Pluri-polarity in almost complex structures

Abstract

J -Holomorphic curves are −∞ sets of J -plurisubharmonic functions, with a singularity of LogLog type, but it is shown that in general they are not −∞ sets of J -plurisubharmonic functions with Logarithmic singularity (i.e. non-zero Lelong number). Some few additional remarks on pluripolarity in almost complex structures are made. Mathematics Subject Classification (2000)32Q60 · 32Q65 · 32U05 0 Introduction J -Plurisubharmonic functions with poles (at which the function is −∞) have already played a role in almost complex analysis. They has been used in the study of the Kobayashi metric [4,5] for getting an efficient control of J -holomorphic discs. In a work in progress with Ivashkovich, applications to uniqueness problems are given. The first pluripolarity result is due to Chirka who showed that if J is a C 1 almost complex structure defined near 0 in C n and if J (0) = J st (the standard complex structure), then for A > 0 large enough log |z| + A|z| is J -plurisubharmonic near 0. A complete proof has been written in [5, Lemma 1.4, page 2400]. The function − log | log |z|| is also J -plurisubharmonic near 0. Although a 'log-log singularity' (zero Lelong number!) is much less interesting and has less applications that a 'log singularity', functions with log-log singularity were introduced in [10] to show pluripolarity of J -holomorphic curves. Later, Elkhadhra [2] generalized the result to show pluri-polarity of J -holomorphic submanifolds, again with a 'loglog' singularity of the function.In [10] there has been an error in stating the smoothness hypotheses, all smoothness requirements in the statements have to be increased by 1 (C k+1 instead of C k ). Indeed on line 11-page 663 it is claimed that because J [Y , J Y ](Z , 0) = 0, one has |J [Y , J Y ](Z , Z )| ≤ C|Z ||Y | 2 . This is correct if J is of class C 2 (or at least C 1,1 ), but C 1 smoothness, hence continuity of [Y , J Y ], is not enough.

Pluripolar Sets, Real Submanifolds and Pseudoholomorphic Discs

“…In particular, we obtain the following far reashing generalization of one of the results of J.-P.Rosay [14]. Corollary 5.3 Let E be a totally real n-dimensional submanifold of an almost complex ndimensional manifold (M, J).…”

Discs and boundary uniqueness for psh functions on almost complex manifold

“…where t = (t 1 , ..., t n ), t j > 0 and c ∈ R n are real parameters. Note that the first and the last terms in the right hand are holomorphic on D. Therefore, any solution of (10) satisfies the Cauchy-Riemann equations (5) i.e. is a J-complex disc.…”

“…In the general case of non-integrable almost complex structures, the development of this theory began recently and many natural questions remain open. Our main motivation arises from the paper by J.-P.Rosay [5] where several interesting properties of (non) pluripolar subsets of almost complex manifolds are established. In particular, he proved that a J-complex curve is a pluripolar set.…”

“…In particular, he proved that a J-complex curve is a pluripolar set. On the other hand, it was sketched in [5] a proof of the fact that a totally real submanifold of dimension 2 can non be a pluripolar subset of an almost complex manifold of complex dimension 2 (a hint for higher dimension argument also was indicated without details).…”

“…It states that a function plurisubharmonic in a wedge with generic totally real edge is equal to −∞ identically if it tends to −∞ approaching the edge. Our proof is completely different from the argument of [5]. Our approach is based on construction of a suitable family of Jcomplex discs and is inspired by the well-known work [3].…”