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Oscar Macia

Publications and source records attributed to Oscar Macia.

13 recordsLinked to original sources
arXiv2026

Transforms of holomorphic maps from flag manifolds into Grassmannians

Building on the generalized do Carmo-Wallach theory, we construct a functorial framework for holomorphic maps from flag manifolds into Grassmannians and quadrics. Through the direct image sheaf and its inverse, we construct Penrose-type transforms that allow both the domain and target to vary. These transforms provide functors between categories of full holomorphic maps satisfying the gauge condition for semi-positive homogeneous bundles. Within this framework, Einstein-Hermitian holomorphic maps form a natural subcategory, stable under all transforms, and the minimality of L^2-norm of the mean curvature operator is preserved. For Grassmannian targets, the moduli of Einstein-Hermitian maps from a flag manifold are identified with r-tuples of non- positive integers modulo symmetry, where r is the rank of the group of holomorphic isometries. For quadric targets, we obtain a complete geometric description of the moduli space. The center of this moduli space corresponds to the Einstein-Hermitian map into the projective space, and the tower of transforms identifies all intermediate moduli spaces with the moduli of holomorphic isometric embeddings at the bottom level. The results yield a unified categorical and geometric description of holomorphic isometric embeddings of flag manifolds.

math.DG↗
arXiv2024

On quaternionic bisectional curvature

In this article we study the concept of quaternionic bisectional curvature introduced by B. Chow and D. Yang for quaternion-Kähler manifolds. We show that non-negative quaternionic bisectional curvature is only realized for the quaternionic projective space. We also show that all symmetric quaternion-Kähler manifolds different from the quaternionic projective space admit quaternionic lines of negative quaternionic bisectional curvature. In particular this implies that non-negative sectional curvature does not imply non-negative quaternionic bisectional curvature. Moreover we give a new and rather short proof of a classification result by A. Gray on compact Kähler manifolds of non-negative sectional curvature.

math.DG↗
arXiv2019

The c-map on groups

We study the projective special Kaehler condition on groups, providing an intrinsic definition of homogeneous projective special Kaehler that includes the previously known examples. We give intrinsic defining equations that may be used without resorting to computations in the special cone, and emphasise certain associated integrability equations. The definition is shown to have the property that the image of such structures under the c-map is necessarily a left-invariant quaternionic Kaehler structure on a Lie group.

math.DG↗
arXiv2017

Moduli of Einstein-Hermitian harmonic mappings of the projective line into quadrics

The present article studies the class of Einstein-Hermitian harmonic maps of constant Kaehler angle from the projective line into quadrics. We provide a description of their moduli spaces up to image, and gauge-equivalence using the language of vector bundles and representation theory. It is shown that the dimension of the moduli spaces is independent of the Einstein-Hermitian constant, and rigidity of the associated real standard, and totally real maps is examined. Finally, certain classical results concerning embeddings of two-dimensional spheres into spheres are rephrased and derived in our formalism.

math.DG↗
arXiv2014

Holomorphic isometric embeddings of the projective line into quadrics

We discuss holomorphic isometric embeddings of the projective line into quadrics using a generalisation of the theorem of do Carmo--Wallach to provide a description of their moduli spaces up to image and gauge--equivalence. Moreover, we show rigidity of the real standard map from the projective line into quadrics.

math.DG↗
arXiv2014

Elementary deformations and the hyperKähler-quaternionic Kähler correspondence

The hyperKähler-quaternionic Kähler correspondence constructs quaternionic Kähler metrics from hyperKähler metrics with a rotating circle symmetry. We discuss how this may be interpreted as a combination of the twist construction with the concept of elementary deformation, surveying results of our forthcoming paper. We outline how this leads to a uniqueness statement for the above correspondence and indicate how basic examples of c-map constructions may be realised in this context.

math.DG↗
arXiv2014

Twist geometry of the c-map

We discuss the geometry of the c-map from projective special Kähler to quaternionic Kähler manifolds using the twist construction to provide a global approach to Hitchin's description. As found by Alexandrov et al. and Alekseevsky et al. this is related to the quaternionic flip of Haydys. We prove uniqueness statements for several steps of the construction. In particular, we show that given a hyperKähler manifold with a rotating symmetry, there is essentially only a one parameter degree of freedom in constructing a quaternionic Kähler manifold of the same dimension. We demonstrate how examples on group manifolds arise from this picture.

math.DG↗
arXiv2009

A Nearly Quaternionic Structure on SU(3)

It is shown that the compact Lie group SU(3) admits an Sp(2)Sp(1)-structure whose distinguished 2-forms $ω_1,ω_2,ω_3$ span a differential ideal. This is achieved by first reducing the structure further to a subgroup isomorphic to SO(3).

math.DG↗
arXiv2009

Finiteness of Ulam Polynomials

A polynomial whose coeffcients are equal to its roots is called a Ulam polynomial. In this paper we show that for a given degree n there exists a finite number of Ulam polynomials of degree n.

math.AG↗
arXiv2006

Real symplectic formulation of local special geometry

We consider a formulation of local special geometry in terms of Darboux special coordinates $P^I=(p^i,q_i)$, $I=1,...,2n$. A general formula for the metric is obtained which is manifestly $\mathbf{Sp}(2n,\mathbb{R})$ covariant. Unlike the rigid case the metric is not given by the Hessian of the real function $S(P)$ which is the Legendre transform of the imaginary part of the holomorphic prepotential. Rather it is given by an expression that contains $S$, its Hessian and the conjugate momenta $S_I=\frac{\partial S}{\partial P^I}$. Only in the one-dimensional case ($n=1$) is the real (two-dimensional) metric proportional to the Hessian with an appropriate conformal factor.

hep-th↗
arXiv2006

Observations on the Darboux coordinates for rigid special geometry

We exploit some relations which exist when (rigid) special geometry is formulated in real symplectic special coordinates $P^I=(p^Λ,q_Λ), I=1,...,2n$. The central role of the real $2n\times 2n$ matrix $M(\Re \mathcal{F},\Im \mathcal{F})$, where $\mathcal{F} = \partial_Λ\partial_ΣF$ and $F$ is the holomorphic prepotential, is elucidated in the real formalism. The property $MΩM=Ω$ with $Ω$ being the invariant symplectic form is used to prove several identities in the Darboux formulation. In this setting the matrix $M$ coincides with the (negative of the) Hessian matrix $H(S)=\frac{\partial^2 S}{\partial P^I\partial P^J}$ of a certain hamiltonian real function $S(P)$, which also provides the metric of the special Kähler manifold. When $S(P)=S(U+\bar U)$ is regarded as a "Kähler potential'' of a complex manifold with coordinates $U^I=\frac12(P^I+iZ^I)$, then it provides a Kähler metric of an hyperkähler manifold which describes the hypermultiplet geometry obtained by c-map from the original n-dimensional special Kähler structure.

hep-th↗