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Rajan Amit Mehta - One of the best experts on this subject based on the ideXlab platform.

  • lie algebroid structures on double vector bundles and representation theory of lie algebroids
    Advances in Mathematics, 2010
    Co-Authors: Alfonso Graciasaz, Rajan Amit Mehta
    Abstract:

    Abstract A VB -algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB -algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct Characteristic Classes, which in special cases reproduce Characteristic Classes constructed by Crainic and Fernandes. We give a complete Classification of regular VB -algebroids, and in the process we obtain another Characteristic Class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids.

Alfonso Graciasaz - One of the best experts on this subject based on the ideXlab platform.

  • lie algebroid structures on double vector bundles and representation theory of lie algebroids
    Advances in Mathematics, 2010
    Co-Authors: Alfonso Graciasaz, Rajan Amit Mehta
    Abstract:

    Abstract A VB -algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB -algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct Characteristic Classes, which in special cases reproduce Characteristic Classes constructed by Crainic and Fernandes. We give a complete Classification of regular VB -algebroids, and in the process we obtain another Characteristic Class of Lie algebroids that does not appear in the ordinary representation theory of Lie algebroids.

Takeshi Saito - One of the best experts on this subject based on the ideXlab platform.

  • Characteristic cycle and the euler number of a constructible sheaf on a surface
    Journal of Mathematical Sciences-the University of Tokyo, 2015
    Co-Authors: Takeshi Saito
    Abstract:

    We define the Characteristic cycle of a constructible sheaf on a smooth surface in the cotangent bundle. We prove that the intersection number with the 0-section equals the Euler number and that the total dimension of vanishing cycles at an isolated character- istic point is also computed as an intersection number. For a constructible sheaf on a smooth algebraic variety in positive char- acteristic, an analogy between the wild ramification of ansheaf and the irregularity of a D-module in Characteristic 0 suggests that the charac- teristic cycle is defined as a cycle of the cotangent bundle. Its intersection product with the 0-section is expected to give the Characteristic Class (4) and the Euler number for a proper variety consequently. At an isolated Characteristic point (see the last paragraph of Section 1 for the definition) of a fibration to a curve, the intersection number with the section defined by a non-vanishing differential form of the curve is expected to be equal to

  • The Characteristic Class and ramification of an l-adic etale sheaf
    Inventiones Mathematicae, 2007
    Co-Authors: Ahmed Abbes, Takeshi Saito
    Abstract:

    We introduce the Characteristic Class of an l-adic etale sheaf using a cohomological pairing due to Verdier (SGA5). As a consequence of the Lefschetz-Verdier trace formula, its trace computes the Euler-Poincare Characteristic of the sheaf. We compare the Characteristic Class to two other invariants arising from ramification theory. One is the Swan Class of Kato-Saito (math.AG/0402010) and the other is the 0-cycle Class defined by Kato for rank 1 sheaves.

Nikolai Neumaier - One of the best experts on this subject based on the ideXlab platform.

  • local ν euler derivations and deligne s Characteristic Class of fedosov star products and star products of special type
    Communications in Mathematical Physics, 2002
    Co-Authors: Nikolai Neumaier
    Abstract:

    In this paper we explicitly construct local ν-Euler derivations $\mathsf E_\alpha = \nu \partial_\nu + \Lie{\xi_\alpha} + \mathsf D_\alpha$ , where the ξα are local, conformally symplectic vector fields and the $\mathsf D_\alpha$ are formal series of locally defined differential operators, for Fedosov star products on a symplectic manifold (M,ω) by means of which we are able to compute Deligne's Characteristic Class of these star products. We show that this Class is given by $\frac{1}{\nu}[\omega]+\frac{1}{\nu} [\Omega]$ , where $\Omega \in \nu Z^2_{{\rm dR}}(M)[[\nu]]$ is a formal series of closed two-forms on M the cohomology Class of which coincides with the one introduced by Fedosov to Classify his star products. Moreover, we consider star products that have additional algebraic structures and compute the effect of these structures on the corresponding Characteristic Classes of these star products. Specifying the constituents of Fedosov's construction we obtain star products with these special properties. Finally, we investigate equivalence transformations between such special star products and prove existence of equivalence transformations being compatible with the considered algebraic structures. Dedicated to the memory of Moshe Flato

Matmat Chahrazade - One of the best experts on this subject based on the ideXlab platform.

  • The Borsuk-Ulam theorem for 3-manifolds
    2021
    Co-Authors: Matmat Chahrazade, Blanchet Christian
    Abstract:

    We study the Borsuk-Ulam theorem for triple (M;\tau; \R^n), where M is a compact, connected, 3-manifold equipped with a fixed-point-free involution \tau. The largest value of n for which the Borsuk-Ulam theorem holds is called the Z_2-index and in our case it takes value 1, 2 or 3. We fully discuss this index according to cohomological operations applied on the Characteristic Class x \in H^1(N; Z_2), where N = M/\tau is the orbit space. In oriented case, we obtain an expression of the index from the linking matrix of a surgery presentation of the orbit space. We illustrate our results with examples, including a non orientable one

  • The Borsuk-Ulam theorem for 3-manifolds
    2020
    Co-Authors: Blanchet Christian, Matmat Chahrazade
    Abstract:

    We study the Borsuk-Ulam theorem for triple $(M, \tau,\R^n)$, where $M$ is a compact, connected, 3-manifold equipped with a fixed-point-free involution $\tau$. We investigate the largest value of $n$ for which the Borsuk-Ulam theorem holds. This number is called the $\Z_2$-index and in our case it takes value $1$, $2$ or $3$. The main ingredients are the Bockstein homomorphism and the triple cup product applied to the Characteristic Class $x\in H^1(N, \Z_2)$, where $N=M/\tau$ is the orbit space. Explicit computations are done, recovering the Classical result for the $3$-sphere $S^3$, the Stolz's theorem for the projective space, and fully discussing cases where the orbit space is a lens space or surgery on an algebraically split link. Finally we apply our results to a non orientable example