arXiv subjects

Yulia Gorginyan

Publications and source records attributed to Yulia Gorginyan.

3 recordsLinked to original sources

arXiv2024

The twistor space of a compact hypercomplex manifold is never Moishezon

Let (X,I,J,K) be a compact hypercomplex manifold, i.e. a smooth manifold X with an action of the quaternion algebra (Id,I,J,K) on the tangent bundle TX, inducing integrable almost complex structures. For any $(a, b, c) \in S^2$, the linear combination $L := aI + bJ + cK$ defines another complex structure on X. This results in a $C P^1$-family of complex structures called the twistor family. Its total space is called the twistor space. We show that the twistor space of a compact hypercomplex manifold is never Moishezon and, moreover, it is never Fujiki class C (in particular, never Kahler and never projective).

math.AG↗

arXiv2023

Flat hypercomplex nilmanifolds are H-solvable

We say that a hypercomplex nilpotent Lie algebra is $\mathbb{H}$-solvable if there exists a sequence of $\mathbb{H}$-invariant subalgebras $\mathfrak{g}_1^{ \mathbb{H}}\supset\mathfrak{g}_2^{ \mathbb{H}}\supset\cdots\supset\mathfrak{g}_{k-1}^{ \mathbb{H}}\supset\mathfrak{g}_k^{ \mathbb{H}}=0,$ such that $[\mathfrak{g}_i^{ \mathbb{H}},\mathfrak{g}_i^{ \mathbb{H}}]\subset\mathfrak{g}^{ \mathbb{H}}_{i+1}.$ Let $N=\Gamma\backslash G$ be a hypercomplex nilmanifold with flat Obata connection and $\mathfrak{g}=Lie(G)$. We prove that the Lie algebra $\mathfrak{g}$ is $ \mathbb{H}$-solvable.

math.DG↗

arXiv2022

Complex curves in hypercomplex nilmanifolds with H-solvable Lie algebras

An operator $I$ on a real Lie algebra $A$ is called a complex structure operator if $I^2=-Id$ and the $\sqrt{-1}$-eigenspace $A^{1,0}$ is a Lie subalgebra in the complexification of $A$. A hypercomplex structure on a Lie algebra $A$ is a triple of complex structures $I,J$ and $K$ on $A$ satisfying the quaternionic relations. We call a hypercomplex nilpotent Lie algebra quaternionic-solvable if there exists a finite filtration by quaternionic-invariant subalgebras with commutative subquotients which converges to zero. We give examples of quaternionic-solvable hypercomplex structures on a nilpotent Lie algebra and conjecture that all hypercomplex structures on nilpotent Lie algebras are quaternionic-solvable. Let $(N,I,J,K)$ be a compact hypercomplex nilmanifold associated to an quaternionic-solvable hypercomplex Lie algebra. We prove that, for a general complex structure $L$ induced by quaternions, there are no complex curves in a complex manifold $(N,L)$.

math.DG↗