The classical Weil-Petersson metric on the Teichmüller space of compact Riemann surfaces is a Kähler metric, which is complete only in the case of elliptic curves [Wo]. It has a natural generalization to the deformation spaces of higher dimansional polarized Kähler-Einstein manifolds. It is still Kähler, and in the case of abelian varieties and K3 surfaces, the Weil-Petersson metric turns out t...

We derive the explicit form of the (g − 2)(g − 3)/2 linearly independent relations among the products of pairs in a basis of holomorphic abelian differentials in the case of canonical curves of genus g ≥ 4. It turns out that Petri’s relations remarkably match in determinantal conditions. We explicitly express the volume form on the moduli space M̂g of canonical curves induced by the Siegel metri...

It is proved that the family of Ahlfors extremal mappings of a multiply connected region in the plane onto the unit disc can be expressed as a rational combination of two fixed Ahlfors mappings in much the same way that the family of Riemann mappings associated to a simply connected region can be expressed in terms of a single such map. The formulas reveal that this family of mappings extends t...

We study the near diagonal asymptotic expansion of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold. We show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two of them. As consequence, we calculate the density of states function of the Bo...

Let f : X −→ S be a smooth projective family and let (L, h) be a singular hermitian line bundle on X with semipositive curvature current. Let Ks := K(Xs, KXs +L | Xs, h | Xs)(s ∈ S) be the Bergman kernel of KXs +L | Xs with respect to h | Xs and let hB the singular hermitian metric on KX + L defined by hB |Xs := 1/Ks. We prove that hB has semipositive curvature. This is a generalization of the ...

We refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between smooth strongly pseudoconvex domains in C are conformal with respect to the sub-Riemannian metric induced by the Levi form. As a corollary we obtain an alternative proof of a result of Fefferman on smooth extensions of biholomorphic mappings between pseudoconvex domains. The proofs are ...

In this article we construct a canonical Kähler-Einstein current on a LC (log canonical) pairs of log general type as the limit of a sequence of canonical Kähler-Einstein currents on KLT(Kawamata log terminal) pairs of log general type. We call the volume form associated with the canonical Kähler-Einstein current the canonical measure of the LC pair. We prove that the relative canonical measure...

We study harmonic Bergman functions on the upper half-space of Rn. Among our main results are: The Bergman projection is bounded for the range 1 < p <∞; certain nonorthogonal projections are bounded for the range 1 ≤ p < ∞; the dual space of the Bergman L1-space is the harmonic Bloch space modulo constants; harmonic conjugation is bounded on the Bergman spaces for the range 1 ≤ p <∞; the Bergma...