ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

5. Differential Geometry and Global Analysis

1991 MS Classification: 53, 14 Konnov, Valery V., Samara State Pedagogical University, Russia Differential geometry of algebraic manifolds The algebraic hypersurface X of degree p in n-dimensional projective space P^{n} is defined by the equation: X_{i_{1}i_{2}...i_{p}}x^{i_{1}}x^{i_{2}} … x^{i_{p}} = 0. The set M of all such surfaces is N-dimensional (N = C_{n+p}^{p} − 1) projective space (P^{N})^{∗}, on which the n-dimensional projective group works. If {U^{0}, U^{1}, …, U^{n}} is a frame in (P^{n})^{∗}, the forms U^{i_{1}i_{2}...i_{p}} = U^{(i_{1}} ⊗ U^{i_{2}} ⊗ … ⊗ U^{i_{p})} will form basis in M, X = X_{i_{1}i_{2}...i_{p}}U^{i_{1}i_{2}...i_{p}}, dU^{i} = −ω_{j}^{i}U^{j}, dω_{j}^{i} = ω_{j}^{k} ∧ ω_{k}^{i}, dU^{i_{1}i_{2}...i_{p}} =

−ω_{j_{1}j_{2}...j_{p}}^{i_{1}i_{2}...i_{p}}U^{j_{1}j_{2}...j_{p}}, ω_{j_{1}j_{2}...j_{p}}^{i_{1}i_{2}...i_{p}} = pω_{(j_{1}}^{(i_{1}}δ_{j_{2}}^{i_{2}} … δ_{j_{p})}^{i_{p})}. The differential geometry of some algebraic submanifolds in manifolds M is studied in the work. The elementary examples: On P^{1}: the variety of all quadrics is the hyperbolic plane; the variety of imaginary quadrics is the Lobachevskian plane; the variety of degenerating quadrics is the quadric-absolute. On P^{2}: the variety of degenerating quadrics is the cubic symmetroid in P^{5}. On P^{n}: the variety of algebraic hypersurfaces of degree p and rank 1 is the Veronese manifold V_{p}^{n} in P^{N}.