A quasi-smooth derived enhancement of a Deligne–Mumford stack naturally endows 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 ℝ⎯⎯⎯⎯redg,n(S;β) whose truncation is the usual stack ⎯⎯⎯⎯g,n(S;β) of pointed stable maps from curves of genus g to S hitting the class β, and such that the inclusion ⎯⎯⎯⎯g(S;β)↪ℝ⎯⎯⎯⎯redg(S;β) induces on ⎯⎯⎯⎯g(S;β) 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.

Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes / T. Schurg; B. Toen; G. Vezzosi. - In: JOURNAL FÜR DIE REINE UND ANGEWANDTE MATHEMATIK. - ISSN 1435-5345. - STAMPA. - Volume 2015:(2015), pp. 1-40. [10.1515/crelle-2013-0037]

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

VEZZOSI, GABRIELE
2015

Abstract

A quasi-smooth derived enhancement of a Deligne–Mumford stack naturally endows 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 ℝ⎯⎯⎯⎯redg,n(S;β) whose truncation is the usual stack ⎯⎯⎯⎯g,n(S;β) of pointed stable maps from curves of genus g to S hitting the class β, and such that the inclusion ⎯⎯⎯⎯g(S;β)↪ℝ⎯⎯⎯⎯redg(S;β) induces on ⎯⎯⎯⎯g(S;β) 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.
2015
Volume 2015
1
40
T. Schurg; B. Toen; G. Vezzosi
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Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/800683
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