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).

Blow-ups and infinitesimal automorphisms of CR-manifolds

For a real-analytic connected CR-hypersurface M of CR-dimension $$n\geqslant 1$$ n ⩾ 1 having a point of Levi-nondegeneracy the following alternative is demonstrated for its symmetry algebra $$\mathfrak {s}={\mathfrak {s}}(M)$$ s = s ( M ) : (i) either $$\dim {\mathfrak {s}}=n^2+4n+3$$ dim s = n 2 + 4 n + 3 and M is spherical everywhere; (ii) or $$\dim {\mathfrak {s}}\leqslant n^2+2n+2+\delta _{2,n}$$ dim s ⩽ n 2 + 2 n + 2 + δ 2 , n and in the case of equality M is spherical and has fixed signature of the Levi form in the complement to its Levi-degeneracy locus. A version of this result is proved for the Lie group of global automorphisms of M. Explicit examples of CR-hypersurfaces and their infinitesimal and global automorphisms realizing the bound in (ii) are constructed. We provide many other models with large symmetry using the technique of blow-up, in particular we realize all maximal parabolic subalgebras of the pseudo-unitary algebras as a symmetry.

The Loewy series of principal indecomposable modules in the regular nilpotent representations of 𝔰𝔩(3,k)

Let [Formula: see text] be a simple Lie algebra of type [Formula: see text] over an algebraically closed field [Formula: see text] of prime characteristic [Formula: see text] and [Formula: see text] be the reduced enveloping algebra of [Formula: see text]. In this paper, when the [Formula: see text]-character [Formula: see text] is regular nilpotent and has standard Levi form, we precisely determine the Lowey series of principal indecomposable [Formula: see text]-modules and the dimensions for the self-extension of irreducible [Formula: see text]-modules.