On the maximal variation problem

In this talk, we study the maximal variation problem for linear systems associated with a very ample line bundle, using Hodge theory and Picard–Lefschetz theory. We provide an affirmative answer to the maximal variation problem for a broad class of smooth projective varieties. This includes varieties $X$ of dimension $n$ at least 2 with $p_g=h^{n,0}(X)>0$ and $H^{n-1,0}(X)=\{0\}$, Enriques surfaces, irregular surfaces with maximal Albanese dimension, smooth hyperkähler varieties, and all the smooth not Fano hypersurfaces in $\mathbb{P}^n$. As a consequence, by a result of Beauville, we establish a Lefschetz property for the Jacobian rings of smooth hypersurfaces in $\mathbb{P}^n$ of degree at least $n+1$. This is based on a joint work in collaboration with Gian Pietro Pirola.