Pluripotential Theory and Monge–Ampère Foliations

A regular, rank one solution u of the complex homogeneous Monge–Ampere equation \({(\partial \overline{\partial }u)}^{n} = 0\) on a complex manifold is associated with the Monge–Ampere foliation, given by the complex curves along which u is harmonic. Monge–Ampere foliations find many applications in complex geometry and the selection of a good candidate for the associated Monge–Ampere foliation is always the first step in the construction of well behaved solutions of the complex homogeneous Monge–Ampere equation. Here, after reviewing some basic notions on Monge–Ampere foliations, we concentrate on two main topics. We discuss the construction of (complete) modular data for a large family of complex manifolds, which carry regular pluricomplex Green functions. This class of manifolds naturally includes all smoothly bounded, strictly linearly convex domains and all smoothly bounded, strongly pseudoconvex circular domains of \({\mathbb{C}}^{n}\). We then report on the problem of defining pluricomplex Green functions in the almost complex setting, providing sufficient conditions on almost complex structures, which ensure existence of almost complex Green pluripotentials and equality between the notions of stationary disks and of Kobayashi extremal disks, and allow extensions of known results to the case of non integrable complex structures.

For $\Omega$ a domain in $\mathbb C^n$, the pluricomplex Green function with poles $a_1, ...,a_N \in \Omega$ is defined as $G(z):=\sup \{u(z): u\in PSH_-(\Omega), u(x)\le \log \|x-a_j\|+C_j \text{when} x \to a_j, j=1,...,N \}$. When there is only one pole, or two poles in the unit ball, it turns out to be equal to the Lempert function defined from analytic disks into $\Omega$ by $L_S (z) :=\inf \{\sum^N_{j=1}\nu_j\log|\zeta_j|: \exists \phi\in \mathcal {O}(\mathbb D,\Omega), \phi(0)=z, \phi(\zeta_j)=a_j, j=1,...,N \}$. It is known that we always have $L_S (z) \ge G_S(z)$. In the more general case where we allow weighted poles, there is a counterexample to equality due to Carlehed and Wiegerinck, with $\Omega$ equal to the bidisk. Here we exhibit a counterexample using only four distinct equally weighted poles in the bidisk. In order to do so, we first define a more general notion of Lempert function "with multiplicities", analogous to the generalized Green functions of Lelong and Rashkovskii, then we show how in some examples this can be realized as a limit of regular Lempert functions when the poles tend to each other. Finally, from an example where $L_S (z) > G_S(z)$ in the case of multiple poles, we deduce that distinct (but close enough) equally weighted poles will provide an example of the same inequality. Open questions are pointed out about the limits of Green and Lempert functions when poles tend to each other.

Using an integrable, homogeneous complex structure on the compact group SO(9), we show that the Hodge-de Rham spectral sequence for this non-Kahler compact complex manifold does not degenerate at Ei, contrary to a well-known conjecture. On any (compact) complex manifold M, the algebra of global complexvalued C°°-differential forms a*(M) has a bigrading given by the Hodge type; and the corresponding decomposition of the de Rham differential d = d + d gives rise to a double complex (a*'*(M),d, d). The spectral sequence corresponding to the first (holomorphic) degree is E = H<*{M, QM) =» HPp(M) with dx = d : this is the Hodge-de Rham (HdR) spectral sequence. When M is compact and Kahler, E = Eoo by Hodge theory. A folklore conjecture of about thirty years' standing says that for any compact complex manifold one should have E2 = E^. We will give an example to show that this conjecture is false. 1. There is an old observation of H. Samelson (see Wang [4]) that every compact Lie group G of even dimension (equivalently even rank) can be made into a complex manifold in such a way that all left-translations by elements g e G are holomorphic maps: we call such a structure an LICS on G (= left invariant, integrable complex structure on G). If G is in addition semisimple, then no LICS can be Kahlerian because H(G : R) = 0. Our example is a particular LICS on SO(9) (equivalently Spin(9): see §3 below), for which E2 has complex dimension 26. Since one knows that E^ has complex dimension 16, we obtain the required nondegeneration of HdR. Received by the editors November 23, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 32C10, 58A14, 55J05, 32M10. 1 Samelson's paper appeared in a Portuguese journal, which is perhaps understandably not available in Indian libraries. ©1989 American Mathematical Society 0273-0979/89 $1.00 + $.25 per page 19

