arXiv subjects

Artour Tomberg

Publications and source records attributed to Artour Tomberg.

4 recordsLinked to original sources

arXiv2021

On vector bundles over hyperkähler twistor spaces

We study the holomorphic vector bundles E over the twistor space Tw(M) of a compact simply connected hyperkähler manifold $M$. We give a characterization of the semistability condition for E in terms of its restrictions to the holomorphic sections of the holomorphic twistor projection π:Tw(M)\rightarrow CP^1. It is shown that if E admits a holomorphic connection, then E is holomorphically trivial and the holomorphic connection on E is trivial as well. For any irreducible vector bundle E on Tw(M) of prime rank, we prove that its restriction to the generic fibre of πis stable. On the other hand, for a K3 surface M, we construct examples of irreducible vector bundles of any composite rank on Tw(M) whose restriction to every fibre of πis non-stable. We have obtained a new method of constructing irreducible vector bundles on hyperkähler twistor spaces; this method is employed in constructing these examples.

math.AG↗

arXiv2019

Families of stable bundles on the fibres of the hyperkähler twistor projection

Given a holomorphic vector bundle $E$ on the twistor space $\mathrm{Tw}(M)$ of a simple hyperkähler manifold $M$, we view it as a family of bundles $\left\{E_I\right\}$ on the fibres $π^{-1}(I)$ of the twistor projection $π: \mathrm{Tw}(M) \to \mathbb{CP}^1$, and study the relationship between stability of $E$ and its fibrewise stability. We verify that the argument of Teleman establishing the Zariski openness of stability and semi-stability in families of bundles applies in the case of the family $\left\{E_I\right\}$. We prove a partial converse to a result of Kaledin and Verbitsky, showing that an irreducible bundle $E$ on $\mathrm{Tw}(M)$ is generically fibrewise stable if the rank of $E$ is 2 or 3, or at least one element of the family $\left\{E_I\right\}$ is a simple bundle, in the sense that $\mathrm{Hom}(E_I, E_I) = \mathbb{C}$.

math.AG↗

arXiv2019

Twistor spaces of hypercomplex manifolds are balanced

A hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations. If, in addition, there exists a metric on the manifold which is Hermitian with respect to the three structures, and such that the corresponding Hermitian forms are closed, the manifold is said to be hyperkaehler. In the paper "Non-Hermitian Yang-Mills connections", Kaledin and Verbitsky proved that the twistor space of a hyperkaehler manifold admits a balanced metric; these were first studied in the article "On the existence of special metrics in complex geometry" by Michelsohn. In the present article, we review the proof of this result and then generalize it and show that twistor spaces of general compact hypercomplex manifolds are balanced.

math.DG↗

arXiv2018

An example of a stable but fibrewise nonstable bundle on the twistor space of a hyperkähler manifold

We construct an explicit example of a stable bundle on the twistor space $\mathrm{Tw}(M)$ of a hyperkähler manifold $M$ whose restrictions to all the fibres of the natural twistor projection $π: \mathrm{Tw}(M) \to \mathbb{CP}^1$ are nonstable. We also describe the relationship between bundles on $\mathrm{Tw}(M)$ that do not have subsheaves of strictly lower rank and bundles that stably restrict to the fibres of $π$, and announce a result whose proof will appear in a forthcoming paper.

math.DG↗