Let D be a simple normal crossing divisor in a smooth complex projective variety X. We show that the existence on X-D of a non-trivial polarized complex variation of Hodge structures with integral monodromy implies that the pair (X,D) has a non-zero logarithmic symmetric differential (a section of a symmetric power of the logarithmic cotangent bundle). When the corresponding period map is generically immersive, we show more precisely that the logarithmic cotangent bundle is big.

We prove a conjecture of Griffiths on the quasi-projectivity of images of period maps using algebraization results arising from o-minimal geometry. Specifically, we first develop a theory of analytic spaces and coherent sheaves that are definable with respect to a given o-minimal structure, and prove a GAGA-type theorem algebraizing definable coherent sheaves on complex algebraic spaces. We then combine this with algebraization theorems of Artin to show that proper definable images of complex algebraic spaces are algebraic. Applying this to period maps, we conclude that the images of period maps are quasi-projective and that the restriction of the Griffiths bundle is ample.

Hélène Esnault asked whether a smooth complex projective variety X with infinite fundamental group must have a nonzero symmetric differential, meaning that H 0 (X, S i Ω 1 X ) = 0 for some i > 0. This was prompted by the second author's work [23]. In fact, Severi had wondered in 1949 about possible relations between symmetric differentials and the fundamental group [34, p. 36]. We know from Hodge theory that the cotangent bundle Ω 1 X has a nonzero section if and only if the abelianization of π 1 X is infinite. The geometric meaning of other symmetric differentials is more mysterious, and it is intriguing that they may have such a direct relation to the fundamental group.In this paper we prove the following result on Esnault's question, in the slightly broader setting of compact Kähler manifolds.Theorem 0.1. Let X be a compact Kähler manifold. Suppose that there is a finitedimensional representation of π 1 X over some field with infinite image. Then X has a nonzero symmetric differential.All known varieties with infinite fundamental group have a finite-dimensional complex representation with infinite image, and so the theorem applies to them. Depending on what we know about the representation, the proof gives more precise lower bounds on the ring of symmetric differentials.Remark 0.2. (1) One reason to be interested in symmetric differentials is that they have implications toward Kobayashi hyperbolicity. At one extreme, if Ω 1 X is ample, then X is Kobayashi hyperbolic [25, Theorem 3.6.21]. (Equivalently, every holomorphic map C → X is constant.) If X is a surface of general type with c 2 1 > c 2 , then Bogomolov showed that Ω 1 X is big and deduced that X contains only finitely many rational or elliptic curves, something which remains open for arbitrary surfaces of general type [3,10]. For any α ∈ H 0 (X, S i Ω 1 X ) with i > 0, the restriction of α to any rational curve in X must be zero (because Ω 1 P 1 is a line bundle of negative degree), and so any symmetric differential gives a first-order algebraic differential equation satisfied by all rational curves in X.A lot is already known about Kobayashi hyperbolicity in the situation of Theorem 0.1. In particular, Yamanoi showed that for any smooth complex projective variety X such that π 1 X has a finite-dimensional complex representation whose image is not virtually abelian, the Zariski closure of any holomorphic map C → X is a proper subset of X [41].(2) Arapura used Simpson's theory of representations of the fundamental group to show that if π 1 X has a non-rigid complex representation, then X has a nonzero symmetric differential [1, Proposition 2.4], which we state as Theorem 4.1.Thus the difficulty for Theorem 0.1 is how to use a rigid representation of the fundamental group. The heart of the proof is a strengthening of Griffiths and 1 Zuo's results on variations of Hodge structure [16,44], from weak positivity of the cotangent bundle (analogous to "pseudoeffectivity" in the case of line bundles) to bigness. As a result, we get many symmetric differentials...

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