Complex Nilmanifolds Research Articles

A Decomposition Theorem for Complex Nilmanifolds

AbstractA complex nilmanifold X is isomorphic to a product X ⋍ ℂp x N/┌, where N is a simply connected nilpotent complex Lie group and ┌ is a discrete subgroup of N not contained in a proper connected complex subgroup of N. The pair (N, ┌) is uniquely determined up to holomorphic group isomorphisms.

Action d'une forme réelle d'un groupe de Lie complexe sur les fonctions plurisousharmoniques

Let G C be a complex Lie group and G R a closed real form of G C . By definition, a pair (G C ,G R ) is pseudo-convex, if there exists on G C a regular function, strictly p.s.h., invariant by G R , and exhaustive on G C /G R . By definition, G R has a purely imaginary specter, if for all X in (G R ), the eigenvalues of adX are purely imaginary. When G C has a simply connected radical, this last property is the same as pseudo-convexity of (G C ,G R ). For (G C ,G R ) pseudo-convex and under a discrete subgroup hypothesis, there exists on an invariant open subset Ω, a strictly p.s.h. invariant function, exhaustive on Ω/G R . With the same hypothesis, we have the following theorem: “Let be Ω a G R -invariant open subset of de X×G C , with connected fibers upon X. His protection on X is Stein, when X is Stein”. We prove also the non existence of an invariant kählerian metric on G C , when the specter of G R is not purely imaginary. We deduce the non existence of a kählerian metric on some non compact complex nilmanifolds.

On the Monodromy of Higher Logarithms

The (multivalued) higher logarithms are interpreted, by studying their monodromy, as giving well-defined maps from ${\mathbf {P}}_{\mathbf {C}}^1 - \left \{ {3\;{\text {points}}} \right \}$ into certain complex nilmanifolds with ${{\mathbf {C}}^ * }$-actions.

On the monodromy of higher logarithms

The (multivalued) higher logarithms are interpreted, by studying their monodromy, as giving well-defined maps from P C 1 − { 3 points } {\mathbf {P}}_{\mathbf {C}}^1 - \left \{ {3\;{\text {points}}} \right \} into certain complex nilmanifolds with C ∗ {{\mathbf {C}}^ * } -actions.