Classification of real moduli spaces over genus 2 curves

We classify, in terms of simple algebraic equations, the fixed point sets of the moduli space of stable bundles over genus 2 curves with anti-holomorphic involutions.Research supported by SRF of University of Missouri.     

On theta characteristics of real algebraic curves

Real theta characteristics of a real algebraic curve are studied. The numbers of even and odd real theta characteristics are calculated. These numbers depend on the topological characteristics of the curve only. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 403–106, September, 1998.     

On theta characteristics of real algebraic curves

Real theta characteristics of a real algebraic curve are studied. The numbers of even and odd real theta characteristics are calculated. These numbers depend on the topological characteristics of the curve only.     

Non-special divisors on real algebraic curves and embeddings into real projective spaces

We show that there is a large class of non-special divisors of relatively small degree on a given real algebraic curve. If the real algebraic curve has many real components, such a divisor gives rise to an embedding (birational embedding, resp.) of the real algebraic curve into the real projective space ℙ ^r for r≥3 (r=2, resp.). We study these embeddings in quite some detail. Received: October 17, 2001?Published online: February 20, 2003     

A formula for the Euler characteristic of a real algebraic manifold

There is given an algebraic formula which expresses the Euler characteristic of a real algebraic manifold in terms of the signature of an appropriate bilinear form. Supported by grant KBN 2/1125/91/01     

Clifford's inequality for real algebraic curves

We improve Clifford's Inequality for real algebraic curves. As an application we improve Harnack's Inequality for real space curves having a certain number of pseudo-lines. Another application involves the number of ovals that a real space curve can have.     

Euler characteristics of the real points of certain varieties of algebraic tori

Let G be a complex connected reductive group which is definedover , let be its Lie algebra, and let be the variety of maximaltori of G. For (), let be the variety of tori in whose Liealgebra is orthogonal to with respect to the Killing form.We show, using the Fourier–Sato transform of conical sheaveson real vector bundles, that the ‘weighted Euler characteristic’of () is zero unless is nilpotent, in which case it equals(–1)^{(dim )/2}. Here ‘weighted Euler characteristic’means the sum of the Euler characteristics of the connectedcomponents, each weighted by a sign ± 1 which dependson the real structure of the tori in the relevant component.This is a real analogue of a result over finite fields whichis connected with the Steinberg representation of a reductivegroup.     

On the Euler characteristic of Kronecker moduli spaces

Combining the MPS degeneration formula for the Poincaré polynomial of moduli spaces of stable quiver representations and localization theory, it turns that the determination of the Euler characteristic of these moduli spaces reduces to a combinatorial problem of counting certain trees. We use this fact in order to obtain an upper bound for the Euler characteristic in the case of the Kronecker quiver. We also derive a formula for the Euler characteristic of some of the moduli spaces appearing in the MPS degeneration formula.     

The geometric complex for algebraic curves with cone-like singularities and admissible Morse functions

In a previous Note the author gave a generalisation of Witten's proof of the Morse inequalities to the model of a singular complex algebraic curve X and a stratified Morse function f. In this Note a geometric interpretation of the complex of eigenforms of the Witten Laplacian corresponding to small eigenvalues is provided in terms of an appropriate subcomplex of the complex of unstable cells of critical points of f. To cite this article: U. Ludwig, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Topology of real algebraic curves

The problem on the existence of an additional first integral of the equations of geodesics on noncompact algebraic surfaces is considered. This problem was discussed as early as by Riemann and Darboux. We indicate coarse obstructions to integrability, which are related to the topology of the real algebraic curve obtained as the line of intersection of such a surface with a sphere of large radius. Some yet unsolved problems are discussed.