Poincaré–Lelong formula, J -analytic subsets and Lelong numbers of currents on almost complex manifolds

Abstract

In this paper, we first establish a Poincaré-Lelong type formula in the almost complex setting. Then, after introducing the notion of J-analytic subsets, we study the restriction of a closed positive current defined in an almost complex manifold (M, J) on a J-analytic subset. Finally, we prove that the Lelong numbers of a plurisubharmonic current defined on an almost complex manifold are independent of the coordinate systems.

Applications of singular connections in symplectic and almost complex geometry

“…where θ is in L 1 loc (X). This generalizes the result of [6] which considers the case of the trivial bundle with flat connection.…”

“…As another special case we refind the result of Elkhadra [6] for sections s holomorphic along the zero set Z = s −1 (0) that is ∂J,L (s) = 0 on Z. Note that Z supposed to be a smooth manifold of real codimensin 2.…”

“…As an application, in the last section we extend Donaldson's result on convergence of currents of integration over zero sets of almost holomorphic sections of line bundles to the case of sections of arbitrary bundles. We note that the first step toward the Lelong-Poincaré formula in the almost complex setting was done in the interesting paper by Elkhadra [6]. He considered the special MSC: 32E20,32E30,32V40,53D12.…”

On tamed almost complex four manifolds

“…By F. Elkhadhra's result (see Theorem 2 in [24] or Lemma B.9 in Appendix B.1), if D is an irreducible J-holomorphic curve in E c (P ), As done in [7], it is always possible to approximate the closed positive current T by smooth real currents admitting a small negative part and that this negative part can be estimated in terms of the Lelong numbers of T and the geometry of (M, g, J, F ) (cf. Theorem C.12 and Remark C.13 in Appendix C.4).…”

“…is valid. In [24], Elkhadhra extended Poincaré-Lelong equation to the almost complex category. Let Ω be an open set of R 2n equipped with an almost complex structure J.…”