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We prove that every rational proper holomorphic map from $\mathbb{B}^2$ to $\mathbb{B}^4$ has degree at most $5$, confirming D'Angelo's conjecture in this case. This sharp bound follows from Meylan's degree estimates and an intersection-theoretic argument ruling out degree $6$.
Let $n$ and $N$ be integers with $2\leq n<N$, and let $F:\mathbb{B}^n\to\mathbb{B}^N$ be a proper holomorphic map that admits a $C^{N-n+1}$-smooth extension to the boundary. We prove that if the CR Gauss map of its boundary restriction is generically degenerate, then $F$ is holomorphically equivalent, under composition...
Let $f:X\to S$ be a smooth projective family, and fix a semisimple flat bundle on a fiber $X_0$. Its isomonodromic deformation determines a holomorphic section $\sigma_{\mathrm{dR}}:S\to M_{\mathrm{dR}}(X/S)$ of the relative de Rham moduli space. Applying the relative non-abelian Hodge correspondence fiberwise gives a...
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