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Intrinsic characterization of projective special complex manifolds

Vicente Cortés, Kazuyuki Hasegawa

TL;DR

This work develops an intrinsic framework linking circle bundles of PSCB type to projective special complex manifolds via conical special complex total spaces, realizing the base as a $\mathbb{C}^{*}$-quotient of a conical manifold. It provides a c-projective perspective on characterizations of PSC manifolds, and uses this to construct a nontrivial $S^1$-family of Obata-Ricci-flat hypercomplex structures on tangent bundles of $\mathbb{C}^{*}$-bundles, tying quaternionic flatness to the vanishing of the base c-projective Weyl tensor. The results unify extrinsic and intrinsic viewpoints, recover projective special Kähler cases, and supply explicit low-dimensional examples, including a local model, thereby enriching the generalized rigid c-map and its quaternionic geometry. Moreover, the work shows that flat quaternionic structures arise precisely when the base c-projective curvature is flat together with a cohomological integrality condition, establishing a deep link between c-projective geometry and hypercomplex/quaternionic structures.

Abstract

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.

Intrinsic characterization of projective special complex manifolds

TL;DR

This work develops an intrinsic framework linking circle bundles of PSCB type to projective special complex manifolds via conical special complex total spaces, realizing the base as a -quotient of a conical manifold. It provides a c-projective perspective on characterizations of PSC manifolds, and uses this to construct a nontrivial -family of Obata-Ricci-flat hypercomplex structures on tangent bundles of -bundles, tying quaternionic flatness to the vanishing of the base c-projective Weyl tensor. The results unify extrinsic and intrinsic viewpoints, recover projective special Kähler cases, and supply explicit low-dimensional examples, including a local model, thereby enriching the generalized rigid c-map and its quaternionic geometry. Moreover, the work shows that flat quaternionic structures arise precisely when the base c-projective curvature is flat together with a cohomological integrality condition, establishing a deep link between c-projective geometry and hypercomplex/quaternionic structures.

Abstract

We define the notion of an -bundle of projective special complex base type and construct a conical special complex manifold from it. Consequently the base space of such an -bundle can be realized as -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 -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 -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 -bundle. Conversely, any c-projectively flat complex manifold satisfying a cohomological integrality condition gives rise to a flat quaternionic structure.
Paper Structure (17 sections, 61 theorems, 205 equations, 1 figure)

This paper contains 17 sections, 61 theorems, 205 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Projective special complex manifolds
  4. Realization of projective special complex manifolds as submanifolds of projective space
  5. The canonical c-projective structure
  6. Local expressions
  7. Characterization of projective special complex manifolds in terms of c-projective structures
  8. $\dim \bar{M} \geq 4$ case
  9. $\dim \bar{M}=2$ case
  10. The special case of projective special Kähler manifolds
  11. Low dimensional examples
  12. $\dim \bar{M}=4$ (products of surfaces)
  13. $\dim \bar{M}=2$
  14. An application to the generalized rigid c-map
  15. An $S^1$-family of hypercomplex structures obtained from a circle bundle of PSCB type
  16. ...and 2 more sections

Key Result

Lemma 2.2

$d^{D}A=0$.

Theorems & Definitions (130)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Proposition 2.6
  • proof
  • ...and 120 more