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Ryushi Goto

Publications and source records attributed to Ryushi Goto.

17 recordsLinked to original sources
arXiv2021

Matsushima-Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kähler manifolds of symplectic type

In Kähler geometry, Fujiki--Donaldson show that the scalar curvature arises as the moment map for Hamiltonian diffeomorphisms. In generalized Kähler geometry, one does not have suitable notions of Levi-Civita connection and curvature, however there still exists a precise framework for a moment map and the scalar curvature is defined as the moment map. Then a fundamental question is to understand the existence or non-existence of generalized Kähler structures with constant scalar curvature. In the paper, we study the Lie algebra of automorphisms of a generalized complex manifold. We assume that $H^{1}(M)=0$. Then we show that the Lie algebra of the automorphisms is a reductive Lie algebra if a generalized complex manifold admits a generalized Kähler structure of symplectic type with constant scalar curvature. This is a generalization of Matsushima and Lichnerowicz theorem in Kähler geometry. We explicitly calculate the Lie algebra of the automorphisms of a generalized complex structure given by a cubic curve on $\Bbb C P^2$. Cubic curves are classified into nine cases (see Figure.$1 -- 9$). In the three cases as in Figures. 7, 8 and 9, the Lie algebra of the automorphisms is not reductive and there is an obstruction to the existence of generalized Kähler structures of symplectic type with constant scalar curvature in the three cases. We also discuss deformations starting from an ordinary Kähler manifold $(X,ω)$ with constant scalar curvature and show that nontrivial generalized Kähler structures of symplectic type with constant scalar curvature arise as deformations if the Lie algebra of automorphisms of $X$ is trivial.

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arXiv2021

Generalized scalar curvature and modified moment map in generalized K\"ahler geometry

We introduce a notion of generalized scalar curvature for generalized K\"ahler manifolds using the pure spinor formalism. For an arbitrary compact generalized K\"ahler manifold, we develop an extended action of generalized Hamiltonians involving an abelian Lie algebra. This leads naturally to the notion of a modified moment map, which satisfies the moment-map condition only modulo the infinitesimal action of this abelian Lie algebra. We prove that this modified moment map is given by the generalized scalar curvature. Thus our result extends the theorem of Fujiki and Donaldson, which realizes the scalar curvature as a moment map in ordinary K\"ahler geometry. We also study explicit examples. Although a compact connected Lie group admits a K\"ahler structure only in the torus case, every connected compact even-dimensional Lie group admits generalized K\"ahler structures with constant generalized scalar curvature. In particular, we explicitly construct such structures on the standard Hopf surface.

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arXiv2020

Kobayashi-Hitchin correspondence of generalized holomorphic vector bundles over generalized Kahler manifolds of symplectic type

In the previous paper \cite{Goto_2017}, the notion of an Einstein-Hermitian metric of a generalized holomorphic vector bundle over a generalized Kahler manifold of symplectic type was introduced from the moment map framework. In this paper we establish a Kobayashi-Hitchin correspondence, that is, the equivalence of the existence of an Einstein-Hermitian metric and $ψ$-polystability of a generalized holomorphic vector bundle over a compact generalized Kahler manifold of symplectic type. Poisson modules provide intriguing generalized holomorphic vector bundles and we obtain $ψ$-stable Poisson modules over complex surfaces which are not stable in the ordinary sense.

math.DG↗
arXiv2018

Scalar curvature as moment map in generalized Kahler geometry

It is known that the scalar curvature arises as the moment map in Kahler geometry. In pursuit of this analogy, we introduce the notion of a moment map in generalized Kahler geometry which gives the definition of a generalized scalar curvature on a generalized Kahler manifold. From the viewpoint of the moment map, we obtain the generalized Ricci form which is a representative of the first Chern class of the anticanonical line bundle. It turns out that infinitesimal deformations of generalized Kahler structures with constant generalized scalar curvature are finite dimensional on a compact manifold. Explicit descriptions of the generalized Ricci form and the generalized scalar curvature are given on a generalized Kahler manifold of type $(0,0)$. Poisson structures constructed from a Kahler action of $T^m$ on a Kahler-Einstein manifold give intriguing deformations of generalized Kahler-Einstein structures. In particular, the anticanical divisor consists of three lines on $C P^2$ in general position yields nontrivial examples of generalized Kahler-Einsein structures

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arXiv2017

Moduli spaces of Einstein-Hermitian generalized connections over generalized Kahler manifolds of symplectic type

