On fundamental groups of algebraic varieties and value distribution theory

Yamanoi, Katsutoshi

doi:10.5802/aif.2532

Cited by 12 publications

(12 citation statements)

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“…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].…”

Section: Symmetric Differentials and The Fundamental Groupmentioning

confidence: 99%

Symmetric differentials and the fundamental group

Brunebarbe¹,

Klingler²,

Totaro³

2013

Duke Math. J.

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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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Section: Symmetric Differentials and The Fundamental Groupmentioning

confidence: 99%

Symmetric differentials and the fundamental group

Brunebarbe¹,

Klingler²,

Totaro³

2013

Duke Math. J.

View full text Add to dashboard Cite

show abstract

“…In this section we show that the image of the fundamental group of a geometrically-special smooth projective variety over C along a linear representation is virtually abelian (Theorem 1.12). Our proof follows closely Yamanoi's strategy [Yam10], and we will indicate as carefully as possible how to adapt Yamanoi's line of reasoning to prove Theorem 1.12.…”

Section: Virtually Abelian Fundamental Groupsmentioning

confidence: 95%

“…Now, let 1 ≤ i < j ≤ ℓ be such that g * k ω ij = 0. Following Yamanoi (see [Yam10] p.557 for details), we consider the Albanese map b : X s → B with respect to ω ij . Also, we let S → b(X s ) be the normalization of b(X s ), so that b factors as…”

Section: Virtually Abelian Fundamental Groupsmentioning

confidence: 99%

“…Then, the image c(Ξ ij ) of Ξ ij is a point P in S. Let L be an ample line bundle on S. Then, Yamanoi constructs (see [Yam10] p.558 for details), for every positive integer n, a positive integer l n and a divisor D n on S such that the following three properties hold.…”

Section: Virtually Abelian Fundamental Groupsmentioning

confidence: 99%

“…Proposition 4.2 (The analogue of 3.1 in [Yam10]). Let X be a smooth projective connected variety over C, and let K be a finite extension of Q p .…”

Section: Virtually Abelian Fundamental Groupsmentioning

confidence: 99%

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