Complex curves in hypercomplex nilmanifolds with [formula omitted]-solvable Lie algebras

The Riemann extension, introduced by E. K. Patterson and A. G. Walker, is a semi-Riemannian metric with a neutral signature on the cotangent bundle T∗M of a smooth manifold M, induced by a symmetric linear connection ∇ on M. In this paper we deal with a natural Riemann extension g¯, which is a generalization (due to M. Sekizawa and O. Kowalski) of the Riemann extension. We construct an almost complex structure J¯ on the cotangent bundle T∗M of an almost complex manifold (M,J,∇) with a symmetric linear connection ∇ such that (T∗M,J¯,g¯) is an almost complex manifold, where the natural Riemann extension g¯ is a Norden metric. We obtain necessary and sufficient conditions for (T∗M,J¯,g¯) to belong to the main classes of the Ganchev–Borisov classification of the almost complex manifolds with Norden metric. We also examine the cases when the base manifold is an almost complex manifold with Norden metric or it is a complex manifold (M,J,∇′) endowed with an almost complex connection ∇′ (∇′J=0). We investigate the harmonicity with respect to g¯ of the almost complex structure J¯, according to the type of the base manifold. Moreover, we define an almost hypercomplex structure (J¯1,J¯2,J¯3) on the cotangent bundle T∗M4n of an almost hypercomplex manifold (M4n,J1,J2,J3,∇) with a symmetric linear connection ∇. The natural Riemann extension g¯ is a Hermitian metric with respect to J¯1 and a Norden metric with respect to J¯2 and J¯3.

Generalized contact bundles are odd-dimensional analogues of generalized complex manifolds. They have been introduced recently and very little is known about them. In this paper we study their local structure. Specifically, we prove a local splitting theorem similar to those appearing in Poisson geometry. In particular, in a neighborhood of a regular point, a generalized contact bundle is either the product of a contact and a complex manifold or the product of a symplectic manifold and a manifold equipped with an integrable complex structure on the gauge algebroid of the trivial line bundle.

Sur l'existence du nombre de Lelong d'un courant positif fermé défini sur une variété presque complexe

