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Paolo Piccinni

Publications and source records attributed to Paolo Piccinni.

18 recordsLinked to original sources
arXiv2020

Clifford systems, Clifford structures, and their canonical differential forms

A comparison among different constructions of the quaternionic $4$-form $Φ_{Sp(2)Sp(1)}$ and of the Cayley calibration $Φ_{Spin(7)}$ shows that one can start for them from the same collections of "Kähler 2-forms", entering in dimension 8 both in quaternion Kähler and in $Spin(7)$ geometry. This comparison relates with the notions of even Clifford structure and of Clifford system. Going to dimension $16$, similar constructions allow to write explicit formulas for the canonical $4$-forms $Φ_{Spin(8)}$ and $Φ_{Spin(7)U(1)}$, associated with Clifford systems related with the subgroups $Spin(8)$ and $Spin(7)U(1)$ of $SO(16)$. We characterize the calibrated $4$-planes of the $4$-forms $Φ_{Spin(8)}$ and $Φ_{Spin(7)U(1)}$, extending in two different ways the notion of Cayley $4$-plane to dimension $16$.

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arXiv2018

The Role of Spin(9) in Octonionic Geometry

Starting from the 2001 Thomas Friedrich's work on Spin(9), we review some interactions between Spin(9) and geometries related to octonions. Several topics are discussed in this respect: explicit descriptions of the Spin(9) canonical 8-form and its analogies with quaternionic geometry as well as the role of Spin(9) both in the classical problems of vector fields on spheres and in the geometry of the octonionic Hopf fibration. Next, we deal with locally conformally parallel Spin(9) manifolds in the framework of intrinsic torsion. Finally, we discuss applications of Clifford systems and Clifford structures to Cayley-Rosenfeld planes and to three series of Grassmannians.

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arXiv2018

On some Grassmannians carrying an even Clifford structure

We give an explicit description of the non-flat parallel even Clifford structures of rank 8, 6, 5 on some real, complex and quaternionic Grassmannians, and discuss the rôle of the octonions in them, in particular for some low dimensional examples.

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arXiv2016

On the cohomology of some exceptional symmetric spaces

This is a survey on the construction of a canonical or "octonionic Kähler" 8-form, representing one of the generators of the cohomology of the four Cayley-Rosenfeld projective planes. The construction, in terms of the associated even Clifford structures, draws a parallel with that of the quaternion Kähler 4-form. We point out how these notions allow to describe the primitive Betti numbers with respect to different even Clifford structures, on most of the exceptional symmetric spaces of compact type.

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arXiv2016

Clifford systems in octonionic geometry

We give an inductive construction for irreducible Clifford systems on Euclidean vector spaces. We then discuss how this notion can be adapted to Riemannian manifolds, and outline some developments in octonionic geometry.

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arXiv2015

The even Clifford structure of the fourth Severi variety

The Hermitian symmetric space $M=\mathrm{EIII}$ appears in the classification of complete simply connected Riemannian manifolds carrying a parallel even Clifford structure. This means the existence of a real oriented Euclidean vector bundle $E$ over it together with an algebra bundle morphism $φ:\mathrm{Cl}^0(E) \rightarrow \mathrm{End}(TM)$ mapping $Λ^2 E$ into skew-symmetric endomorphisms, and the existence of a metric connection on $E$ compatible with $φ$. We give an explicit description of such a vector bundle $E$ as a sub-bundle of $\mathrm{End}(TM)$. From this we construct a canonical differential 8-form on $\mathrm{EIII}$, associated with its holonomy $\mathrm{Spin}(10) \cdot \mathrm{U}(1) \subset \mathrm{U}(16)$, that represents a generator of its cohomology ring. We relate it with a Schubert cycle structure by looking at $\mathrm{EIII}$ as the smooth projective variety $V_{(4)} \subset \mathbb{C}P^{26}$ known as the fourth Severi variety.

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arXiv2013

Spin(9) geometry of the octonionic Hopf fibration

We deal with Riemannian properties of the octonionic Hopf fibration S^{15}-->S^8, in terms of the structure given by its symmetry group Spin(9). In particular, we show that any vertical vector field has at least one zero, thus reproving the non-existence of S^1 subfibrations. We then discuss Spin(9)-structures from a conformal viewpoint and determine the structure of compact locally conformally parallel Spin(9)-manifolds. Eventually, we give a list of examples of locally conformally parallel Spin(9)-manifolds.

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arXiv2012

Spheres with more than 7 vector fields: all the fault of Spin(9)

We give an interpretation of the maximal number of linearly independent vector fields on spheres in terms of the Spin(9) representation on R^16. This casts an insight on the role of Spin(9) as a subgroup of SO(16) on the existence of vector fields on spheres, parallel to the one played by complex, quaternionic and octonionic structures on R^2, R^4 and R^8, respectively.

math.DG↗
arXiv2011

Spin(9) and almost complex structures on 16-dimensional manifolds

For a Spin(9)-structure on a Riemannian manifold M^16 we write explicitly the matrix psi of its Kähler 2-forms and the canonical 8-form Phi. We then prove that Phi coincides up to a constant with the fourth coefficient of the characteristic polynomial of psi. This is inspired by lower dimensional situations, related to Hopf fibrations and to Spin(7). As applications, formulas are deduced for Pontrjagin classes and integrals of Phi and Phi^2 in the special case of holonomy Spin(9).

