We review some constructions and properties of complex manifolds admitting pluriclosed and balanced metrics. We prove that for a 6-dimensional solvmanifold endowed with an invariant complex structure J having holomorphically trivial canonical bundle the pluriclosed flow has a long time solution for every invariant init…
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48 results for “6-dimensional solvmanifolds”
The article classifies 6D flat solvmanifolds by analyzing conjugacy classes of matrices.
problem Classifying 6-dimensional splittable flat solvmanifolds.
method Analyzing conjugacy classes of integer matrices of finite order in dimensions 4 and 5.
result Classification of 6-dimensional splittable flat solvmanifolds.
Constructs mirror pairs for solvmanifolds using Lie groups.
problem Finding mirror pairs for non-Kaehler solvmanifolds.
method Left-invariant affine structures on Lie groups.
result Explicitly finds SYZ mirror symmetric partners for all known compact 6D solvmanifolds.
We construct a compact 6-dimensional solvmanifold endowed with a non-trivial invariant generalized Kähler structure and which does not admit any Kähler metric. This is in contrast with the case of nilmanifolds which cannot admit any invariant generalized Kähler structure unless they are tori.
Study Gauduchon connections on Kähler-like metrics in 6D solvmanifolds.
problem Investigate Hermitian metrics with Gauduchon connections having symmetries similar to Levi-Civita and Chern connections.
method Analyze -dimensional solvmanifolds with invariant complex structures and trivial canonical bundles.
result Evidence supports two conjectures about Kähler-like conditions for different Gauduchon connections.
Study computes invariants on six-dimensional solvmanifolds, providing symplectic structure obstructions.
problem Computing invariants on six-dimensional solvmanifolds.
method Computed almost-complex and almost-Hermitian invariants on families of solvmanifolds.
result Provides obstructions to symplectic structures on compact almost-complex manifolds.
The article classifies six-dimensional solvmanifolds with non-invariant trivializing sections of their canonical bundle.
problem Classifying six-dimensional solvmanifolds with non-invariant trivializing sections of their canonical bundle.
method Complete classification of six-dimensional solvable strongly unimodular Lie algebras admitting complex structures, identifying those with non-invariant holomorphic sections of their canonical bundle.
result Construction of a new six-dimensional solvmanifold with non-invariant holomorphic sections of its 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 Kähler with torsion (SKT), generalized Gauduchon, balanced and strongly Gauduchon metrics is studied.…
Starting from a 6-dimensional nilpotent Lie group N endowed with an invariant SU(3) structure, we construct a homogeneous conformally parallel G_2-metric on an associated solvmanifold. We classify all half-flat SU(3) structures that endow the rank-one solvable extension of N with a conformally parallel G_2 structure. B…
Bott-Chern formality and Massey products on strong Kähler with torsion and Kähler solvmanifoldsmath.DG
Study geometric formal metrics and Massey products on Kähler manifolds with torsion.
problem Interplay between geometrically-Bott-Chern-formal metrics and SKT metrics on Kähler manifolds.
method Analyzing nilmanifolds and Kähler solvmanifolds, proving conditions for existence of SKT metrics and Massey products.
result Any Kähler solvmanifold is geometrically formal, and explicit constructions of lattices with non-vanishing Massey products.
Study LCS structures on Lie algebras of type I, proving trivial Morse-Novikov cohomology and constructing solvmanifolds.
problem Locally conformal symplectic structures on Lie algebras of type I.
method Analyzing Lie algebras of type I, proving trivial Morse-Novikov cohomology, and constructing solvmanifolds.
result LCS structures on Lie algebras of type I are of the first kind and can be used to construct compact solvmanifolds.
Classifies 6D homogeneous spaces with holomorphically trivial canonical bundle.
problem Classifying 6D homogeneous spaces with specific geometric properties.
method Analyzing Lie algebras and homogeneous spaces, studying connections and instantons.
result Compact non-Kähler homogeneous spaces are unique in their solutions to the Hull-Strominger system.
Study of Type IIA flow on symplectic Lie algebras for geometric structures.
problem Detecting geometric structures like Lagrangian torus fibrations and harmonic almost complex structures.
method Investigate -harmonic forms and the long-time behavior of the Type IIA flow.
result The Type IIA flow helps in detecting desired geometric structures.
Study of Monge-Ampère volumes on hermitian manifolds, focusing on plurisigned metrics.
problem Behavior of Monge-Ampère volumes under hermitian variations.
method Analysis of hermitian forms and cohomology, studying boundedness and positivity.
result Existence and non-existence of plurisigned hermitian metrics on specific manifolds.
