I'm trying to read the proof of the Lefschetz hyperplane theorem from Griffiths-Harris. They prove the theorem (on pages 156-157) using the Kodaira vanishing theorem. I have a basic question regarding their strategy of proof.
They begin by noting that we have Hodge decompositions for both the Kähler manifold $M$ and the positive hypersurface $V$, and so, they claim that to prove the statement about the restriction map $H^r(M,\mathbb C)\to H^r(V,\mathbb C)$, it is enough to prove the corresponding statement for the restriction maps $H^q(M,\Omega^p_M)\to H^q(V,\Omega_V^p)$.
It is not clear to me why this is sufficient, i.e., why the restriction map from the Dolbeault cohomology of $M$ to that of $V$, and the restriction map on de Rham cohomology are compatible with the Hodge decompositions of $M$ and $V$. Could someone please explain why this is the case? I think this would follow if the restrictions of harmonic forms on $M$ are harmonic on $V$, but I don't see how to prove that either.