2006/12/14 by Paul Bressler, Bressler, P., Mikhail Kapranov +5 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.math/0612410
If E is a C^∞ complex vector bundle on an oriented C^∞ manifold Σ, diffeomorphic to a circle, then the space of sections of E has a canonical polarization in the sense of Pressley and Segal and so one has its determinantal gerbe with lien C^*, the group of nonzero complex numbers. If q:Σ-->B is a smooth family of circles as above and E is a vector bundle on Σ, then the smooth direct image q_*(E) is an infinite-dimensional bundle with fibers as above and so we have its determinantal gerbe on B with lien being the sheaf of invertible complex valued C^∞ functions, it gives a class in H3(B, Z). In this paper we consider a family q:Σ-->B as above but with fibers being compact oriented C^∞ manifolds of dimension d. For a bundle E on Σone expects q_*(E) to possess a determinantal d-gerbe and hence to give a class in Hd+2(B, Z). We construct directly, by means of a version of the Chern-Weil theory, the real version of this would be class. We further prove a real version of the Grothendieck-Riemann-Roch theorem describing this class as a direct image of a certain characteristic class of E.