Subbundle

Abstract: A CR structure on a real manifold M is a distinguished complex subbundle Z of the complex tangent bundle C TM, satisfying X n Z = 0 and [k, X] c X For example, the complex structure of Cn+I induces a natural biholomorphically invariant CR structure on any real hypersurface: *' is the space of vectors in the span of a/lz1, / , aaZn+l which are tangent to the hypersurface An abstract CR manifold M is said to be of hypersurface type if dimR M = 2n + I and dim, X = n; all our CR manifolds will be of this type If M is oriented, then there is a globally defined real one-form 0 that annihilates *' and The Levi form, given by Lo (V, W) = -2idO (V A W), is a hermitian form on X We will assume that the CR structure is strictly pseudoconvex: for some choice of 0, the Levi form Lo is positive definite on Z In this case 0 defines a contact structure on M and we call 0 a contact form associated with the CR structure The Levi form plays a role similar to that of the metric in Riemannian geometry However, the CR structure only determines the Levi form up to a conformal multiple; this multiple is fixed by the choice of a contact form A CR structure with a given choice of contact form is called a pseudohermitian structure Thus there is an analogy between pseudohermitian and CR manifolds on the one hand and Riemannian and conformal manifolds on the other In particular, Webster [W1, W2] and Tanaka [T] have defined a pseudohermitian scalar curvature associated to Lo The CR Yamabe problem is: given a compact, strictly pseudoconvex CR manifold, find a choice of contact form for which the pseudohermitian scalar curvature is constant Suppose M is a strictly pseudoconvex CR manifold of dimension 2n + 1 Solutions to the CR Yamabe problem are precisely the critical points of the CR

Abstract: These notes are an expanded version of lectures delivered at the AMS Summer School on Algebraic Geometry, held at Santa Cruz in July 1995 The main goal of the notes is to study complex varieties (mostly compact or projective algebraic ones), through a few geometric questions related to hyperbolicity in the sense of Kobayashi A convenient framework for this is the category of “directed manifolds”, that is, the category of pairs (X, V ) whereX is a complex manifold and V a holomorphic subbundle of TX If X is compact, the pair (X, V ) is hyperbolic if and only if there are no nonconstant entire holomorphic curves f : C → X tangent to V (Brody’s criterion) We describe a construction of projectivized kjet bundles PkV , which generalizes a construction made by Semple in 1954 and allows to analyze hyperbolicity in terms of negativity properties of the curvature More precisely, πk : PkV → X is a tower of projective bundles over X and carries a canonical line bundle OPkV (1) ; the hyperbolicity of X is then conjecturally equivalent to the existence of suitable singular hermitian metrics of negative curvature on OPkV (−1) for k large enough The direct images (πk)⋆OPkV (m) can be viewed as bundles of algebraic differential operators of order k and degree m, acting on germs of curves and invariant under reparametrization Following an approach initiated by Green and Griffiths, we establish a basic Ahlfors-Schwarz lemma in the situation when OPkV (−1) has a (possibly singular) metric of negative curvature, and we infer that every nonconstant entire curve f : C → V tangent to V must be contained in the base locus of the metric This basic result is then used to obtain a proof of the Bloch theorem, according to which the Zariski closure of an entire curve in a complex torus is a translate of a subtorus Our hope, supported by explicit Riemann-Roch calculations and other geometric considerations, is that the Semple bundle construction should be an efficient tool to calculate the base locus Necessary or sufficient algebraic criteria for hyperbolicity are then obtained in terms of inequalities relating genera of algebraic curves drawn on the variety, and singularities of such curves We finally describe some techniques introduced by Siu in value distribution theory, based on a use of meromorphic connections These techniques have been developped later by Nadel to produce elegant examples of hyperbolic surfaces of low degree in projective 3-space; thanks to a suitable concept of “partial projective projection” and the associated Wronskian operators, substantial improvements on Nadel’s degree estimate will be achieved here 2 J-P Demailly, Kobayashi hyperbolic projective varieties and jet differentials

Abstract: This paper concerns the development and application of the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations. In this theory, solutions of a PDE are sections of a fiber bundle $Y$ over a base manifold $X$ of dimension $n$$+$1, typically taken to be spacetime. Given a connection on $Y$, a covariant Hamiltonian density ${\mathcal H}$ is then intrinsically defined on the primary constraint manifold $P_{\mathcal L}$, the image of the multisymplectic version of the Legendre transformation. One views $P_{\mathcal L}$ as a subbundle of $J^1(Y)^\star$, the affine dual of $J^1(Y)$, the first jet bundle of $Y$. A canonical multisymplectic ($n$$+$2)-form $\Omega_{\mathcal H}$ is then defined, from which we obtain a multisymplectic Hamiltonian system of differential equations that is equivalent to both the original PDE as well as the Euler-Lagrange equations of the corresponding Lagrangian. We show that the $n$$+$1 2-forms $\omega^{(\mu)}$ defined by Bridges [1997] are a particular coordinate representation for a single multisymplectic ($n$$+$2)-form, and in the presence of symmetries, can be assembled into $\Omega_{\mathcal H}$. A generalized Hamiltonian Noether theory is then constructed which recovers the vanishing of the divergence of the vector of $n$$+$1 distinct momentum mappings defined in Bridges [1997] and, when applied to water waves, recovers Whitham's conservation of wave action. We also show the utility of this theory in the study of periodic pattern formation and wave instability.

Abstract: Introduction The Evans function and the stability index Tracking the fast subbundle The slow subbundle Calculation of the stability index Concluding remarks Bibliography