We classify all positive integers $n$ and $r$ such that (stably) nonrational 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 ${\mathbb{P}}_{\mathbb{C}}^n$ are not stably rational if $r\in [2^{n-1}-1,2^n-2]$. In our proofs we introduce a generalization of the specialization method of Voisin and of Colliot-Thélène and Pirutka which avoids universally ${\mathrm{CH}}_0$-trivial resolutions of singularities.

In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. First, we provide a systematic recipe for translating from a Weinstein Lefschetz bifibration to a Legendrian handlebody. Then we present several new applications of this technique to symplectic topology. This includes the detection of flexibility and rigidity for several families of Weinstein manifolds and the existence of closed, exact Lagrangian submanifolds. In particular, we prove that the Koras–Russell cubic is Stein deformation-equivalent to ${\mathbb{C}}^3$, and we verify the affine parts of the algebraic mirrors of two Weinstein $4$-folds.

We prove Szegő–Widom asymptotics for the Chebyshev polynomials of a compact subset of $\mathbb{R}$ which is regular for potential theory and obeys the Parreau–Widom and DCT conditions.

The main result of this article is to reduce a proof of the conjecture to a statement about principal bundles on affine line over a regular local scheme. This reduction is obtained via a theory of nice triples, which goes back to the ideas of Voevodsky. As an application, an unpublished result due to Gabber is proved.