A characterization of graded symplectic structures (original) (raw)
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Given a symplectic form and a pseudo-Riemannian metric on a manifold, a nondegenerate even Poisson bracket on the algebra of differential forms is defined and its properties are studied. A comparison with the Koszul–Schouten bracket is established. Mathematics Subject Classifications (2000): Primary: 58A50, 58F05; Secondary: 58A10, 53C15.
Symplectic structures on the tangent bundle of a smooth manifold
We give a method to lift (2,0)(2,0)(2,0)-tensors fields on a manifold MMM to build symplectic forms on TMTMTM. Conversely, we show that any symplectic form Om\OmOm on TMTMTM is symplectomorphic, in a neighborhood of the zero section, to a symplectic form built naturally from three (2,0)(2,0)(2,0)-tensor fields associated to Om\OmOm.
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Commentarii Mathematici Helvetici, 1995
One-to-one correspondences are established between the set of all nondegenerate graded Jacobi operators of degree 1 defined on the graded algebra (M) of differential forms on a smooth, oriented, Riemannian manifold M, the space of bundle isomorphisms L: T M ! T M , and the space of nondegenerate derivations of degree 1 having null square. Derivations with this property, and Jacobi structures of odd Z2-degree are also studied through the action of the automorphism group of (M).
Publications mathématiques de l'IHÉS, 2013
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
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2002
In this note the long standing problem of the definition of a Poisson bracket in the framework of a multisymplectic formulation of classical field theory is solved. The new bracket operation can be applied to forms of arbitary degree. Relevant examples are discussed and important properties are stated with proofs sketched.
Generalized n-Poisson brackets on a symplectic manifold
1999
On a symplectic manifold a family of generalized Poisson brackets associated with powers of the symplectic form is studied. The extreme cases are related to the Hamiltonian and Liouville dynamics. It is shown that the Dirac brackets can be obtained in a similar way.
Tamed symplectic forms and generalized geometry
Journal of Geometry and Physics, 2013
We show that symplectic forms taming complex structures on compact manifolds are related to special types of almost generalized Kähler structures. By considering the commutator Q of the two associated almost complex structures J ± , we prove that if either the manifold is 4dimensional or the distribution Im Q is involutive, then the manifold can be expressed locally as a disjoint union of twisted Poisson leaves.
Shifted symplectic structures on derived Quot-stacks I: Differential graded manifolds
2019
A theory of dg schemes is developed so that it becomes a homotopy site, and the corresponding infinity category of stacks is equivalent to the infinity category of stacks, as constructed by Toën and Vezzosi, on the site of dg algebras whose cohomologies have finitely many generators in each degree. Stacks represented by dg schemes are shown to be derived schemes under this correspondence. MSC codes: 14A20, 14J35, 14J40, 14F05