171 research outputs found
Maximal slope of tensor product of Hermitian vector bundles
We give an upper bound for the maximal slope of the tensor product of several
non-zero Hermitian vector bundles on the spectrum of an algebraic integer ring.
By Minkowski's theorem, we need to estimate the Arakelov degree of an arbitrary
Hermitian line subbundle of the tensor product. In the case where the
generic fiber of is semistable in the sense of geometric invariant theory,
the estimation is established by constructing, through the classical invariant
theory, a special polynomial which does not vanish on the generic fibre of .
Otherwise we use an explicte version of a result of Ramanan and Ramanathan to
reduce the general case to the former one
Positive degree and arithmetic bigness
We establish, for a generically big Hermitian line bundle, the convergence of
truncated Harder-Narasimhan polygons and the uniform continuity of the limit.
As applications, we prove a conjecture of Moriwaki asserting that the
arithmetic volume function is actually a limit instead of a sup-limit, and we
show how to compute the asymptotic polygon of a Hermitian line bundle, by using
the arithmetic volume function
Convergence of Harder-Narasimhan polygons
We establish in this article convergence results of normalized
Harder-Narasimhan polygons both in geometric and in arithmetic frameworks by
introducing the Harder-Narasimhan filtration indexed by and the
associated Borel probability measure
Differentiability of the arithmetic volume function
We introduce the positive intersection product in Arakelov geometry and prove
that the arithmetic volume function is continuously differentiable. As
applications, we compute the distribution function of the asymptotic measure of
a Hermitian line bundle and several other arithmetic invariants
Algebraic dynamical systems and Dirichlet's unit theorem on arithmetic varieties
In this paper, we study obstructions to the Dirichlet property by two
approaches: density of non-positive points and functionals on adelic
R-divisors. Applied to the algebraic dynamical systems, these results provide
examples of nef adelic arithmetic R-Cartier divisor which does not have the
Dirichlet property. We hope the obstructions obtained in the article will give
ways toward criteria of the Dirichlet property.Comment: 36 page
On subfiniteness of graded linear series
Hilbert's 14th problem studies the finite generation property of the
intersection of an integral algebra of finite type with a subfield of the field
of fractions of the algebra. It has a negative answer due to the counterexample
of Nagata. We show that a subfinite version of Hilbert's 14th problem has a
confirmative answer. We then establish a graded analogue of this result, which
permits to show that the subfiniteness of graded linear series does not depend
on the function field in which we consider it. Finally, we apply the
subfiniteness result to the study of geometric and arithmetic graded linear
series
Concerning the semistability of tensor products in Arakelov geometry
We study the semistability of the tensor product of hermitian vector bundles
by using the -tensor product and the geometric (semi)stability of
vector subspaces in the tensor product of two vector spaces
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