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

Riccardo Caniato, Tristan Rivière

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.

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

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 -cycles in arbitrary almost complex manifolds.

Paper Structure

This paper contains 16 sections, 25 theorems, 278 equations.

Table of Contents

  1. Introduction
  2. Weakly holomorphic and locally approximable maps
  3. Statement of the main results and previous literature
  4. Key ideas to face the non-integrable case in higher dimensions
  5. Final comments and open problems
  6. Almost monotonicity formula and tangent cones
  7. Smoothness at points with small density
  8. The fundamental Morrey type estimate
  9. Some technical lemmata
  10. Proof of the fundamental Morrey type estimate
  11. Almost pseudo-holomorphic foliations
  12. Proof of the main theorem
  13. A model problem
  14. The general case
  15. Recovering uniqueness of tangent maps for $u$
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $m,n\in\mathbb{N}_0$ be such that $m\geqslant 2$. Let $M$ be a smooth $2m$-dimensional manifold without boundary and let $J$ be any Lipschitz almost complex structure on $M$. Let $N\subset\mathbb{CP}^n$ be a projective algebraic variety. Assume that $u\in W_{loc}^{1,2}(M,N)$ is weakly $(J,j_n)$-

Theorems & Definitions (57)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Example 1.1
  • Definition 1.3: Semicalibrated currents
  • Theorem 1.2
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Corollary 2.1
  • ...and 47 more