Complex Nilmanifolds Research Articles

COMPATIBILITY BETWEEN NON-KÄHLER STRUCTURES ON COMPLEX (NIL)MANIFOLDS

We study the interplay between the following types of special non-Kähler Hermitian metrics on compact complex manifolds (locally conformally Kähler, k-Gauduchon, balanced, and locally conformally balanced) and prove that a locally conformally Kähler compact nilmanifold carrying a balanced or a left-invariant k-Gauduchon metric is necessarily a torus. Combined with the main result in [FV16], 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, we construct a family of compact nilmanifolds in any dimension carrying both balanced and locally conformally balanced metrics and finally we show a compact complex nilmanifold does not support a left-invariant locally conformally hyper-Kähler structure.

Complex nilmanifolds with constant holomorphic sectional curvature

A well known conjecture in complex geometry states that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kahler 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 that case.

On the Real Homotopy Type of Generalized Complex Nilmanifolds

We prove that for any n≥4, there are infinitely many real homotopy types of 2n-dimensional nilmanifolds admitting generalized complex structures of every type k, for 0≤k≤n.

TWISTED DOLBEAULT COHOMOLOGY OF NILPOTENT LIE ALGEBRAS

It is well known that the cohomology of any non-trivial 1-dimensional local system on a nilmanifold vanishes (this result, due to J. Dixmier, was also announced and proved in some particular case by 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 Dixmier’s and Alaniya’s theorem, showing that the Dolbeault cohomology $$ {H}^{0,p}\left(\mathfrak{g},L\right) $$ 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. This implies that the twisted version of Console–Fino theorem is false (Console–Fino proved that the Dolbeault cohomology of a complex nilmanifold is equal to the Dolbeault cohomology of its Lie algebra, when the complex structure is rational). As an application, we give a new proof of a theorem due to H. Sawai, who obtained an explicit description of LCK nilmanifolds. An LCK structure on a manifold M is a Kahler structure on its cover $$ \tilde{M} $$ such that the deck transform map acts on $$ \tilde{M} $$ by homotheties. We show that any complex nilmanifold admitting an LCK structure is Vaisman, and is obtained as a compact quotient of the product of a Heisenberg group and the real line.

Complex nilmanifolds and Kähler-like connections

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 second case and a partial description in the first and 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.

Dolbeault cohomology of complex nilmanifolds foliated in toroidal groups

Abstract 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 generalizes previous methods, where the existence of a holomorphic fibration was a crucial ingredient.

A Family of Complex Nilmanifolds with in finitely Many Real Homotopy Types

Abstract We find a one-parameter family of non-isomorphic nilpotent Lie algebras ga, with a > [0,∞), 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.

Algebraic dimension of complex nilmanifolds

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)⩽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 Kähler manifolds.

On generalized Gauduchon nilmanifolds

We construct invariant generalized Gauduchon metrics on the product of two complex nilmanifolds that do not necessarily admit this kind of metrics. In particular, we prove that the product of a locally conformal Kähler nilmanifold and a balanced nilmanifold admits a generalized Gauduchon metric. In complex dimension 4, generalized Gauduchon nilmanifolds with (the highest possible) nilpotency step s=5 are given, as well as 3-step and 4-step examples for which the center of their underlying Lie algebras does not contain any non-trivial J-invariant ideal. These examples show strong differences between the SKT and the generalized Gauduchon geometries of nilmanifolds.

On the cone of strong Kähler with torsion metrics

This paper examines special metrics on compact complex manifolds, and it is notably focused on the notion of super strong Kahler with torsion metric. This condition is related to the strong Kahler with torsion one in the same manner as the strongly Gauduchon condition is related to the Gauduchon one. Moreover, we provide sufficient and necessary conditions so that every strong Kahler with torsion metric on a compact complex manifold is in fact super strong Kahler with torsion. We prove that these conditions are verified on compact complex manifolds satisfying $$\partial \overline{\partial }$$ -lemma but not on 6-dimensional compact complex nilmanifolds.

