Vanishing Theorems and Complex Structures on Non-Classical Flag Domains

We prove that every nontrivial line bundle on a compact quotient of a non-classical flag domain has no nonzero global sections. The proof first establishes the Green--Griffiths--Kerr conjecture by showing that the curvature of every nontrivial locally homogeneous line bundle has a negative direction, and then extends this property to arbitrary line bundles by decomposing their curvature into a homogeneous part and a seminegative correction term. We also establish several equivalent geometric and root-theoretic characterizations of non-classical flag domains. As consequences, their compact quotients are not in Fujiki class $\mathcal C$, contain no nonzero effective divisors, admit no nonconstant meromorphic functions, and have algebraic dimension zero. When $D=G_\R/V$ is non-classical and $G_\R$ is of Hermitian type, we construct another natural $G_\R$-invariant complex structure on the underlying differentiable manifold of $D$. The resulting classical flag domain has projective compact quotients. Thus the same differentiable manifold admits two invariant complex structures with opposite algebro-geometric behavior: one gives a projective manifold, whereas the other gives a non-classical quotient with the vanishing and non-algebraicity properties above.