We show that all 6-dimensional nilmanifolds admit generalized complex structures. This includes the five classes of nilmanifold which admit no known complex or symplectic structure. Furthermore, we classify all 6-dimensional nilmanifolds according to which of the four types of left-invariant generalized complex structure they admit. We also show that the two components of the left-invariant complex moduli space for the Iwasawa manifold are no longer disjoint when they are viewed in the generalized complex moduli space. Finally, we provide an 8-dimensional nilmanifold which admits no left-invariant generalized complex structure.
We prove that for any $n\geq 4$ there are infinitely many real homotopy types of $2n$-dimensional nilmanifolds admitting generalized complex structures of every type $k$, for $0 \leq k \leq n$. This is in deep contrast to the $6$-dimensional case.
We study generalized complex cohomologies of generalized complex structures constructed from certain symplectic fibre bundles over complex manifolds. We apply our results in the case of left-invariant generalized complex structures on nilmanifolds and to their space of small deformations.
We prove that a compact nilmanifold admits a Sasakian structure if and only if it is a compact quotient of the generalized Heisenberg group of odd dimension by a co-compact discrete subgroup.
We study generalized complex structures and $T$-duality (in the sense of Bouwknegt, Evslin, Hannabuss and Mathai) on Lie algebras and construct the corresponding Cavalcanti and Gualtieri map. Such a construction is called "Infinitesimal $T$-duality". As an application we deal with the problem of finding symplectic structures on 2-step nilpotent Lie algebras. We also give a criteria for the intregability of the infinitesimal $T$-duality of Lie algebras to topological $T$-duality of the associated nilmanifolds.
Let $(X,J)$ be a nilmanifold with a left-invariant nilpotent complex structure. We study the existence of $p$-Kähler structures (which include Kähler and balanced metrics) on $X$. More precisely, we determine an optimal $p$ such that there are no $p$-Kähler structures on $X$. Finally, we show that, contrarily to the Kähler case, on compact complex manifolds there is no relation between the existence of balanced metrics and the degeneracy step of the Frölicher spectral sequence. More precisely, on balanced manifolds the degeneracy step can be arbitrarily large.
In this note, we analyze the question of when will a complex nilmanifold have Kähler-like Strominger (also known as Bismut), Chern, or Riemannian connection, in the sense that the curvature of the connection obeys all the symmetries of that of a Kähler metric. We give a classification in the first two cases and a partial description in the third case. It would be interesting to understand these questions for all Lie-Hermitian manifolds, namely, Lie groups equipped with a left invariant complex structure and a compatible left invariant metric.
We find a one-parameter family of non-isomorphic nilpotent Lie algebras $\mathfrak{g}_a$, with $a \in [0,\infty)$, of real dimension eight with (strongly non-nilpotent) complex structures. By restricting $a$ to take rational values, we arrive at the existence of infinitely many real homotopy types of $8$-dimensional nilmanifolds admitting a complex structure. Moreover, balanced Hermitian metrics and generalized Gauduchon metrics on such nilmanifolds are constructed.
We use Bott-Chern cohomology to measure the non-Kählerianity of 6-dimensional nilmanifolds endowed with the invariant complex structures in M. Ceballos, A. Otal, L. Ugarte, and R. Villacampa's classification, [Invariant Complex Structures on 6-Nilmanifolds: Classification, Frölicher Spectral Sequence and Special Hermitian Metrics, J. Geom. Anal. (2014)]. We investigate the existence of pluriclosed metric in connection with such a classification.
A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompact lattice. A hypercomplex structure on a manifold is a triple of complex structure operators satisfying the quaternionic relations. A hypercomplex nilmanifold is a compact quotient of a nilpotent Lie group equipped with a left-invariant hypercomplex structure. Such a manifold admits a whole 2-dimensional sphere $S^2$ of complex structures induced by quaternions. We prove that for any hypercomplex nilmanifold $M$ and a generic complex structure $L\in S^2$, the complex manifold $(M,L)$ has algebraic dimension 0. A stronger …
A well known conjecture in complex geometry states that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler if the constant is non-zero and must be Chern flat if the constant is zero. The conjecture is confirmed in complex dimension $2$, by the work of Balas-Gauduchon in 1985 (when the constant is zero or negative) and by Apostolov-Davidov-Muskarov in 1996 (when the constant is positive). For higher dimensions, the conjecture is still largely unknown. In this article, we restrict ourselves to the class of complex nilmanifolds and confirm the conjecture in …
On a complex manifold an Hermitian metric which is simultaneously SKT and balanced has to be necessarily Kähler. It has been conjectured that if a compact complex manifold (M,J) has an SKT metric and a balanced metric both compatible with J, then (M, J) is necessarily Kähler. We show that the conjecture is true for nilmanifolds.
