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Kazuyuki Hasegawa

Publications and source records attributed to Kazuyuki Hasegawa.

5 recordsLinked to original sources
arXiv2026

Quaternionic complex manifolds and fixed-point sets of $S^{1}$-actions

In this paper, we study fixed-point sets of $S^{1}$-actions and compatible complex structures on quaternionic manifolds. We obtain an equation involving the first Chern classes of the fixed-point set and of a quaternionically flat manifold with compatible complex structure of closed type. In addition, if the first Chern class of the fixed-point set is not trivial, then the quaternionic manifold does not admit hypercomplex structures containing given compatible complex structure on any open set containing the fixed-point set. Moreover, we determine the connected components of the fixed-point set arising from quaternionic $S^{1}$-actions on the quaternionic projective space. We apply these results to Pontecorvo's example $\mathrm{SO}^{\ast}(2n+2)/\mathrm{SO}^{\ast}(2n) \times \mathrm{SO}^{\ast}(2)$.

math.DG↗
arXiv2025

Intrinsic characterization of projective special complex manifolds

We define the notion of an $S^1$-bundle of projective special complex base type and construct a conical special complex manifold from it. Consequently the base space of such an $S^{1}$-bundle can be realized as $\mathbb{C}^{\ast}$-quotient of a conical special complex manifold. As a corollary, we give an intrinsic characterization of a projective special complex manifold generalizing Mantegazza's characterization of a projective special Kähler manifold. Our characterization is in the language of c-projective structures. As an application, a non-trivial $S^1$-family of Obata-Ricci-flat hypercomplex structures (given by a generalization of the rigid c-map) on the tangent bundle of the total space of a $\mathbb{C}^*$-bundle over a complex manifold with certain kind of c-projective structure is constructed. Finally, we show that the quaternionic structure underlying any of these hypercomplex structures is in general not flat and that its flatness implies the vanishing of the c-projective Weyl tensor of the base of the $\mathbb{C}^*$-bundle. Conversely, any c-projectively flat complex manifold satisfying a cohomological integrality condition gives rise to a flat quaternionic structure.

math.DG↗
arXiv2022

The H/Q-correspondence and a generalization of the supergravity c-map

Given a hypercomplex manifold with a rotating vector field (and additional data), we construct a conical hypercomplex manifold. As a consequence, we associate a quaternionic manifold to a hypercomplex manifold of the same dimension with a rotating vector field. This is a generalization of the HK/QK-correspondence. As an application, we show that a quaternionic manifold can be associated to a conical special complex manifold of half its dimension. Furthermore, a projective special complex manifold (with a canonical c-projective structure) associates with a quaternionic manifold. The latter is a generalization of the supergravity c-map. We do also show that the tangent bundle of any special complex manifold carries a canonical Ricci-flat hypercomplex structure, thereby generalizing the rigid c-map.

math.DG↗
arXiv2019

The quaternionic/hypercomplex-correspondence

Given a quaternionic manifold $M$ with a certain $\mathrm{U}(1)$-symmetry, we construct a hypercomplex manifold $M'$ of the same dimension. This construction generalizes the quaternionic Kähler/hyper-Kähler-correspondence. As an example of this construction, we obtain a compact homogeneous hypercomplex manifold which does not admit any hyper-Kähler structure. Therefore our construction is a proper generalization of the quaternionic Kähler/hyper-Kähler-correspondence.

math.DG↗