The homomorphism induced by holomorphic map preserves Hodge decomposition

Question

Let $f:X\to Y$ be a holomorphic map between two compact Kähler manifolds. Then the induced map $f^*:H^1(Y,\mathbb{C})\to H^1(X,\mathbb{C})$ preserves the Hodge decomposition. Is there a reference for a proof of this statement? Or Can you give a proof?

In particular, can you explain what exactly does “preserves Hodeg decomposition” mean?

Answer 1

The Hodge decomposition is the isomorphism $H^1(Y; \mathbb{C}) \cong H^{1,0}_{\bar{\partial}}(Y)\oplus H^{0,1}_{\bar{\partial}}(Y)$. The statement just means that $f^* : H^{1,0}_{\bar{\partial}}(Y) \to H^{1,0}_{\bar{\partial}}(X)$ and $f^* : H^{0,1}_{\bar{\partial}}(Y) \to H^{0,1}_{\bar{\partial}}(X)$.