International Journal of Mathematics

For domains in ℝⁿ we construct the Bergman kernel on the diagonal using solutions of the Dirichlet problem. Starting from this, in a natural way we obtain an algebra 𝔸_n of dimension n(n - 1)/2 + 1 over ℝ and a class of holomorphic functions valued in 𝔸_n. Of course 𝔸₂ is the field of complex numbers, and it turns out that 𝔸₃ is the algebra of quaternions, whereas for n ≥ 4, 𝔸_n is non-associative. Holomorphic functions can be written as f + ω, where f is a (real-valued) function and ω a differential 2-form such that d* ω = df and dω = 0. We investigate the main properties of the obtained objects, especially from the analytic point of view.

We study the Brill–Noether loci for semistable torsionfree sheaves of small slopes on a nodal curve. We determine conditions for nonemptiness and irreducibility of these loci.

We obtain in this work, in an intrinsic way, the First classification of Pfaffian systems of rank two and three in dimension five, as well as a complete classification up to dimension four. For the general case, we prove that there exists a one-to-one correspondence, which is explicitly given, between Pfaffian systems of rank two and Pfaffian systems of rank three. That is, for a Pfaffian system of rank two in dimension five, the covariant system is constructed.

An example about non-convergence of Perron–Bremermann envelopes on the unit ball in Cⁿ is constructed. Under additional conditions, we show that convergence of envelopes is possible. We also deal with a representation of envelope in the style of Poletsky on domains which are not necessarily strictly pseudoconvex.

Three new examples of 4-dimensional irreducible symplectic V-manifolds are constructed. Two of them are relative compactified Prymians of a family of genus-3 curves with involution, and the third one is obtained from a Prymian by Mukai's flop. They have the same singularities as two of Fujiki's examples, namely, 28 isolated singular points analytically equivalent to the Veronese cone of degree 8, but a different Euler number. The family of curves used in this construction forms a linear system on a K3 surface with involution. The structure morphism of both Prymians to the base of the family is a Lagrangian fibration in abelian surfaces with polarization of type (1,2). No example of such fibration is known on nonsingular irreducible symplectic varieties.

We introduce a new skein module for three manifolds based on properly embedded surfaces and their relations introduced by Bar-Natan in [3], and modified by Khovanov [6]. We compute the structure of the modules for some manifolds, including Seifert fibered manifolds.