We study extremal properties of the determinant of the Laplacian in the Bergman metric on the moduli space of compact genus two Riemann surfaces. By a combination of analytical and numerical methods we identify four non-degenerate critical points of this function and compute the signature of the Hessian at these points. The curve with the maximal number of automorphisms (the Burnside curve) tur...

(Strictly speaking, one should put ( —1)"2/2 in front of £)'> but this is not essential in the following discussion.) Suppose F is ample in the following sense: (A.l). For every z in M, there exists an/ in F which does not vanish at 2. (A.2). For every holomorphic vector Z at z, there exists an/ in F such that / vanishes at z and Z(f *) ^0, where /=/ * dz1 A ■ ■ • Adzn with respect to a local c...

The purpose of this article is to establish Jackson-type inequality in the polydiscs UN of C for holomorphic spaces X, such as Bergman-type spaces, Hardy spaces, polydisc algebra and Lipschitz spaces. Namely, E k(f,X) (−→ 1/k, f,X ) , where E k(f,X) is the deviation of the best approximation of f ∈ X by polynomials of degree at most kj about the jth variable zj with respect to the X-metric and ...

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 L be a (semi)-positive line bundle over a Kähler manifold, X , fibered over a complex manifold Y . Assuming the fibers are compact and non-singular we prove that the hermitian vector bundle E whose fibers are the space of global sections to L⊗KX/Y endowed with the L-metric is (semi)-positive in the sense of Nakano. As an application we prove a partial result on a conjecture of Griffiths on ...

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...

One of the main purpose of this paper is to compare those well-known canonical and complete metrics on the Teichmüller and the moduli spaces of Riemann surfaces. We use as bridge two new metrics, the Ricci metric and the perturbed Ricci metric. We will prove that these metrics are equivalent to those classical complete metrics. For this purpose we study in detail the asymptotic behaviors and th...

In this short note we are concerned with the Kähler-Einstein metrics near cone type log canonical singularities. By two different approaches, construct a metric negative scalar curvature in neighborhood of over Calabi-Yau manifold which is complete towards vertex. This provides local model for further study global on singular varieties. first approach, show that singularity uniformized by compl...

For semigroups and for bounded operators we introduce the new notion of Bergman distance. Systems with a finite Bergman distance share the same stability properties, and the Bergman distance is preserved under the Cayley transform. This way, we get stability results in continuous and discrete time. As an example, we show that bounded perturbations lead to pairs of semigroups with finite Bergman...

The Umehara algebra is studied with motivation on the problem of non-existence common complex submanifolds. In this paper, we prove some new results in and obtain applications. particular, if a manifolds admits holomorphic polynomial isometric immersion to one indefinite space form, then it cannot another form different type. Other consequences include submanifolds for projective or hyperbolic ...