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  • $\begingroup$ Do you know a reference for the following statement (proved in your answer)? Let $\varphi\colon X\to Y$ be a surjective morphism of smooth $\Bbb C$-varieties. Then $\varphi$ is smooth if and only if the map on $\Bbb C$-points is a submersion, that is, $\varphi$ induces surjections on the tangent spaces. $\endgroup$ Commented Apr 25, 2021 at 19:46
  • $\begingroup$ I don't know a precise reference off the top of my head, but the place I'd look first would be Griffiths and Harris' Principles of Algebraic Geometry, which develops a lot of the theory of algebraic geometry from a complex-analytic perspective. I don't have a copy with me right now, so can't say for certain whether they treat smoothness and submersions. Let me know if this doesn't work, and I can ask a few colleagues for other suggestions. $\endgroup$ Commented Apr 26, 2021 at 7:43
  • $\begingroup$ @MikhailBorovoi Hartshorne, Algebraic geometry, III Proposition 10.4. $\endgroup$ Commented Mar 21, 2023 at 15:43
  • $\begingroup$ @red_trumpet: Thank you! it is very helpful. $\endgroup$ Commented Mar 21, 2023 at 16:11