Minimal models of nilmanifolds

Abstract. We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical Kähler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of C n by a discrete group of complex isometries.The purpose of this article is to give an account on recent developments concerning the classification of compact aspherical Kähler manifolds whose fundamental groups contain a solvable subgroup of finite index. This problem stirred some interest among differential geometers for quite some time. It traces back to the classification of Kähler Lie groups and became popular through a conjecture of Benson and Gordon on Kähler quotients of solvable Lie groups. Only recently satisfying answers to the original Benson-Gordon conjecture became available when methods from complex geometry were introduced into the subject. The story as it presents itself now may be seen as an interesting show case where research questions from differential geometry, complex geometry and topology meet.Outline of the paper. The first section of the paper has a survey character. We introduce the Benson-Gordon conjecture and describe its relation to the classification of Kähler Lie groups, Kähler groups and Kähler manifolds. In the course we also sketch the proof of a theorem of Arapura and Nori on polycyclic Kähler groups. This leads us to the following rigidity result for Kähler infra-solvmanifolds, which, in particular, contains the solution to the conjecture of Benson and Gordon.Theorem A. Let M be an infra-solvmanifold which admits a Kähler metric. Then M is diffeomorphic to a flat Riemannian manifold.In section 2 we develop generalisations of the previously described results. These concern the classification of Kähler structures on general smooth aspherical manifolds with polycyclic fundamental group. One new key point is to establish that a

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“…Together with the results of Bieri [13], Benson-Gordon [11], Hasegawa [32], Arapura-Nori [4], Theorem 2 implies:…”

Section: Theorem 2 Let X Be a Compact Complex Manifold Which Is Kählsupporting

confidence: 55%

Aspherical Kähler manifolds with solvable fundamental group

Baues

Cortés

2006

Geom Dedicata

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“…Since n * is generated by elements of degree 1, n * is 1-minimal and hence it is also the 1-minimal model of A * (Γ\N ). It is proven in [11] that the DGA n * is formal if and only if the Lie algebra n is Abelian. Hence, A * (Γ\N ) is formal if and only if the nilmanifold Γ\N is a torus.…”

Section: Theorem 34 ([16]mentioning

confidence: 99%

Mixed Hodge structures and Sullivan’s minimal models of Sasakian manifolds

Kasuya¹

2017

Ann. inst. Fourier

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“…We shall repeatedly choose a basis satisfying both (13) and (14), and then exploit the freedom of choice. In particular, we are at liberty to apply a conformal transformation…”

Section: Choice Of Basesmentioning

confidence: 99%

“…In this case, the differential graded algebra A of left-invariant forms on G is isomorphic to the de Rham algebra of M , and the Betti numbers b i of M coincide with the dimensions of the Lie algebra cohomology groups of g [21]. Property (1) then implies that A provides a minimal model for M in the sense of Sullivan, and A cannot be formal unless G is abelian and M is a torus [9,3,13,5]. Our interest arises from the imposition of extra geometrical structures on M .…”

Section: Introductionmentioning

confidence: 99%