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We construct new twistorial examples of non-Kähler almost Hermitian manifolds with Hermitian Ricci tensor by means of a natural almost Hermitian structures on the twistor space of an almost Hermitian four manifold. [ABSTRACT FROM AUTHOR]

A trisymplecti cstructure on a complex $2n$-manifold is a three-dimensional space ${\rm\Omega}$ of closed holomorphic forms such that any element of ${\rm\Omega}$ has constant rank $2n$, $n$ or zero, and degenerate forms in ${\rm\Omega}$ belong to a non-degenerate quadric hypersurface. We show that a trisymplectic manifold is equipped with a holomorphic 3-web and the Chern connection of this 3-web is holomorphic, torsion-free, and preserves the three symplectic forms. We construct a trisymplectic structure on the moduli of regular rational curves in the twistor space of a hyperkähler manifold, and define a trisymplectic reduction of a trisymplectic manifold, which is a complexified form of a hyperkähler reduction. We prove that the trisymplectic reduction in the space of regular rational curves on the twistor space of a hyperkähler manifold $M$ is compatible with the hyperkähler reduction on $M$. As an application of these geometric ideas, we consider the ADHM construction of instantons and show that the moduli space of rank $r$, charge $c$ framed instanton bundles on $\mathbb{C}\mathbb{P}^{3}$ is a smooth trisymplectic manifold of complex dimension $4rc$. In particular, it follows that the moduli space of rank two, charge $c$ instanton bundles on $\mathbb{C}\mathbb{P}^{3}$ is a smooth complex manifold dimension $8c-3$, thus settling part of a 30-year-old conjecture. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we prove that the CR manifold induced from the canonical parabolic geometry of a quaternionic contact (qc) manifold via a Fefferman-type construction is equivalent to the CR twistor space of the qc manifold defined by O. Biquard. [Copyright &y& Elsevier]

In this article, we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein-Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein-Weyl structure on the space of smooth rational curves on a complex surface, given by Hitchin. As geometric objects naturally associated to Einstein-Weyl structure, we investigate null surfaces and geodesics on the Severi varieties. Also, we see that if the projective surface has an appropriate real structure, then the real locus of the Severi variety becomes a positive definite Einstein-Weyl manifold. Moreover, we construct various explicit examples of rational surfaces having 3-dimensional Severi varieties of rational curves. [ABSTRACT FROM AUTHOR]

Abstract: Let M be a -manifold. We consider an almost CR-structure on the sphere bundle of unit tangent vectors on M, called the CR twistor space. This CR-structure is integrable if and only if M is a holonomy -manifold. We interpret -instanton bundles as CR-holomorphic bundles on its twistor space. [Copyright &y& Elsevier]

Abstract: We prove that the twistor lifts of certain twistor holomorphic surfaces in four-dimensional manifolds are weakly stable harmonic sections. As a corollary, if ambient spaces are self-dual Einstein manifolds with nonnegative scalar curvature, then the twistor lifts of twistor holomorphic surfaces are weakly stable. Moreover, for certain surfaces in four-dimensional hyperkähler manifolds, we show that the surfaces are twistor holomorphic if their twistor lifts are weakly stable harmonic sections. In particular, we characterize twistor holomorphic surfaces in four-dimensional Euclidean space by weak stability of the twistor lifts. [Copyright &y& Elsevier]

Abstract: A set of canonical paraHermitian connections on an almost paraHermitian manifold is defined. ParaHermitian version of the Apostolov–Gauduchon generalization of the Goldberg–Sachs theorem in General Relativity is given. It is proved that the Nijenhuis tensor of a Nearly paraKähler manifolds is parallel with respect to the canonical connection. Salamon''s twistor construction on quaternionic manifold is adapted to the paraquaternionic case. A hyper-paracomplex structure is constructed on Kodaira–Thurston (properly elliptic) surfaces as well as on the Inoe surfaces modeled on . A locally conformally flat hyper-paraKähler (hypersymplectic) structure with parallel Lee form on Kodaira–Thurston surfaces is obtained. Anti-self-dual non-Weyl flat neutral metric on Inoe surfaces modeled on is presented. An example of anti-self-dual neutral metric which is not locally conformally hyper-paraKähler is constructed. [Copyright &y& Elsevier]