2,786 research outputs found
Low-momentum ring diagrams of neutron matter at and near the unitary limit
We study neutron matter at and near the unitary limit using a low-momentum
ring diagram approach. By slightly tuning the meson-exchange CD-Bonn potential,
neutron-neutron potentials with various scattering lengths such as
and are constructed. Such potentials are renormalized
with rigorous procedures to give the corresponding -equivalent
low-momentum potentials , with which the low-momentum
particle-particle hole-hole ring diagrams are summed up to all orders, giving
the ground state energy of neutron matter for various scattering lengths.
At the limit of , our calculated ratio of to that of
the non-interacting case is found remarkably close to a constant of 0.44 over a
wide range of Fermi-momenta. This result reveals an universality that is well
consistent with the recent experimental and Monte-Carlo computational study on
low-density cold Fermi gas at the unitary limit. The overall behavior of this
ratio obtained with various scattering lengths is presented and discussed.
Ring-diagram results obtained with and those with -matrix
interactions are compared.Comment: 9 pages, 7 figure
Effective algebraic degeneracy
We prove that any nonconstant entire holomorphic curve from the complex line
C into a projective algebraic hypersurface X = X^n in P^{n+1}(C) of arbitrary
dimension n (at least 2) must be algebraically degenerate provided X is generic
if its degree d = deg(X) satisfies the effective lower bound: d larger than or
equal to n^{{(n+1)}^{n+5}}
On the cohomology of pseudoeffective line bundles
The goal of this survey is to present various results concerning the
cohomology of pseudoeffective line bundles on compact K{\"a}hler manifolds, and
related properties of their multiplier ideal sheaves. In case the curvature is
strictly positive, the prototype is the well known Nadel vanishing theorem,
which is itself a generalized analytic version of the fundamental
Kawamata-Viehweg vanishing theorem of algebraic geometry. We are interested
here in the case where the curvature is merely semipositive in the sense of
currents, and the base manifold is not necessarily projective. In this
situation, one can still obtain interesting information on cohomology, e.g. a
Hard Lefschetz theorem with pseudoeffective coefficients, in the form of a
surjectivity statement for the Lefschetz map. More recently, Junyan Cao, in his
PhD thesis defended in Grenoble, obtained a general K{\"a}hler vanishing
theorem that depends on the concept of numerical dimension of a given
pseudoeffective line bundle. The proof of these results depends in a crucial
way on a general approximation result for closed (1,1)-currents, based on the
use of Bergman kernels, and the related intersection theory of currents.
Another important ingredient is the recent proof by Guan and Zhou of the strong
openness conjecture. As an application, we discuss a structure theorem for
compact K{\"a}hler threefolds without nontrivial subvarieties, following a
joint work with F.Campana and M.Verbitsky. We hope that these notes will serve
as a useful guide to the more detailed and more technical papers in the
literature; in some cases, we provide here substantially simplified proofs and
unifying viewpoints.Comment: 39 pages. This survey is a written account of a lecture given at the
Abel Symposium, Trondheim, July 201
Finite Generation of Canonical Ring by Analytic Method
In the 80th birthday conference for Professor LU Qikeng in June 2006 I gave a
talk on the analytic approach to the finite generation of the canonical ring
for a compact complex algebraic manifold of general type. This article is my
contribution to the proceedings of that conference from my talk. In this
article I give an overview of the analytic proof and focus on explaining how
the analytic method handles the problem of infinite number of interminable
blow-ups in the intuitive approach to prove the finite generation of the
canonical ring. The proceedings of the LU Qikeng conference will appear as
Issue No. 4 of Volume 51 of Science in China Series A: Mathematics
(www.springer.com/math/applications/journal/11425)
Section Extension from Hyperbolic Geometry of Punctured Disk and Holomorphic Family of Flat Bundles
The construction of sections of bundles with prescribed jet values plays a
fundamental role in problems of algebraic and complex geometry. When the jet
values are prescribed on a positive dimensional subvariety, it is handled by
theorems of Ohsawa-Takegoshi type which give extension of line bundle valued
square-integrable top-degree holomorphic forms from the fiber at the origin of
a family of complex manifolds over the open unit 1-disk when the curvature of
the metric of line bundle is semipositive. We prove here an extension result
when the curvature of the line bundle is only semipositive on each fiber with
negativity on the total space assumed bounded from below and the connection of
the metric locally bounded, if a square-integrable extension is known to be
possible over a double point at the origin. It is a Hensel-lemma-type result
analogous to Artin's application of the generalized implicit function theorem
to the theory of obstruction in deformation theory. The motivation is the need
in the abundance conjecture to construct pluricanonical sections from flatly
twisted pluricanonical sections. We also give here a new approach to the
original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the
punctured open unit 1-disk to reduce the original theorem of Ohsawa-Takegoshi
to a simple application of the standard method of constructing holomorphic
functions by solving the d-bar equation with cut-off functions and additional
blowup weight functions
Unitarity potentials and neutron matter at the unitary limit
We study the equation of state of neutron matter using a family of unitarity
potentials all of which are constructed to have infinite scattering
lengths . For such system, a quantity of much interest is the ratio
where is the true ground-state energy of the system,
and is that for the non-interacting system. In the limit of
, often referred to as the unitary limit, this ratio is
expected to approach a universal constant, namely . In the
present work we calculate this ratio using a family of hard-core
square-well potentials whose can be exactly obtained, thus enabling us to
have many potentials of different ranges and strengths, all with infinite
. We have also calculated using a unitarity CDBonn potential
obtained by slightly scaling its meson parameters. The ratios given by
these different unitarity potentials are all close to each other and also
remarkably close to 0.44, suggesting that the above ratio is indifferent
to the details of the underlying interactions as long as they have infinite
scattering length. A sum-rule and scaling constraint for the renormalized
low-momentum interaction in neutron matter at the unitary limit is discussed.Comment: 7.5 pages, 7 figure
Local syzygies of multiplier ideals
In recent years, multiplier ideals have found many applications in local and
global algebraic geometry. Because of their importance, there has been some
interest in the question of which ideals on a smooth complex variety can be
realized as multiplier ideals. Other than integral closure no local
obstructions have been known up to now, and in dimension two it was established
by Favre-Jonsson and Lipman-Watanabe that any integrally closed ideal is
locally a multiplier ideal. We prove the somewhat unexpected result that
multiplier ideals in fact satisfy some rather strong algebraic properties
involving higher syzygies. It follows that in dimensions three and higher,
multiplier ideals are very special among all integrally closed ideals.Comment: 8 page
Positivity of relative canonical bundles and applications
Given a family of canonically polarized manifolds, the
unique K\"ahler-Einstein metrics on the fibers induce a hermitian metric on the
relative canonical bundle . We use a global elliptic
equation to show that this metric is strictly positive on , unless
the family is infinitesimally trivial.
For degenerating families we show that the curvature form on the total space
can be extended as a (semi-)positive closed current. By fiber integration it
follows that the generalized Weil-Petersson form on the base possesses an
extension as a positive current. We prove an extension theorem for hermitian
line bundles, whose curvature forms have this property. This theorem can be
applied to a determinant line bundle associated to the relative canonical
bundle on the total space. As an application the quasi-projectivity of the
moduli space of canonically polarized varieties
follows.
The direct images , , carry natural hermitian metrics. We prove an
explicit formula for the curvature tensor of these direct images. We apply it
to the morphisms that are induced by the Kodaira-Spencer map and obtain a differential
geometric proof for hyperbolicity properties of .Comment: Supercedes arXiv:0808.3259v4 and arXiv:1002.4858v2. To appear in
Invent. mat
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