Representations of fundamental groups of algebraic manifolds and their restrictions to fibers of a fibration

We consider a surjective morphism f : X → Y from a smooth projective variety X onto a smooth projective variety Y with connected fibers, henceforth called a fibration for short. Typically, if a certain geometric object on X like a cohomology class has a certain property, then its restriction to a smooth fiber of f trivially satisfies the same property. The converse question is of more interest: if the restriction of such a geometric object to a smooth fiber enjoys a certain property, is this property also valid for the object on X itself? A prototype is the geometric version of the (p, q) component theorem of Griffiths saying that if a class H(X,C) is of pure Hodge type (p, q) at some smooth fiber f(y), then it has Hodge type (p, q) at any smooth fiber. In the so-called nonabelian cohomology, instead of classes in H(X,C), one considers representations ρ : π1(x) → G into some linear algebraic group G. In the same way as one associates a harmonic form to a cohomology class, one finds a ρ-equivariant harmonic map h : X → G K into the symmetric space of noncompact type obtained as a homogeneous space for G. This harmonic map turns out to be pluriharmonic, meaning that its restriction to any subvariety is harmonic itself. We may thus reformulate the question indicated in the title of our paper, namely what one can infer about a representation of π1(X) if one knows a relevant property of the induced representation on π1(f (y)), or, more generally, of the one on π1(Z), Z a generic subvariety of X, as the question of what we can deduce about a pluriharmonic map from its restriction to f(y) or Z. Let us start with some easy observations in this direction before formulating our actual results. A harmonic map into G K is constant on any rational variety. On