Logarithmic operator intervals in the boundary theory of critical percolation

We consider the sub-sector of the $c=0$ logarithmic conformal field theory (LCFT) generated by the boundary condition changing (bcc) operator in two dimensional critical percolation. This operator is the zero weight Kac operator $\phi_{1,2}$, identified with the growing hull of the SLE$_6$ process. We identify percolation configurations with the significant operators in the theory. We consider operators from the first four bcc operator fusions: the identity and bcc operator; the stress tensor and its logarithmic partner; the derivative of the bcc operator and its logarithmic partner; and the pre-logarithmic operator $\phi_{1,3}$. We construct several intervals in the percolation model, each associated to one of the LCFT operators we consider, allowing us to calculate crossing probabilities and expectation values of crossing cluster numbers. We review the Coulomb gas, which we use as a method of calculating these quantities when the number of bcc operator makes a direct solution to the system of differential equations intractable. Finally we discuss the case of the six-point correlation function, which applies to crossing probabilities between the sides of a conformal hexagon. Specifically we introduce an integral result that allows one to identify the probability that a single percolation cluster touches three alternating sides a hexagon with free boundaries. We give results of the numerical integration for the case of a regular hexagon.Comment: 32 pages, 20 figure

On asymptotics for the Mabuchi energy functional

If $M$ is a projective manifold in $P^N$, then one can associate to each one parameter subgroup $H$ of $SL(N+1)$ the Mumford $\mu$ invariant. The manifold $M$ is Chow-Mumford stable if $\mu$ is positive for all $H$. Tian has defined the notion of K-stability, and has shown it to be intimately related to the existence of K\"ahler-Einstein metrics. The manifold $M$ is K-stable if $\mu'$ is positive for all $H$, where $\mu'$ is an invariant which is defined in terms of the Mabuchi K-energy. In this paper we derive an explicit formula for $\mu'$ in the case where $M$ is a curve. The formula is similar to Mumford's formula for $\mu$, and is likewise expressed in terms of the vertices of the Newton diagram of a basis of holomorphic sections for the hyperplane line bundle.Comment: 14 page

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

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 $C^0$ norm

Stability, energy functionals, and K\"ahler-Einstein metrics

An explicit seminorm ||f||_{#} on the vector space of Chow vectors of projective varieties is introduced, and shown to be a generalized Mabuchi energy functional for Chow varieties. The singularities of the Chow varieties give rise to currents supported on their singular loci, while the regular parts are shown to reproduce the Mabuchi energy functional of the corresponding projective variety. Thus the boundedness from below of the Mabuchi functional, and hence the existence of K\"ahler-Einstein metrics, is related to the behavior of the current $[Y_s]$ and the seminorm ||f||_{#} along the orbits of $SL(N+1,{\bf C})$.Comment: PlainTEX file, 28 page

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

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

Test Configurations for K-Stability and Geodesic Rays

Let $X$ be a compact complex manifold, $L\to X$ an ample line bundle over $X$, and ${\cal H}$ the space of all positively curved metrics on $L$. We show that a pair $(h_0,T)$ consisting of a point $h_0\in {\cal H}$ and a test configuration $T=({\cal L}\to {\cal X}\to {\bf C})$, canonically determines a weak geodesic ray $R(h_0,T)$ in ${\cal H}$ which emanates from $h_0$. Thus a test configuration behaves like a vector field on the space of K\"ahler potentials ${\cal H}$. We prove that $R$ is non-trivial if the ${\bf C}^\times$ action on $X_0$, the central fiber of $\cal X$, is non-trivial. The ray $R$ is obtained as limit of smooth geodesic rays $R_k\subset{\cal H}_k$, where ${\cal H}_k\subset{\cal H}$ 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 $C^{1,\alpha}, 0<\alpha<1$. 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

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