Some relations between Hodge numbers and invariant complex structures on compact nilmanifolds

Abstract Let N be a simply connected real nilpotent Lie group, n its Lie algebra, and € a lattice in N. If a left-invariant complex structure on N is Γ-rational, then HƏ̄ s,t(Γ/N) ≃ HƏ̄ s,t(nC) for each s; t. We can construct different left-invariant complex structures on one nilpotent Lie group by using the complexification and the scalar restriction. We investigate relationships to Hodge numbers of associated compact complex nilmanifolds.

Duality of Hodge numbers of compact complex nilmanifolds

Abstract A compact K¨ahlerian manifoldM of dimension n satisfies hp,q(M) = hq,p(M) for each p, q.However, a compact complex manifold does not satisfy the equations in general. In this paper, we consider duality of Hodge numbers of compact complex nilmanifolds.

Cohomological decomposition of complex nilmanifolds

We study \emph{pureness} and \emph{fullness} of invariant complex structures on nilmanifolds. We prove that in dimension six, apart from the complex torus, there exist only two non-isomorphic complex structures satisfying both properties, which live on the real nilmanifold underlying the Iwasawa manifold. We also show that the product of two almost complex manifolds which are pure and full is not necessarily full.

Balanced and strongly Gauduchon cones on solvmanifolds

We show conditions under which the balanced and strongly Gauduchon cones of a complex solvmanifold are non-degenerate. These cones are explicitly described on the complex nilmanifolds with underlying Lie algebra [Formula: see text].

The Pluriclosed Flow on Nilmanifolds and Tamed Symplectic Forms

We study the evolution of strong Kahler with torsion (SKT) structures via the pluriclosed flow on complex nilmanifolds, i.e., on compact quotients of simply connected nilpotent Lie groups by discrete subgroups endowed with an invariant complex structure. Adapting to our case the techniques introduced by Jorge Lauret for studying Ricci flow on homogeneous spaces, we show that for SKT Lie algebras the pluriclosed flow is equivalent to a bracket flow, and we prove a long time existence result in the nilpotent case. Finally, we introduce a natural flow for evolving symplectic forms taming a complex structure, by considering the evolution of symplectic forms via the flow induced by the Bismut Ricci form.

ON COHOMOLOGICAL DECOMPOSITION OF BALANCED MANIFOLDS

We study complex-pure-and-fullness at the first stage on compact complex nilmanifolds M of complex dimension 3, with special attention to the case of existence of a balanced Hermitian metric on M.

The Cohomologies of the Iwasawa Manifold and of Its Small Deformations

We prove that, for some classes of complex nilmanifolds, the Bott-Chern cohomology is completely determined by the Lie algebra associated to the nilmanifold with the induced complex structure. We use these tools to compute the Bott-Chern and Aeppli cohomologies of the Iwasawa manifold and of its small deformations, completing the computations in arXiv:0709.3528v1 [math.AG] by M. Schweitzer.

On deformations of compact balanced manifolds

We study suitable deformations of the complex structure on a compact balanced manifold , naturally associated with cohomology classes in . We construct curves of balanced structures on a compact holomorphically parallelizable -dimensional complex nilmanifold and of half-flat structures on a compact complex -dimensional solvmanifold respectively.

Canonical bundles of complex nilmanifolds, with applications to hypercomplex geometry

A nilmanifold is a quotient of a nilpotent group $G$ by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a $G$-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle. This is used to study hypercomplex nilmanifolds (nilmanifolds with a triple of $G$-invariant complex structures which satisfy quaternionic relations). We prove that a hypercomplex nilmanifold admits an HKT (hyperk\ahler with torsion) metric if and only if the underlying hypercomplex structure is abelian. Moreover, any $G$-invariant HKT-metric on a nilmanifold is balanced with respect to all associated complex structures.

A general description of the terms in the Frölicher spectral sequence

An explicit description of the Frölicher spectral sequence is given that is useful for calculations for compact nilmanifolds. We use the description to exhibit examples of compact complex nilmanifolds of (complex) dimension 3 for which the Frölicher spectral sequence does not degenerate at the second term.