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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.

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    $\begingroup$ Another approach to the Hodge decomposition of the singular cohomology with complex coefficients of a projective, complex manifold is the Hodge-to-de Rham spectral sequence. This arises from a "standard" spectral sequence on the hypercohomology of the holomorphic de Rham complex. Via the holomorphic Poincar'e lemma, the holomorphic de Rham complex is quasi-isomorphic to the sheaf of locally constant functions to $\mathbb{C}$. For a holomorphic map of complex manifolds, the pullback map on holomorphic $p$-forms induces a pullback map on hypercohomology, thus a map of Hodge structures. $\endgroup$
    – Jason Starr
    Dec 23, 2017 at 19:41

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The map doesn’t have to preserve the direct sum decomposition coming from harmonic representatives, but it does preserve the filtration on deRham cohomology by subspaces for different $p$ of classes with representatives which are sums of $(n-k,k)$ forms for $k\leq p$. Doulbeaut cohomology is the associated graded of this filtration, compatibly with the pullback map. It’s a standard result that if a map of associated gradeds is surjective, the original map is.

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