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New examples of Stein manifolds with volume density property

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Abstract

In the present paper, we shall provide new examples of Stein manifolds enjoying the (algebraic) volume density property and compute their homology groups.

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Notes

  1. The continuous version of this theorem was proven by Vaserstein in [29]. The algebraic version of it (for a polynomial map f of n variables), apart from the trivial case \(n=1\) and Cohn’s well-known counterexample in case \(n=k=2\) (see [6]), for \(k\ge 3\) and any n is based upon a deep result of Suslin (see [25]): any matrix in \({{\,\mathrm{SL}\,}}_k(\mathbb {C}[\mathbb {C}^n])\) decomposes as a finite product of unipotent (and equivalently elementary) matrices.

  2. We would like to thank the referees for the proof in the above remark concerning the uniqueness of the volume form each variety \(X_n\) can be endowed with.

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Funding

Funding was provided by Schweizerischer Nationalfonds zur Förderung der wissenschaftlichen Forschung (SNF). Project No. 200021:178730.

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De Vito, G. New examples of Stein manifolds with volume density property. Complex Anal Synerg 6, 9 (2020). https://doi.org/10.1007/s40627-020-00043-y

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