Six dimensional solvmanifolds with holomorphically trivial canonical bundle

We determine the 6-dimensional solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong K\"ahler with torsion (SKT), generalized Gauduchon, balanced and strongly Gauduchon metrics is studied. As an application we construct a holomorphic family $(M,J_a)$ of compact complex manifolds such that $(M,J_a)$ satisfies the $\partial\bar\partial$-Lemma and admits a balanced metric for any $a\not=0$, but the central limit neither satisfies the $\partial\bar\partial$-Lemma nor admits balanced metrics.Comment: 32 pages; to appear in IMR

Complex structures of splitting type

We study the six-dimensional solvmanifolds that admit complex structures of splitting type classifying the underlying solvable Lie algebras. In particular, many complex structures of this type exist on the Nakamura manifold $X$, and they allow us to construct a countable family of compact complex non-$\partial\overline\partial$ manifolds $X_k$, $k\in\mathbb{Z}$, that admit a small holomorphic deformation $\{(X_{k})_{t}\}_{t\in\Delta_k}$ satisfying the $\partial\overline\partial$-Lemma for any $t\in\Delta_k$ except for the central fibre. Moreover, a study of the existence of special Hermitian metrics is also carried out on six-dimensional solvmanifolds with splitting-type complex structures

Six dimensional homogeneous spaces with holomorphically trivial canonical bundle

We classify all the 6-dimensional unimodular Lie algebras gadmitting a complex structure with non-zero closed (3, 0)-form. This gives rise to 6-dimensional compact homogeneous spaces M= \G, where is a lattice, admitting an invariant complex structure with holomorphically trivial canonical bundle. As an application, in the balanced Hermitian case, we study the instanton condition for any metric connection ∇ε,ρ in the plane generated by the Levi-Civita connection and the Gauduchon line of Hermitian connections. In the setting of the Hull-Strominger system with connection on the tangent bundle being HermitianYang-Mills, we prove that if a compact non-Kähler homogeneous space M= \Gadmits an invariant solution with respect to some non-flat connection ∇in the family ∇ε,ρ, then Mis a nilmanifold with underlying Lie algebra h3, a solvmanifold with underlying algebra g7, or a quotient of the semisimple group SL(2, C). Since it is known that the system can be solved on these spaces, our result implies that they are the unique compact non-Kähler balanced homogeneous spaces admitting such invariant solutions. As another application, on the compact solvmanifold underlying the Nakamura manifold, we construct solutions, on any given balanced Bott-Chern class, to the heterotic equations of motion taking the Chern connection as (flat) instanton

