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Mikiya Masuda

Publications and source records attributed to Mikiya Masuda.

At least 55 records · Page 3Linked to original sources
arXiv2015

Volume polynomials and duality algebras of multi-fans

We introduce a theory of volume polynomials and corresponding duality algebras of multi-fans. Any complete simplicial multi-fan $Δ$ determines a volume polynomial $V_Δ$ whose values are the volumes of multi-polytopes based on $Δ$. This homogeneous polynomial is further used to construct a Poincare duality algebra $\mathcal{A}^*(Δ)$. We study the structure and properties of $V_Δ$ and $\mathcal{A}^*(Δ)$ and give applications and connections to other subjects, such as Macaulay duality, Novik--Swartz theory of face rings of simplicial manifolds, generalizations of Minkowski's theorem on convex polytopes, cohomology of torus manifolds, computations of volumes, and linear relations on the powers of linear forms. In particular, we prove that the analogue of the $g$-theorem does not hold for multi-polytopes.

math.CO↗
arXiv2015

Root systems and symmetries of torus manifolds

We associate a root system to a finite set in a free abelian group and prove that its irreducible subsystem is of type A, B or D. We apply this general result to a torus manifold, where a torus manifold is a $2n$-dimensional connected closed smooth manifold with a smooth effective action of an $n$-dimensional compact torus having a fixed point, and show that if the torus action extends to a smooth action of a connected compact Lie group $G$, then a simple factor of the Lie algebra of $G$ is of type A, B or D. This gives an alternative proof to Wiemeler's theorem. We also discuss a similar problem for a torus manifold with an invariant stably complex structure. In this case only type A appears.

math.GT↗
arXiv2014

The equivariant cohomology rings of regular nilpotent Hessenberg varieties in Lie type A: a research announcement

Let $n$ be a fixed positive integer and $h: \{1,2,...,n\} \rightarrow \{1,2,...,n\}$ a Hessenberg function. The main result of this manuscript is to give a systematic method for producing an explicit presentation by generators and relations of the equivariant and ordinary cohomology rings (with $\mathbb{Q}$ coefficients) of any regular nilpotent Hessenberg variety $\mathrm{Hess}(h)$ in type A. Specifically, we give an explicit algorithm, depending only on the Hessenberg function $h$, which produces the $n$ defining relations $\{f_{h(j),j}\}_{j=1}^n$ in the equivariant cohomology ring. Our result generalizes known results: for the case $h=(2,3,4,...,n,n)$, which corresponds to the Peterson variety $\mathrm{Pet}_n$, we recover the presentation of $H^*_S(\mathrm{Pet}_n)$ given previously by Fukukawa, Harada, and Masuda. Moreover, in the case $h=(n,n,...,n)$, for which the corresponding regular nilpotent Hessenberg variety is the full flag variety $\mathrm{Flags}(\mathbb{C}^n)$, we can explicitly relate the generators of our ideal with those in the usual Borel presentation of the cohomology ring of $\mathrm{Flags}(\mathbb{C}^n)$. The proof of our main theorem includes an argument that the restriction homomorphism $H^*_T(\mathrm{Flags}(\mathbb{C}^n)) \to H^*_S(\mathrm{Hess}(h))$ is surjective. In this research announcement, we briefly recount the context and state our results; we also give a sketch of our proofs and conclude with a brief discussion of open questions. A manuscript containing more details and full proofs is forthcoming.

math.AG↗
arXiv2014

Toric origami structures on quasitoric manifolds

We construct quasitoric manifolds of dimension 6 and higher which are not equivariantly homeomorphic to any toric origami manifold. All necessary topological definitions and combinatorial constructions are given and the statement is reformulated in discrete geometrical terms. The problem reduces to existence of planar triangulations with certain coloring and metric properties.