From a view point of the moment map, we shall introduce the notion of Einstein-Hermitian generalized connections over a generalized Kähler manifold of symplectic type. We show that moduli spaces of Einstein-Hermitian generalized connections arise as the Kähler quotients. The deformation complex of Einstein-Hermitian generalized connections is an elliptic complex and it turns out that the smooth part of the moduli space is a finite dimensional Kähler manifold. The canonical line bundle over a generalized Kähler manifold of symplectic type has the canonical generalized connection and its curvature coincides with "the scalar curvature as the moment map" which is defined in the previous paper [Goto_2016]. Kähler-Ricci solitons provide examples of Einstein-Hermitian generalized connections and Einstein Hermitian co-Higgs bundles are also discussed.

math.DG↗
arXiv2015

Unobstructed deformations of generalized complex structures induced by $C^\infty$ logarithmic symplectic structures and logarithmic Poisson structures

We shall introduce the notion of $C^\infty$ logarithmic symplectic structures on a differentiable manifold which is an analog of the one of logarithmic symplectic structures in the holomorphic category. We show that the generalized complex structure induced by a $C^\infty$ logarithmic symplectic structure has unobstructed deformations which are parametrized by an open set of the second de Rham cohomology group of the complement of type changing loci if the type changing loci are smooth. Complex surfaces with smooth effective anti-canonical divisors admit unobstructed deformations of generalized complex structures such as del pezzo surfaces and Hirzebruch surfaces. We also give some calculations of Poisson cohomology groups on these surfaces. Generalized complex structures ${\cal J}_m$ on the connected sum $(2k-1)\Bbb C P^2\# (10k-1)\ol {\Bbb C P^2}$ as in \cite{Cavalcanti_Gualtieri_2006}, \cite{Goto_Hayano} are induced by $C^\infty$ logarithmic symplectic structures modulo the action of $b$-fields and it turns out that {generalized complex structure}s ${\cal J}_m$ have unobstructed deformations of dimension $12k+2m-3$.}

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arXiv2013

C^\infty-logarithmic transformations and generalized complex structures

Applying logarithmic transformations along 2-tori, we construct a generalized complex structure J_n with n type changing luci for every $n\geq 0$ on genus 1-Lefschetz fibrations with a cusp neighborhood, which include elliptic surfaces with non-zero euler characteristic. Applying a technique of broken Lefschetz fibrations, we further obtain twisted generalized complex structures with arbitrary large numbers of connected components of type changing loci on the manifold which is obtained from a symplectic manifold by logarithmic transformations of multiplicity 0 on a symplectic 2-torus with trivial normal bundle. The connected sums $(2m+1)S^2\times S^2$ for $m\geq 0$, $(2n-1)\C P^2# (10n-1)\ol{\C P^2}$ and $S^1\times S^3$ admit twisted generalized complex structures J_n with n type changing luci for arbitrary large n.

math.DG↗
arXiv2011

Calabi-Yau structures and Einstein-Sasakian structures on crepant resolutions of isolated singularities

Let $X_0$ be an affine variety with only normal isolated singularity $p$ and $π: X\to X_0$ a smooth resolution of the singularity with trivial canonical line bundle $K_X$. If the complement of the affine variety $X_0\backslash\{p\}$ is the cone $C(S)=\Bbb R_{>0}\times S$ of an Einstein-Sasakian manifold $S$, we shall prove that the crepant resolution $X$ of $X_0$ admits a complete Ricci-flat Kähler metric in every Kähler class in $H^2(X)$. We apply the continuity method for solving the Monge-Ampère equation to obtain a relevant existence theorem and a uniqueness theorem of Ricci-flat conical Kähler metrics. By using the vanishing theorem on the crepant resolution $X$ and the Hodge and Lefschetz decompositions of the basic cohomology groups on the Sasakian manifold $S$, we construct an initial Kähler metric in every Kähler class on which the existence theorem can be applied.We show there are many examples of Ricci-flat complete Kähler manifolds arising as crepant resolutions.

