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Keizo Hasegawa

Publications and source records attributed to Keizo Hasegawa.

14 recordsLinked to original sources
arXiv2026

Invariant forms compute the Dolbeault cohomology of complex nilmanifolds

We prove that the inclusion of left-invariant forms into the Dolbeault complex of a compact nilmanifold $M$ endowed with a left-invariant complex structure $J$ induces an isomorphism in cohomology in every bidegree, settling a long-standing conjecture. As consequences, we show that small deformations of $J$ are still invariant, as conjectured by Hasegawa. We also prove that Bott--Chern, Aeppli, and Frölicher invariants are computed by invariant forms and are independent of the lattice, settling a conjecture of Angella on Bott--Chern cohomology.

math.DG↗
arXiv2025

Cartan Flat Non-degenerate CR Lie Groups

In this paper we determine all the simply connected non-degenerate CR Lie groups, which are flat with respect to the Cartan connection: in terms of associated Lie algebras, we assert that the only Cartan flat non-degenerate CR Lie algebras are $\mathfrak{su}(2)$, $\mathfrak{sl}(2,\mathbb{R})$, $\mathfrak{aff}(\mathbb{R}) \oplus \mathbb{R}$, and $\mathfrak{h}_{2m+1}$ with its modifications, where $\mathfrak{aff}(\mathbb{R})$ is the affine Lie algebra of dimension 2 and $\mathfrak{h}_{2m+1}$ is the Heisenberg Lie algebra of dimension $2m+1$. Furthermore, we determine all the (flat and non-flat) non-degenerate CR structures on each of these Lie groups.

math.DG↗
arXiv2023

A note on locally conformally Kaehler structures and small deformations of Hopf manifolds

A Hopf manifold is a compact complex manifold of which the universal covering is C^n\{0}. In this note we show that any Hopf manifold admits a locally conformally Kaehler structure (shortly lcK structure), by constructing a complex analytic family around a Hopf manifold of diagonal type, which admits a lcK potential, and applying a well known fact (due to Ornea and Verbitsky) that the property of lcK potential is preserved under a complex analytic family over a sufficiently small parameter space.

math.DG↗
arXiv2021

Unimodular Sasaki and Vaisman Lie groups

This is a continuation of our study on homogeneous locally conformally Kaehler and Sasaki manifolds. In a recent work, applying the technique of modification we have determined all homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups, up to modifications. In this paper we determine all such modifications explicitly for the case of Lie groups, obtaining a complete classification of unimodular Sasaki and Vaisman Lie groups. Furthermore, we determine the biholomorphism type of a simply connected unimodular Vaisman Lie group of each type.

math.DG↗
arXiv2020

Homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups

A Vaisman manifold is a special kind of locally conformally Kaehler manifold, which is closely related to a Sasaki manifold. In this paper we show a basic structure theorem of simply connected homogeneous Sasaki and Vaisman manifods of unimodular Lie groups, up to holomorphic isometry. For the case of unimodular Lie groups, we obtain a complete classification of simply connected Sasaki and Vaisman unimodular Lie groups, up to modification.

math.DG↗
arXiv2015

Homogeneous locally conformally Kaehler and Sasaki manifolds

We prove various classification results for homogeneous locally conformally symplectic manifolds. In particular, we show that a homogeneous locally conformally Kaehler manifold of a reductive group is of Vaisman type, if the normalizer of the isotropy group is compact. We also show that such a result does not hold in the case of noncompact normalizer and determine all left-invariant locally conformally Kaehler structures on reductive Lie groups.

math.DG↗
arXiv2014

Locally conformally Kaehler structures on homogeneous spaces

We will discuss in this paper homogeneous locally conformally Keahler (or shortly homogeneous l.c.K.) manifolds and locally homogeneous l.c.K. manifolds from various aspects of study in the field of l.c.K. geometry. We will provide a survey of known results along with some new results and observations; in particular we make a complete classification of 4-dimensional homogeneous and locally homogeneous l.c.K. manifolds in terms of Lie algebras.

math.DG↗
arXiv2013

Compact Homogeneous Locally Conformally Kaehler Manifolds

In this paper we show as main results two structure theorems of a compact homogeneous locally conformally Kaehler (or shortly l.c.K.) manifold, a holomorphic structure theorem asserting that it has a structure of holomorphic principal fiber bundle over a flag manifold with fiber a 1-dimensional complex torus, and a metric structure theorem asserting that it is necessarily of Vaisman type. We also discuss and determine l.c.K. reductive Lie groups and compact locally homogeneous l.c.K. manifolds of reductive Lie groups.

math.CV↗
arXiv2008

Complex and Kaehler structures on compact homogeneous manifolds - their existence, classification and moduli problem

The existence of some complex geometrical structures on a compact manifold such as complex structures, Kaehler (pseudo-Kaehler) structures often impose certain restrictions on its underling topological or differentiable manifold. In this article we survey recent developments in the study of the existence, classification and moduli problems of such structures on compact homogeneous manifolds.

math.CV↗
arXiv2008

Complex and Kaehler structures on compact 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).

math.CV↗
arXiv2008

Small deformations and non-left-invariant complex structures on a compact solvmanifold

We observed in our previous paper that all the complex structures on four-dimensional compact solvmanifolds, including tori, are left-invariant. In this paper we will give an example of a six-dimensional compact solvmanifold which admits a continuous family of non-left-invariant complex structures. Furthermore, we will make a complete classification of three-dimensional compact homogeneous complex solvmanifolds; and determine which of them admit pseudo-Kaehler structures.

math.CV↗
arXiv2004

A note on compact solvmanifolds with Kaehler structures

A solvmanifold is a compact differentiable manifold M on which a connected solvable Lie group G acts transitively. As the main result, we will see, applying a result of Arapura and Nori on solvable Kaehler groups and some of the author's previous results, that a compact solvmanifold admits a Kaehler structure if and only if it is a finite quotient of a complex torus which has a structure of a complex torus bundle over a complex torus. There are also quite a few comments and remarks on and around this result.

math.CV↗
arXiv2004

Four-dimensional compact solvmanifolds with and without complex analytic structures

We classify four-dimensional compact solvmanifolds up to diffeomorphism, while determining which of them have complex analytic structures. In particular, we shall see that a four-dimensional compact solvmanifold S can be written, up to double covering, as G/L where G is a simply connected solvable Lie group and L is a lattice of G, and every complex structure J on S is the canonical complex structure induced from a left-invariant complex structure on G. We also give a complete list of all the complex structures on four-dimensional compact homogeneous spaces, referring to their corresponding complex surfaces.

math.CV↗