34 research outputs found
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
The K\"ahler-Ricci flow on surfaces of positive Kodaira dimension
The existence of K\"ahler-Einstein metrics on a compact K\"ahler manifold has
been the subject of intensive study over the last few decades, following Yau's
solution to Calabi's conjecture. The Ricci flow, introduced by Richard Hamilton
has become one of the most powerful tools in geometric analysis.
We study the K\"ahler-Ricci flow on minimal surfaces of Kodaira dimension one
and show that the flow collapses and converges to a unique canonical metric on
its canonical model. Such a canonical is a generalized K\"ahler-Einstein
metric. Combining the results of Cao, Tsuji, Tian and Zhang, we give a metric
classification for K\"aher surfaces with a numerical effective canonical line
bundle by the K\"ahler-Ricci flow. In general, we propose a program of finding
canonical metrics on canonical models of projective varieties of positive
Kodaira dimension
Manifolds with 1/4-pinched flag curvature
We say that a nonnegatively curved manifold has quarter pinched flag
curvature if for any two planes which intersect in a line the ratio of their
sectional curvature is bounded above by 4. We show that these manifolds have
nonnegative complex sectional curvature. By combining with a theorem of Brendle
and Schoen it follows that any positively curved manifold with strictly quarter
pinched flag curvature must be a space form. This in turn generalizes a result
of Andrews and Nguyen in dimension 4. For odd dimensional manifolds we obtain
results for the case that the flag curvature is pinched with some constant
below one quarter, one of which generalizes a recent work of Petersen and Tao
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
Existence of Kähler–Einstein metrics and multiplier ideal sheaves on del Pezzo surfaces
We apply Nadel’s method of multiplier ideal sheaves to show that every complex del Pezzo surface of degree at most six whose automorphism group acts without fixed points has a Kähler–Einstein metric. In particular, all del Pezzo surfaces of degree 4, 5, or 6 and certain special del Pezzo surfaces of lower degree are shown to have a Kähler–Einstein metric. These existence statements are not new, but the proofs given in the present paper are less involved than earlier ones by Siu, Tian and Tian–Yau
Differential inequalities on complete Riemannian manifolds and applications
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46226/1/208_2005_Article_BF01455859.pd
Laser light-scattering study of ultrasonicated and thermosonicated soy proteins
In the food industry, detailed information on the relation between processing conditions and size of aggregates formed is desirable to permit optimization of some manufacturing processes. Ultrasonication (US) and thermosonication (TS, the combination of ultrasound and heat) have been used to modify the structure and function of various food proteins. Detailed investigation on the mechanisms requires accurate determination of the molecular weight and size distribution of proteins. The objective of this study was to investigate the structural characteristic of soy proteins after US and TS treatments using size-exclusion chromatography combined with on-line multi-angle laser light scattering and dynamic quasi-elastic light scattering (SEC-MALLS-QELS). SPI solutions (100 mg/mL) were sonicated for 0, 10, 30, 60, 180 and 300 seconds at power level of 72.02 W/cm2. For thermosonicated samples, protein solutions were first brought up to 25, 40, 50 and 70oC in a temperature-controlled water bath and then sonicated with the same processor for 10 seconds at 72.02 W/cm2. Samples were then analyzed by the SEC-MALLS-QELS system. US treatment caused aggregation of SPI and 7S globulins. Prolonged period of treatment also led to aggregation of 11S globulin to form larger soluble macromolecules. Changes in weight-average molecular weight (Mw) and hydrodynamic radii (rh) suggest that initial formation of large aggregates followed by dissociation. Upon TS treatment, SPI and 7S globulins were associated to form large macromolecules whereas 11S globulin was dissociated. Increases in Mw in TS treated SPI and 7S globulins suggest that heat and ultrasound treatments acted synergistically to promote protein aggregation. TS treatment also led to a slight decrease in Mw of 11S globulin, suggesting protein dissociation. The present results showed that the SEC-MALLS-QELS system is suitable for monitoring changes in molecular weight/size and conformation of soy proteins during ultrasonic treatments