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Nobuhiro Honda

Publications and source records attributed to Nobuhiro Honda.

At least 19 recordsLinked to original sources
arXiv2026

Fibrations on the 6-sphere and Clemens threefolds

Let $Z$ be a compact, connected $3$-dimensional complex manifold with vanishing first and second Betti numbers and non-vanishing Euler characteristic. We prove that there is no surjective holomorphic mapping from $Z$ onto any $2$-dimensional complex space. In other words, $Z$ can only possibly fiber over a curve. This result applies in particular to a class of threefolds, known as Clemens threefolds, which are diffeomorphic to a connected sum $k \# (S^3 \times S^3)$ for $k \geq 2$. This result also gives a new restriction on any hypothetical complex structure on the $6$-sphere $S^6$.

math.AG↗
arXiv2026

On the twistor spaces of ALE gravitational instantons of type $A_{\rm odd}$

We study the twistor spaces of toric ALE gravitational instantons of type $A_{2n-1}$ and the associated non-standard minitwistor spaces introduced by Hitchin. By analyzing the base locus of the linear system that induces the quotient meromorphic map from the compactified twistor space, we explicitly determine the images of certain distinguished twistor lines as hyperplane sections of the minitwistor space. Using this family of special minitwistor lines as boundary data, we describe the $3$-dimensional family of real minitwistor lines arising from the instanton. The central sphere in the gravitational instanton appears naturally throughout the analysis.

math.DG↗
arXiv2025

Hyperelliptic curves, minitwistors, and spacelike Zoll spaces

We construct a compact minitwistor space from a hyperelliptic curve with real structure and show that it yields a lot of new Lorentzian Einstein-Weyl spaces all of which are diffeomorphic to the 3-dimensional deSitter space. These structures are real analytic, admit a circle symmetry and moreover, all their spacelike geodesics are closed and simple. The number of the nodes of minitwistor lines on the minitwistor space is equal to the genus of the hyperelliptic curve and is taken arbitrarily. These Einstein-Weyl structures deform as the hyperelliptic curves deform, and so have $(2g-1)$-dimensional moduli space, where $g$ is the genus of the hyperelliptic curve. A relationship between the minitwistor spaces recently obtained by Hitchin from ALE gravitational instantons is also given for A$_{\rm odd}$-type.

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arXiv2024

The Einstein-Weyl spaces associated to Segre quartic surfaces

We find explicit examples of compact minitwistor spaces of genus one, whose Einstein-Weyl spaces have a connected component that is diffeomorphic to the de Sitter space. The induced Einstein-Weyl structure on it is Lorenzian, real-analytic, whose spacelike geodesics are all closed and simple. The identity component of the automorphism group of the Einstein-Weyl structure is the circle and therefore the structure is not isomorphic to the standard de Sitter structure. We show that these Einstein-Weyl structures deform as the Segre surfaces deform and converge to the standard de Sitter structure. The minitwistor spaces we study are the so-called Segre quartic surfaces. They have a real pair of nodes, which play a crucial role in proving the above results. These singularities also allow us to construct explicit examples of non-compact complex surfaces that do not admit any compactification.

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arXiv2021

On the cuspidal locus in the dual varieties of Segre quartic surface

Motivated by a kind of Penrose correspondence, we investigate the space of hyperplane sections of Segre quartic surfaces which have an ordinary cusp. We show that the space of such hyperplane sections is empty for two kinds of Segre surfaces, and it is a connected surface for all other kinds of Segre surfaces. We also show that when it is non-empty, the closure of the space is either birational to the surface itself or birational to a double covering of the surface, whose branch divisor consists of some specific lines on the surface.

math.AG↗
arXiv2020

Segre quartic surfaces and minitwistor spaces

Segre surfaces in the title mean quartic surfaces in $\mathbb{CP}^4$ which are the images of weak del Pezzo surfaces of degree four under the anti-canonical map. We first show that minimal minitwistor spaces with genus one are exactly Segre quartic surfaces. By a kind of Penrose correspondence, Zariski open subsets of the projective dual varieties of these surfaces admit Einstein-Weyl structure. We investigate structures of these dual varieties in detail. In particular, we determine the degrees of these varieties (namely the classes of the Segre surfaces), as well as structure of several components of the boundary divisors which are the complements of the Einstein-Weyl spaces in the projective dual varieties.

math.AG↗
arXiv2018

Twistors, quartics, and del Pezzo fibrations

We investigate the structure of a variety of new Moishezon twistor spaces, by utilizing the pluri-half-anti-canonical map from the twistor spaces. Each of these twistor spaces is bimeromorphic to a double covering of a scroll of planes over a rational normal curve, and the branch divisor of the double cover is a cut of the scroll by a quartic hypersurface. In particular, the double covering has a pencil of Del Pezzo surfaces of degree two. Correspondingly, the twistor spaces have a pencil of rational surfaces with big anti-canonical class. The base locus of the last pencil is a cycle of rational curves, and it is an anti-canonical curve on smooth members of the pencil. These twistor spaces are naturally classified into four types according to the type of singularities of the branch divisor, or equivalently, those of the Del Pezzo surfaces in the pencil. We also show that the quartic hypersurface satisfies a strong constraint and as a result the defining polynomial of the quartic hypersurface has to be of a specific form. Together with our previous result, the present result completes a classification of Moishezon twistor spaces whose half-anti-canonical system is a pencil. Twistor spaces whose half-anti-canonical system is larger than pencil have been understood for a long time before. In the opposite direction, no example is known of a Moishezon twistor space whose half-anti-canonical system is smaller than a pencil.

