Related papers: A constructive approach to a conjecture by Voskres…

The study of the birational properties of algebraic $k$-tori began in the sixties and seventies with work of Voskresenkii, Endo, Miyata, Colliot-Th\'el\`ene and Sansuc. There was particular interest in determining the rationality of a given…

We investigate the stable and retract rationality of multinorm one tori associated to finite {\'e}tale algebras. Our results are organized according to the greatest common divisor $d$ of the degrees of the factors. We show that these tori…

We study normal compact K\"ahler spaces whose rational cohomology ring is isomorphic to that of a complex torus. We call them rational cohomology tori. We classify, up to dimension three, those with rational singularities. We then give…

We prove a conjecture of V. V. Shokurov which in particular implies that the fibers of a resolution of a variety with divisorial log terminal singularities are rationally chain connected.

We show that any quiver Grassmannian associated with a rigid representation of a quiver is a rational variety using torus localization techniques.

A resolution-free definition of rational singularities is introduced, and it is proved that for a variety admitting a resolution of singularities, so in particular in characteristic zero, this is equivalent to the usual definition. It is…

We consider Knizhnik-Zamolodchikov system of linear differential equations. The coefficients of this system are rational functions. We prove that under some conditions the solution of KZ system is rational too. This assertion confirms…

It is believed arXiv:0808.2762, arXiv:math/9904055 that, among the coefficients entering Kontsevich's formality quasi-isomorphism arXiv:q-alg/9709040, there are irrational (possibly even transcendental) numbers. In this paper, we prove that…

We give a stably birational classification for algebraic tori of dimensions $3$ and $4$ over a field $k$. First, we define the weak stably equivalence of algebraic tori and show that there exist $13$ (resp. $128$) weak stably equivalent…

We present theorems which provide the existence of invariant whiskered tori in finite-dimensional exact symplectic maps and flows. The method is based on the study of a functional equation expressing that there is an invariant torus. We…

In this note we derive a formalism for describing equivariant sheaves over toric varieties. This formalism is a generalization of a correspondence due to Klyachko, which states that equivariant vector bundles on toric varieties are…

In recent years, there has been a development in approaching rationality problems through motivic methods (cf. [Kontsevich--Tschinkel'19], [Nicaise--Shinder'19], [Nicaise--Ottem'21]). This method requires the explicit construction of…

We study the existence of whiskered tori in a family $f_\mu$ of conformally symplectic maps depending on parameters $\mu$. Whiskered tori are tori on which the motion is a rotation, but they have as many expanding/contracting directions as…

In this paper, combining the works of Miyanishi-Tsunoda and Keel-McKernan, we prove the log Castelnuovo's rationality criterion for smooth quasiprojective surfaces over complex numbers.

We introduce the notion of rational links in the solid torus. We show that rational links in the solid torus are fully characterized by rational tangles, and hence by the continued fraction of the rational tangle. Furthermore, we generalize…

We prove a bilinear Strichartz type estimate for irrational tori via a decoupling type argument, \cite{bourgain2014proof}, recovering and generalizing the result of \cite{de2006global}. As a corollary, we derive a global well-posedness…

For any prime number $p$ and field $k$, we characterize the $p$-retract rationality of an algebraic $k$-torus in terms of its character lattice. We show that a $k$-torus is retract rational if and only if it is $p$-retract rational for…

The Tarskian classical relevant logic TR arises from Tarski's work on the foundations of the calculus of relations and on first-order logic restricted to finitely many variables, presented by Tarski and Givant their book, A Formalization of…

We show how the techniques of Voevodsky's proof of the Milnor conjecture and the Voevodsky- Rost proof of its generalization the Bloch-Kato conjecture can be used to study counterexamples to the classical L\"uroth problem. By generalizing a…

We prove the existence of periodic orbits for steady $C^\omega$ Euler flows on all Riemannian solid tori. By using the correspondence theorem from part I of this series, we reduce the problem to the Weinstein Conjecture for solid tori. We…