It is conjectured that the Dolbeault cohomology of a complex nilmanifold $X$ is computed by left-invariant forms. We prove this under the assumption that $X$ is suitably foliated in toroidal groups and deduce that the conjecture holds in real dimension up to six. Our approach generalises previous methods, where the existence of a holomorphic fibration was a crucial ingredient.
The aim of this article is to study the existence of invariant SKT structures on nilmanifolds. More precisely, we give a negative answer to the question of whether there exist a $k$-step ($k>2$) complex nilmanifold admitting an invariant SKT metric. We also provide a construction which serves as a tool to generate examples of invariant SKT structures on $2$-step nilmanifolds in arbitrary dimensions.
We classify invariant complex structures on 6-dimensional nilmanifolds up to equivalence. As an application, the behaviour of the associated Frölicher sequence is studied as well as its relation to the existence of strongly Gauduchon metrics. We also show that the strongly Gauduchon property and the balanced property are not closed under holomorphic deformation.
A cohomology theory associated to a holomorphic Poisson structure is the hypercohomology of a bi-complex where one of the two operators is the classical $\overline\partial$-operator, while the other operator is the adjoint action of the Poisson bivector with respect to the Schouten-Nijenhuis bracket. The first page of the associated spectral sequence is the Dolbeault cohomology with coefficients in the sheaf of germs of holomorphic polyvector fields. In this note, the authors investigate the conditions for which this spectral sequence degenerates on the first page when the underlying complex m…
Let M be a complex nilmanifold, that is, a compact quotient of a nilpotent Lie group endowed with an invariant complex structure by a discrete lattice. A holomorphic differential on M is a closed, holomorphic 1-form. We show that $a(M)\leq k$, where $a(M)$ is the algebraic dimension $a(M)$ (i.e. the transcendence degree of the field of meromorphic functions) and $k$ is the dimension of the space of holomorphic differentials. We prove a similar result about meromorphic maps to Kahler manifolds.
The subject for investigation in this note is concerned with holomorphic Poisson structures on nilmanifolds with abelian complex structures. As a basic fact, we establish that on such manifolds, the Dolbeault cohomology with coefficients in holomorphic polyvector fields is isomorphic to the cohomology of invariant forms with coefficients in invariant polyvector fields. We then quickly identify the existence of invariant holomorphic Poisson structures. More important, the spectral sequence of the Poisson bi-complex associated to such holomorphic Poisson structure degenerates at $E_2$. We will a…
We study the interplay between the following types of special non-Kähler Hermitian metrics on compact complex manifolds: it locally conformally Kähler, $k$-Gauduchon, balanced and locally conformally balanced and prove that a locally conformally Kähler compact nilmanifold carrying a balanced or a $k$-Gauduchon metric is necessarily a torus. Combined with a result of Fino and Vezzoni from 2016, this leads to the fact that a compact complex 2-step nilmanifold endowed with whichever two of the following types of metrics: balanced, pluriclosed and locally conformally Kähler is a torus. Moreover, w…
It is proved that on nilmanifolds with abelian complex structure, there exists a canonically constructed non-trivial holomorphic Poisson structure. We identify the necessary and sufficient condition for its associated cohomology to be isomorphic to the cohomology associated to trivial (zero) holomorphic Poisson structure. We also identify a sufficient condition for this isomorphism to be at the level of Gerstenhaber algebras.
We describe the generalized Kuranishi spaces of solvmanifolds with left-invariant complex structures. By using such description, we study the stability of left-invariantness of deformed generalized complex structures and smoothness of generalized Kuranishi spaces on certain classes of solvmanifolds. We also give explicit finite dimensional cohcain complexes which computes the holomorphic Poisson cohomology of nilmanifolds and solvmanifolds.
$p$-Kähler structures on nilmanifolds and holomorphically parallelizable manifolds
Dipartimento di Scienze Matematiche, Fisiche e Informatiche Unità di Matematica e Informatica Università degli Studi di Parma（帕尔马大学数学、物理与信息科学系数学与信息单元）；Dipartimento di Matematica e Informatica Universita degli Studi di Ferrara（费拉拉大学数学与信息学系）；IMAR（罗马尼亚数学研究所）
It is well known that cohomology of any non-trivial 1-dimensional local system on a nilmanifold vanishes (this result is due to L. Alaniya). A complex nilmanifold is a quotient of a nilpotent Lie group equipped with a left-invariant complex structure by an action of a discrete, co-compact subgroup. We prove a Dolbeault version of Alaniya's theorem, showing that the Dolbeault cohomology of a nilpotent Lie algebra with coefficients in any non-trivial 1-dimensional local system vanishes. Note that the Dolbeault cohomology of the corresponding local system on the manifold is not necessarily zero. …
