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

Abstract

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 ∈ ℕ, and β ≠ 0 in H₂(S,ℤ) a curve class, we construct a derived stack ℝ𝐌¯g,nred(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¯gred(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. 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.