Characterizations of toric varieties via polarized endomorphisms

In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the context of complete toric varieties fit nicely with a generalized Hirzebruch‐Riemann‐Roch theorem. As important special cases, we obtain new (or recover well‐known) formulae for the Baum‐Fulton‐MacPherson Todd (or MacPherson's Chern) classes of toric varieties, as well as for the Thom‐Milnor L‐classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of the torus‐orbit decomposition and the motivic properties of the motivic Chern and respectively homology Hirzebruch classes to express the latter in terms of dualizing sheaves and respectively the (dual) Todd classes of closures of orbits. This method even applies to torus‐invariant subspaces of a given toric variety. The obtained formula is then applied to weighted lattice‐point counting in lattice polytopes and their subcomplexes, yielding generalized Pick‐type formulae. Second, in the case of simplicial toric varieties, we compute our characteristic classes by using the Lefschetz‐Riemann‐Roch theorem of Edidin‐Graham in the context of the geometric quotient description of such varieties. In this setting, we define mock Hirzebruch classes of simplicial toric varieties (which specialize to the mock Chern, mock Todd, and mock L‐classes of such varieties) and investigate the difference between the (actual) homology Hirzebruch class and the mock Hirzebruch class. We show that this difference is localized on the singular locus, and we obtain a formula for it in which the contribution of each singular cone is identified explicitly. Finally, the two methods of computing characteristic classes are combined for proving several characteristic class formulae originally obtained by Cappell and Shaneson in the early 1990s.© 2015 Wiley Periodicals, Inc.

Let [Formula: see text] be a [Formula: see text]-equivariant algebraic principal [Formula: see text]-bundle over a normal complex affine variety [Formula: see text] equipped with an action of [Formula: see text], where [Formula: see text] and [Formula: see text] are complex linear algebraic groups. Suppose [Formula: see text] is contractible as a topological [Formula: see text]-space with a dense orbit, and [Formula: see text] is a [Formula: see text]-fixed point. We show that if [Formula: see text] is reductive, then [Formula: see text] admits a [Formula: see text]-equivariant isomorphism with the product principal [Formula: see text]-bundle [Formula: see text], where [Formula: see text] is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal [Formula: see text]-bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal [Formula: see text]-bundles over any complex toric variety, generalizing the main result of [A classification of equivariant principal bundles over nonsingular toric varieties, Internat. J. Math. 27(14) (2016)].

Arithmetic purity of strong approximation for homogeneous spaces

A root system $R$ of rank $n$ defines an $n$-dimensional smooth projective toric variety $X(R)$ associated with its fan of Weyl chambers. We give a simple description of the functor of $X(R)$ in terms of the root system $R$ and apply this result in the case of root systems of type $A$ to give a new proof of the fact that the toric variety $X(A_n)$ is the fine moduli space $\overline{L}_{n+1}$ of stable $(n+1)$-pointed chains of projective lines investigated by Losev and Manin.

We establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in Bruzzo and Grassi (Commun Anal Geom 28: 1773–1786, 2020). We also show that the Hodge Conjecture holds asymptotically for suitable quasi-smooth hypersurface in the Noether–Lefschetz locus, where “asymptotically” means that the degree of the hypersurface is big enough, under the assumption that the ambient variety {{mathbb {P}}}_Sigma ^{2k+1} has Picard group {mathbb {Z}}. This extends to a class of toric varieties Otwinowska’s result in Otwinowska (J Alg Geom 12: 307–320, 2003).

Let \mathbb K be an algebraically closed field of characteristic zero, \mathbb G_m=(\mathbb K\setminus{0},\times) be its multiplicative group, and \mathbb G_a=(\mathbb K,+) be its additive group. Consider a commutative linear algebraic group \mathbb G=(\mathbb G_m)^r\times\mathbb G_a . We study equivariant \mathbb G -embeddings, i.e. normal \mathbb G -varieties X containing \mathbb G as an open orbit. We prove that X is a toric variety and all such actions of \mathbb G on X correspond to Demazure roots of the fan of X . In these terms, the orbit structure of a \mathbb G -variety X is described.

