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Globalizing L-infinity automorphisms of the Schouten algebra of polyvector fields
Stockholm University, Faculty of Science, Department of Mathematics.
(English)In: Differential geometry and its applications (Print), ISSN 0926-2245, E-ISSN 1872-6984Article in journal (Refereed) Accepted
Abstract [en]

Recently, Willwacher showed that the Grothendieck-Teichmuller group GRT acts by L-infinity-automorphisms on the Schouten algebra of polyvector fields T_poly(R^d) on affine space R^d. In this article, we prove that a large class of L-infinity-automorphisms on the Schouten algebra, including Willwacher's, can be globalized. That is, given an L-infinity-automorphism of T_poly(R^d) and a general smooth manifold M with the choice of a torsion-free connection, we give an explicit construction of an L-infinity-automorphism of the Schouten algebra T_poly(M) on the manifold M, depending on the chosen connection. The method we use is the Fedosov trick.

National Category
Geometry
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-87356OAI: oai:DiVA.org:su-87356DiVA: diva2:602958
Available from: 2013-02-04 Created: 2013-02-04 Last updated: 2017-12-06Bibliographically approved
In thesis
1. Topics in Computational Algebraic Geometry and Deformation Quantization
Open this publication in new window or tab >>Topics in Computational Algebraic Geometry and Deformation Quantization
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of two parts, a first part on computations in algebraic geometry, and a second part on deformation quantization. More specifically, it is a collection of four papers. In the papers I, II and III, we present algorithms and an implementation for the computation of degrees of characteristic classes in algebraic geometry. Paper IV is a contribution to the field of deformation quantization and actions of the Grothendieck-Teichmüller group.

In Paper I, we present an algorithm for the computation of degrees of Segre classes of closed subschemes of complex projective space. The algorithm is based on the residual intersection theorem and can be implemented both symbolically and numerically.

In Paper II, we describe an algorithm for the computation of the degrees of Chern-Schwartz-MacPherson classes and the topological Euler characteristic of closed subschemes of complex projective space, provided an algorithm for the computation of degrees of Segre classes. We also explain in detail how the algorithm in Paper I can be implemented numerically. Together this yields a symbolical and a numerical version of the algorithm.

Paper III describes the Macaulay2 package CharacteristicClasses. It implements the algorithms from papers I and II, as well as an algorithm for the computation of degrees of Chern classes.

In Paper IV, we show that L-infinity-automorphisms of the Schouten algebra T_poly(R^d) of polyvector fields on affine space R^d which satisfy certain conditions can be globalized. This means that from a given L-infinity-automorphism of T_poly(R^d) an L-infinity-automorphism of T_poly(M) can be constructed, for a general smooth manifold M. It follows that Willwacher's action of the Grothendieck-Teichmüller group on T_poly(R^d) can be globalized, i.e., the Grothendieck-Teichmüller group acts on the Schouten algebra T_poly(M) of polyvector fields on a general manifold M.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2013. 20 p.
Keyword
Segre classes, Chern-Schwartz-MacPherson classes, topological Euler characteristic, computational algebraic geometry, numerical algebraic geometry, numerical homotopy methods, deformation quantization, polyvector fields, Fedosov quantization, Grothendieck-Teichmüller group
National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-87399 (URN)978-91-7447-623-1 (ISBN)
Public defence
2013-03-11, room 14, house 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 2: Manuscript. Paper 3: Manuscript. Paper 4: Accepted.

Available from: 2013-02-14 Created: 2013-02-05 Last updated: 2013-02-11Bibliographically approved

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