Degeneration of families of projective hypersurfaces and Hodge conjecture

Abstract

We prove by induction on dimension the Hodge conjecture for smooth complex projective varieties. Let $X$ be a smooth complex projective variety. Then $X$ is birational to a possibly singular projective hypersurface, hence to a smooth projective variety $E_0$ which is a component of a normal crossing divisor $E=\cup_{i=0}^rE_i\subset Y$ which is the singular fiber of a pencil $f:Y\to\mathbb A^1$ of smooth projective hypersurfaces. Using the smooth hypersurface case by a previous result of the autor, the nearby cycle functor on mixed Hodge module with rational de Rham factor, and the induction hypothesis, we prove that an Hodge class of $E_0$ is absolute Hodge, more precisely the locus of Hodge classes inside the algebraic vector bundle given the De Rham cohomology the rational deformation of $E_0$ is defined over $\mathbb Q$. By another previous result of the autor, we get the Hodge conjecture for $E_0$. By the induction hypothesis we also have the Hodge conjecture for $X$.