math.AT↗
arXiv2014

The equivariant cohomology rings of Peterson varieties in all Lie types

Let G be a complex semisimple linear algebraic group and let Pet be the associated Peterson variety in the flag variety G/B. The main theorem of this note gives an efficient presentation of the equivariant cohomology ring H^*_S(Pet) of the Peterson variety as a quotient of a polynomial ring by an ideal J generated by quadratic polynomials, in the spirit of the Borel presentation of the cohomology of the flag variety. Here the group S \cong \mathbb{C}^* is a certain subgroup of a maximal torus T of G. Our description of the ideal J uses the Cartan matrix and is uniform across Lie types. In our arguments we use the Monk formula and Giambelli formula for the equivariant cohomology rings of Peterson varieties for all Lie types, as obtained in the work of Drellich. Our result generalizes a previous theorem of Fukukawa-Harada-Masuda, which was only for Lie type A.

math.AG↗
arXiv2014

Invariance of Pontrjagin classes for Bott manifolds

A Bott manifold is the total space of some iterated $\mathbb C P^1$-bundle over a point. We prove that any graded ring isomorphism between the cohomology rings of two Bott manifolds preserves their Pontrjagin classes. Moreover, we prove that such an isomorphism is induced from a diffeomorphism if the Bott manifolds are $\mathbb Z/2$-trivial, where a Bott manifold is called $\mathbb Z/2$-trivial if its cohomology ring with $\mathbb Z/2$-coefficient is isomorphic to that of a product of $\mathbb C P^1$'s.

math.AT↗
arXiv2013

The equivariant cohomology rings of Peterson varieties

The main result of this note is an efficient presentation of the $S^1$-equivariant cohomology ring of Peterson varieties (in type $A$) as a quotient of a polynomial ring by an ideal $\mathcal{J}$, in the spirit of the well-known Borel presentation of the cohomology of the flag variety. Our result simplifies previous presentations given by Harada-Tymoczko and Bayegan-Harada. In particular, our result gives an affirmative answer to a conjecture of Bayegan and Harada that the defining ideal $\mathcal{J}$ is generated by quadratics.

math.AG↗
arXiv2013

Toric origami manifolds and multi-fans

The notion of a toric origami manifold, which weakens the notion of a symplectic toric manifold, was introduced by Cannas da Silva-Guillemin-Pires \cite{ca-gu-pi11} and they show that toric origami manifolds bijectively correspond to origami templates via moment maps, where an origami template is a collection of Delzant polytopes with some folding data. Like a fan is associated to a Delzant polytope, a multi-fan introduced in \cite{ha-ma03} and \cite{masu99} can be associated to an oriented origami template. In this paper, we discuss their relationship and show that any simply connected compact smooth 4-manifold with a smooth action of $T^2$ can be a toric origami manifold. We also characterize products of even dimensional spheres which can be toric origami manifolds.

math.SG↗
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

Topology of iterated $S^1$-bundles

In this paper we investigate what kind of manifolds arise as the total spaces of iterated $S^1$-bundles. A real Bott tower studied in \cite{CMO}, \cite{KM} and \cite{KN} is an example of an iterated $S^1$-bundle. We show that the total space of an iterated $S^1$-bundle is homeomorphic to an infra-nilmanifold. A real Bott manifold, which is the total space of a real Bott tower, provides an example of a closed flat Riemannian manifold. We also show that real Bott manifolds are the only closed flat Riemannian manifolds obtained from iterated $\bbr{P}^1$-bundles. Finally we classify the homeomorphism types of the total spaces of iterated $S^1$-bundles in dimension 3.

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↗
arXiv2009

Classification of Q-trivial Bott manifolds

A Bott manifold is a closed smooth manifold obtained as the total space of an iterated $\C P^1$-bundle starting with a point, where each $\C P^1$-bundle is the projectivization of a Whitney sum of two complex line bundles. A \emph{$\Q$-trivial Bott manifold} of dimension $2n$ is a Bott manifold whose cohomology ring is isomorphic to that of $(\CP^1)^n$ with $\Q$-coefficients. We find all diffeomorphism types of $\Q$-trivial Bott manifolds and show that they are distinguished by their cohomology rings with $\Z$-coefficients. As a consequence, we see that the number of diffeomorphism classes in $\Q$-trivial Bott manifolds of dimension $2n$ is equal to the number of partitions of $n$. We even show that any cohomology ring isomorphism between two $\Q$-trivial Bott manifolds is induced by a diffeomorphism.

