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arXiv · 2609.15698

Sharp degree bound for rational proper maps from $\mathbb B^2$ to $\mathbb B^4$

Tianzhi Hu·Mai Shi·Pingsan Yuan
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Abstract

We prove D'Angelo's degree conjecture for rational proper holomorphic maps from $\mathbb{B}^2$ to $\mathbb{B}^4$, establishing the sharp degree bound of five. Suppose, to the contrary, that a rational proper map of degree six exists. We associate to the map a characteristic number measuring the degeneracy of its projective differential data. A global intersection-theoretic computation determines this number exactly, while a local analysis along the degeneracy locus yields a strictly larger lower bound for the same quantity. This contradiction excludes degree six and proves the conjectured bound.

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Tianzhi Hu, Mai Shi, Pingsan Yuan. 2026-09-14. Sharp degree bound for rational proper maps from $\mathbb B^2$ to $\mathbb B^4$. https://arxiv.org/abs/2609.15698

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