0 L 2 – Riemann – Roch Inequalities for Covering Manifolds

Symplectic Fibrations and Riemann-roch Numbers of Reduced Spaces

In this article we give formulas for the Riemann-Roch number of a symplectic quotient arising as the reduced space of a coadjoint orbit OΛ (for Λ ∈ g ∗ close to 0) as an evaluation of cohomology classes over the reduced space at 0. This formula exhibits the dependence of the Riemann-Roch number on Λ. We also express the formula as a sum over the components of the fixed point set of the maximal ...

Vector Bundles on Riemann Surfaces

1. Differentiable Manifolds 2 2. Complex Manifolds 3 2.1. Riemann Surfaces of Genus One 4 2.2. Constructing Riemann Surfaces as Curves in P 6 2.3. Constructing Riemann Surfaces as Covers 9 2.4. Constructing Riemann Surfaces by Glueing 10 3. Topological Vector Bundles 11 3.1. The Tangent and Cotangent Bundles 13 3.2. Interlude: Categories, Complexes and Exact Sequences 14 3.3. Metrics on Vector ...

Riemann-roch Theorems for Differentiable Manifolds

A Parametrized Index Theorem for the Algebraic

Riemann-Roch theorems assert that certain algebraically defined wrong way maps (transfers) in algebraic K–theory agree with topologically defined ones [BaDo]. Bismut and Lott [BiLo] proved such a Riemann–Roch theorem where the wrong way maps are induced by the projection of a smooth fiber bundle, and the topologically defined transfer map is the Becker–Gottlieb transfer. We generalize and refin...

On the Betti Numbers of Irreducible Compact Hyperkähler Manifolds of Complex Dimension Four

The study of higher dimensional hyperkähler manifolds has attracted much attention: we have [Wk], [Bg1,2,3,4], [Fj1,2], [Bv1], [Vb1,2], [Sl1,2], [HS], [Huy], [Gu3,4,5] etc. It is evident that there are only a few known examples of these manifolds and the obvious question is: can we classify them as in the case of complex dimension 4? The Riemann-Roch formula plays an important role in the surfa...