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arXiv2005

Locally conformal parallel $G_2$ and $Spin(7)$ manifolds

We characterize compact locally conformal parallel $G_2$ (respectively, $Spin(7)$) manifolds as fiber bundles over $S^1$ with compact nearly Kähler (respectively, compact nearly parallel $G_2$) fiber. A more specific characterization is provided when the local parallel structures are flat.

math.DG↗
arXiv2005

Reduction of Vaisman structures in complex and quaternionic geometry

We consider locally conformal Kaehler geometry as an equivariant (homothetic) Kaehler geometry: a locally conformal Kaehler manifold is, up to equivalence, a pair (K,Γ) where K is a Kaehler manifold and Γa discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kaehler manifold (K,Γ) as the rank of a natural quotient of Γ, and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kaehler reduction of compact Vaisman manifolds produces Vaisman manifolds and is equivalent to a Sasakian reduction. Moreover we define locally conformal hyperkaehler reduction as an equivariant version of hyperkaehler reduction and in the compact case we show its equivalence with 3-Sasakian reduction. Finally we show that locally conformal hyperkaehler reduction induces hyperkaehler with torsion (HKT) reduction of the associated HKT structure and the two reductions are compatible, even though not every HKT reduction comes from a locally conformal hyperkaehler reduction.

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arXiv2003

Toric self-dual Einstein metrics as quotients

We use the quaternion Kahler reduction technique to study old and new self-dual Einstein metrics of negative scalar curvature with at least a two-dimensional isometry group, and relate the quotient construction to the hyperbolic eigenfunction Ansatz. We focus in particular on the (semi-)quaternion Kahler quotients of (semi-)quaternion Kahler hyperboloids, analysing the completeness and topology, and relating them to the self-dual Einstein Hermitian metrics of Apostolov-Gauduchon and Bryant.

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arXiv2000

On some Moment Maps and Induced Hopf Bundles in the Quaternionic Projective Space

We describe a diagram containing the zero sets of the moment maps associated to the diagonal U(1) and Sp(1) actions on the quaternionic projective space HP^n. These sets are related both to focal sets of submanifolds and to Sasakian-Einstein structures on induced Hopf bundles. As an application, we construct a complex structure on the Stiefel manifolds V_2 (C^{n+1}) and V_4 (R^{n+1}), the one on the former manifold not being compatible with its known hypercomplex structure.

math.DG↗
arXiv2000

Complex Structures on some Stiefel Manifolds

We discuss conditions for the integrability of an almost complex structure defined on the total space of an induced Hopf S^3-bundle over a Sasakian manifold . As an application, we obtain an uncountable family of inequivalent complex structures on the Stiefel manifolds of orthonormal 2-frames in C^{n+1}, non compatible with its standard hypercomplex structure. Similar families of complex structures are constructed on the Stiefel manifold of oriented orthonormal 4-frames in R^{n+1}, as well as on some special Stiefel manifolds related to the groups G_2 and Spin(7).

math.DG↗
arXiv2000

3-Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients

Using 3-Sasakian reduction techniques we obtain infinite families of new 3-Sasakian manifolds $\scriptstyle{{\cal M}(p_1,p_2,p_3)}$ and $\scriptstyle{{\cal M}(p_1,p_2,p_3,p_4)}$ in dimension 11 and 15 respectively. The metric cone on $\scriptstyle{{\cal M}(p_1,p_2,p_3)}$ is a generalization of the Kronheimer hyperkähler metric on the regular maximal nilpotent orbit of $\scriptstyle{{\Got s}{\Got l}(3,\bbc)}$ whereas the cone on $\scriptstyle{{\cal M}(p_1,p_2,p_3,p_4)}$ generalizes the hyperkähler metric on the 16-dimensional orbit of $\scriptstyle{{\Got s}{\Got o}(6,\bbc)}$. These are first examples of 3-Sasakian metrics which are neither homogeneous nor toric. In addition we consider some further $\scriptstyle{U(1)}$-reductions of $\scriptstyle{{\cal M}(p_1,p_2,p_3)}$. These yield examples of non-toric 3-Sasakian orbifold metrics in dimensions 7. As a result we obtain explicit families $\scriptstyle{{\cal O}(Θ)}$ of compact self-dual positive scalar curvature Einstein metrics with orbifold singularities and with only one Killing vector field.

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arXiv2000

Foliations with Transversal Quaternionic Structures

We consider manifolds equipped with a foliation $\cal F$ of codimension $4q$, and an almost quaternionic structure $Q$ on the transversal bundle of ${\cal F}$. After discussing conditions of projectability and integrability of $Q$, we study the transversal twistor space $Z{\cal F}$ which, by definition, consists of the $Q$-compatible almost complex structures. We show that $Z{\cal F}$ can be endowed with a lifted foliation ${\hat {\cal F}}$ and two natural almost complex structures $J_1$, $J_2$ on the transversal bundle of $\hat{\cal F}$. We establish the conditions which ensure the projectability of $J_1$ and $J_2$, and the integrability of $J_{1}$ ($J_{2}$ is never integrable).

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