math.AG↗
arXiv2017

Algebraic dimension of twistor spaces whose fundamental system is a pencil

We show that the algebraic dimension of a twistor space over n#CP^2 cannot be two if n>4 and the fundamental system (i.e. the linear system associated to the half-anti-canonical bundle, which is available on any twistor space) is a pencil. This means that if the algebraic dimension of a twistor space on n#CP^2, n>4, is two, then the fundamental system either is empty or consists of a single member. The existence problem for a twistor space on n#CP^2 with algebraic dimension two is open for n>4.

math.DG↗
arXiv2015

Geometry of some twistor spaces of algebraic dimension one

It is shown that there exists a twistor space on the $n$-fold connected sum of complex projective planes $n\mathbb{CP}^2$, whose algebraic dimension is one and whose general fiber of the algebraic reduction is birational to an elliptic ruled surface or a K3 surface. The former kind of twistor spaces are constructed over $n\mathbb{CP}^2$ for any $n\ge 5$, while the latter kind of example is constructed over $5\mathbb{CP}^2$. Both of these seem to be the first such example on $n\mathbb{CP}^2$. The algebraic reduction in these examples is induced by the anti-canonical system of the twistor spaces. It is also shown that the former kind of twistor spaces contain a pair of non-normal Hopf surfaces.

math.DG↗
arXiv2014

Scalar Flat Kähler Metrics on Affine Bundles over $\mathbb{CP}^1$

We show that the total space of any affine $\mathbb{C}$-bundle over $\mathbb{CP}^1$ with negative degree admits an ALE scalar-flat Kähler metric. Here the degree of an affine bundle means the negative of the self-intersection number of the section at infinity in a natural compactification of the bundle, and so for line bundles it agrees with the usual notion of the degree.

math.DG↗
arXiv2012

Deformation of LeBrun's ALE metrics with negative mass

In this article we investigate deformations of a scalar-flat Kähler metric on the total space of complex line bundles over CP^1 constructed by C. LeBrun. In particular, we find that the metric is included in a one-dimensional family of such metrics on the four-manifold, where the complex structure in the deformation is not the standard one.

math.DG↗
arXiv2012

Toric LeBrun metrics and Joyce metrics

We show that, on the connected sum of complex projective planes, any toric LeBrun metric can be identified with a Joyce metric admitting a semi-free circle action through an explicit conformal equivalence. A crucial ingredient of the proof is an explicit connection form for toric LeBrun metrics.

math.DG↗
arXiv2011

Moishezon twistor spaces on 4CP^2

In this paper we classify all Moishezon twistor spaces on 4CP^2. The classification is given in terms of the structure of the anticanonical system of the twistor spaces. We show that the anticanonical map satisfies one of the following three properties: (a) birational over the image, (b) two to one over the image, or (c) the image is two-dimensional. We determine structure of the images for each case in explicit forms. Then we intensively investigate structure of the twistor spaces in the case (b), and determine the defining equation of the branch divisor of the anticanonical map.

math.DG↗
arXiv2011

Double solid twistor spaces II: general case

In this paper we investigate Moishezon twistor spaces which have a structure of double covering over a very simple rational threefold. These spaces can be regarded as a direct generalization of the twistor spaces studied by Poon and Kreussler-Kurke to the case of arbitrary signature. In particular, the branch divisor of the double covering is a cut of the rational threefold by a single quartic hypersurface. A defining equation of the hypersurface is determined in an explicit form. We also show that these twistor spaces interpolate LeBrun twistor spaces and the twistor spaces constructed in math.DG/0701278.

math.DG↗
arXiv2011

Geometry of generic Moishezon twistor spaces on 4CP^2, II: degenerate cases

We continue to study twistor spaces on the connected sum of four complex projective planes, whose anticanonical map is of degree two over the image. In particular, we determine the defining equation of the branch divisor of the anticanonical map in an explicit form. Together with previous two articles (arXiv:1009.3153 and arXiv:0705.0060), this completes explicit description of all such twistor spaces.

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arXiv2011

Classification of Moishezon twistor spaces on 4CP^2

In this paper we provide a classification of all Moishezon twistor spaces on the connected sum of four complex projective planes. This is given by means of the anticanonical system of the twistor spaces. In particular, we show that the anticanonical map is birational, two to one over the image, or otherwise the image of the anticanonical map is a rational surface. We also obtain the structure of the images of the anticanonical map in each of the three cases in explicit forms.

math.DG↗
arXiv2010

Geomety of generic Moishezon twistor spaces on 4CP^2

In this paper we investigate a family of Moishezon twistor spaces on the connected sum of 4 complex projective planes, which can be regarded as a direct generalization of the twistor spaces on 3CP^2 of double solid type studied by Poon and Kreussler-Kurke. These twistor spaces have a natural structure of double covering over a scroll of 2-planes over a conic. We determine the defining equations of the branch divisors in an explicit form, which are very similar to the case of 3CP^2. Using these explicit description we compute the dimension of the moduli spaces of these twistor spaces. Also we observe that similarly to the case of 3CP^2, these twistor spaces can also be considered as generic Moishezon twistor spaces on 4CP^2. We obtain these results by analyzing the anticanonical map of the twistor spaces in detail, which enables us to give an explicit construction of the twistor spaces, up to small resolutions.

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