2011/02/06 by Timo Schürg, Schürg, Timo, Bertrand Toën +3
Mathematics · #14A20 #14J10 #14J28 #14N35 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #K-Theory and Homology (math.KT)
paper · pdf · doi:10.48550/arxiv.1102.1150
openalex publication_date 2011/02/06 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
We show how a quasi-smooth derived enhancement of a Deligne-Mumford stack X\nnaturally endows X with a functorial perfect obstruction theory in the sense of\nBehrend-Fantechi. This result is then applied to moduli of maps and perfect\ncomplexes on a smooth complex projective variety. For moduli of maps, we\nconsider X=S an algebraic K3-surface, g\≥ 0, and \β a curve class, and\nwe construct a derived stack whose truncation is the usual stack of pointed\nstable maps from curves of genus g to S hitting the class \β, and such\nthat the inclusion of the trunaction induces on a perfect obstruction theory\nwhose tangent and obstruction spaces coincide with the corresponding reduced\nspaces of Okounkov-Maulik-Pandharipande-Thomas. We give two further\napplications to moduli of complexes. For a K3-surface S we show that the stack\nof simple perfect complexes on S is smooth. This result was proved with\ndifferent methods by Inaba for the corresponding coarse moduli space. Finally,\nwe construct a map from the derived stack of stable embeddings of curves (into\na smooth complex projective variety X) to the derived stack of simple perfect\ncomplexes on X with vanishing negative Ext's, and show how this map induces a\nmorphism of the corresponding obstruction theories when X is a Calabi-Yau\nthreefold. An important ingredient of our construction is a perfect determinant\nmap from the derived stack of perfect complexes to the derived stack of line\nbundles whose tangent morphism is, pointwise, Illusie's trace map for perfect\ncomplexes. We expect that this determinant map might be useful in other\ncontexts as well.\n