Tangent cones to positive-(1,1) De Rham currents

TL;DR

This paper proves the uniqueness of tangent cones for positive-(1,1) De Rham currents in almost complex manifolds under a density continuity condition, extending classical geometric blow-up techniques without relying on plurisubharmonic potentials.

Contribution

It establishes a new uniqueness result for tangent cones in almost complex manifolds, adapting classical blow-up methods to a more general setting without plurisubharmonic functions.

Findings

Uniqueness of tangent cones under density continuity

Counterexamples exist without the density assumption

Extension of blow-up techniques to almost complex settings

Abstract

We consider positive-(1,1) De Rham currents in arbitrary almost complex manifolds and prove the uniqueness of the tangent cone at any point where the density does not have a jump with respect to all of its values in a neighbourhood. Without this assumption, counterexamples to the uniqueness of tangent cones can be produced already in C^n, hence our result is optimal. The key idea is an implementation, for currents in an almost complex setting, of the classical blow up of curves in algebraic or symplectic geometry. Unlike the classical approach in C^n, we cannot rely on plurisubharmonic potentials.