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 $f:X\to X $ be a dominant self-morphism of an algebraic variety. Consider the set $\Sigma _{f^{\infty }}$ of $f$-periodic subvarieties of small dynamical degree (SDD), the subset $S_{f^{\infty }}$ of maximal elements in $\Sigma _{f^{\infty }}$, and the subset $S_f$ of $f$-invariant elements in $S_{f^{\infty }}$. When $X$ is projective, we prove the finiteness of the set $P_f$ of $f$-invariant prime divisors with SDD and give an optimal upper bound $$\begin{align*} &\sharp P_{f^n}\le d_1(f)^n(1+o(1))\end{align*}$$as $n\to \infty $, where $d_1(f)$ is the 1st dynamic degree. When $X$ is an algebraic group (with $f$ being a translation of an isogeny), or a (not necessarily complete) toric variety, we give an optimal upper bound $$\begin{align*} &\sharp S_{f^n}\le d_1(f)^{n\cdot\dim(X)}(1+o(1))\end{align*}$$as $n \to \infty $, which slightly generalizes a conjecture of S.-W. Zhang for polarized $f$.

Toric varieties with huge Grothendieck group

Given a finite set of lattice points $\mathcal{A}$, we consider the associated homogeneous binomial ideal $I_\mathcal{A}$ and projective toric variety $X_\mathcal{A}$. We give a concise combinatorial description of all linear subspaces contained in the variety $X_\mathcal{A}$, or, equivalently, all solutions in linear forms to the system of binomial equations determined by $I_\mathcal{A}$. More precisely, we study the Fano scheme $\mathbf{F}_k(X_\mathcal{A})$ whose closed points correspond to $k$-dimensional linear spaces contained in $X_\mathcal{A}$. We show that the irreducible components of $\mathbf{F}_k(X_\mathcal{A})$ are in bijection to maximal Cayley structures for $\mathcal{A}$ of length at least $k$. We explicitly describe these irreducible components and their intersection behavior, characterize when $\mathbf{F}_k(X_\mathcal{A})$ is connected, and prove that if $X_\mathcal{A}$ is smooth in dimension $k$, then every component of $\mathbf{F}_k(X_\mathcal{A})$ is smooth in its reduced structure. Furthermore, in the special case $k=\dim X_\mathcal{A}-1$, we describe the nonreduced structure of $\mathbf{F}_k(X_\mathcal{A})$.

We prove an explicit formula for the first non-zero entry in the n-th row of the graded Betti table of an n-dimensional projective toric variety associated to a normal polytope with at least one interior lattice point. This applies to Veronese embeddings of projective space where we prove a special case of a conjecture of Ein and Lazarsfeld. We also prove an explicit formula for the entire n-th row when the interior of the polytope is one-dimensional. All results are valid over an arbitrary field k.

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.

Computations of Gromov–Witten invariants of toric varieties

An arc space of an affine cone over a projective toric variety is known to be nonreduced in general.It was demonstrated recently that the reduced scheme structure of arc spaces is very meaningful from algebro-geometric, representation-theoretic and combinatorial points of view.In this paper we develop a general machinery for the description of the reduced arc spaces of affine cones over toric varieties.We apply our techniques to a number of classical cases and explore some connections with representation theory of current algebras.

We construct geometric realizations—projective algebraic versions of cobordisms—for birational maps between Mori Dream Spaces. We show that these geometric realizations are Mori Dream Spaces, as well, and that they can be constructed so that they induce factorizations of the original birational maps as compositions of wall-crossings. In the case of toric birational maps between normal Q -factorial, projective toric varieties, we provide several SageMath functions to work with C ∗ -actions and birational geometry; in particular we show how to explicitly construct a moment polytope of a toric geometric realization. Moreover, by embedding Mori Dream Spaces in toric varieties, we obtain geometric realizations of birational maps of Mori Dream Spaces as restrictions of toric geometric realizations. We also provide examples and discuss when a geometric realization is Fano.