We establish disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space. This function is also known as the pluricomplex Green function with logarithmic growth or a logarithmic pole at infinity. We extend Lempert’s formula for this function from the convex case to the connected case. Introduction The Siciak-Zahariuta extremal function VX of a subset X of complex affine space C is defined as the supremum of all entire plurisubharmonic functions u of minimal growth with u|X ≤ 0. It is also called the pluricomplex Green function of X with logarithmic growth or a logarithmic pole at infinity (although this is a bit of a misnomer if X is not bounded). A plurisubharmonic function u on C is said to have minimal growth (and belong to the class L) if u− log ‖·‖ is bounded above on C. If X is open and nonempty, then VX ∈ L. More generally, if X is not pluripolar, then the upper semicontinuous regularization V ∗ X of VX is in L, and if X is pluripolar, then V ∗ X =∞. Siciak-Zahariuta extremal functions play a fundamental role in pluripotential theory and have found important applications in approximation theory, complex dynamics, and elsewhere. For a detailed account of the basic theory, see [K, Chapter 5]. For an overview of some recent developments, see [Pl]. The extremal functions of pluripotential theory are usually defined as suprema of classes of plurisubharmonic functions with appropriate properties. The theory of disc functionals, initiated by Poletsky in the late 1980s [P1, PS], offers a different approach to extremal functions, realizing them as envelopes of disc functionals. A disc functional on a complex manifold Y is a map H into R = [−∞,∞] from the set of analytic discs in Y , that is, holomorphic maps from the open unit disc D into Y . We usually restrict ourselves to analytic discs that extend holomorphically to a neighbourhood of the closed unit disc. 2000 Mathematics Subject Classification. Primary: 32U35. The first-named author was supported in part by the Natural Sciences and Engineering Research Council of Canada. First version 22 April 2005. Second, expanded version 6 July 2005. Typeset by AMS-TEX 1 The envelope EH of H is the map Y → R that takes a point x ∈ Y to the infimum of the values H(f) for all analytic discs f in Y with f(0) = x. Disc formulas have been proved for such extremal functions as largest plurisubharmonic minorants, including relative extremal functions, and pluricomplex Green functions of various sorts, and used to establish properties of these functions that had proved difficult to handle via the supremum definition. Some of this work has been devoted to extending to arbitrary complex manifolds results that were first proved for domains in C. See for instance [BS, E, EP, LS1, LS2, LLS, P2, P3, R, RS]. In the convex case, there is a disc formula for the Siciak-Zahariuta extremal function due to Lempert [M, Appendix]. The main motivation for the present work was to generalize Lempert’s formula. Because of the growth condition in the definition of the Siciak-Zahariuta extremal function, we did not see how to fit it into the theory of disc functionals until we realized, from a remark of Guedj and Zeriahi [GZ], that minimal growth is nothing but quasi-plurisubharmonicity with respect to the current of integration along the hyperplane at infinity. This observation is implicit in the proof of Theorem 1, which presents a family of new disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of affine space. Theorem 2 contains more such formulas. Our main result, Theorem 3, establishes Lempert’s formula, in the following slightly modified form, for every connected open subset of affine space. The formula is easily seen to fail for disconnected sets in general. Theorem. The Siciak-Zahariuta extremal function VX of a connected open subset X of C is given by the disc formula

It is well known that any (paracompact) differentiable manifold M has a complexification, i.e., a complex manifold X ⊃ M, dimc X = dimℝ M, such that M is totally real in X (see Ref. 8). It is also known that a small neighborhood U of M in X is diffeomorphic to the tangent bundle TM of M. Thus, the tangent bundle TM of any differentiable manifold carries a complex manifold structure. This complex structure is, of course, not unique. One way of finding a “canonical” complex structure is to endow M with some extra structure and require that the complex structure on TM interact with the structure of M. Here we consider smooth (meaning infinitely differentiable) Riemannian manifolds M. When M = ℝ, there is a natural identification Tℝ ≅ ℂ given by $${{T}_{\sigma }}\mathbb{R} \mathrel\backepsilon \tau \frac{\partial }{{\partial \sigma }} \leftrightarrow \sigma + i\tau \in \mathbb{C},$$ (1.1) and this endows Tℝ with a complex structure. In (1.1) σ denotes the coordinate on R. This coordinate depends on the algebraic structure of the identification (1.1); however, the complex structure on Tℝ depends only on the metric of ℝ. In other words, an isometry of ℝ induces a biholomorphic mapping on Tℝ.

In this paper, the existence of almost complex structures on connected sums of almost complex manifolds and complex projective spaces are investigated. Firstly, we show that if M is a 2n-dimensional almost complex manifold, then so is $$M\sharp \overline{{\mathbb {C}}P^{n}}$$ , where $$\overline{{\mathbb {C}}P^{n}}$$ is the n-dimensional complex projective space with the reversed orientation. Secondly, for any positive integer $$\alpha $$ and any 4n-dimensional almost complex manifolds $$M_{i}, ~1\le i \le \alpha $$ , we prove that $$\left( \sharp _{i=1}^{\alpha } M_{i}\right) \sharp (\alpha {-}1) {\mathbb {C}}P^{2n}$$ must admit an almost complex structure. At last, as an application, we obtain that $$\alpha {\mathbb {C}}P^{2n}\sharp ~\beta \overline{{\mathbb {C}}P^{2n}}$$ admits an almost complex structure if and only if $$\alpha $$ is odd.

