An Analytic Approach to the Theorems of Riemann-roch and Abel

Riemann-roch and Abel-jacobi Theory on a Finite Graph

It is well-known that a finite graph can be viewed, in many respects, as a discrete analogue of a Riemann surface. In this paper, we pursue this analogy further in the context of linear equivalence of divisors. In particular, we formulate and prove a graph-theoretic analogue of the classical Riemann-Roch theorem. We also prove several results, analogous to classical facts about Riemann surfaces...

Riemann-roch for Algebraic Stacks:iii Virtual Structure Sheaves and Virtual Fundamental Classes

In this paper we apply the Riemann-Roch and Lefschetz-Riemann-Roch theorems proved in our earlier papers to define virtual fundamental classes for the moduli stacks of stable curves in great generality and establish various formulae for them.

Math8811: Complex Analysis

1. Classical Topics 2 1.1. Complex numbers 2 1.2. Differentiability 2 1.3. Cauchy-Riemann Equations 3 1.4. The Riemann Sphere 4 1.5. Möbius transformations 5 1.6. Integration 6 1.7. Cauchy’s Theorems and Applications 7 1.8. Properties of Analytic Functions 9 1.9. The Winding Number 11 1.10. Singularities 12 1.11. Laurent Series 13 1.12. Residue Theory 15 1.13. Normal Families 18 1.14. The Riema...

Riemann-roch Theorems for Differentiable Manifolds

Bredon-style homology, cohomology and Riemann–Roch for algebraic stacks

One of the main obstacles for proving Riemann–Roch for algebraic stacks is the lack of cohomology and homology theories that are closer to the K-theory and G-theory of algebraic stacks than the traditional cohomology and homology theories for algebraic stacks. In this paper we study in detail a family of cohomology and homology theories which we call Bredon-style theories that are of this type ...