 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 3-fold. 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 given by Illusie's trace map for perfect complexes.
- 出版日期2015-5
