Frölicher spectral sequence of compact complex manifolds with special Hermitian metrics

Garcia-Fernandez M., Jordan J., Streets J.

Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kähler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct complex manifolds in every dimension with vanishing first Chern class which do not admit Bismut Hermitian-Einstein metrics. This reformulation also leads to a new description of pluriclosed flow in terms of Hermitian metrics on holomorphic Courant algebroids, implying new global existence results, in particular on all complex non-Kähler surfaces of Kodaira dimension κ≥0. On complex manifolds which admit Bismut-flat metrics we show global existence and convergence of pluriclosed flow to a Bismut-flat metric, which in turn gives a classification of generalized Kähler structures on these spaces. Les métriques Hermitiennes plurifermées avec forme de Bismut-Ricci nulle donnent une extension naturelle des métriques de Calabi-Yau au cadre des variétés complexes non-Kähler, et apparaissent indépendamment en physique mathématique. Nous réinterprétons cette condition en termes de l'équation d'Hermite-Einstein sur un algébroide de Courant holomorphe associé, et nous appellerons donc ces solutions des métriques de Bismut Hermite-Einstein. Cela donne des obstructions de stabilité de pente de Mumford-Takemoto, et en utilisant celles-ci, nous mettons en évidence, pour toute dimension, une infinité de variétés complexes topologiquement distinctes avec première classe de Chern nulle qui n'admettent pas de métrique Bismut Hermite-Einstein. Cette reformulation conduit également à une nouvelle description du flot plurifermé en termes de métriques hermitienne sur les algébroides de Courant holomorphes, impliquant de nouveaux résultats d'existence globale, en particulier sur toutes les surfaces complexes non-Kähler de dimension de Kodaira κ≥0. Sur les variétés complexes qui admettent une métrique Bismut-plate, nous montrons l'existence globale et la convergence d'un flux plurifermé vers une métrique Bismut-plate, qui à son tour donne une classification des structures de Kähler généralisées sur ces espaces.

Latorre A., Ugarte L., Villacampa R.

We classify the nilpotent Lie algebras of real dimension eight and minimal center that admit a complex structure. Furthermore, for every such nilpotent Lie algebra g , we describe the space of complex structures on g up to isomorphism. As an application, the nilpotent Lie algebras having a non-trivial abelian J -invariant ideal are classified up to eight dimensions.

Stelzig J.

We prove that there are no unexpected universal integral linear relations and congruences between Hodge, Betti and Chern numbers of compact complex manifolds and determine the linear combinations of such numbers which are bimeromorphic or topological invariants. This extends results in the Kähler case by Kotschick and Schreieder. We then develop a framework to tackle the more general questions taking into account ‘all’ cohomological invariants (e.g. the dimensions of the higher pages of the Frölicher spectral sequence, Bott-Chern and Aeppli cohomology). This allows us to reduce the general questions to specific construction problems. We solve these problems in many cases. In particular, we obtain full answers to the general questions concerning universal relations and bimeromorphic invariants in low dimensions.

Sferruzza T., Tardini N.

Let (X, J) be a nilmanifold with an 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.

Arroyo R.M., Nicolini M.

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.

ORNEA L., OTIMAN A.-., STANCIU M.

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.

Popovici D., Stelzig J., Ugarte L.

Abstract For every positive integer r, we introduce two new cohomologies, that we call E r {E_{r}} -Bott–Chern and E r {E_{r}} -Aeppli, on compact complex manifolds. When r = 1 {r\kern-1.0pt=\kern-1.0pt1} , they coincide with the usual Bott–Chern and Aeppli cohomologies, but they are coarser, respectively finer, than these when r ≥ 2 {r\geq 2} . They provide analogues in the Bott–Chern–Aeppli context of the E r {E_{r}} -cohomologies featuring in the Frölicher spectral sequence of the manifold. We apply these new cohomologies in several ways to characterise the notion of page- ( r - 1 ) {(r-1)} - ∂ ⁡ ∂ ¯ {\partial\bar{\partial}} -manifolds that we introduced very recently. We also prove analogues of the Serre duality for these higher-page Bott–Chern and Aeppli cohomologies and for the spaces featuring in the Frölicher spectral sequence. We obtain a further group of applications of our cohomologies to the study of Hermitian-symplectic and strongly Gauduchon metrics for which we show that they provide the natural cohomological framework.

Stelzig J.

We study consequences and applications of the folklore statement that every double complex over a field decomposes into so-called squares and zigzags. This result makes questions about the associated cohomology groups and spectral sequences easy to understand. We describe a notion of `universal' quasi-isomorphism, investigate the behaviour of the decomposition under tensor product and compute the Grothendieck ring of the category of bounded double complexes over a field with finite cohomologies up to such quasi-isomorphism (and some variants). Applying the theory to the double complexes of smooth complex valued forms on compact complex manifolds, we obtain a Poincar\'e duality for higher pages of the Fr\"olicher spectral sequence, construct a functorial three-space decomposition of the middle cohomology, give an example of a map between compact complex manifolds which does not respect the Hodge filtration strictly, compute the Bott-Chern and Aeppli cohomology for Calabi-Eckmann manifolds, introduce new numerical bimeromorphic invariants, show that the non-K\"ahlerness degrees are not bimeromorphic invariants in dimensions higher than three and that the $\partial\overline{\partial}$-lemma and some related properties are bimeromorphic invariants if, and only if, they are stable under restriction to complex submanifolds.

Milivojević A.

AbstractSerre’s duality theorem implies a symmetry between the Hodge numbers, hp,q = hn−p,n−q, on a compact complex n–manifold. Equivalently, the first page of the associated Frölicher spectral sequence satisfies \dim E_1^{p,q} = \dim E_1^{n - p,n - q} for all p, q. Adapting an argument of Chern, Hirzebruch, and Serre [3] in an obvious way, in this short note we observe that this “Serre symmetry” \dim E_k^{p,q} = \dim E_k^{n - p,n - q} holds on all subsequent pages of the spectral sequence as well. The argument shows that an analogous statement holds for the Frölicher spectral sequence of an almost complex structure on a nilpotent real Lie group as considered by Cirici and Wilson in [4].

Wang Q., Yang B., Zheng F.

In this paper, we give a classification of all compact Hermitian manifolds with flat Bismut connection. We show that the torsion tensor of such a manifold must be parallel, thus the universal cover of such a manifold is a Lie group equipped with a bi-invariant metric and a compatible left invariant complex structure. In particular, isosceles Hopf surfaces are the only Bismut flat compact non-Kähler surfaces, while central Calabi-Eckmann threefolds are the only simply-connected compact Bismut flat threefolds.

Fino A., Vezzoni L.

Rollenske S., Tomassini A., Wang X.

In this paper, we obtain a vertical–horizontal decomposition formula of Laplacians on manifolds with a special foliation structure. Two Nomizu-type theorems for cohomologies of nilmanifolds follow as applications.

Popovici D.

Latorre A., Ugarte L., Villacampa R.

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.

Latorre A., Ugarte L., Villacampa R.

We obtain several restrictions on the terms of the ascending central series of a nilpotent Lie algebra g \mathfrak {g} under the presence of a complex structure J J . In particular, we find a bound for the dimension of the center of g \mathfrak {g} when it does not contain any non-trivial J J -invariant ideal. Thanks to these results, we provide a structural theorem describing the ascending central series of 8-dimensional nilpotent Lie algebras g \mathfrak {g} admitting this particular type of complex structure J J . Since our method is constructive, it allows us to describe the complex structure equations that parametrize all such pairs ( g , J ) (\mathfrak {g}, J) .