Abstract:
Let Γ⊂SL(2,C) be a cocompact lattice and X=Γ∖SL(2,C) the associated compact complex parallelizable manifold. We show that any non-constant
holomorphic map f:M→X from a compact Riemann surface M into X decomposes as
f=t∘h∘α, where α:M→Alb(M) is the Albanese map,
h:Alb(M)→X=Γ∖SL(2,C) has its image in a maximal torus
T=Γ∩A∖A≅Z∖C∗ in X (A being a maximal torus in SL(2,C))
defining an algebraic group homomorphism
h:Alb(M)→T=(A∩Γ)∖A,
and finally t is a right translation by some element of SL(2,C).
The proof is based on Bishop's measure theoretic criterion of analyticity of sets combined with
a simple observation in hyperbolic geometry.
I will discuss some applications of this rigidity.