Shifted symplectic structures
Tony Pantev, Bertrand Toën, Michel Vaquié & Gabriele Vezzosi
Publications mathématiques de l'IHÉS de l' IHES
ISSN 0073-8301
Publ.math.IHES DOI 10.1007/s10240-013-0054-1
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SHIFTED SYMPLECTIC STRUCTURES , by TONY PANTEV ,BERTRAND TOËN ,MICHEL VAQUIÉ , and , GABRIELE VEZZOSI
ABSTRACT
This is the first of a series of papers about quantization in the context of derived algebraic geometry.Inthisfirstpart, we introduce the notion of n-shifted symplectic structures (n-symplectic structures for short), a generalization of the notion of symplectic structures on smooth varieties and schemes, meaningful in the setting of derived Artin n-stacks (see Toën and Vezzosi in Mem. Am. Math. Soc. 193, 2008 and Toën in Proc. Symp. Pure Math. 80:435–487, 2009). We prove that clas- sifying stacks of reductive groups, as well as the derived stack of perfect complexes, carry canonical 2-symplectic structures. Our main existence theorem states that for any derived Artin stack F equipped with an n-symplectic structure, the derived mapping stack Map(X, F) is equipped with a canonical (n − d)-symplectic structure as soon a X satisfies a Calabi-Yau condition in dimension d. These two results imply the existence of many examples of derived moduli stacks equipped with n-symplectic structures, such as the derived moduli of perfect complexes on Calabi-Yau varieties, or the derived moduli stack of perfect complexes of local systems on a compact and oriented topological manifold. We explain how the known symplectic structures on smooth moduli spaces of simple objects (e.g. simple sheaves on Calabi-Yau surfaces, or simple representations of π1 of compact Riemann surfaces) can be recovered from our results, and that they extend canonically as 0-symplectic structures outside of the smooth locus of simple objects. We also deduce new existence statements, such as the existence of a natural (−1)-symplectic structure (whose formal counterpart has been previously constructed in (Costello, arXiv:1111.4234, 2001) and (Costello and Gwilliam, 2011) on the derived mapping scheme Map(E, T∗X), for E an ellip- tic curve and T∗X is the total space of the cotangent bundle of a smooth scheme X. Canonical (−1)-symplectic structures are also shown to exist on Lagrangian intersections, on moduli of sheaves (or complexes of sheaves) on Calabi-Yau 3-folds, and on moduli of representations of π1 of compact topological 3-manifolds. More generally, the moduli sheaves on higher dimensional varieties are shown to carry canonical shifted symplectic structures (with a shift depending on the dimension).
CONTENTS
Introduction...... p-Forms, closed p-formsandsymplecticformsinthederivedsetting...... Existenceresults...... Examplesandapplications...... Futureworksandopenquestions...... Relatedworks...... Notationsandconventions...... 1.Definitionsandproperties...... 1.1.Gradedmixedcomplexes...... 1.2. p-Forms, closed p-forms and n-shiftedsymplecticstructures...... 2.Existenceofshiftedsymplecticstructures...... 2.1.Mappingstacks...... 2.2. Lagrangian intersections ...... 2.3. 2-Shifted symplectic structure on RPerf ...... 3.Examplesandapplications...... 3.1.0-Shiftedsymplecticstructuresonmoduliofsheavesoncurvesandsurfaces...... 3.2. (−1)-Shiftedsymplecticstructuresandsymmetricobstructiontheories...... Acknowledgements...... References......