Let $X$ be a compact K\"ahler manifold. We are interested in the effect of the complexity of the fundamental group $\pi_1(X)=\Gamma$ on the complex structure of $X$. A guiding principle is that $X$ is conjecturally of positive algebraic dimension if $\Gamma$ is sufficiently large. More precisely, we expect that in this situation, replacing $X$ by some modification if necessary, $X$ can be holomorphically mapped onto a projective-algebraic manifold of the general type.
We propose a new method of constructing meromorphic fibrations of compact K\"ahler manifolds over projective-algebraic varieties of the general type by introducing the notion of semi-K\"ahler structures. Roughly speaking a semi-K\"ahler structure $(X,{\scr F},\omega)$ is a meromorphic foliation together with a compatible transverse (singular) K\"ahler form $\omega$. Here a meromorphic foliation on $X$ means a holomorphic foliation ${\scr F}$ on $X-V$ for some complex-analytic subvariety $V\subset X$ of codimension $\geq 2$. Our approach consists of (1) constructing semi-K\"ahler structures on $X$ arising from linear representations of the fundamental group, (2) proving that all leaves $L$ of the holomorphic foliations on $X-V$ are closed and that the set-theoretical closures $L$ in $X$ are complex-analytic subvarieties (in which case we will say that $(X,{\scr F},\omega)$ is factorizable); and (3) proving that on a nonsingular model $Z$ of the leaf space of ${\scr F}$ the canonical line bundle $K$ is equipped with a singular Hermitian metric of nonnegative and generically positive curvature. The upshot is that we can then use the K\"ahler case of the Grauert-Riemennschneider conjecture to prove that $Z$ is of the general type.
From finite-dimensional reductive linear representations of the fundamental group we can construct semi-K\"ahler structures by using equivariant harmonic maps into Riemannian symmetric spaces of nonpositive sectional curvature. The semi-K\"ahler structures, with semi-K\"ahler forms $\omega$, are constructed using the Bochner-Kodaira formula of Siu [Si] and Sampson [Sa]. They are of nonpositive bisectional curvature in the sense that on a local holomorphic submanifold $S$ of $X$ transverse to ${\scr F}$ on which $\omega|S$ is a bona fide K\"ahler form, $(S,\omega|S)$ is of nonpositive bisectional curvature. In the discrete and semisimple case by studying these semi-K\"ahler structures and those arising from Ricci forms, we proved in Mok [M]
Theorem 1
Let $X$ be a compact K\"ahler manifold such that the fundamental group $\Gamma$ admits a noncompact semisimple discrete representation $\Phi$ in a real linear group. Then, replacing $X$ by some finite unramified covering if necessary, there exists a modification $\hat X$ of $X$, a porjective-algebraic manifold $Z$ of the general type, a surjective holomorphic map $\sigma\colon \hat X\to Z$ and a linear representation $\Xi$ on $\pi_1(Z)$ such that the representation $\Phi$ on $\pi_1(X)\cong \pi_1(\hat X)$ factorizes into $\Phi=\Xi\circ\sigma_*$.
Theorem 1 is obtained by proving that the semi-K\"ahler structures arising from the semi-K\"ahler form $\omega$ and from the Ricci form are factorizable. $Z$ is then obtained as a leaf space and is equipped with a singular K\"ahler metric which is of nonpositive bisectional curvature and which is of negative Ricci curvature almost everywhere. As a consequenc e of Theorem 1 by imposing homological conditions on $\Phi$ we obtained also the following result asserting the algebraicity of $X$.
Theorem 2
Let $\Omega$ be a bounded symmetric domain and $\Gamma\subset{\rm Aut}(\Omega)$ be a torsion-free discrete group of automorphisms. Let $X$ be a compact K\"ahler manifold admitting a continuous map $F\colon X\to \Omega/\Gamma$ such that the image of the fundamental class of $X$ is nontrivial. Then, $X$ is of the general type and hence projective-algebraic. In particular, compact K\"ahler manifolds homotopic to complex submanifolds of $\Omega/\Gamma$ are of the general type and projective-algebraic.
References
[M] Mok, N. ``Factorization of semisimple discrete representations of K\"ahler groups'', Invent. Math., {\bf 110} (1992), 557--615.
[Sa] Sampson, J. ``Applications of harmonic maps to K\"ahler geometry'', Contemp. Math. {\bf 49} (1986), 125--133.
[Si] Siu, Y.-T. ``The complex analyticity of harmonic maps and the strong rigidity of compact K\"ahler manifolds'', Ann. Math. {\bf 112} (1980), 73--111.