Let ψ be a conformal map on D with ψ(0)=0 and let F_α={z∈D:|ψ(z)|=α} for α>0. Denote by H^p(D) the classical Hardy space with exponent p>0 and by h(ψ) the Hardy number of ψ. Consider the limits L:=lim_{α→+∞} (log ω_D(0,Fα)^{−1} / logα), μ:=lim_{α→+∞}(d_D(0,Fα) / logα), where ω_D(0,Fα) denotes the harmonic measure at 0 of F_α and d_D(0,Fα) denotes the hyperbolic distance between 0 and F_α in D. We study a problem posed by P. Poggi-Corradini. What is the relation between L, μ and h(ψ)? Motivated by the result of Kim and Sugawa that h(ψ)=lim inf_{α→+∞} (log ω_D(0,Fα)^{−1} logα), we show that h(ψ)=lim inf_{α→+∞} (d_D(0,Fα) / logα). We also provide conditions for the existence of L and μ and for the equalities L=μ=h(ψ). Poggi-Corradini proved that ψ∉H^μ(D) for a wide class of conformal maps ψ. We present an example of ψ such that ψ∈H^μ(D).

We discuss the Kodaira problem for uniruled Kähler spaces. Building on a construction due to Voisin, we give an example of a uniruled Kähler space X such that every run of the K_X-MMP immediately terminates with a Mori fibre space, yet X does not admit an algebraic approximation. Our example also shows that for a Mori fibration, approximability of the base does not imply approximability of the total space.

The Separatrix Theorem of C. Camacho and P. Sad says that there exists at least one invariant curve (separatrix) passing through the singularity of a germ of holomorphic foliation on complex surface, when the surface underlying the foliation is smooth or when it is singular and the dual graph of resolution surface singularity is a tree. Under some assumptions, we obtain existence of separatrix even when the resolution dual graph of the surface singular point is not a tree. It will be necessary to require an extra condition of the foliation, namely, absence of saddle-node in its reduction of singularities.

We give an example of a dominant rational self-map of the projective plane whose dynamical degree is a transcendental number.

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