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Hiroaki Ishida

Publications and source records attributed to Hiroaki Ishida.

16 recordsLinked to original sources
arXiv2026

GIT stability and biquotients of $SU(3)$

We study double-sided actions of $(\mathbb{C}^*)^2$ on $SL(3,\mathbb{C})/U$ and the associated quotients, where $U$ is a maximal unipotent subgroup of $SL(3,\mathbb{C})$. The main results of this paper are a sufficient condition for the double-sided quotient to agree with the quotient in terms of the geometric invariant theory (GIT), and an explicit necessary and sufficient condition for $SL(3,\mathbb{C})/U$ to agree with the $χ$-stable locus in its affine closure. We apply this result to characterize certain complex structures on $SU(3)$ which are not left invariant by means of the GIT quotient.

math.AG↗
arXiv2024

Double sided torus actions and complex geometry on $SU(3)$

We construct explicit complex structures and transversely Kähler holomorphic foliations on $SU(3)$ corresponding to variations of real quadratic equations on a complex quadric in $\mathbb{C}^{6}$ as generalizations of left-invariant complex structures on $SU(3) $ and an invariant Kähler structure on the flag variety $SU(3)/T$.Consequently, we obtain orbifold variants of the flag variety $SU(3)/T$ as quotients of double sided torus actions.

math.CV↗
arXiv2020

Basic cohomology of canonical holomorphic foliations on complex moment-angle manifolds

We describe the basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology ring of a complete simplicial toric variety due to Danilov and Jurkiewicz. This settles a question of Battaglia and Zaffran, who previously computed the basic Betti numbers for the canonical holomorphic foliation in the case of a shellable fan. Our proof uses an Eilenberg-Moore spectral sequence argument; the key ingredient is the formality of the Cartan model for the torus action on a moment-angle manifold. We develop the concept of transverse equivalence as an important tool for studying smooth and holomorphic foliated manifolds. For an arbitrary complex manifold with a maximal torus action, we show that it is transverse equivalent to a moment-angle manifold and therefore has the same basic cohomology.

math.DG↗
arXiv2018

Towards transverse toric geometry

We study certain foliated complex manifolds that behave similarly to complete nonsingular toric varieties. We classify them by combinatorial objects that we call marked fans. We describe the basic cohomology algebras of them in terms of corresponding marked fans. We also study the basic Dolbeault cohomology algebras of them.

math.AG↗
arXiv2018

Transverse Kähler structures on central foliations of complex manifolds

For a compact complex manifold, we introduce holomorphic foliations associated with certain abelian subgroups of the automorphism group. Such foliations are generalizations of holomorphic principal torus bundles. If there exists a transverse Kähler structure on such a foliation, then we obtain a nice differential graded algebra which is quasi-isomorphic to the de Rham complex and a nice differential bigraded algebra which is quasi-isomorphic to the Dolbeault complex like the formality of compact Kähler manifolds. Moreover, under certain additional condition, we can develop Morgan's theory of mixed Hodge structures as similar to the study on smooth algebraic varieties.

math.CV↗
arXiv2015

Torus invariant transverse Kähler foliations

In this paper, we show the convexity of the image of a moment map on a transverse symplectic manifold equipped with a torus action under a certain condition. We also study properties of moment maps in the case of transverse Kähler manifolds. As an application, we give a positive answer to the conjecture posed by Cupit-Foutou and Zaffran.

math.CV↗
arXiv2015

Complex manifolds with maximal torus actions

In this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds, in terms of combinatorial objects, which are triples $(Δ, \mathfrak{h}, G)$ of nonsingular complete fan $Δ$ in $\mathfrak{g}$, complex vector subspace $\mathfrak{h}$ of $\mathfrak{g}^{\mathbb{C}}$ and compact torus $G$ satisfying certain conditions. We also give an equivalence of categories with suitable definitions of morphisms in these families, like toric geometry. We obtain several results as applications of our equivalence of categories; complex structures on moment-angle manifolds, classification of holomorphic nondegenerate $\mathbb{C}^n$-actions on compact connected complex manifolds of complex dimension $n$, and construction of concrete examples of non-Kähler manifolds.

math.CV↗
arXiv2013

The cohomology ring of the GKM graph of a flag manifold of classical type

If a closed smooth manifold $M$ with an action of a torus $T$ satisfies certain conditions, then a labeled graph $\mG_M$ with labeling in $H^2(BT)$ is associated with $M$, which encodes a lot of geometrical information on $M$. For instance, the "graph cohomology" ring $\mHT^*(\mG_M)$ of $\mG_M$ is defined to be a subring of $\bigoplus_{v\in V(\mG_M)}H^*(BT)$, where $V(\mG_M)$ is the set of vertices of $\mG_M$, and is known to be often isomorphic to the equivariant cohomology $H^*_T(M)$ of $M$. In this paper, we determine the ring structure of $\mHT^*(\mG_M)$ with $\Z$ (resp. $\Z[1/2]$) coefficients when $M$ is a flag manifold of type A, B or D (resp. C) in an elementary way.

math.AT↗
arXiv2012

Todd genera of complex torus manifolds

In this paper, we prove that the Todd genus of a compact complex manifold $X$ of complex dimension $n$ with vanishing odd degree cohomology is one if the automorphism group of $X$ contains a compact $n$-dimensional torus $\Tn$ as a subgroup. This implies that if a quasitoric manifold admits an invariant complex structure, then it is equivariantly homeomorphic to a compact smooth toric variety, which gives a negative answer to a problem posed by Buchstaber-Panov.

math.AT↗
arXiv2012

Topological toric manifolds

We introduce the notion of a topological toric manifold and a topological fan and show that there is a bijection between omnioriented topological toric manifolds and complete non-singular topological fans. A topological toric manifold is a topological analogue of a toric manifold and the family of topological toric manifolds is much larger than that of toric manifolds. A topological fan is a combinatorial object generalizing the notion of a simplicial fan in toric geometry. Prior to this paper, two topological analogues of a toric manifold have been introduced. One is a quasitoric manifold and the other is a torus manifold. One major difference between the previous notions and topological toric manifolds is that the former support a smooth action of an $S^1$-torus while the latter support a smooth action of a $\C^*$-torus. We also discuss their relation in details.

math.AT↗
arXiv2010

(Filtered) cohomological rigidity of Bott towers

A Bott tower is an iterated $\CP ^1$-bundle over a point, where each $\CP ^1$-bundle is the projectivization of a rank $2$ decomposable complex vector bundle. For a Bott tower, the filtered cohomology is naturally defined. We show that isomorphism classes of Bott towers are distinguished by their filtered cohomology rings. We even show that any filtered cohomology ring isomorphism between two Bott towers is induced by an isomorphism of the Bott towers.

math.AT↗
arXiv2010

Symplectic real Bott manifolds

A real Bott manifold is the total space of an iterated $\RP ^1$-bundles over a point, where each $\RP^1$-bundle is the projectivization of a Whitney sum of two real line bundles. In this paper, we characterize real Bott manifolds which admit a symplectic form. In particular, it turns out that a real Bott manifold admits a symplectic form if and only if it is cohomologically symplectic. In this case, it admits even a Kähler structure. We also prove that any symplectic cohomology class of a real Bott manifolds can be represented by a symplectic form. Finally, we study the flux of a symplectic real Bott manifold.

math.SG↗