Representations of elementary abelian p-groups and bundles on Grassmannians

We initiate the study of representations of elementary abelian p-groups via restrictions to truncated polynomial subalgebras of the group algebra generated by r nilpotent elements, $k[t_1,\dots ,t_r]/(t_1^p,\dots ,t_r^p)$. We introduce new geometric invariants based on the behavior of modules upon restrictions to such subalgebras. We also introduce modules of constant radical and socle type generalizing modules of constant Jordan type and provide several general constructions of modules with these properties. We show that modules of constant radical and socle type lead to families of algebraic vector bundles on Grassmannians and illustrate our theory with numerous examples.     

Vector bundles over Grassmannians and the skew-symmetric curvature operator

A pseudo-Riemannian manifold is said to be spacelike Jordan IP if the Jordan normal form of the skew-symmetric curvature operator depends upon the point of the manifold, but not upon the particular spacelike 2-plane in the tangent bundle at that point. We use methods of algebraic topology to classify connected spacelike Jordan IP pseudo-Riemannian manifolds of signature (p,q), where q?11, and where the set {q,…,q+p} does not contain a power of 2.     

Cohomological characterization of vector bundles on Grassmannians of lines

We introduce a notion of regularity for coherent sheaves on Grassmannians of lines. We use this notion to prove some extension of Evans–Griffith criterion to characterize direct sums of line bundles. We also give, in the line of previous results by Costa and Miró-Roig, a cohomological characterization of exterior and symmetric powers of the universal bundles of the Grassmannian.     

Principal bundles and topological quantization of charges

The space of admissible particle velocities is assumed to be a four-dimensional nonholonomic distribution on a principal or associated bundle. Equations for the horizontal geodesics of this distribution coincide with the equations of motion of charged particles in general relativity theory. It is proved that, if the Lie group of the standard model of elementary particle physics is augmented by the 4-torus, then the wave functions are eigenfunctions of charge operators and the horizontal lift does not depend on the coupling constants. These wave functions satisfy the well-known Dirac equation and its generalizations. For such wave functions, the topological quantization of electric, lepton, and baryon charges takes place.     

Blocking sets in line Grassmannians

In this paper the most natural questions concerning the blocking sets in the line Grassmannian of PG(n,q) are partially answered. In particular, the following Bose-Burton type theorems are proved: if n is odd or n=4, then the blocking sets of minimum size are precisely the linear complexes with singular subspace of minimum dimension.     

Cohomology groups of deformations of line bundles on complex tori

The cohomology groups of line bundles over complex tori (or abelian varieties) are classically studied invariants of these spaces. In this article, we compute the cohomology groups of line bundles over various holomorphic, non-commutative deformations of complex tori. Our analysis interpolates between two extreme cases. The first case is a calculation of the space of (cohomological) theta functions for line bundles over constant, commutative deformations. The second case is a calculation of the cohomologies of non-commutative deformations of degree-zero line bundles.     

Some extensions of horrocks criterion to vector bundles on Grassmannians and quadrics

Summary In this paper we prove that a vector bundle E on a grassmannian (resp. on a quadric) splits as a direct sum of line bundles if and only if certain cohomology groups involving E and the quotient bundle (resp. the spinor bundle) are zero. When rank E=2 a better criterion is obtained considering only finitely many suitably chosen cohomology groups.This paper has been written while the author was enrolled in the Research Doctorate of the University of Florence. Partially supported by MPI 40% funds.     

Vector bundles over Grassmannians and the spectral geometry of the Riemann tensor

Methods from algebraic topology are often used to relate the algebraic properties of the Riemann curvature tensor to the geometry and topology of the underlying manifold. This paper provides a study of vector bundles over Grassmannians suitable for analyzing the spectral geometry of the Riemann tensor. Primarily, we study bundles over Grk(m), k?3, which are sub-bundles of the trivial bundle of rank m.     

Coisotropic and polar actions on complex Grassmannians

The main result of the paper is the complete classification of the compact connected Lie groups acting coisotropically on complex Grassmannians. This is used to determine the polar actions on the same manifolds.

     

Vanishing and nonvanishing theorems for numerically effective line bundles on complex spaces

Summary We prove some vanishing and nonvanishing theorems for numerically effective line bundles, generalizing to complex spaces and morphisms between complex spaces results of Kawamata, Viehweg and Shokurov.A Guido Zappa per il suo settantesimo compleannoSupported by SFB 170 «Geometrie und Analysis», Göttingen (FRG), and by Ministero Pubblica Istruzione.     

Vector bundles as direct images of line bundles

LetX be a smooth irreducible projective variety over an algebraically closed fieldK andE a vector bundle onX. We prove that, if dimX ≥ 1, there exist a smooth irreducible projective varietyZ overK, a surjective separable morphismf:Z →X which is finite outside an algebraic subset of codimension ≥ 3 inX and a line bundleL onX such that the direct image ofL byf is isomorphic toE. WhenX is a curve, we show thatZ, f, L can be so chosen thatf is finite and the canonical mapH ¹(Z, O) →H ¹(X, EndE) is surjective. Dedicated to the memory of Professor K G Ramanathan     

Semi-parallel real hypersurfaces in complex two-plane Grassmannians

We prove that there does not exist any semi-parallel real hypersurface in complex two-plane Grassmannians. With this result, the nonexistence of recurrent real hypersurfaces in complex two-plane Grassmannians can also be proved.     

Real minimal hypersurfaces in complex two-plane Grassmannians

We give a pinching condition for compact minimal hypersurfaces in complex two-plane Grassmannians G ₂(? ^m+2) in terms of sectional curvature and the squared norm of the shape operator.     

Recurrent real hypersurfaces in complex two-plane Grassmannians

Summary We introduce the notion of recurrent shape operator for a real hypersurface M in the complex two-plane Grassmannians G₂(C^m+2) and give a non-existence property of real hypersurfaces in G₂(C^m+2) with the recurrent shape operator.     

Pseudo-holomorphic curves of constant curvature in complex Grassmannians

In this paper we consider pseudo-holomorphic curves in complex Grassmiannians. Let φ ₀, φ ₁, ?, $\varphi _{\alpha _0 } $ : S ² → G _k,n be a linearly full non-degenerate pseudo-holomorphic harmonic sequence, and let degφ_α and K _α be the degree and the Gauss curvature of φ_α (α = 0, 1, ?, α ₀) respectively. Assume that φ ₀, φ ₁, ?, $\varphi _{\alpha _0 } $ is totally unramified. Then we prove that (i) degφ_α for all α = 0, 1, ?, α ₀; (ii) $K_\alpha = \tfrac{4}{{k(\alpha _0 + 2\alpha (\alpha _0 - \alpha ))}}$ if K _α is constant for some α = 0, 1, ?, α ₀,. We also give some conditions for pseudo-holomorphic curves with constant Kähler angle in complex Grassmiannians to be of constant curvature.