Let me give a sketch of proof for Gromov's claim in the case where $X$ is Kähler. More precisely, let me prove the following ${}$

**Proposition [G03, p.223].** Let $X$, $Y$ be two complex manifolds (not necessarily compact) of the same dimension and having the same even Betti numbers. If $X$ is Kähler, then every proper surjective holomorphic map $f \colon X \to Y$ is finite-to-one.

*Proof.* By a result of Wells [**W74**, Theorem 3.1], the map $f \colon X \to Y$ induces an injection $$f^* \colon H^{r}(Y, \, \mathbb{C}) \to H^{r}(X, \, \mathbb{C})$$ for all $r$. Using the duality $$H_r(X, \, \mathbb{C})=\mathrm{Hom}(H^r(X, \, \mathbb{C}), \, \mathbb{C})$$ we infer that the induced map in homology $$f_* \colon H_{r}(X, \, \mathbb{C}) \to H_{r}(Y, \, \mathbb{C})$$ is surjective for all $r$.

If $f \colon X \to Y$ contracts a subvariety $Z$ of positive complex dimension $i$ to a point, then its fundamental class would give an element $[Z]$ in the kernel of $$f_* \colon H_{2i}(X, \, \mathbb{C}) \to H_{2i}(Y, \, \mathbb{C})$$

Now $Z$ is compact by the properness assumption and so, pairing with the Kähler

form on $X$, we can deduce that $[Z]$ is non-zero. Hence $b_{2i}(X) > b_{2i}(Y)$, contradiction. $\square$

**Remark.** The first version of this answer did not assume that $X$ was Kähler (in fact, Gromov does not make this assumption). However, if $X$ is not Kähler it may happen that the fundamental class of a compact subvariety of $X$ is homologically trivial, see the example of Hopf's surface given in Michael Albanese answer to **[MSE]**: in this case, $X$ contains a torus $C$ whose fundamental class $[C]$ is zero in $H_2(X, \, \mathbb{C})$, simply because the last group is trivial. So, if $X$ is not Kähler, the last part of the proof breaks down.

In fact, I do not know how to make the argument work without the Kähler assumption for $X$, so I asked a new MO question about this problem.

**References.**

**[MSE]** https://math.stackexchange.com/q/1556561/456212

**[G03]** Gromov, M: On the entropy of holomorphic maps, *Enseign. Math.* II. Sér. **49**, No. 3-4, 217-235 (2003). ZBL1080.37051.

**[W74]** Wells, R. O. jun: Comparison of de Rham and Dolbeault cohomology for proper surjective mappings, *Pac. J. Math.* **53**, 281-300 (1974). ZBL0261.32005.

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