Let k be an uncountable field of characteristic different from two. We show that a very general hypersurface X ⊂ P N +1 k of dimension N ≥ 3 and degree at least log 2 N + 2 is not stably rational over the algebraic closure of k.

We classify all positive integers n and r such that (stably) non-rational complex r-fold quadric bundles over rational n-folds exist. We show in particular that for any n and r, a wide class of smooth r-fold quadric bundles over P n C are not stably rational if r ∈ [2 n−1 − 1, 2 n − 2]. In our proofs we introduce a generalization of the specialization method of Voisin and Colliot-Thélène-Pirutka which avoids universally CH 0 -trivial resolutions of singularities.

We determine the structure of the Hodge ring, a natural object encoding the Hodge numbers of all compact Kähler manifolds. As a consequence of this structure, there are no unexpected relations among the Hodge numbers, and no essential differences between the Hodge numbers of smooth complex projective varieties and those of arbitrary Kähler manifolds. The consideration of certain natural ideals in the Hodge ring allows us to determine exactly which linear combinations of Hodge numbers are birationally invariant, and which are topological invariants. Combining the Hodge and unitary bordism rings, we are also able to treat linear combinations of Hodge and Chern numbers. In particular, this leads to a complete solution of a classical problem of Hirzebruch's.

We prove a general specialization theorem which implies stable irrationality for a wide class of quadric surface bundles over rational surfaces. As an application, we solve, with the exception of two cases, the stable rationality problem for any very general complex projective quadric surface bundle over P 2 , given by a symmetric matrix of homogeneous polynomials. Both exceptions degenerate over a plane sextic curve and the corresponding double cover is a K3 surface.

Abstract. For any symmetric collection (h p,q ) p+q=k of natural numbers, we construct a smooth complex projective variety X whose weight k Hodge structure has Hodge numbers h p,q (X) = h p,q ; if k = 2m is even, then we have to impose that h m,m is bigger than some quadratic bound in m. Combining these results for different weights, we solve the construction problem for the truncated Hodge diamond under two additional assumptions. Our results lead to a complete classification of all nontrivial dominations among Hodge numbers of Kähler manifolds.