We show that supersymmetric flux vacua with intermediate SU(2) structure is closely related to some special classes of half-flat structures. More concretely, solutions of the SUSY equations IIA possess a symplectic half-flat structure, whereas solutions of the SUSY equations IIB admit a half-flat structure which is in …
Study of complex and Hermitian structures on specific Lie groups.
problem Classifying Lie groups with specific geometric structures.
method Analysis of left-invariant structures on almost abelian Lie groups.
result Classification of six-dimensional generalized Kähler almost abelian Lie groups.
This note summarizes the talk by the author at the workshop "Geometry and Computer Science" held in Pescara in February 2017. We present how SageMath can help in research in Complex and Differential Geometry, with two simple applications, which are not intended to be original. We consider two "classification problems" …
In this paper, we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple Lie algebras. These solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type. We show that the Ricci curvatures of our solvmanifolds coincide with the restrictions of the Ricci curva…
New Einstein metric found on non-standard solvmanifold.
problem Finding indefinite invariant Einstein metrics on solvmanifolds.
method Described an example of a non-standard solvmanifold with an indefinite invariant Einstein metric.
result Found a new type of Einstein metric on a non-standard solvmanifold.
The study of holonomy groups in flat solvmanifolds, proving finite abelian groups can be holonomy groups and describing dimensions.
problem Understanding the holonomy groups of flat solvmanifolds.
method Elementary proof and construction of examples for various dimensions.
result Finite abelian groups can be holonomy groups of flat solvmanifolds, and specific dimensions and holonomy groups are described.
We obtain new examples of non-symmetric Einstein solvmanifolds by combining two techniques. In \cite{T2}, H. Tamaru constructs new {\em attached} solvmanifolds, which are submanifolds of the solvmanifolds corresponding to noncompact symmetric spaces, endowed with a natural metric. Extending this construction, we apply …
Proves Vaisman solvmanifolds are finite quotients of Kodaira-Thurston manifolds.
problem Characterizing Vaisman solvmanifolds and their properties.
method Analyzing fundamental groups and quotient structures.
result Every Vaisman solvmanifold is a finite quotient of a Kodaira-Thurston manifold.
The paper constructs pseudo-Iwasawa solvmanifolds with Killing spinors.
problem Constructing Killing spinors on pseudo-Riemannian solvmanifolds.
method Using nilsolitons and pseudo-Iwasawa condition, the paper constructs families of pseudo-Iwasawa solvmanifolds with Killing spinors.
result All pseudo-Iwasawa solvmanifolds admitting a Killing spinor belong to a specific family.
Study compact symplectic solvmanifolds' hard Lefschetz property.
problem Compact symplectic solvmanifolds' hard Lefschetz property.
method Analysis of compact symplectic solvmanifolds as quotients of solvable Lie groups by lattices.
result Characterization of conditions for the hard Lefschetz property.
Survey on LCK structures on solvmanifolds, focusing on invariant and Vaisman structures.
problem Understanding LCK structures on solvmanifolds.
method Review of recent results and presentation of new findings.
result New insights into the canonical bundle of solvmanifolds with Vaisman structures.
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 explic…
Characterizes hypercomplex Lie groups and their solvmanifolds.
problem Understanding hypercomplex structures on Lie groups and their solvmanifolds.
method Characterization of almost abelian Lie groups with hypercomplex structures, analysis of Obata and Bismut connections, classification of hypercomplex Lie groups, and construction of solvmanifolds.
result Classification of hypercomplex almost abelian Lie groups in dimension 8 and properties of their solvmanifolds.
We obtain new supersymmetric flux vacua of type II supergravities on four-dimensional Minkowski times six-dimensional solvmanifolds. The orientifold O4, O5, O6, O7, or O8-planes and D-branes are localized. All vacua are in addition not T-dual to a vacuum on the torus. The corresponding solvmanifolds are proven to be Ca…
We derive a curvature identity that holds on any 6-dimensional Riemannian manifold, from the Chern-Gauss-Bonnet theorem for a 6-dimensional closed Riemannian manifold. We also introduce some applications of this curvature identity.
New Einstein solvmanifolds created from non-flat Ricci solitons.
problem Creating new Einstein solvmanifolds from non-flat Ricci solitons.
method Constructing a one-parameter family of expanding, gradient Ricci solitons with cohomogeneity one isometric action.
result Exactly one metric in the family is Einstein and has orbits isometric to the original solvmanifold.
Ricci flow on solvmanifolds leads to unique solvsolitons.
problem Understanding Ricci flow on specific solvmanifolds.
method Applying Ricci flow to homogeneous solvmanifolds of real type.
result Ricci flow converges to unique solvsolitons on solvmanifolds.
We study the homotopical minimal periods for maps on infra-solvmanifolds of type (R) using the density of the homotopical minimal period set in the natural numbers. This extends the result of [10] from flat manifolds to infra-solvmanifolds of type (R). Applying our main result we will list all possible maps on infra-so…
Einstein solvmanifolds embedded in symmetric spaces.
problem Finding faithful representations of Lie groups as isometric embeddings.
method Investigating special maps for solvable Lie groups, including Einstein and Ricci soliton solvmanifolds.
result All Einstein solvmanifolds can be realized as submanifolds of a symmetric space.