We give a construction of integrable complex structures on the total space of a smooth principal bundle over a complex manifold, with an even dimensional compact Lie group as structure group, under certain conditions. This generalizes the constructions of complex structure on compact Lie groups by Samelson and Wang, and on principal torus bundles by Calabi-Eckmann and others. It also yields large classes of new examples of non-Kähler compact complex manifolds. Moreover, under suitable restrictions on the base manifold, the structure group, and characteristic classes, the total space of the principal bundle admits SKT metrics. This generalizes recent results of Grantcharov et al. We study the Picard group and the algebraic dimension of the total space in some cases. We also use a slightly generalized version of the construction to obtain (non-Kähler) complex structures on tangential frame bundles of complex orbifolds.

Let (g, [•, •]) be a Lie algebra with an integrable complex structure J. The ±i eigenspaces of J are complex subalgebras of gC isomorphic to the algebra (g, [*]J) with bracket [X * Y]J = 2 ([X, Y] - [JX, JY]). We consider here the case where these subalgebras are nilpotent and prove that the original (g, [•, •]) Lie algebra must be solvable. We consider also the 6-dimensional case and determine explicitly the possible nilpotent Lie algebras (g, [*]J). Finally we produce several examples illustrating different situations, in particular we show that for each given s there exists g with complex structure J such that (g, [*]J) is s-step nilpotent. Similar examples of hypercomplex structures are also built.

An integral condition involving [formula omitted]-harmonic (0,1)-forms

It is well known that the tensor field [Formula: see text] of type [Formula: see text] on the manifold [Formula: see text] is an almost complex structure if [Formula: see text] is an identity tensor field and the manifold [Formula: see text] is called the complex manifold. LetkM be the [Formula: see text] order extended complex manifold of the manifold [Formula: see text]. A tensor field [Formula: see text] onkM is called extended almost complex structure if [Formula: see text]. This paper aims to study the higher order complete and vertical lifts of the extended almost complex structures on an extended complex manifoldkM. The proposed theorems on the Nijenhuis tensor of an extended almost complex structure [Formula: see text] on the extended complex manifoldkM are proved. Also, a tensor field [Formula: see text] of type [Formula: see text] is introduced and shows that it is an extended almost complex structure. Furthermore, the Lie derivative concerning higher-order lifts is studied and basic results on the almost analytic complex vector concerning an extended almost complex structure onkM are investigated. Finally, for more detailed explanation and better understanding a tensor field [Formula: see text] of type [Formula: see text] is introduced onkM, proving that it is a metallic structure onkM. A study of a golden structure, which is a type of metallic structure, is also carried out.

On a smooth manifold $$M$$ , generalized complex (generalized paracomplex) structures provide a notion of interpolation between complex (paracomplex) and symplectic structures on $$M$$ . Given a complex manifold $$\left( M,j\right) $$ , we define six families of distinguished generalized complex or paracomplex structures on $$M$$ . Each one of them interpolates between two geometric structures on $$M$$ compatible with $$j$$ , for instance, between totally real foliations and Kähler structures, or between hypercomplex and $$\mathbb {C}$$ -symplectic structures. These structures on $$M$$ are sections of fiber bundles over $$M$$ with typical fiber $$G/H$$ for some Lie groups $$G$$ and $$H$$ . We determine $$G$$ and $$H$$ in each case. We proceed similarly for symplectic manifolds. We define six families of generalized structures on $$\left( M,\omega \right) $$ , each of them interpolating between two structures compatible with $$\omega $$ , for instance, between a $$\mathbb {C}$$ -symplectic and a para-Kähler structure (aka bi-Lagrangian foliation).