# Transversality of holomorphic maps into hyperquadrics

## TL;DR

This paper investigates the transversality of holomorphic maps from certain real hypersurfaces into hyperquadrics, providing conditions under which these maps are transversal or flat.

## Contribution

The paper introduces new conditions for transversality and flatness of holomorphic maps into hyperquadrics with specific signature constraints.

## Key findings

- If the signatures are equal, the map is either CR transversal or maps a neighborhood into the hyperquadric.
- When the target signature is greater, non-transversal maps must be transversally flat.
- The transversally flat result is shown to be optimal.

## Abstract

We study holomorphic maps F from a smooth Levi non-degenerate real hypersurface \documentclass[12pt]{minimal}
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\begin{document}$$ M_{\ell }\subset {\mathbb {C}}^n $$\end{document}Mℓ⊂Cn into a hyperquadric \documentclass[12pt]{minimal}
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\begin{document}$$ {\mathbb {H}}_{\ell '}^N $$\end{document}Hℓ′N with signatures \documentclass[12pt]{minimal}
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\begin{document}$$ \ell \le (n-1)/2 $$\end{document}ℓ≤(n-1)/2 and \documentclass[12pt]{minimal}
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\begin{document}$$ \ell '\le (N-1)/2,$$\end{document}ℓ′≤(N-1)/2, respectively. Assuming that \documentclass[12pt]{minimal}
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\begin{document}$$ N - n < n - 1,$$\end{document}N-n<n-1, we prove that if \documentclass[12pt]{minimal}
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\begin{document}$$ \ell = \ell ',$$\end{document}ℓ=ℓ′, then F is either CR transversal to \documentclass[12pt]{minimal}
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\begin{document}$$ {\mathbb {H}}_{\ell }^N $$\end{document}HℓN at every point of \documentclass[12pt]{minimal}
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\begin{document}$$ M_{\ell },$$\end{document}Mℓ, or it maps a neighborhood of \documentclass[12pt]{minimal}
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\begin{document}$$ M_{\ell } $$\end{document}Mℓ in \documentclass[12pt]{minimal}
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\begin{document}$$ {\mathbb {C}}^n $$\end{document}Cn into \documentclass[12pt]{minimal}
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\begin{document}$$ {\mathbb {H}}_{\ell }^N.$$\end{document}HℓN. Furthermore, in the case where \documentclass[12pt]{minimal}
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\begin{document}$$ \ell ' > \ell ,$$\end{document}ℓ′>ℓ, we show that if F is not CR transversal at \documentclass[12pt]{minimal}
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\begin{document}$$0\in M_\ell ,$$\end{document}0∈Mℓ, then it must be transversally flat. The latter is best possible.