Quaternionic differential forms and symplectic Pontrjagin classes

Sunto Si considerano fibrati vettoriali quaternionali dotati di connessione quaternionale e si determinano forme differenziali rappresentanti delle classi simplettiche di Pontrjagin.

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Lavoro eseguito nell'ambito del gruppo G.N.S.A.G.A. del C.N.R.; esposto in una comunicazione a Oberwolfach, 7–13 Giugno 1981.     

Gysin maps and cycle classes for Hodge cohomology

Iff:X→Y is a projective morphism between regular varieties over a field, we construct Gysin maps $$f_ * :H^i \left( {X,\Omega _{X/Z}^j } \right) \to H_{f(x)}^{i + d} \left( {X,\Omega _{Y/Z}^j } \right)$$ for the Hodge cohomology groups, whered-dimY-dimX. These Gysin maps have the expected properties, and in particular may be used to construct a cycle class map $$Cl_X :CH^i \left( {X,S} \right) \to H^i \left( {X,\Omega _{X/Z}^i } \right)$$ whereX is quasi-projective over a field,S is the singular locus, andCH ⁱ(X, S) is the relative Chow group of codimension-i cycles modulo rational equivalence. Simple properties of this cycle map easily imply the infinite dimensionality theorem for the Chow group of zero cycles of a normal projective varietyX overC with \(H^n \left( {X,\mathcal{O}_X } \right) \ne 0\) , wheren=dimX. One also recovers examples of Nori of affinen-dimensional varieties which support indecomposable vector bundles of rankn.     

The locus of Hodge classes in an admissible variation of mixed Hodge structure

We generalize the theorem of E. Cattani, P. Deligne, and A. Kaplan to admissible variations of mixed Hodge structure.     

Quaternionic analyticity

Complex analyticity is generalized to hypercomplex functions, quaternion or octonion, in such a manner that it includes the standard complex definition and does not reduce analytic functions to a trivial class. A brief comparison with other definitions is presented.     

Quaternionic structures

Any oriented 4-dimensional real vector bundle is naturally a line bundle over a bundle of quaternion algebras. In this paper we give an account of modules over bundles of quaternion algebras, discussing Morita equivalence, characteristic classes and K-theory. The results have been used to describe obstructions for the existence of almost quaternionic structures on 8-dimensional Spin^c manifolds in ?adek et al. (2008) [5] and may be of some interest, also, in quaternionic and algebraic geometry.     

Quaternionic Starters

Let m be an integer, m 2 and set n = 2^m. Let G be a non-cyclic group of order 2n admitting a cyclic subgroup of order n. We prove that G always admits a starter and so there exists a one–factorization of K_2n admitting G as an automorphism group acting sharply transitively on vertices. For an arbitrary even n > 2 we also show the existence of a starter in the dicyclic group of order 2n.Research performed within the activity of INdAM–GNSAGA with the financial support of the Italian Ministry MIUR, project Strutture Geometriche, Combinatoria e loro Applicazioni     

Fundaments of Quaternionic Clifford Analysis I: Quaternionic Structure

Introducing a quaternionic structure on Euclidean space, the fundaments for quaternionic and symplectic Clifford analysis are studied in detail from the viewpoint of invariance for the symplectic group action.     

Congruence Criteria for Finite Subsets of Quaternionic Elliptic and Quaternionic Hyperbolic Spaces

We investigate congruence classes of m-tuples of points in the quaternionic elliptic space ?P ⁿ . We establish a canonical bijection between the set of congruence classes of m-tuples of points in ?P ⁿ and the set of equivalence classes of positive semidefinite Hermitian m×m matrices of rank at most n+1 with the 1's on the diagonal. We show that with each m-tuple of points in ?P ⁿ is associated a tuple of points on the real unit sphere S ². Then we get that the congruence class of an m-tuple of points in ?P ⁿ is determined by the congruence classes of all its triangles and by the direct congruence class of the associated tuple on the sphere S ² provided that no pair of points of the m-tuple has distance π/2. Finally we carry out the same kind of investigation for the quaternionic hyperbolic space ?H ⁿ . Most of the results are completely analogous, although there are also some interesting differences.     

Quaternionic analytic torsion

We define an (equivariant) quaternionic analytic torsion for anti-self-dual vector bundles on quaternionic Kähler manifolds, using ideas by Leung and Yi. We do so by constructing a Laplace operator associated to a complex defined by Salamon. We compute this torsion for vector bundles on quaternionic homogeneous spaces with respect to any isometry in the component of the identity, in terms of roots and Weyl groups.     

Hodge Theory and Geometry

Hodge theory for a smooth algebraic curve includes both theHodge structure (period matrix) on cohomology and the use ofthat Hodge structure to study the geometry of the curve, viathe Jacobian variety. Hodge extended the theory of the periodmatrix to smooth algebraic varieties of any dimension, definingin general a Hodge structure on the cohomology of the variety.He gave a few applications to the geometry of the variety, butthese did not attain the richness of the Jacobian variety. Inrecent years, Hodge theory has been successfully extended toarbitrary varieties, and to families of varieties. In this expositorypaper, some of these developments are reviewed, with specialemphasis on instances where these extensions can be used tostudy the geometry – especially the algebraic cycles –on the variety. 2000 Mathematics Subject Classification 14CDFJ.     

Quaternionic compositum genus

A quaternionic field over the rationals contains three quadratic subfields with a compositum genus relation of the type described in the author's paper in Volume 9 of this journal, involving the representation of a prime as norm in these subfields. These representations had previously been only partially exlored by the transfer of class structure from the rational to the quadratic fields. Here a full exposition is given by constructing the Artin characters when the subfields are $\text{Q}$(2^1/2), $\text{Q}$(q^1/2), and $\text{Q}$(2q)^1/2 (q prime). A special role belongs to q = A² + 32b².     

Quaternionic lie algebras

Lie algebras which are isomorphic to central quotients of quaternion division algebras are investigated.