Oleg Mushkarov

Twistorial construction of generalized Kähler manifolds

Abstract The twistor method is applied for obtaining examples of generalized Kahler structures which are not yielded by Kahler structures.

Twistor spaces of generalized complex structures

Abstract The twistor construction is applied for obtaining examples of generalized complex structures (in the sense of Hitchin) that are not induced by a complex or a symplectic structure.

Generalized Pseudo-Kähler Structures

In this paper we consider pseudo-bihermitian structures – pairs of complex structures compatible with a pseudo-Riemannian metric. We establish relations of these structures with generalized (pseudo-) Kähler geometry and holomorphic Poisson structures similar to that in the positive definite case. We provide a list of compact complex surfaces which could admit pseudo-bihermitian structures and give examples of such structures on some of them. We also consider a naturally defined null plane distribution on a generalized pseudo-Kähler 4-manifold and show that under a mild restriction it determines an Engel structure.

Compact complex surfaces with geometric structures related to split quaternions

Abstract We study the problem of existence of geometric structures on compact complex surfaces that are related to split quaternions. These structures, called para-hypercomplex, para-hyperhermitian and para-hyperkahler, are analogs of the hypercomplex, hyperhermitian and hyperkahler structures in the definite case. We show that a compact 4-manifold carries a para-hyperkahler structure iff it has a metric of split signature together with two parallel, null, orthogonal, pointwise linearly independent vector fields. Every compact complex surface admitting a para-hyperhermitian structure has vanishing first Chern class and we show that, unlike the definite case, many of these surfaces carry infinite-dimensional families of such structures. We provide also compact examples of complex surfaces with para-hyperhermitian structures which are not locally conformally para-hyperkahler. Finally, we discuss the problem of non-existence of para-hyperhermitian structures on Inoue surfaces of type S 0 and provide a list of compact complex surfaces which could carry para-hypercomplex structures.

Harmonicity of the Atiyah–Hitchin–Singer and Eells–Salamon almost complex structures

In this paper, we describe the oriented Riemannian four-manifolds M for which the Atiyah–Hitchin–Singer or Eells–Salamon almost complex structure on the twistor space

Erratum: Convolution filters for polygons and the Petr Douglas Neumann theorem (American Mathematical Monthly 54 (2013) (701-708))

{\mathcal {Z}}

Geometric Problems on Maxima and Minima

Z of M determines a harmonic map from