Shifted symplectic structures
Gabriele Vezzosi2013, Publications mathématiques de l'IHÉS
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-symplectic structures, a generalization of the notion of symplectic structures on smooth varieties and schemes, meaningful in the setting of derived Artin n-stacks (see [HAG-II, 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 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 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 the cotangent space of a smooth scheme X. Canonical (−1)-symplectic structures are also shown to exist on Lagrangian intersections, moduli of sheaves (or complexes of sheaves) on Calabi-Yau 3-folds, and moduli of representations of π 1 of compact topological 3-manifolds. More generally, moduli sheaves on higher dimensional varieties are shown to carry canonical n-symplectic structures (with n depending on the dimension). * Partially supported by NSF RTG grant DMS-0636606 and NSF grants DMS-0700446 and DMS-1001693. † Partially supported by the ANR grant ANR-09-BLAN-0151 (HODAG). 3. Let M be a compact oriented topological manifold of dimension d. Then, the (derived) moduli stack of perfect complexes of local systems on X admits a canonical (2 − d)shifted symplectic structure. Future parts of this work will be concerned with the dual notion of Poisson (and n-Poisson) structures in derived algebraic geometry, formality (and n-formality) theorems, and finally with quantization. p-Forms, closed p-forms and symplectic forms in the derived setting A symplectic form on a smooth scheme X (over some base ring k, of characteristic zero), is the datum of a closed 2-form ω ∈ H 0 (X, Ω 2,cl X/k), which is moreover required to be non-degenerate, i.e. it induces an isomorphism Θ ω : T X/k ≃ Ω 1 X/k between the tangent and cotangent bundles. In our context X will no longer be a scheme, but rather a derived Artin stack in the sense of [HAG-II, To2], the typical example being an X that is the solution to some derived moduli problem (e.g. of sheaves, or complexes of sheaves on smooth and proper schemes, see [To-Va, Corollary 3.31], or of maps between proper schemes as in [HAG-II, Corollary 2.2.6.14]). In this context, differential 1-forms are naturally sections in a quasi-coherent complex L X/k , called the cotangent complex (see [Il, To2]), and the quasi-coherent complex of p-forms is defined to be ∧ p L X/k. The p-forms on X are then naturally defined as sections of ∧ p L X/k , i.e. the set of p-forms on X is defined to be the (hyper)cohomology group H 0 (X, ∧ p L X/k). More generally, elements in H n (X, ∧ p L X/k) are called p-forms of degree n on X (see Definition 1.11 and Proposition 1.13). The first main difficulty is to define the notion of closed p-forms and of closed p-forms of degree n in a meaningful manner. The key idea of this work is to interpret p-forms, i.e. sections of ∧ p L X/k , as functions on the derived loop stack LX of [To2, To-Ve-1, Ben-Nad] by means of the HKR theorem of [To-Ve-2] (see also [Ben-Nad]), and to interpret closedness as the condition of being S 1-equivariant. One important aspect here is that S 1-equivariance must be understood in the sense of homotopy theory, and therefore closedness defined as above is not simply a property of p-form but consists of an extra structure (see Definition 1.9). This picture is accurate (see Remark 1.8 and 1.15), but technically difficult to work with 1. We have therefore chosen a different presentation, by introducing local constructions for affine derived schemes, that are then glued over X to obtain global definitions for any derived Artin stack X. To each commutative dg-algebra A over k, we define a graded complex, called the weighted negative cyclic complex of A over k, explicitly constructed using the derived de Rham complex of A. Elements of weight p and of degree n − p of this complex are by definition closed p-forms of degree n on Spec A (Definition 1.7). For a general derived Artin stack X, closed p-forms are defined by smooth descent techniques (Definition 1.11). This definition of closed p-forms has a more explicit local nature, but can be shown to coincide with the original idea of S 1-equivariant functions on the loop stack LX (using, for instance, results from [To-Ve-2, Ben-Nad]). By definition a closed p-form ω of degree n on X has an underlying p-form of degree n (as we already mentioned this underlying p-form does not determine the closed p-form
