Anticanonical divisors and curve classes on Fano manifolds

TL;DR

This paper investigates whether integral Hodge classes of degree 2n-2 on rationally connected complex projective n-folds are generated by classes of curves, providing affirmative results for Fano fourfolds using advanced geometric techniques.

Contribution

It proves that on a large class of manifolds, including Fano fourfolds, integral Hodge classes of degree 2n-2 are generated by classes of curves, using singularity and Hodge structure methods.

Findings

Integral Hodge classes are generated by curve classes on many Fano manifolds.

Existence of anticanonical divisors with isolated canonical singularities on Fano fourfolds.

Affirmative answer to a refined Hodge conjecture for certain rationally connected varieties.

Abstract

It is well known that the Hodge conjecture with rational coefficients holds for degree 2n-2 classes on complex projective n-folds. In this paper we study the more precise question if on a rationally connected complex projective n-fold the integral Hodge classes of degree 2n-2 are generated over $\mathbb Z$ by classes of curves. We combine techniques from the theory of singularities of pairs on the one hand and infinitesimal variation of Hodge structures on the other hand to give an affirmative answer to this question for a large class of manifolds including Fano fourfolds. In the last case, one step in the proof is the following result of independent interest: There exist anticanonical divisors with isolated canonical singularities on a smooth Fano fourfold.