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arXiv2010

On the stability of locally conformal Kaehler structures

In this article we develop a new approach to the problem of the stability of locally conformally Kähler structures (l.c.k structures) under small deformations of complex structures and deformations of flat line bundles. We show that under the certain cohomological condition the stability of l.c.k structures does hold. We apply our approach to Hopf manifolds and its generalizations to obtain the stability of l.c.k structures which do not have potential in general. We give an explicit description of the cohomological obstructions of the stability of l.c.k structures on Inoue surfaces with $b_2=0$.

math.DG↗
arXiv2010

Unobstructed K-deformations of Generalized Complex Structures and Bihermitian Structures

We introduce K-deformations of generalized complex structures on a compact Kahler manifold $M=(X, J)$ with an effective anti-canonical divisor and show that obstructions to K-deformations of generalized complex structures on $M$ always vanish. Applying the stability theorem of generalized Kahler structures, together with unobstructed K-deformations, we construct deformations of bihermitian structures in the form $(J, J^-_t, h_t)$ on a compact Kahler surface with a non-zero holomorphic Poisson structure. Then we prove that a compact Kahler surface $S$ admits a non-trivial bihermitian structure if and only if $S$ has a non-zero holomorphic Poisson structure.

math.DG↗
arXiv2009

Deformations of generalized complex and generalized Kahler structures

In this paper we obtain a stability theorem of generalized Kahler structures with one pure spinor under small deformations of generalized complex structures. (This is analogous to the stability theorem of Kahler manifolds by Kodaira-Spencer.) We apply the stability theorem to a class of compact Kahler manifolds which admits deformations to generalized complex manifolds and obtain non-trivial generalized Kahler structures on Fano surfaces and toric Kahler manifolds. In particular, we show that holomorphic Poisson structures on a Kahler manifold induce deformations of generalized Kahler structures.

math.DG↗
arXiv2009

Deformations of Generalized Kahler Structures and Bihermitian Structures

Let $(X, J)$ be a compact Kahler manifold with a non-zero holomorphic Poisson structure $β$. If the obstruction space for deformations of generalized complex structures on $(X, J)$ vanishes, we obtain a family of deformations of non-trivial bihermitian structures $(J, J^-_t, h_t)$ on $X$ by using $β$. In addition, if the class $[β\cdot ω]$ does not vanish for a Kähler form $ω$, then the complex structure $J_t^-$ is not equivalent to $J$ for small $t\neq 0$ under diffeomorphisms. Our method is based on the construction of generalized complex and Kahler structures developed in \cite{Go1} and \cite{Go2}. As applications, we obtain such deformations of bihermitian structures on del Pezzo surfaces, the Hirtzebruch surfaces $F_2, F_3$ and degenerate del Pezzo surfaces. Further we show that del Pezzo surfaces $S_n (5\leq n\leq 8)$, $F_2$ and degenerate del Pezzo surfaces admit bihermitian structures for which $(X, J^-_t)$ is not biholomorphic to $(X, J)$ for small $t\neq 0$.

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arXiv2008

Poisson structures and generalized Kahler structures

Let X be a compact Kahler manifold with a non-trivial holomorphic Poisson structure. Then there exist deformations of non-trivial generalized Kahler structures with one pure spinor on X. We prove that every Poisson submanifold of X is a generalized Kahler submanifold with respect to the deformed generalized Kahler structures and provide non-trivial examples of generalized Kahler submanifolds arising as holomorphic Poisson submanifolds. We also obtain unobstructed deformations of bi-Hermitian structures constructed from Poisson structures.

math.DG↗
arXiv2005

On deformations of generalized Calabi-Yau, hyperKähler, $G_2$ and Spin$(7)$ structures I

In this paper we will introduce a new notion of geometric structures defined by systems of closed differential forms in term of the Clifford algebra of the direct sum of the tangent bundle and the cotangent bundle on a manifold. We develop a unified approach of a deformation problem and establish a criterion of unobstructed deformations of the structures from a cohomological point of view. We construct the moduli spaces of the structures by using the action of b-fields and show that the period map of the moduli space is locally injective under a cohomological condition (the local Torelli type theorem). We apply our approach to generalized Calabi-Yau (metrical) structures and obtain an analog of the theorem by Bogomolov-Tian-Todorov. Further we prove that deformations of generalized Spin(7) structures are unobstructed.

math.DG↗
arXiv2002

Deformations of calibrations, {Calabi-Yau, HyperKähler, $G_2$ and Spin(7) structures}

We shall develop a new deformation theory of geometric structures in terms of closed differential forms. This theory is a generalization of Kodaira -Spencer theory and further we obtain a criterion of unobstructed deformations. We apply this theory to certain geometric structures: Calabi-Yau, HyperKähler, $\G$ and $\Spin$ structures and show that these deformation spaces are smooth in a systematic way.

math.DG↗
arXiv2001

Moduli spaces of special Lagrangians and Kähler-Einstein structures

We shall construct a moduli space of pairs of Kähler-Einstein structures and special lagrangians and obtain smoothness of the moduli space of these pairs. Further we show that the moduli space of these pairs is locally embedded in a certain relative cohomology group.

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