Induced isomorphism between $H^{1,0}(X)$ and $H^{1,0}(Y)$ implies the induced isomorphism between $H^{0,1}(X)$ and $H^{0,1}(Y)$?

Question

Let $X,Y$ be two compact Kahler manifolds and $f:X\to Y$ is a holomorphic map. If the induced map $f^*:H^1(X)\to H^1(Y)$'s restriction $f^*|_{H^{1,0}}$ an isomorphism from $H^{1,0}(X)\to H^{1,0}(Y)$, can we also conclude that $f^*$'s restriction $f^*|_{H^{0,1}}$ is an isomorphism from $H^{0,1}(X)\to H^{0,1}(Y)$?

Answer 1

The conjugation map on differential forms induces an isomorphism $H^{p,q} = H^{q,p}$ for complex manifolds. As long as $f$ respects the complex structures this isomorphism is natural in $f$, you can check it commutes locally.