G. Giorgadze

Institute of Cibernetics, Georgian Academy of Sciences Tbilisi, Georgia

ON ONE PROPERTY OF A 2-RANK HOLOMORPHIC VECTOR BUNDLE ON THE RIEMANN SPHERE

Let S = {s_{1}, s_{2}, …, s_{m}} be a set of marked points on CP^{1} and

df = ( ∑_{i=1}^{m} \frac{A_{i}}{z − s_{i}} dz)f  (1)

be the system of ODE’s which is induced by the representation

ρ : π_{1}(CP^{1}\S, z_{0}) → GL(n, C),

where the matrices A_{i} satisfy the condition ∑_{i=1}^{m} A_{i} = 0.

For every Fuchsian equation there exists a Fuchsian system (1), which has the same singular points and monodromy. In particular, for the hypergeometric equation

y′′ + \frac{γ + (α + β + 1)}{z(z − 1)}y′ − \frac{αβy}{z(z − 1)} = 0

the aforementioned system will be:

df = ( (\begin{matrix}0 & 0 \\ −αβ & −γ \end{matrix}) \frac{dz}{z} + (\begin{matrix}0 & 1 \\ 0 & γ − (α + β) \end{matrix}) \frac{dz}{z − 1})f  (2)

Fix a new representation ρ for any Riemann data (CP^{1}, S, ρ) and denote by Ω_{ρ} the set of Fuchsian systems corresponding to this data. The number γ_{ρ} = min_{Ω_{ρ}} γ_{ω} is called the Fuchsian weight for the representation ρ.

Let E → CP^{1} be the holomorphic vector bundle induced by the representation ρ and (k_{1}, k_{2}) be its splitting type. Then γ_{ρ} = k_{1} − k_{2}.

Every rank two holomorphic bundle on CP^{1} is holomorphically equivalent to any bundle F → CP^{1}, which is obtained by the extension of the bundle induced by an irreducible representation. So, for every rank 2 holomorphic bundle there exists an irreducible connexion, which is holomorphic except for the finite number n_{ω} of points, where it has simple poles. Denote by Ω^{irr} the space of irreducible Fuchsian connexions. Let p = min_{ω∈Ω^{irr}} n_{ω}. Then the identity p = k_{1} − k_{2} + 2 is satisfied.

From this follows a criterion of stability of such bundles.

Proposition. A rank two vector bundle F → CP^{1} is stable if and only if it is induced by the system (2).