This is a review of the results on Hermitian Yang–Mills equation and their generalization called the deformed Hermitian Yang–Mills equation. Solutions of Hermitian Yang–Mills equation in complex dimension 2 are given by antiself-dual connections so this equation may be considered as a multidimensional generalization of the instanton equations. Deformed Hermitian Yang–Mills equation reduce to the Hermitian Yang–Mills equation in the large volume limit.

We study the boundary regularity of proper holomorphic mappings between strictly pseudoconvex domains with boundaries of low regularity.

This paper was motivated by Donaldson’s paper [2] on the Yang–Mills equations on complex manifolds. He proved there the existence of a unique solution of the Dirichlet problem for Yang–Mills fields on a Hermitian vector bundle over a Kähler manifold with smooth boundary. One consequence of this result is a theorem on extending CR-structures on a Hermitian vector bundle over the boundary of a strictly pseudoconvex domain with smooth boundary in the interior of the domain, to a connection associated with a Hermitian Yang–Mills metric. In [2] this was proved in dimension 2. We show that Donaldson’s result holds in each dimension (as was conjectured in [2]). Let E be a holomorphic vector bundle over a complex manifold Z and let H be a fibrewise Hermitian metric on E. It is known that there exists a unique unitary connection on E that is compatible with the complex structure. Let FH denote its curvature. Fix a local holomorphic trivialization of E by sections (sj). In it H is given by a Hermitian matrix with entries Hij = (si, sj)H . Then the connection has the matrix H−1∂H, and its curvature is given by

We establish an effective criterion for a dicritical singularity of a real analytic Levi-flat hypersurface. The criterion is stated in terms of Segre varieties. As an application, we obtain a structure theorem for a certain class of currents in the non-dicritical case.

Absorption spectra of liquid water H2O are investigated in the region 4500–5600 cm–1 at temperatures from –53.0 to +20.3°C. The absorption band is decomposed into component modes at (I) 4700, (II) 4890, (III) 5080, and (IV) 5200 cm–1 whose centers are shifted slightly with temperature. Modes (II), (III), and (IV) are present in the spectrum of liquid water, whereas modes (I), (II), and (III) are registered in the spectrum of ice. The appearance of the ordered crystalline lattice is characterized by the occurrence of low-frequency mode (I) and by the disappearance of high-frequency component (IV) of the water absorption band in the region 4500–5600 cm–1. The spectral regions and temperature ranges in which the structure is changed during phase transition are determined.

А. Б. Сухов О преобразованиях аналитических CR-структурУстановлена связь между CR-геометрией вещественно-аналитических под многообразий в С п и геометрией дифференциальных уравнений.Подход основан на рассмотрении биголоморфизмов невырожденного по Л еви вещественно-анали тического CR-многообразия Ж как точечных симметрии Ли для голоморфной системы дифференциальных уравнений второго порядка, определяющей семей ство Сегре многообразия Ж. Это позволяет изучать группу биголоморфизмов многообразия Ж средствами геометрической теории дифференциальных уравне ний.Приведены несколько примеров и приложений к CR-геометрии: результаты о конечномерности группы биголоморфизмов и точные оценки ее размерности, а также явная параметризация алгебры Ли инфинитезимальных автоморфизмов.Библиография: 37 наименований.

A theorem on boundary continuity of holomorphic mappings between domains in is proved under purely local restrictions on both the mapping and the domains.Bibliography: 36 titles.

A theorem is proved concerning the boundary continuity of holomorphic mappings between domains with piecewise smooth strictly pseudoconvex boundaries in . This theorem is used to establish results on nonexistence of proper holomorphic mappings between certain classes of domains.Bibliography: 31 titles.

A sufficient condition for algebraicity of complex analytic sets is established.