$I$-Maximal Variation of Hodge Structure and Jacobian Rings

Abstract

We investigate higher--order variation of Hodge structure for families of smooth hypersurfaces and complete intersections through the notion of $I$--maximal variation. Using Griffiths' description of primitive cohomology, we interpret the infinitesimal variation of Hodge structure and the $n$--fold Yukawa coupling as graded multiplication maps in the Jacobian ring. Our main result shows that the Strong Lefschetz property of the Jacobian ring provides the algebraic mechanism ensuring $I$--maximal variation. In particular, we prove that smooth hypersurfaces of degree $d\ge n+2$ and smooth complete intersections with $\kappa>0$ exhibit $I$--maximal variation. We further establish that for complete intersections of general type the infinitesimal Torelli property is equivalent to the nondegeneracy of the Yukawa coupling. Finally, we analyze degenerations and show that the failure of the Strong Lefschetz property leads to degeneration of the Yukawa coupling and the loss of $I$--maximal variation. These results identify the Lefschetz property of the Jacobian ring as the fundamental algebraic structure governing maximal variation of Hodge structure.