math.AT↗
arXiv2009

Buchstaber invariants of skeleta of a simplex

A moment-angle complex $\mathcal{Z}_K$ is a compact topological space associated with a finite simplicial complex $K$. It is realized as a subspace of a polydisk $(D^2)^m$, where $m$ is the number of vertices in $K$ and $D^2$ is the unit disk of the complex numbers $\C$, and the natural action of a torus $(S^1)^m$ on $(D^2)^m$ leaves $\mathcal{Z}_K$ invariant. The Buchstaber invariant $s(K)$ of $K$ is the maximum integer for which there is a subtorus of rank $s(K)$ acting on $\mathcal{Z}_K$ freely. The story above goes over the real numbers $\R$ in place of $\C$ and a real analogue of the Buchstaber invariant, denoted $s_\R(K)$, can be defined for $K$ and $s(K)\leqq s_\R(K)$. In this paper we will make some computations of $s_\R(K)$ when $K$ is a skeleton of a simplex. We take two approaches to find $s_\R(K)$ and the latter one turns out to be a problem of integer linear programming and of independent interest.

math.AT↗
arXiv2009

Symmetry of a symplectic toric manifold

The action of a torus group $T$ on a symplectic toric manifold $(M,ω)$ often extends to an effective action of a (non-abelian) compact Lie group $G$. We may think of $T$ and $G$ as compact Lie subgroups of the symplectomorphism group $Symp(M,ω)$ of $(M,ω)$. On the other hand, $(M,ω)$ is determined by the associated moment polytope $P$ by the result of Delzant. Therefore, the group $G$ should be estimated in terms of $P$ or we may say that a maximal compact Lie subgroup of $Symp(M,ω)$ containing the torus $T$ should be described in terms of $P$. In this paper, we introduce a root system $R(P)$ associated to $P$ and prove that any irreducible subsystem of $R(P)$ is of type A and the root system $Δ(G)$ of the group $G$ is a subsystem of $R(P)$ (so that $R(P)$ gives an upper bound for the identity component of $G$ and any irreducible factor of $Δ(G)$ is of type A). We also introduce a homomorphism from the normalizer of $T$ in $G$ to an automorphism group $Aut(P)$ of $P$, which detects the connected components of $G$. Finally we find a maximal compact Lie subgroup $G_{\max}$ of $Symp(M,ω)$ containing the torus $T$.

math.SG↗
arXiv2008

Quasitoric manifolds over a product of simplices

A quasitoric manifold (resp. a small cover) is a $2n$-dimensional (resp. an $n$-dimensional) smooth closed manifold with an effective locally standard action of $(S^1)^n$ (resp. $(\mathbb Z_2)^n$) whose orbit space is combinatorially an $n$-dimensional simple convex polytope $P$. In this paper we study them when $P$ is a product of simplices. A generalized Bott tower over $\F$, where $\F=\C$ or $\R$, is a sequence of projective bundles of the Whitney sum of $\F$-line bundles starting with a point. Each stage of the tower over $\F$, which we call a generalized Bott manifold, provides an example of quasitoric manifolds (when $\F=\C$) and small covers (when $\F=\R$) over a product of simplices. It turns out that every small cover over a product of simplices is equivalent (in the sense of Davis and Januszkiewicz \cite{DJ}) to a generalized Bott manifold. But this is not the case for quasitoric manifolds and we show that a quasitoric manifold over a product of simplices is equivalent to a generalized Bott manifold if and only if it admits an almost complex structure left invariant under the action. Finally, we show that a quasitoric manifold $M$ over a product of simplices is homeomorphic to a generalized Bott manifold if $M$ has the same cohomology ring as a product of complex projective spaces with $\Q$ coefficients.

math.AT↗