Let $M$ be a compact Kahler manifold with $c_{1}(M)\lt 0$. It seems that I can prove the following claim: there is no nontrivial holomorphic vector field on $M$.
Here is my proof: let $T^{1,0}M$ be the holomorphic tangent bundle, $\Omega_{M}$ be its dual bundle, then $$c_{1}(\wedge^{n}\Omega_{M})=-c_{1}(M)>0$$ hence
$$\Gamma(M,T^{1,0}M)=H^{0}(M,T^{1,0}M) =H^{0.0}(M,\Omega_{M}^{*})=H^{n,n}(M,\Omega_{M})^{*}\\ =H^{n}(M,\wedge^{n}\Omega_{M}\otimes \Omega_{M})^{*}=H^{1,n}(M,\wedge^{n}\Omega_{M})^{*}=0 $$
where the third = is due to Serre duality, the last = is due to Kodaira vanishing thm.
I wonder if there is any mistake in my argument, and I don't find any reference about this question. Is it true?
The statement is right, you can refer to the fourth Theorem in the following article, saying that the claim you wrote is due to S. Kobayashi. See Ono, Kaoru Some remarks on group actions in symplectic geometry. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 35 (1988), no. 3, 431–437.