A streamlined proof that the Bergman kernel associated to a quadrature domain in the plane must be algebraic will be given. A byproduct of the proof will be that the Bergman kernel is a rational function of z and one other explicit function known as the Schwarz function. Simplified proofs of several other well known facts about quadrature domains will fall out along the way. Finally, Bergman re...

(Ivan Arzhantsev) In the talk of Hirokaki Ishida it was proved that a torus manifold with an (S)-invariant complex structure is equivariantly biholomorphic to a toric manifold (a joint work with Yael Karshon). After the talk I asked whether every action of a compact torus T = (S) on a compact complex manifold M of complex dimension n by holomorphic automorphisms may be extended to an action of ...

Let X be a complex Banach space with norm ∥ · ∥, B be the unit ball in X. In this paper, we introduce a class of holomorphic mappings Mg on B. Let f(x) be a normalized locally biholomorphic mappings on B such that (Df(x))−1f(x) ∈ Mg and x = 0 is the zero of order k + 1 of f(x)− x. We investigate the growth theorem for f(x). As applications, the distortion theorems for the Jacobian matrix Jf (z)...

Let X be a closed, 1-dimensional, complex subvariety of C and let B be a closed ball in C −X. Then there exists a Fatou-Bieberbach domain Ω with X ⊆ Ω ⊆ C − B such that Ω − X is Kobayashi hyperbolic. In particular, there exists a biholomorphic map Φ : Ω → C such that C−Φ(X) is Kobayashi hyperbolic. As corollaries, there is an embedding of the plane in C whose complement is hyperbolic, and there...

Example 1.2. (a) Suppose X is a Riemann surface. Let Y ⊂ X be a (connected) open subset. Then Y is a Riemann surface, whose complex structure is given by taking all U ⊂ Y from charts of X. (b) Let P = C ∪ {∞}, homeomorphic to the real sphere. Take U1 = P\{∞} = C, U2 = P\{0} = C∗ ∪ {∞}. Define φ1(z) = z, φ2(z) = 1/z for z 6= ∞ and φ2(∞) = 0. Then φ2 ◦ φ−1 1 : C∗ → C∗ is given by z 7→ 1/z, which ...

In this note we provide a direct proof of the following: Any compact KRS with positive bisectional curvature is CP. As a corollary, we obtain an alternative proof of the Frankel conjecture by using the Kähler-Ricci flow. The purpose of this note is to give a proof of the following theorem, which does not rely on the solution of the Frankel conjecture: Theorem 1. Every compact complex manifold a...

Applying a well known result for attracting fixed points of biholomorphisms [4, 6], we observe that one immediately obtains the following result: if (Mn, g) is a complete non-compact gradient Kähler-Ricci soliton which is either steady with positive Ricci curvature so that the scalar curvature attains its maximum at some point, or expanding with non-negative Ricci curvature, then M is biholomor...

The boundary bD of a bounded domain D in C is called piecewise smooth if there exists a neighborhood U of D and ρk ∈ C (U), 1 ≤ k ≤ m, such that D = {q ∈ U ; ρk(q) < 0, 1 ≤ k ≤ m} and dρk1∧. . .∧dρkl 6= 0 on ∩ l j=1Skj for any distinct k1, . . . , kl ∈ {1, . . . , m}, where Sj = {q ∈ U ; ρj(q) = 0}. It is called generic piecewise smooth if ∂ρk1∧. . .∧∂ρkl 6= 0 on ∩j=1Skj . The boundary bD is ca...