The Unique Tangent Cone Property for Weakly Holomorphic Maps into Projective Algebraic Varieties

TL;DR

This paper proves the uniqueness of tangent maps for weakly holomorphic maps into projective algebraic varieties and extends the tangent cone uniqueness to certain pseudo-holomorphic cycles, providing new insights and proofs.

Contribution

It establishes the tangent map uniqueness for weakly holomorphic maps into projective varieties and offers a new proof for tangent cone uniqueness of positive cycles in almost complex manifolds.

Findings

Uniqueness of tangent maps for weakly holomorphic maps.

Unique tangent cone property for certain pseudo-holomorphic cycles.

New proof of tangent cone uniqueness for positive integral cycles.

Abstract

In the present paper, we establish the uniqueness of tangent maps for general weakly holomorphic and locally approximable maps from an arbitrary almost complex manifold into projective algebraic varieties. As a byproduct of the approach and the techniques developed we also obtain the unique tangent cone property for a special class of non-rectifiable positive pseudo-holomorphic cycles. This approach gives also a new proof of the main result by C. Bellettini on the uniqueness of tangent cones for positive integral $(p,p)$-cycles in arbitrary almost complex manifolds.