Cohomological restrictions on Kahler groups

Journal de Mathématiques Pures et Appliquées, 2019

The generalized Franchetta conjecture for hyper-Kähler varieties predicts that an algebraic cycle on the universal family of certain polarized hyper-Kähler varieties is fiberwise rationally equivalent to zero if and only if it vanishes in cohomology fiberwise. We establish Franchetta-type results for certain low (Hilbert) powers of low degree K3 surfaces, for the Beauville-Donagi family of Fano varieties of lines on cubic fourfolds and its relative square, and for 0-cycles and codimension-2 cycles for the Lehn-Lehn-Sorger-van Straten family of hyper-Kähler eightfolds. We also draw many consequences in the direction of the Beauville-Voisin conjecture as well as Voisin's refinement involving coisotropic subvarieties. In the appendix, we establish a new relation among tautological cycles on the square of the Fano variety of lines of a smooth cubic fourfold and provide some applications. Contents Recently, Bergeron and Li [8, Theorem 8.1.1] have proven the cohomological version of the generalized Franchetta conjecture 1.3 for relative 0-cycles when the second Betti number is sufficiently large, which is an important support in favor of the conjecture, at least for 0-cycles. Let us also mention that Conjecture 1.3 is closely related to the so-called Beauville-Voisin conjecture and its refinement (see Conjectures 2.3 and 2.4). On the one hand, the proof of some of our main results actually uses some known cases of the Beauville-Voisin conjecture (especially [48]) ; on the other hand, the generalized Franchetta conjecture implies the part of the Beauville-Voisin conjecture involving only Chern classes and the polarization, see Proposition 2.5. We outline the main results of the paper, which provide more evidence for the generalized Franchetta conjecture. 1.1. Powers and Hilbert powers of some K3 surfaces. We can establish Franchetta-type results for the relative squares and cubes, as well as the relative Hilbert squares and Hilbert cubes, of the universal family of K3 surfaces which are complete intersections in projective spaces. Theorem 1.4. Let M be the moduli stack of smooth K3 surfaces of genus = 3, 4 or 5, and let S → M be the universal family. Let X be

Proceedings of the Japan Academy, Series A, Mathematical Sciences, 1980

Introduction. The structure of algebraic threefolds with nonpositive Kodaira dimension has been studied by Ueno [8], [9] and Viehweg [10]. Their results are based on the semi-positivity theorem of the direct image sheaf of the relative canonical sheaf of a fibre space ([4]). This is a consequence of the theory of variation of Hodge structure. Therefore it is easy to show that the similar results hold for compact Kihler manifolds of dimension three with _ 0. On the other hand, Atiyah [1] and Blanchard [2] showed that the semi-positivity theorem does not necessarily hold for non-Khler fibre spaces. Hence it is expected that the structure of non-Khler manifolds is different from that of K/ihler manifolds. The main purpose of the present note is to announce structure theorems of compact complex manifolds of dimension three with _0 which have non-trivial Albanese tori. Contrary to the case of analytic surfaces, we have i.nteresting new phenomena. 1. Preliminaries. In the present note, by an analytic threefold M we mean a compact complex manifold of dimension three. We use

Communications in Contemporary Mathematics, 2012

be an exact sequence of finitely presented groups where Q is infinite and not virtually cyclic, and is the fundamental group of some closed 3-manifold.

Mathematische Zeitschrift, 2004

Annales de l'Institut Fourier, 2018

Using the Minimal model program, any degeneration of Ktrivial varieties can be arranged to be in a Kulikov type form, i.e. with trivial relative canonical divisor and mild singularities. In the hyper-Kähler setting, we can then deduce a finiteness statement for the monodromy acting on H 2 , once one knows that one component of the central fiber is not uniruled. Independently of this, using deep results from the theory of hyper-Kähler manifolds, we prove that a finite monodromy projective degeneration of hyper-Kähler manifolds has a smooth filling (after base change and birational modifications). As a consequence of these two results, we prove a generalization of Huybrechts' theorem about birational versus deformation equivalence, allowing singular central fibers. As an application, we give simple proofs for the deformation type of certain explicit models of projective hyper-Kähler manifolds. In a slightly different direction, we establish some basic properties (dimension and rational homology type) for the dual complex of a Kulikov type degeneration of hyper-Kähler manifolds.

Annali di matematica pura ed applicata, 2024

A p-Kähler structure on a complex manifold of complex dimension n is given by a d-closed transverse real ( p, p)-form. In the paper, we study the existence of p-Kähler structures on compact quotients of simply connected Lie groups by discrete subgroups endowed with an invariant complex structure. In particular, we discuss the existence of p-Kähler structures on nilmanifolds, with a focus on the case p = 2 and complex dimension n = 4. Moreover, we prove that a (n -2)-Kähler almost abelian solvmanifold of complex dimension n ≥ 3 has to be Kähler. Keywords p-Kähler structure • Nilmanifold • Almost abelian solvmanifold B Anna Fino

We give a list of six dimensional flat Kähler manifolds. Moreover, we present an example of eight dimensional flat Kähler manifold M with finite Out(π1(M) group.

2009

We consider hyper-Kähler manifolds of complex dimension 4 which are fibrations. It is known that the fibers are abelian varieties and the base is P. We assume that the general fiber is isomorphic to a product of two elliptic curves. We prove that such a hyper-Kähler manifold is deformation equivalent to a Hilbert scheme of two points on a K3 surface. 1. Preliminaries First we define our main objects of study, irreducible symplectic manifolds or hyperKähler manifolds. Definition 1.1. A compact complex Kähler manifold X is called irreducible symplectic if it is simply connected and if H(X, ΩX) is spanned by an everywhere non-degenerate 2-form ω. Any holomorphic two-form σ induces a homomorphism TX → ΩX . The two-form is everywhere non-degenerate if and only if TX → ΩX is bijective. The last condition in the definition implies that h(X) = h(X) = 1 and KX ∼= OX , i.e., c1(X) = 0. Definition 1.2. A compact connected 4n-dimensional Riemannian manifold (M, g) is called irreducible hyper-Kä...

Advances in Mathematics, 2010

Simply connected compact Kähler manifolds of dimension up to three with elliptic homotopy type are characterized in terms of their Hodge diamonds. This is applied to classify the simply connected Kähler surfaces and Fano threefolds with elliptic homotopy type.

Journal de l'École polytechnique, 2021

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