arXiv subjects
Publications and source records attributed to Takayuki Moriyama.
In this paper, we define a differential operator as a modified Dirac operator. Using the operator, we introduce a quaternionic $k$-vector field on a quaternionic Kähler manifold and show that any quaternionic $k$-vector field corresponds to a holomorphic $k$-vector field on the twistor space. We calculate the dimension of the space of quaternionic $k$-vector fields on $\mathbb{H}P^n$.
A Poisson structure is represented by a bivector whose Schouten bracket vanishes. We study a global Poisson structure on $S^4$ associated with a holomorphic Poisson structure on $\mathbb{CP}^3$. The space of the Poisson structures on $S^4$ is a real algebraic variety in the space of holomorphic Poisson structures on $\mathbb{CP}^3$. We generalize it to $\mathbb{HP}^n$ by using the twistor method. Furthermore, we provide examples of Poisson structures on $S^4$ associated with codimension one holomorphic foliations of degree 2 on $\mathbb{CP}^3$.
In this paper we study the deformation theory of submanifolds characterized by a system of differential forms and provide a criterion for deformations of such submanifolds to be unobstructed. We apply this deformation theory to special Legendrian submanifolds in Sasaki-Einstein manifolds. In general, special Legendrian deformations have the obstruction. However, we show that the deformation space of special Legendrian submanifolds is the intersection of two larger smooth deformation spaces of different types. We also prove that any special Legendrian submanifold admits smooth deformations, which are not special Legendrian deformations, given by harmonic $1$-forms.
A complex contact structure $γ$ is defined by a system of holomorphic local $1$-forms satisfying the completely non-integrability condition. The contact structure induces a subbundle ${\rm Ker}\, γ$ of the tangent bundle and a line bundle $L$. In this paper, we prove that the sheaf of holomorphic $k$-vectors on a complex contact manifold splits into the sum of $\mathcal{O}(\bigwedge^{k}{\rm Ker}\, γ)$ and $\mathcal{O}(L\otimes \bigwedge^{k-1} {\rm Ker}\, γ)$ as sheaves of {\it $\mathbb{C}$-module}. The theorem induces the short exact sequence of cohomology of holomorphic $k$-vectors, and we obtain vanishing theorems for the cohomology of $\mathcal{O}(\bigwedge^{k} \ker γ)$.
We show that every toric Sasaki-Einstein manifold $S$ admits a special Legendrian submanifold $L$ which arises as the link ${\rm fix}(τ)\cap S$ of the fixed point set ${\rm fix}(τ)$ of an anti-holomorphic involution $τ$ on the cone $C(S)$. In particular, an irregular toric Sasaki-Einstein manifold $S^{2}\times S^{3}$ has a special Legendrian torus $S^{1}\times S^{1}$. Moreover, we also obtain a special Legendrian submanifold in $\sharp m(S^{2}\times S^{3})$ for each $m\ge 1$.