This is partly a survey and partly a research article. Some known results and open problems about Kaehler groups (fundamental groups of compact Kaehler manifolds) are discussed. A new notion of Kaehler homomorphism is introduced. This is a homomorphism induced by a holomorphic map between these types of manifolds. Some obstructions for a homomorphism to be Kaehler are discussed. Among these is the main result on the vanishing of a certain cohomology class associated to such map. This is reduced to the decomposition theorem for perverse sheaves suitably extended to Kaehler orbifolds.
We describe the BNS invariant of Kaehler groups. As an application we prove that if the fundamental group of a Kaehler manifold is solvable, it is virtually nilpotent.
We define the C^*-action on moduli spaces of reductive representations of fundamental groups of quasi-compact Kaehler manifolds by solving Hermitian-Yang-Mills equation. As applications in algebraic geometry we show a non-abelian Hodge (p,q)-type theorem for families of quasi-projective manifolds. We also prove that any discrete sub group of a Lie group of rank >1 that is not of Hodge type can't be the fundamental group of a quasi-compact Kaehler manifold.
This is the topological part of two papers on the cohomology of Kaehler groups. In this paper we show that if a linear duality group of dimension larger than 6 is the fundamental group of a compact Kaehler manifold then its second or its fourth Betti number is non-zero. As a corollary a cocompact p-adic lattice of rank larger than 6 is never Kaehler.
This is the geometric part of two papers on the cohomology of Kaehler groups. Using non-Abelian Hodge theory we show that if a finitely presented group with an unbounded complex linear morphism is the fundamental group of a compact Kaehler manifold then its second or its fourth Betti number does not vanish. Combined with our first paper this shows that a cocompact lattice in a real simple Lie group G of sufficiently large real rank is Kaehler if and only if G is of Hermitian type (a conjecture of Carlson and Toledo).
We consider Fano manifolds M that admit a collection of finite automorphism groups G_1, ..., G_k, such that the quotients M/G_i are smooth Fano manifolds possessing a Kaehler-Einstein metric. Under some numerical and smoothness assumptions on the ramification divisors, we prove that M admits a Kaehler-Einstein metric too.
Given a compact Kaehler manifold X, it is shown that pairs of the form (E, D), where E is a trivial holomorphic vector bundle on X, and D is an integrable holomorphic connection on $E$, produce a neutral Tannakian category. The corresponding pro-algebraic affine group scheme is studied. In particular, it is shown that this pro-algebraic affine group scheme for a compact Riemann surface determines uniquely the isomorphism class of the Riemann surface.
Based on recent work of S. K. Donaldson and T. Mabuchi, we prove that any extremal Kaehler metric in the sense of E. Calabi, defined on the product of polarized compact complex projective manifolds is the product of extremal Kaehler metrics on each factor, provided that the integral Futaki invariants of the polarized manifold vanish or its automorphism group satisfies a constraint. This extends a result of S.-T. Yau about the splitting of a Kaehler-Einstein metric on the product of compact complex manifolds to the more general setting of extremal Kaehler metrics.
Given a compact Kaehler manifold, we consider the complement U of a divisor with normal crossings. We study the variety of unitary representations of the fundamental group of U with certain restrictions related to the divisor. We show that the possible singularities of this variety as well as of the corresponding moduli space of irreducible representations are quadratic. In the course of our proof we exhibit a differential graded Lie algebra of which reflects our deformation problem.
Generalizing the theorem of Green--Lazarsfeld and Gromov, we classify Kaehler groups of deficiency at least two. As a consequence we see that there are no Kaehler groups of even and strictly positive deficiency. With the same arguments we prove that Kaehler groups that are non-Abelian and are limit groups in the sense of Sela are surface groups.
We prove that if the fundamental group of an arbitrary three-manifold -- not necessarily closed, nor orientable -- is a Kaehler group, then it is either finite or the fundamental group of a closed orientable surface.
Let $ 1 \rightarrow N \rightarrow G \rightarrow Q \rightarrow 1$ 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. If G is Kaehler, we show that Q is either the 3-dimensional Heisenberg group or the fundamental group of the Cartesian product of a closed oriented surface of positive genus and the circle. As a corollary, we obtain a new proof of a theorem of Dimca and Suciu by taking N to be the trivial group, If G is the fundamental group of a compact complex surface, we show that Q must be the…
We prove two results relating 3-manifold groups to fundamental groups occurring in complex geometry. Let N be a compact, connected, orientable 3-manifold. If N has non-empty, toroidal boundary, and π_1(N) is a Kaehler group, then N is the product of a torus with an interval. On the other hand, if N has either empty or toroidal boundary, and π_1(N) is a quasi-projective group, then all the prime components of N are graph manifolds.
The aim of this paper is to study the pro-algebraic fundamental group of a compact Kaehler manifold. Following work by Simpson, the structure of this group's pro-reductive quotient is already well understood. We show that Hodge-theoretic methods can also be used to establish that the pro-unipotent radical is quadratically presented. This generalises both Deligne et al.'s result on the de Rham fundamental group, and Goldman and Millson's result on deforming representations of Kaehler groups, and can be regarded as a consequence of formality of the schematic homotopy type. New examples are given…
We survey some properties of the automorphism groups of compact Kaehler manifolds. In particular, we present recent results by Keum, Oguiso and Zhang on the structure of these groups from the Tits alternative point of view. Several other related results will be also discussed. This paper is written for the VIASM Annual Meeting 2012.
