A maximum rank problem for degenerate elliptic fully nonlinear equations

The solutions to the Dirichlet problem for two degenerate elliptic fully nonlinear equations in $n+1$ dimensions, namely the real Monge-Amp\`ere equation and the Donaldson equation, are shown to have maximum rank in the space variables when $n \leq 2$. A constant rank property is also established for the Donaldson equation when $n=3$

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

Calogero-Moser and Toda Systems for Twisted and Untwisted Affine Lie Algebras

The elliptic Calogero-Moser Hamiltonian and Lax pair associated with a general simple Lie algebra \G are shown to scale to the (affine) Toda Hamiltonian and Lax pair. The limit consists in taking the elliptic modulus $\tau$ and the Calogero-Moser couplings $m$ to infinity, while keeping fixed the combination $M = m e^{i \pi \delta \tau}$ for some exponent $\delta$. Critical scaling limits arise when $1/\delta$ equals the Coxeter number or the dual Coxeter number for the untwisted and twisted Calogero-Moser systems respectively; the limit consists then of the Toda system for the affine Lie algebras \G^{(1)} and (\G ^{(1)})^\vee. The limits of the untwisted or twisted Calogero-Moser system, for $\delta$ less than these critical values, but non-zero, consists of the ordinary Toda system, while for $\delta =0$, it consists of the trigonometric Calogero-Moser systems for the algebras \G and \G^\vee respectively.Comment: 34 pages, Plain TeX, minor typos correcte

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

Lectures on Supersymmetric Yang-Mills Theory and Integrable Systems

We present a series of four self-contained lectures on the following topics: (I) An introduction to 4-dimensional 1\leq N \leq 4 supersymmetric Yang-Mills theory, including particle and field contents, N=1 and N=2 superfield methods and the construction of general invariant Lagrangians; (II) A review of holomorphicity and duality in N=2 super-Yang-Mills, of Seiberg-Witten theory and its formulation in terms of Riemann surfaces; (III) An introduction to mechanical Hamiltonian integrable systems, such as the Toda and Calogero-Moser systems associated with general Lie algebras; a review of the recently constructed Lax pairs with spectral parameter for twisted and untwisted elliptic Calogero-Moser systems; (IV) A review of recent solutions of the Seiberg-Witten theory for general gauge algebra and adjoint hypermultiplet content in terms of the elliptic Calogero-Moser integrable systems.Comment: 124 pages, 1 figure, LaTe

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

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

Complex Geometry and Supergeometry

Complex geometry and supergeometry are closely entertwined in superstring perturbation theory, since perturbative superstring amplitudes are formulated in terms of supergeometry, and yet should reduce to integrals of holomorphic forms on the moduli space of punctured Riemann surfaces. The presence of supermoduli has been a major obstacle for a long time in carrying out this program. Recently, this obstacle has been overcome at genus 2, which is the first loop order where it appears in all amplitudes. An important ingredient is a better understanding of the relation between geometry and supergeometry, and between holomorphicity and superholomorphicity. This talk provides a survey of these developments and a brief discussion of the directions for further investigation.Comment: 42 page

Lax Pairs and Spectral Curves for Calogero-Moser and Spin Calogero-Moser Systems

We summarize recent results on the construction of Lax pairs with spectral parameter for the twisted and untwisted elliptic Calogero-Moser systems associated with arbitrary simple Lie algebras, their scaling limits to Toda systems, and their role in Seiberg-Witten theory. We extend part of this work by presenting a new parametrization for the spectral curves for elliptic spin Calogero-Moser systems associated with SL(N).Comment: 18 pages, no figures, Plain TeX; Contribution to "Regular and Chaotic Dynamics" dedicated to J. Moser; several references 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