On the Applications of Bochner-Kodaira-Morrey-Kohn Identity

This paper is devoted to studying some applications of the Bochner-Kodaira-Morrey-Kohn identity. For this study, we define a condition which is called (Hq) condition which is related to the Levi form on the complex manifold. Under the (Hq) condition and combining with the basic Bochner-Kodaira-Morrey-Kohn identity, we study the L2 ∂ Cauchy problems on domains in ℂn, Kähler manifold and in projective space. Also, we study this problem on a piecewise smooth strongly pseudoconvex domain in a complex manifold. Furthermore, the weighted L2 ∂ Cauchy problem is studied under the same condition in a Kähler manifold with semi-positive holomorphic bisectional curvature. On the other hand, we study the global regularity and the L2 theory for the ∂-operator with mixed boundary conditions on an annulus domain in a Stein manifold between an inner domain which satisfy (Hn−q−1) and an outer domain which satisfy (Hq).

Connecting Homomorphism and Separating Cycles

We discuss the construction of a long semi-exact Mayer–Vietoris sequence for the homology of any finite union of open subspaces. This sequence is used to obtain topological conditions of representation of the integral of a meromorphic n-form on an n-dimensional complex manifold in terms of Grothendieck residues. For such a representation of the integral to exist, it is necessary that the cycle of integration separates the set of polar hypersurfaces of the form. The separation condition in a number of cases turns out to be a sufficient condition for representing the integral as a sum of residues. Earlier, when describing such cases (in the works of Tsikh, Yuzhakov, Ulvert, etc.), the key was the condition that the manifold be Stein. The main result of this article is the relaxation of this condition

Dolbeault and J-Invariant Cohomologies on Almost Complex Manifolds

In this paper we relate the cohomology of J-invariant forms to the Dolbeault cohomology of an almost complex manifold. We find necessary and sufficient condition for the inclusion of the former into the latter to be true up to isomorphism. We also extend some results obtained by J. Cirici and S. O. Wilson about the computation of the left-invariant cohomology of nilmanifolds to the setting of solvmanifolds. Several examples are given.

On non-proper intersections and local intersection numbers

Given equidimensional (generalized) cycles $$\mu _1$$ μ 1 and $$\mu _2$$ μ 2 on a complex manifold Y we introduce a product $$\mu _1\diamond _{Y} \mu _2$$ μ 1 ⋄ Y μ 2 that is a generalized cycle whose multiplicities at each point are the local intersection numbers at the point. If Y is projective, then given a very ample line bundle $$L\rightarrow Y$$ L → Y we define a product $$\mu _1{\bullet _L}\mu _2$$ μ 1 ∙ L μ 2 whose multiplicities at each point also coincide with the local intersection numbers. In addition, provided that $$\mu _1$$ μ 1 and $$\mu _2$$ μ 2 are effective, this product satisfies a Bézout inequality. If $$i:Y\rightarrow {\mathbb P}^N$$ i : Y → P N is an embedding such that $$i^*\mathcal O(1)=L$$ i ∗ O ( 1 ) = L , then $$\mu _1{\bullet _L}\mu _2$$ μ 1 ∙ L μ 2 can be expressed as a mean value of Stückrad–Vogel cycles on $${\mathbb P}^N$$ P N . There are quite explicit relations between $${\diamond }_Y$$ ⋄ Y and $${\bullet _L}$$ ∙ L .

Complex hyperkähler structures defined by Donaldson–Thomas invariants

The notion of a Joyce structure was introduced in Bridgeland (Geometry from Donaldson–Thomas invariants, preprint arXiv:1912.06504) to describe the geometric structure on the space of stability conditions of a $$\hbox {CY}_3$$ CY 3 category naturally encoded by the Donaldson-Thomas invariants. In this paper we show that a Joyce structure on a complex manifold defines a complex hyperkähler structure on the total space of its tangent bundle, and give a characterisation of the resulting hyperkähler metrics in geometric terms.

Equivariant embeddings of strongly pseudoconvex Cauchy–Riemann manifolds

Let X be a CR manifold with transversal, proper CR action of a Lie group G. We show that the quotient X/G is a complex space such that the quotient map is a CR map. Moreover the quotient is universal, i.e. every invariant CR map into a complex manifold factorizes uniquely over a holomorphic map on X/G. We then use this result and complex geometry to prove an embedding theorem for (non-compact) strongly pseudoconvex CR manifolds with transversal $$G \rtimes S^1$$ G ⋊ S 1 -action. The methods of the proof are applied to obtain a projective embedding theorem for compact CR manifolds.

Chern-Yamabe problem and Chern-Yamabe soliton

Let [Formula: see text] be a compact complex manifold of complex dimension [Formula: see text] endowed with a Hermitian metric [Formula: see text]. The Chern-Yamabe problem is to find a conformal metric of [Formula: see text] such that its Chern scalar curvature is constant. In this paper, we prove that the solution to the Chern-Yamabe problem is unique under some conditions. On the other hand, we obtain some results related to the Chern-Yamabe soliton.

Non-gravitational Effects of the Metric Field over Complex Manifolds

Objective of this work is to study whether some of the known non-gravitational phenomena can be explained as motion on a straight line as gravity is treated within General Relativity. To do that, we explore a metric field on a complexified manifold with holomorphic coordinates. Specifically we look into the behaviour of geodesics on such a smooth complex manifold and the path traced out by its real component. This yields a family of equations of motions in real coordinates which is shown to have deviations from usual geodesic equation and in that way expands the geodesic to capture contributions from additional fields and interactions beyond the mere gravitational ones as a function of the metric field.

RUANG PROYEKTIF KOMPLEKS 〖CP〗^n ADALAH MANIFOLD KOMPLEKS

<p>The purpose of this paper to determine the complex projective space as a complex manifold is to calculate the cohomology of the coherent sheaves of . The research method in this paper is to construct an -dimensional complex projective space, namely and then the n-dimensional complex projective space, namely , is a complex manifold. The result of this research is the -dimensional complex projective space, namely is a complex and compact manifold.</p>