We study the analogue of the Goldberg conjecture on non-compact solvmanifolds. In contrast to the almost-Kähler case we prove that a 7-dimensional solvmanifold cannot admit any left-invariant calibrated -structure such that the induced metric is Einstein, unless is flat.…
Homogeneous Einstein metrics on Euclidean spaces are shown to be Einstein solvmanifolds.
problem Characterizing homogeneous Einstein metrics on Euclidean spaces.
method Using periodic, integrally minimal foliations and geometric flow induced by the orbit-Einstein condition.
result Homogeneous Einstein metrics on Euclidean spaces are proven to be Einstein solvmanifolds.
Study complex solvmanifolds with trivial canonical bundle and hypercomplex geometry.
problem Characterize and produce examples of complex solvmanifolds with trivial canonical bundle.
method Characterize invariant trivializing sections using Koszul 1-form, provide algebraic obstructions, and exhibit specific examples.
result New examples of complex solvmanifolds with trivial canonical bundle and algebraic obstructions for triviality.
In this paper we study some properties of almost abelian solvmanifolds using minimal models associated to a fibration. In particular we state a necessary and sufficient condition to formality and a method for finding symplectic strucures of this kind of solvmanifolds.
We discuss our recent results on the existence and classification problem of complex and Kaehler structures on compact solvmanifolds. In particular, we determine in this paper all the complex surfaces which are diffeomorphic to compact solvmanifolds (and compact homogeneous manifolds in general).
Kähler Solvmanifoldsmath.DG
Compact Kähler solvmanifolds are classified up to biholomorphism. A proof of a conjecture Benson and Gordon, that completely solvable compact Kähler solvmanifolds are tori is deduced from this. The main ingredient in the proof is a restriction theorem for polycyclic Kähler groups proved by Nori and the author.
In this paper, we continue studying the 6-dimensional pseudo-Riemannian space V^6(g_{ij}) with signature [++--], which admits projective motions, i. e. continuous transformation groups preserving geodesics. In particular, we determine a necessary and sufficient condition that the 6-dimensional rigid h-spaces have const…
We study Einstein manifolds admitting a transitive solvable Lie group of isometries (solvmanifolds). It is conjectured that these exhaust the class of noncompact homogeneous Einstein manifolds. J. Heber has showed that under certain simple algebraic condition called standard (i.e. the orthogonal complement of the deriv…
Study the rank of Nijenhuis tensor on parallelizable almost complex manifolds.
problem Understanding the rank of Nijenhuis tensor on parallelizable almost complex manifolds.
method Reduction of computations to solving PDEs, explicit solutions on specific manifolds, analysis of curve of almost complex structures, classification of Lie algebras.
result Classification of Lie algebras admitting almost complex structures with specific Nijenhuis tensor ranks.
Surveying mean curvature flow on solvmanifolds, focusing on translating solutions.
problem Understanding translating solutions on solvmanifolds.
method Exploration of mean curvature flow, introduction of translating solutions, discussion of solvmanifolds.
result Equations describing translating solutions on the family of 3D solvmanifolds.
We study rank flat bundles over solvmanifolds whose cohomologies are non-trivial. By using Hodge theoretical properties for all topologically trivial rank flat bundles, we represent the structure theorem of Kähler solvmanifolds as extensions of Hasegawa's result and Benson-Gordon's result for nilmanifolds.
The paper calculates the noncommutative residue for a rescaled Dirac operator on 6D manifolds.
problem Computing the noncommutative residue for a specific Dirac operator on 6D manifolds.
method Calculations and proofs for the rescaled Dirac operator fDh on 6D compact manifolds.
result Proof of the Kastler-Kalau-Walze type theorem for the rescaled Dirac operator on 6D compact manifolds with boundary.
New Einstein solvmanifolds constructed without using nilsolitons.
problem Constructing Einstein solvmanifolds not based on nilsolitons.
method Using nonsymmetric derivations and a computer algorithm to classify and find solutions up to dimension 9.
result Einstein solvmanifolds of dimensions ≤ 5 are not isometric to standard extensions of nilsolitons.
The study classifies flat solvmanifolds and finds -structures on them.
problem Classifying and finding -structures on flat solvmanifolds.
method Classification of flat splittable solvmanifolds and search for compatible -structures.
result Examples of compact flat manifolds with specific -structures.
We study the symplectic Bott-Chern cohomology by L.-S. Tseng and S.-T. Yau for solvmanifolds endowed with left-invariant symplectic structures. Our results are applicable to cohomology with values in local systems. Studying symplectic Bott-Chern cohomology of solvmanifolds with values in local systems, we give some rem…