Solvmanifolds with holomorphically trivial canonical budle

As it is well-known Calabi-Yau manifolds constitute one of the most important classes in geometry. These manifolds, which can be thought as higher-dimensional analogues of K3 surfaces, are compact complex manifolds (M,J) of complex dimension n endowed with an SU(n) structure (F,\Psi) such that the fundamental 2-form F is closed and the (n,0)-form \Psi is holomorphic. Thus, the holonomy of the metric g reduces to a subgroup of SU(n), so that g is a Ricci-flat Kähler metric, and the canonical bundle of (M,J) is holomorphically trivial. The above conditions defining a Calabi-Yau manifold have been weakened in different directions so that the resulting geometries still play an important role in several aspects of complex geometry. In this thesis we focus our attention in the geometry of compact complex manifolds (M,J) with holomorphically trivial canonical bundle endowed with special Hermitian metrics which are less restrictive than the Kähler ones. Concerning compact complex manifolds with holomorphically trivial canonical bundle, we recall that in complex dimension 2 the possibilities, up to isomorphism, are a K3 surface, a torus or a Kodaira surface, where the first two are Kähler and the latter is an example of a nilmanifold M = G/ \Gamma, i.e. a compact quotient of a simply connected nilpotent Lie group G by a lattice \Gamma of maximal rank in G. However, there are no classifications in complex dimension 3 or higher, so it is natural to begin by studying such complex geometry on some particular classes of compact manifolds of real dimension 6. A good candidate is the class consisting of nilmanifolds endowed with an invariant complex structure, as Salamon proved that any such complex nilmanifold has holomorphically trivial canonical bundle. In (real) dimension 6 a classification of nilmanifolds admitting this kind of complex structures is also provided by Salamon, where the Iwasawa (nil)manifold is a classical example which plays a relevant role in complex geometry. Although the complex geometry of nilmanifolds provides an important source of examples in differential geometry, these spaces never satisfy the ddbar-lemma because they are not formal except for tori. However, the investigation of some properties in complex geometry requires compact complex manifolds satisfying the ddbar-lemma, so one needs to consider a broader class of homogeneous spaces M = G/ \Gamma. The first natural generalization of nilmanifolds is given by compact quotients of Lie groups G which are solvable instead of nilpotent. For instance, the Nakamura manifold, whose complex geometry is very rich, is an example of this type. This class of manifolds, known as solvmanifolds, is the central object of study in this thesis. More concretely, we describe the 6-dimensional solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle, as well as we obtain a classification of such invariant structures. As we mentioned above, another goal in this thesis is the study of special Hermitian metrics which are less restrictive than the Kähler ones. It is well-known that the existence of a Kähler metric on a compact manifold imposes strong topological obstructions. In contrast, Gauduchon proves that on a compact complex manifold (M,J) of complex dimension n there always exists a Gauduchon metric, defined by ddbar F^{n-1}=0, in the conformal class of any given Hermitian metric. Between the Kähler class and the Gauduchon class other interesting classes of special Hermitian metrics have been considered in relation to different problems in differential geometry such as balanced, strongly Gauduchon, strong Kähler with torsion and k-th Gauduchon. Associated to any compact complex manifold (M,J) there exist several complex invariants which measure some specific aspects of (M,J). Among them, we distinguish the Dolbeault, the Bott-Chern and the Aeppli cohomologies, and the Frölicher spectral sequence {E_r(M)}, with r>=1, relating the Dolbeault to the de Rham cohomology of the manifold. If M is a compact Kähler manifold then all these complex invariants coincide because M satisfies the ddbar-lemma, however the Frölicher sequence may not degenerate at the first step for arbitrary compact complex manifolds. A problem of interest in complex geometry is to study the behaviour of these invariants. In the case of 6-dimensional nilmanifolds a complete picture of the behaviour of the sequence {E_r(M)} is given in this thesis, and for solvmanifolds of dimension 6 endowed with an invariant complex structure of splitting type (in the sense of Kasuya) with holomorphically trivial canonical bundle we use the results by Kasuya and Angella and by Angella and Tomassini to find when the ddbar-lemma is satisfied. Finally, motivated by some recent results obtained by Popovici, we also explore in this thesis the relations among the degeneration of the Frölicher spectral sequence, the ddbar-lemma and the existence of balanced or strongly Gauduchon metrics, as well as their behaviour under small holomorphic deformations of the complex structure

Solutions of the Laplacian flow and coflow of a locally conformal parallel G2-structure

We study the Laplacian flow of a G 2-structure where this latter structure is claimed to be locally conformal parallel. The first examples of long time solutions of this flow with the locally conformal parallel condition are given. All of the solutions are ancient and Laplacian soliton of shrinking type. These examples are one-parameter families of locally conformal parallel G2-structures on rank-one solvable extensions of six-dimensional nilpotent Lie groups. The found solutions are used to construct long time solutions to the Laplacian coflow starting from a locally conformal parallel structure. We also study the behavior of the curvature of the solutions obtaining that for one of the examples the induced metric is Einstein along all the flow (resp. coflow)

Laplacian coflow for warped G2-structures

We consider the Laplacian coflow of a G2-structure on warped products of the form M7=M6×fS1 with M6 a compact 6-manifold endowed with an SU(3)-structure. We give an explicit reinterpretation of this flow as a set of evolution equations of the differential forms defining the SU(3)-structure on M6 and the warping function f. Necessary and sufficient conditions for the existence of solution for this flow are given. Finally we describe new solutions for this flow where the SU(3)-structure on M6 is nearly Kähler, symplectic half-flat or balanced. © 2020 Elsevier B.V