Supersymmetry on noncompact manifolds and complex geometry

The special properties of the realizations of supersymmetry on noncompact manifolds are discussed. On the basis of the supersymmetric scattering theory and the supersymmetric trace formulas the absolute or relative Euuler characteristic of an obstacle in $R^N$ may be obtained from scattering data for the Laplace operator on forms with absolute or relative boundary conditions.
The analytic expression for the topological index of the map from the stationary curve of antihomorphici involution on compact Riemann surface to the real circle on the Riemann sphere generated by real meromorphic function is obtained from supersymmetric quantum mechanics with meromorphic superpotetial on Klein surface. Bibliography: 27 titles.