506 research outputs found
Test Configurations for K-Stability and Geodesic Rays
Let be a compact complex manifold, an ample line bundle over
, and the space of all positively curved metrics on . We show
that a pair consisting of a point and a test
configuration , canonically determines a
weak geodesic ray in which emanates from . Thus a
test configuration behaves like a vector field on the space of K\"ahler
potentials . We prove that is non-trivial if the
action on , the central fiber of , is non-trivial. The ray is
obtained as limit of smooth geodesic rays , where is the subspace of Bergman metrics.Comment: 27 pages, no figure; references added; typos correcte
Regularity of geodesic rays and Monge-Ampere equations
It is shown that the geodesic rays constructed as limits of Bergman geodesics
from a test configuration are always of class . An
essential step is to establish that the rays can be extended as solutions of a
Dirichlet problem for a Monge-Ampere equation on a Kaehler manifold which is
compact
The Dirichlet problem for degenerate complex Monge-Ampere equations
The Dirichlet problem for a Monge-Ampere equation corresponding to a
nonnegative, possible degenerate cohomology class on a Kaehler manifold with
boundary is studied. C^{1,\alpha} estimates away from a divisor are obtained,
by combining techniques of Blocki, Tsuji, Yau, and pluripotential theory. In
particular, C^{1,\alpha} geodesic rays in the space of Kaehler potentials are
constructed for each test configuratio
Partial Legendre transforms of non-linear equations
The partial Legendre transform of a non-linear elliptic differential equation
is shown to be another non-linear elliptic differential equation. In
particular, the partial Legendre transform of the Monge-Amp\`ere equation is
another equation of Monge-Amp\`ere type. In 1+1 dimensions, this can be applied
to obtain uniform estimates to all orders for the degenerate Monge-Amp\`ere
equation with boundary data satisfying a strict convexity condition.Comment: 12 pages, no figur
Algebraic estimates, stability of local zeta functions, and uniform estimates for distribution functions
A method of ``algebraic estimates'' is developed, and used to study the
stability properties of integrals of the form \int_B|f(z)|^{-\d}dV, under small
deformations of the function f. The estimates are described in terms of a
stratification of the space of functions \{R(z)=|P(z)|^{\e}/|Q(z)|^{\d}\} by
algebraic varieties, on each of which the size of the integral of R(z) is given
by an explicit algebraic expression. The method gives an independent proof of a
result on stability of Tian in 2 dimensions, as well as a partial extension of
this result to 3 dimensions. In arbitrary dimensions, combined with a key lemma
of Siu, it establishes the continuity of the mapping c\ra
\int_B|f(z,c)|^{-\d}dV_1\cdots dV_n when f(z,c) is a holomorphic function of
(z,c). In particular the leading pole is semicontinuous in f, strengthening
also an earlier result of Lichtin.Comment: 53 pages, published versio
On stability and the convergence of the K\"ahler-Ricci flow
Assuming uniform bounds for the curvature, the exponential convergence of the
K\"ahler-Ricci flow is established under two conditions which are a form of
stability: the Mabuchi energy is bounded from below, and the dimension of the
space of holomorphic vector fields in an orbit of the diffeomorphism group
cannot jump up in the limit.Comment: 18 pages, no figur
On Pointwise Gradient Estimates for the Complex Monge-Ampere Equation
In this note, a gradient estimate for the complex Monge-Ampere equation is
established. It differs from previous estimates of Yau, Hanani, Blocki, P.
Guan, B. Guan - Q. Li in that it is pointwise, and depends only on the infimum
of the solution instead of its norm
Lectures on Two-Loop Superstrings
In these lectures, recent progress on multiloop superstring perturbation
theory is reviewed. A construction from first principles is given for an
unambiguous and slice-independent two-loop superstring measure on moduli space
for even spin structure. A consistent choice of moduli, invariant under local
worldsheet supersymmetry is made in terms of the super-period matrix. A variety
of subtle new contributions arising from a careful gauge fixing procedure are
taken into account.
The superstring measure is computed explicitly in terms of genus two
theta-functions and reveals the importance of a new modular object of weight 6.
For given even spin structure, the measure exhibits a behavior under
degenerations of the worldsheet that is consistent with physical principles.
The measure allows for a unique modular covariant GSO projection. Under this
GSO projection, the cosmological constant, the 1-, 2- and 3- point functions of
massless supergravitons vanish pointwise on moduli space. A certain
disconnected part of the 4-point function is shown to be given by a convergent
integral on moduli space. A general consistent formula is given for the
two-loop cosmological constant in compactifications with central charge c=15
and with N=1 worldsheet supersymmetry. Finally, some comments are made on
possible extensions of this work to higher loop order.Comment: 37 pages, 3 figures, Lectures delivered at Hangzhou and Beijing 200
Lectures on Stability and Constant Scalar Curvature
An introduction is provided to some current research trends in stability in
geometric invariant theory and the problem of Kaehler metrics of constant
scalar curvature. Besides classical notions such as Chow-Mumford stability, the
emphasis is on several new stability conditions, such as K-stability,
Donaldson's infinite-dimensional GIT, and conditions on the closure of orbits
of almost-complex structures under the diffeomorphism group. Related analytic
methods are also discussed, including estimates for energy functionals,
Tian-Yau-Zelditch approximations, estimates for moment maps, complex
Monge-Ampere equations and pluripotential theory, and the Kaehler-Ricci flowComment: 85 pages, minor corrections adde
Seiberg-Witten Theory and Integrable Systems
We summarize recent results on the resolution of two intimately related
problems, one physical, the other mathematical. The first deals with the
resolution of the non-perturbative low energy dynamics of certain N=2
supersymmetric Yang-Mills theories. We concentrate on the theories with one
massive hypermultiplet in the adjoint representation of an arbitrary gauge
algebra G. The second deals with the construction of Lax pairs with spectral
parameter for certain classical mechanics Calogero-Moser integrable systems
associated with an arbitrary Lie algebra G. We review the solution to both of
these problems as well as their interrelation.Comment: 30 pages, Based on Lectures delivered at Edinburgh and Kyot
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