We give a simple proof of a result originally due to Dimca and Suciu: a group that is both Kaehler and the fundamental group of a closed three-manifold is finite. We also prove that a group that is both the fundamental group of a closed three-manifold and of a non-Kaehler compact complex surface is infinite cyclic or the direct product of an infinite cyclic group and a group of order two.
Given a Kaehler group $G$ and a primitive class $ϕ\in H^1(G;Z)$, we show that the rank gradient of $(G;ϕ)$ is zero if and only if Ker $ϕ$ is finitely generated. Using this approach, we give a quick proof of the fact (originally due to Napier and Ramachandran) that Kaehler groups are not properly ascending or descending HNN extensions. Further investigation of the properties of Bieri-Neumann-Strebel invariants of Kaehler groups allows us to show that a large class of groups of orientation-preserving PL homeomorphisms of an interval, which generalize Thompson's group $F$, are not Kaehler.
We prove that a one-relator group $G$ is Kähler if and only if either $G$ is finite cyclic or $G$ is isomorphic to the fundamental group of a compact orbifold Riemann surface of genus $g > 0$ with at most one cone point of order $n$: $$< a_1\, b_1\, \,...\, a_g\, b_g\, \mid\, (\prod_{i=1}^g [a_i\, b_i])^n>\, .$$
A group morphism is constructed, which can be realized as the induced morphism of fundamental groups from a holomorphic map between compact Kahler manifolds, but can not be realized by a holomorphic map between smooth projective varieties. And it is also proved that there exists no such example between abelian groups.
Let X be a compact Kaehler manifold of complex dimension n. Let G be a connected solvable subgroup of the automorphism group Aut(X), and let N(G) be the normal subgroup of G of elements of null entropy. One of the goals of this paper is to show that G/N(G) is a free abelian group of rank r(G) less than or equal to n-1 and that the rank estimate is optimal. This gives an alternative proof of the conjecture of Tits type. In addition, we show some non-obvious implications on the structure of solvable automorphism groups of compact Kaehler manifolds. Furthermore, we also show that if the rank r(G)…
We characterize those unipotent representations of the fundamental group $π_1(X,x)$ of a compact Kaehler manifold $X$, which correspond to a Higgs bundle whose underlying Higgs field is equal to zero. The characterization is parallel to the one that R. Hain gave of those unipotent representations of $π_1(X,x)$ that can be realized as the monodromy of a flat connection on the holomorphically trivial vector bundle. We see that Hain's result and ours follow from a careful study of Simpson's correspondence between Higgs bundles and local systems.
In this note we prove that the kernel of the push-forward homomorphism on $d$-cycles modulo rational equivalence, induced by the closed embedding of an ample divisor linearly equivalent to some multiple of the theta divisor inside the Jacobian variety $J(C)$ is trivial. Here $C$ is a smooth projective curve of genus $g$.
In this paper we study abstract group homomorphisms between the groups of rational points of linear algebraic groups which are not necessarily reductive. One of our main goal is to obtain results on homomorphisms from the groups of rational points of linear algebraic groups defined over certain specific fields to the groups of rational points of linear algebraic groups over number fields and non-archimedean local fields of characteristic zero; in this set-up we deal with the unexplored topic of abstract homomorphisms from the groups of rational points of anisotropic groups over non-archimedean…
Let $X$ be a complex toric Fano $n$-fold and ${\cal N}(T)$ the normalizer of a maximal torus $T$ in the group of biholomorphic authomorphisms $Aut(X)$. We call $X$ {\em symmetric} if the trivial character is a single ${\cal N}(T)$-invariant algebraic character of $T$. Using an invariant $α_G(X)$ introduced by Tian, we show that all symmetric toric Fano $n$-folds admit an Einstein-Kähler metric. We remark that so far one doesn't know any example of a toric Fano $n$-fold $X$ such that $Aut(X)$ is reductive, the Futaki character of $X$ vanishes, but $X$ is not symmetric.
This paper is an overview of my recent work on abstract homomorphisms of algebraic groups. It is based on a talk given at the Conference on Group Actions and Applications in Geometry, Topology, and Analysis held in Kunming in July 2012.
In this text we prove that if an abelian variety $A$ admits of an embedding into the Jacobian of a smooth projective curve $C$, and if we consider $\Th_A$ to be the divisor $\Th_C\cap A$, where $\Th_C$ denotes the theta divisor of $J(C)$, then the embedding of $\Th_A$ into $A$ induces an injective push-forward homomorphism at the level of Chow groups. We show that this is the case for every principally polarized abelian varieties.
We give a geometric setup in which the connecting homomorphism in the localization long exact sequence for Witt groups decomposes as the pull-back to the exceptional fiber of a suitable blow-up followed by a push-forward.
The aim of this paper is to give verifiable criteria for the existence of {\em irreducible} homomorphisms of $π_{1}(\mathbb P^1 - \mathcal R)$ into compact semisimple groups, for a finite subset $\mathcal R$ such that the conjugacy classes of the images of lassos around the marked points are fixed. By a theorem in \cite{bs}, this question reduces into one of giving verifiable criteria for the existence of stable $\mathcal G$-torsors on $\mathbb{P}^1_{\mathbb{C}}$, where $\mathcal{G} \to \mathbb{P}^1_\mathbb{C}$ is a a Bruhat-Tits group scheme.
Geometric properties of homomorphisms between the absolute Galois groups of mixed-characteristic complete discrete valuation fields with perfect residue fields
