IntroductionReminders on abstract algebraic geometry The setting Linear and commutative algebra in a symmetric monoidal model category Geometric stacks Infinitesimal theory Higher Artin stacks (after C. Simpson) Derived algebraic geometry: D − -stacks Complicial algebraic geometry: D-stacks Brave new algebraic geometry: S-stacks Relations with other works Acknowledgments Notations and conventions Part 1. General theory of geometric stacks vii viii CONTENTS 1.3.6. Properties of morphisms 1.3.7. Quasi-coherent modules, perfect modules and vector bundles Chapter 1.4. Geometric stacks: Infinitesimal theory 1.4.1. Tangent stacks and cotangent complexes 1.4.2. Obstruction theory 1.4.3. Artin conditions Part 2. Applications Introduction to Part 2 Chapter 2.1. Geometric n-stacks in algebraic geometry (after C. Simpson) 2.1.1. The general theory 2.1.

This is the first of a series of papers about quantization in the context of derived algebraic geometry. In this first part, 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 To2]). We prove that classifying 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 CalabiYau 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 π 1 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 [Co, Co-Gw]) on the derived mapping scheme Map(E, T * X), for E an elliptic 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 π 1 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).

The main purpose of this work is to study the homotopy theory of dg-categories up to quasi-equivalences. Our main result is a description of the mapping spaces between two dg-categories C and D in terms of the nerve of a certain category of (C, D)-bimodules. We also prove that the homotopy category Ho(dg − Cat) possesses internal Hom's relative to the (derived) tensor product of dg-categories. We use these two results in order to prove a derived version of Morita theory, describing the morphisms between dg-categories of modules over two dg-categories C and D as the dg-category of (C, D)-bi-modules. Finally, we give three applications of our results. The first one expresses Hochschild cohomology as endomorphisms of the identity functor, as well as higher homotopy groups of the classifying space of dgcategories (i.e. the nerve of the category of dg-categories and quasi-equivalences between them). The second application is the existence of a good theory of localization for dgcategories, defined in terms of a natural universal property. Our last application states that the dg-category of (continuous) morphisms between the dg-categories of quasi-coherent (resp. perfect) complexes on two schemes (resp. smooth and proper schemes) is quasi-equivalent to the dg-category of quasi-coherent (resp. perfect) complexes on their product.