First, we shall formulate and prove Theorem of Lie-Kolchin type for a cone and derive some algebro-geometric consequences. Next, inspired by a recent result of Dinh and Sibony we pose a conjecture of Tits type for a group of automorphisms of a complex variety and verify its weaker version. Finally, applying Theorem of Lie-Kolchin type for a cone, we shall confirm the conjecture of Tits type for complex tori, hyperkähler manifolds, surfaces, and minimal threefolds.
We prove a theorem of Tits type for automorphism groups of projective varieties over an algebraically closed field of arbitrary characteristic, which was first conjectured by Keum, Oguiso and Zhang for complex projective varieties.
In this article we construct a new motivic measure called the ${\it Jacques}$ ${\it Tits}$ ${\it motivic}$ ${\it measure}$. As a first main application of the Jacques Tits motivic measure, we prove that two Severi-Brauer varieties (or, more generally, two twisted Grassmannian varieties), associated to $2$-torsion central simple algebras, have the same class in the Grothendieck ring of varieties if and only if they are isomorphic. In addition, we prove that if two Severi-Brauer varieties, associated to central simple algebras of period $\{3, 4, 5, 6\}$, have the same class in the Grothendieck r…
We first present an intersection theory of partial differential varieties with quasi-generic differential hypersurfaces. Then based on the generic intersection theory, we define the partial differential Chow form for an irreducible partial differential variety $V$ of Kolchin polynomial $ω_V(t)=(d+1){t+m\choose m}-{t+m-s\choose m}$. And we establish for the partial differential Chow form most of the basic properties of the ordinary differential Chow form. Furthermore, we prove the existence of a type of partial differential Chow varieties.
Given a toric affine algebraic variety $X$ and a collection of one-parameter unipotent subgroups $U_1,\ldots,U_s$ of $\mathop{\rm Aut}(X)$ which are normalized by the torus acting on $X$, we show that the group $G$ generated by $U_1,\ldots,U_s$ verifies the following alternative of Tits' type: either $G$ is a unipotent algebraic group, or it contains a non-abelian free subgroup. We deduce that if $G$ is $2$-transitive on a $G$-orbit in $X$, then $G$ contains a non-abelian free subgroup, and so, is of exponential growth.
We study for each fixed integer $g \ge 2$, for all primes $\ell$ and $p$ with $\ell \neq p$, finite regular directed graphs associated with the set of equivalence classes of $\ell$-marked principally polarized superspecial abelian varieties of dimension $g$ in characteristic $p$, and show that the adjacency matrices have real eigenvalues with spectral gaps independent of $p$. This implies a rapid mixing property of natural random walks on the family of isogeny graphs beyond the elliptic curve case and suggests a potential construction of the Charles-Goren-Lauter type cryptographic hash functio…
We study the prime-to-$p$ Hecke action on the projective limit of the sets of connected components of Shimura varieties with fixed parahoric or Bruhat--Tits level at $p$. In particular, we construct infinitely many Shimura varieties for CM unitary groups in odd variables for which the considering actions are not transitive. We prove this result by giving negative examples on the question of Bruhat--Colliot-Thélène--Sansuc--Tits or its variant, which is related to the weak approximation on tori over $\mathbb{Q}$.
Making use of the recent theory of noncommutative motives, we construct a new motivic measure, which we call the Tits' motivic measure. As a first application, we prove that two Severi-Brauer varieties (or more generally twisted Grassmannian varieties), associated to central simple algebras of period 2, have the same Grothendieck class if and only if they are isomorphic. As a second application, we show that if two Severi-Brauer varieties, associated to central simple algebras of period 2, 3, 4, 5 or 6, have the same Grothendieck class, then they are necessarily birational. As a third applicat…
Motivic equivalence for algebraic groups was recently introduced in [9], where a characterization of motivic equivalent groups in terms of higher Tits indexes is given. As a consequence, if the quadrics associated to two quadratic forms have the same Chow motives with coefficients in F_2, this remains true for any two projective homogeneous varieties of the same type under the orthogonal groups of those two quadratic forms. Our main result extends this to all groups of classical type, and to some exceptional groups, introducing a notion of critical variety. On the way, we prove that motivic eq…
