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- Title
Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes.
- Authors
Schürg, Timo; Toën, Bertrand; Vezzosi, Gabriele
- Abstract
A quasi-smooth derived enhancement of a Deligne-Mumford stack 풳 naturally endows 풳 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 ∈ ℕ, and β ≠ 0 in H2( S,ℤ) a curve class, 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 induces on 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.
- Subjects
ALGEBRAIC geometry; OBSTRUCTION theory; ISOMORPHISM (Mathematics); GROMOV-Witten invariants; HOMOTOPY theory
- Publication
Journal für die Reine und Angewandte Mathematik, 2015, Vol 2015, Issue 702, p1
- ISSN
0075-4102
- Publication type
Article
- DOI
10.1515/crelle-2013-0037