117 research outputs found
Entire pluricomplex Green functions and Lelong numbers of projective currents
Let be a positive closed current of bidimension (1,1) and unit mass on
the complex projective space . We prove that the set
of points where has Lelong number larger than is contained in a
complex line if , and for some
complex line if . We also prove that in dimension 2 and
if , then for some conic
.Comment: 9 page
Overinterpolation
In this paper we study the consequences of overinterpolation, i.e., the
situation when a function can be interpolated by polynomial, or rational, or
algebraic functions in more points that normally expected. We show that in many
cases such a function has specific forms.Comment: 14 page
Equidistribution results for singular metrics on line bundles
Let L be a holomorphic line bundle with a positively curved singular
Hermitian metric over a complex manifold X. One can define naturally the
sequence of Fubini-Study currents associated to the space of square integrable
holomorphic sections of the p-th tensor powers of L. Assuming that the singular
set of the metric is contained in a compact analytic subset of X and that the
logarithm of the Bergman kernel function associated to the p-th tensor power of
L (defined outside the singular set) grows like o(p) as p tends to infinity, we
prove the following:
1) the k-th power of the Fubini-Study currents converge weakly on the whole X
to the k-th power of the curvature current of L.
2) the expectations of the common zeros of a random k-tuple of square
integrable holomorphic sections converge weakly in the sense of currents to to
the k-th power of the curvature current of L.
Here k is so that the codimension of the singular set of the metric is
greater or equal as k. Our weak asymptotic condition on the Bergman kernel
function is known to hold in many cases, as it is a consequence of its
asymptotic expansion. We also prove it here in a quite general setting. We then
show that many important geometric situations (singular metrics on big line
bundles, Kaehler-Einstein metrics on Zariski-open sets, artihmetic quotients)
fit into our framework.Comment: 40 page
Invariant currents and dynamical Lelong numbers
Let be a polynomial automorphism of of degree ,
whose rational extension to maps the hyperplane at infinity to a
single point. Given any positive closed current on of bidegree
(1,1), we show that the sequence converges in the
sense of currents on to a linear combination of the Green current
of and the current of integration along the hyperplane at infinity.
We give an interpretation of the coefficients in terms of generalized Lelong
numbers with respect to an invariant dynamical current for .Comment: 15 page
Pade interpolation by F-polynomials and transfinite diameter
We define -polynomials as linear combinations of dilations by some
frequencies of an entire function . In this paper we use Pade interpolation
of holomorphic functions in the unit disk by -polynomials to obtain
explicitly approximating -polynomials with sharp estimates on their
coefficients. We show that when frequencies lie in a compact set
then optimal choices for the frequencies of interpolating
polynomials are similar to Fekete points. Moreover, the minimal norms of the
interpolating operators form a sequence whose rate of growth is determined by
the transfinite diameter of .
In case of the Laplace transforms of measures on , we show that the
coefficients of interpolating polynomials stay bounded provided that the
frequencies are Fekete points. Finally, we give a sufficient condition for
measures on the unit circle which ensures that the sums of the absolute values
of the coefficients of interpolating polynomials stay bounded.Comment: 16 page
Stable algebras of entire functions
Suppose that and belong to the algebra \B generated by the rational
functions and an entire function of finite order on and that
has algebraic polar variety. We show that either h/g\in\B or
, where is a polynomial and are rational functions.
In the latter case, belongs to the algebra generated by the rational
functions, and .Comment: 11 page
Polynomial estimates, exponential curves and Diophantine approximation
Let and . If is a polynomial of degree in
, normalized by , we obtain sharp estimates for
in terms of , where is the closed unit bidisk.
For most , we show that .
However, for in a subset of the Liouville numbers,
has bigger order of growth. We give a precise
characterization of the set and study its properties.Comment: 12 pages. To appear in Mathematical Research Letter
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