Uniqueness of Tangent Cones to Positive-(p,p) Integral Cycles

TL;DR

This paper proves the uniqueness of tangent cones for integral cycles semi-calibrated by a specific form in almost hermitian manifolds, using an algebraic blow-up technique adapted to the almost complex setting.

Contribution

It establishes the uniqueness of tangent cones for positive-(p,p) integral cycles in almost hermitian manifolds, extending previous results to a broader geometric context.

Findings

Unique tangent cones at every point for semi-calibrated integral cycles

Development of an algebraic blow-up method for almost complex manifolds

Extension of tangent cone uniqueness results beyond closed calibrations

Abstract

Let $(M, \om)$ be a symplectic manifold, endowed with a compatible almost complex structure J and the associated metric g . For any p \in {1, 2, ... (dim M)/2} the form $\Om := \frac{\om^p}{p!}$ is a calibration. More generally, dropping the closedness assumption on $\om$, we get an almost hermitian manifold $(M, \om, J, g)$ and then $\Om$ is a so-called semi-calibration. We prove that integral cycles of dimension 2p (semi-)calibrated by $\Om$ possess at every point a unique tangent cone. The argument relies on an algebraic blow up perturbed in order to face the analysis issues of this problem in the almost complex setting.