On equivariant Serre problem for principal bundles

An affine algebraic group

We describe maximal commutative unipotent subgroups of the automorphism group \mathrm{Aut}(X) of an irreducible affine variety X . Furthermore, we show that a group isomorphism \mathrm{Aut}(X) \allowbreak\to \mathrm{Aut}(Y) maps unipotent elements to unipotent elements, where Y is irreducible and affine. Using this result, we show that the automorphism group detects sphericity and the weight monoid.As an application, we show that an affine toric variety different from an algebraic torus is determined by its automorphism group among normal irreducible affine varieties, and we show that a smooth affine spherical variety different from an algebraic torus is determined by its automorphism group (up to an automorphism of the base field) among smooth irreducible affine varieties. This generalizes results obtained by Cantat, Kraft, Liendo, Urech, Xie and the authors.

We study 2-cocycle twists, or equivalently Zhang twists, of semigroup algebras over a field ${\mathbb K}$ . If the underlying semigroup is affine, that is abelian, cancellative and finitely generated, then $\mathsf {Spec}~{\mathbb K}[S]$ is an affine toric variety over ${\mathbb K}$ , and we refer to the twists of ${\mathbb K}[S]$ as quantum affine toric varieties. We show that every quantum affine toric variety has a “dense quantum torus”, in the sense that it has a localization isomorphic to a quantum torus. We study quantum affine toric varieties and show that many geometric regularity properties of the original toric variety survive the deformation process.

Let X be an affine toric variety and let D(X) be the set of weights of all root subgroups. It is known that D(X) together with its embedding into the character group determines X as a toric variety. In this article we prove that X is already determined by the abstract set D(X) together with some additional combinatorial data.

Let X be an affine toric variety. The total coordinates on X provide a canonical presentation $$\bar X \to X$$ of X as a quotient of a vector space $$\bar X$$ by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the Luna strata defined by the canonical quotient presentation.

Let [Formula: see text] be the Whitney sum of complex line bundles over a topological space [Formula: see text]. Then, the projectivization [Formula: see text] of [Formula: see text] is called a projective bundle over [Formula: see text]. If [Formula: see text] is a nonsingular complete toric variety, then so is [Formula: see text]. In this paper, we show that the cohomology ring of a nonsingular projective toric variety [Formula: see text] determines whether it admits a projective bundle structure over a nonsingular complete toric surface. In addition, we show that two [Formula: see text]-dimensional projective bundles over [Formula: see text]-dimensional quasitoric manifolds are diffeomorphic if their cohomology rings are isomorphic as graded rings. Furthermore, we study the smooth classification of higher dimensional projective bundles over [Formula: see text]-dimensional quasitoric manifolds.

Let M be a smooth complex projective toric variety equipped with an action of a torus T , such that the complement D of the open T-orbit in M is a simple normal crossing divisor.Let G be a complex reductive affine algebraic group.We prove that an algebraic principal G-bundle E G M admits a T-equivariant structure if and only if E G admits a logarithmic connection singular over D. If E H M is a T-equivariant algebraic principal H-bundle, where H is any complex affine algebraic group, then E H in fact has a canonical integrable logarithmic connection singular over D.

Let $X$ be a normal projective variety and $f:X\to X$ a non-isomorphic polarized endomorphism. We give two characterizations for $X$ to be a toric variety. First we show that if $X$ is $\mathbb{Q}$-factorial and $G$-almost homogeneous for some linear algebraic group $G$ such that $f$ is $G$-equivariant, then $X$ is a toric variety. Next we give a geometric characterization: if $X$ is of Fano type and smooth in codimension 2 and if there is an $f^{-1}$-invariant reduced divisor $D$ such that $f|_{X\backslash D}$ is quasi-\'etale and $K_X+D$ is $\mathbb{Q}$-Cartier, then $X$ admits a quasi-\'etale cover $\widetilde{X}$ such that $\widetilde{X}$ is a toric variety and $f$ lifts to $\widetilde{X}$. In particular, if $X$ is further assumed to be smooth, then $X$ is a toric variety.

