Automorphisms of hypersurfaces in polarized varieties

Let $F$ be a projective variety, let $H$ be a normally generated line bundle on $F$, and let $X\in \mathrm{\mid }{H}^{\otimes d}\mathrm{\mid }$ be a general projective variety. We establish a uniform criterion, formulated in terms of the graded Betti numbers of the section ring $R(F,H)$, that provides sufficient conditions for every automorphism of $X$ to extend to an automorphism of $F$. This extends the classical results of Matsumura-Monsky on automorphisms of hypersurfaces in projective spaces to a broader class of polarized varieties $(F,H)$, including rational homogeneous varieties of Picard number one, smooth Fano threefolds with their anticanonical polarization, and examples of hyperkähler and Calabi-Yau varieties.