Let $X\subseteq {\mathbb{P}}^N$ be an $n$-dimensional complex projective manifold. Denote by $\pi\colon U\to X$ the affine cone of $X$ with the vertex removed; it is a $\mathbb{C}^*$-bundle over $X$. If the canonical divisor of $X$ is cut by a degree $m$ hypersurface in ${\mathbb{P}}^N$, then the relative tangent sheaf exact sequence $$ 0 \rightarrow \Theta_{U/X} \rightarrow \Theta_{U} \stackrel{d\pi}{\rightarrow} \pi^*(\Theta_X) \rightarrow 0 $$ induces canonical morphisms $$ \lambda\colon H^{n-1,i}(X) \to H^{n,i+1}(X). $$ Namely, the Euler vector field on $U$ gives a trivialization of $\Theta_{U/X}$ and so an identification $\Theta_{U/X}\cong \mathcal{O}_U\cong \pi^*\mathcal{O}_X$. With this in mind, the connecting homomorphisms in the long exact cohomology sequence associated with the relative tangent sheaf exact sequence twisted by $\pi^*\mathcal{O}_X(m)$ read $$ H^i(X, \Theta_X(m)) \to H^{i+1}(X, \mathcal O_X(m))). $$ Finally, since $\omega_X\simeq \mathcal{O}_X(m)$, one uses the Serre duality isomorphisms $H^i(X, \Theta_X(m))\simeq H^{n-1,i}(X)$ and $H^{i+1}(X, \mathcal O_X(m))\simeq H^{n,i+1}(X)$.
Now the question is: are the homomorphism $\lambda$ the Lefschetz homomorphisms? (i.e., are they given by the cap product with an hyperplane in ${\mathbb{P}}^N$). Since the construction sketched above is completely canonical and only depends on the projective embedding of $X$ I suspect the answer to be yes, but I've not been able to work out a completely clean proof of this, yet, nor to locate this statement in the literature. Since I strongly suspect the result to be known, I'm asking here for a reference.
