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On the Similarity of Holomorphic Matrices

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Abstract

Guralnick (Linear Algebr Appl 99:85–96, 1988) proved that two holomorphic matrices on a noncompact connected Riemann surface, which are locally holomorphically similar, are globally holomorphically similar. We generalize this to (possibly, nonsmooth) one-dimensional Stein spaces. For Stein spaces of arbitrary dimension, we prove that global \(\mathcal {C}^\infty \) similarity implies global holomorphic similarity, whereas global continuous similarity is not sufficient.

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Notes

  1. Here and in the following we use the convention that statements like “\(f=g\) on \(\emptyset \)” or “\(f:=g\) on \(\emptyset \)” have to be omitted.

  2. Indeed, let f be a \(\mathcal {C}_X^G\) cocycle, and let B be the principal G-bundle defined by f. Then (by definition of B) the \(\mathcal {C}_X^G\)-triviality of f (which we have to prove) is equivalent to the existence of a global continuous section of B, and the existence of such a global continuous section follows, e.g., from [21, Theorem 11.5 and §29.1].

  3. Possibly, some of the zeros in this block matrix have to be omitted.

  4. Lemma 6.3 is not explicitly stated in [22], but it follows immediately from Lemma 1 of [22]. Also, in [22], X is a domain in the complex plane, but the proof given there works also in the general case.

  5. Below we explain this in detail in the case of Lemmas 7.4 and 7.5, each of which is stronger than Lemma 7.1.

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Acknowledgements

The author wants to thank F. Forstneric̆, F. Kutzschebauch and J. Ruppenthal for helpful discussions (in particular, see Remark 5.3).

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  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, 12489, Berlin, Germany

    Jürgen Leiterer

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  1. Jürgen Leiterer
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Correspondence to Jürgen Leiterer.

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Dedicated to the memory of Gennadi Henkin, teacher and coauthor.

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Some of the results of this text are contained already in the preprint “Local and global similarity of holomorphic matrices” sent to some colleagues in September 2016 and later posted in the arXiv [15] (partially with different proofs).

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Leiterer, J. On the Similarity of Holomorphic Matrices. J Geom Anal 30, 2731–2757 (2020). https://doi.org/10.1007/s12220-018-0008-4

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