Let X=spec A be a normal affine variety over an algebraically closed field k of characteristic 0 endowed with an effective action of a torus T of dimension n. Let also D be a homogeneous locally nilpotent derivation on the normal affine Z^n-graded domain A, so that D generates a k_+-action on X that is normalized by the T-action. We provide a complete classification of pairs (X,D) in two cases: for toric varieties (n=\dim X) and in the case where n=\dim X-1. This generalizes previously known results for surfaces due to Flenner and Zaidenberg. As an application we compute the homogeneous Makar-Limanov invariant of such varieties. In particular we exhibit a family of non-rational varieties with trivial Makar-Limanov invariant.

Any map of schemes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X right-arrow upper Y"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X\to Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> defines an equivalence relation <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R equals upper X times Subscript upper Y Baseline upper X right-arrow upper X times upper X"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>=</mml:mo> <mml:mi>X</mml:mi> <mml:msub> <mml:mo> × </mml:mo> <mml:mi>Y</mml:mi> </mml:msub> <mml:mi>X</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mi>X</mml:mi> <mml:mo> × </mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">R=X\times _Y X\to X\times X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the relation of “being in the same fiber”. We have shown elsewhere that not every equivalence relation has this form, even if it is assumed to be finite. By contrast, we prove here that every toric equivalence relation on an affine toric variety does come from a morphism and that quotients by finite toric equivalence relations always exist in the affine case. In special cases, this result is a consequence of the vanishing of the first cohomology group in the Amitsur complex associated to a toric map of toric algebras. We prove more generally the exactness of the Amitsur complex for maps of commutative monoid rings.

Let $X$ be a scheme of finite type over a finite field $k$, and let ${\cal L} X$ denote its arc space; in particular, ${\cal L} X(k)=X(k[[t]])$. Using the theory of Grinberg, Kazhdan, and Drinfeld on the finite-dimensionality of singularities of ${\cal L} X$ in the neighborhood of non-degenerate arcs, we show that a canonical ``basic function'' can be defined on the non-degenerate locus of ${\cal L} X(k)$, which corresponds to the trace of Frobenius on the stalks of the intersection complex of any finite-dimensional model. We then proceed to compute this function when $X$ is an affine toric variety or an ``$L$-monoid''. Our computation confirms the expectation that the basic function is a generating function for a local unramified $L$-function; in particular, in the case of an $L$-monoid we prove a conjecture formulated by the second author.

Consider an algebraic semigroup S and its closed subscheme of idempotents, E(S). When S is commutative, we show that E(S) is finite and reduced; if in addition S is irreducible, then E(S) is contained in a smallest closed irreducible subsemigroup of S, and this subsemigroup is an affine toric variety. It follows that E(S) (viewed as a partially ordered set) is the set of faces of a rational polyhedral convex cone. On the other hand, when S is an irreducible algebraic monoid, we show that E(S) is smooth, and its connected components are conjugacy classes of the unit group.

Arithmetic purity of strong approximation for homogeneous spaces

We study M-theory on a Calabi-Yau fourfold with a smooth surface $S$ of $A_{N-1}$ singularities. The resulting three-dimensional theory has a $\mathcal{N}=2$ $SU(N)$ gauge theory sector, which we obtain from a twisted dimensional reduction of a seven-dimensional $\mathcal{N}=1$ $SU(N)$ gauge theory on the surface $S$. A variant of the Vafa-Witten equations governs the moduli space of the gauge theory, which, for a trivial $SU(N)$ principal bundle over $S$, admits a Coulomb and a Higgs branch. In M-theory these two gauge theory branches arise from a resolution and a deformation to smooth Calabi-Yau fourfolds, respectively. We find that the deformed Calabi-Yau fourfold associated to the Higgs branch requires for consistency a non-trivial four-form background flux in M-theory. The flat directions of the flux-induced superpotential are in agreement with the gauge theory prediction for the moduli space of the Higgs branch. We illustrate our findings with explicit examples that realize the Coulomb and Higgs phase transition in Calabi-Yau fourfolds embedded in weighted projective spaces. We generalize and enlarge this class of examples to Calabi-Yau fourfolds embedded in toric varieties with an $A_{N-1}$ singularity in codimension two.

In this chapter, we come to those results on affine varieties and algebraic groups which rely in an essential way on the hypothesis that the ground field k is algebraically closed. We will see that, under this hypothesis, the image of a morphism between affine varieties is a relatively ‘thick’ subset, more precisely, it contains an open subset of its closure. This fact will turn out to be a key tool.