Toric varieties with huge Grothendieck group

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.

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).

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 $\mathbf{F}_k(X_\mathcal{A})$.

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.

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 first investigate torus‐equivariant motivic characteristic classes of toric varieties, and then apply them via the equivariant Riemann–Roch formalism to prove very general Euler–Maclaurin‐type formulae for full‐dimensional simple lattice polytopes. We consider ‐equivariant versions and of the motivic Chern and, resp., Hirzebruch characteristic classes of a toric variety (with corresponding torus ), and extend many known results from the non‐equivariant context to the equivariant setting. For example, the equivariant motivic Chern class is computed as the sum of the equivariant Grothendieck classes of the ‐equivariant sheaves of Zariski ‐forms weighted by . Using the motivic, as well as the characteristic class nature of , the corresponding generalized equivariant Hirzebruch ‐genus of a ‐invariant Cartier divisor on is also calculated. Further global formulae for are obtained in the simplicial context based on the Cox construction and the equivariant Lefschetz–Riemann–Roch theorem of Edidin–Graham. Alternative proofs of all these results are given via localization techniques at the torus fixed points in ‐equivariant ‐ and, resp., homology theories of toric varieties, due to Brion–Vergne and, resp., Brylinski–Zhang. These localization results apply to any toric variety with a torus fixed point. In localized ‐equivariant ‐theory, we extend a classical formula of Brion for a full‐dimensional lattice polytope to a weighted version. We also generalize the Molien formula of Brion–Vergne for the localized class of the structure sheaf of a simplicial toric variety to the context of . Similarly, we calculate the localized Hirzebruch class in localized ‐equivariant homology, extending the corresponding results of Brylinski–Zhang for the localized Todd class (fitting with the equivariant Hirzebruch class for ). As main applications of our equivariant characteristic class formulae, we provide a geometric perspective on several weighted Euler–Maclaurin‐type formulae for full‐dimensional simple lattice polytopes (corresponding to simplicial toric varieties), coming from the equivariant toric geometry via the equivariant Hirzebruch–Riemann–Roch (for an ample torus invariant Cartier divisor). Our main results even provide generalizations to arbitrary equivariant coherent sheaf coefficients , including algebraic geometric proofs of (weighted versions of) the Euler–Maclaurin formulae of Cappell–Shaneson, Brion–Vergne, Guillemin, and so forth (all of which correspond to the choice of the structure sheaf), via the equivariant Hirzebruch–Riemann–Roch formalism. In particular, we give a first complete proof of the Euler–Maclaurin formula of Cappell–Shaneson. Our approach, based on motivic characteristic classes , allows us to obtain such Euler–Maclaurin formulae also for (the interior of) a face, as well as for the polytope with several facets (i.e., codimension one faces) removed, for example, for the interior of the polytope (as well as for equivariant characteristic class formulae for locally closed ‐invariant subsets of a toric variety). Moreover, we prove such results also in the weighted context, as well as for ‐Minkowski summands of the given full‐dimensional lattice polytope (corresponding to globally generated torus invariant Cartier divisors in the toric context). Some of these results are extended to local Euler–Maclaurin formulae for the tangent cones at the vertices of the given full‐dimensional lattice polytope (fitting with the localization at the torus fixed points in equivariant ‐theory and equivariant (co)homology). Finally, we also give an application of our abstract Euler–Maclaurin formula to generalized reciprocity for Dedekind sums.

β-Density function on the class group of projective toric varieties

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.

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.

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.

Computations of Gromov–Witten invariants of toric varieties

Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects — the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra — which have been loosely referred to as “positive geometries”. The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic singularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. The structures seen in the physical setting of the Amplituhedron are both rigid and rich enough to motivate an investigation of the notions of “positive geometries” and their associated “canonical forms” as objects of study in their own right, in a more general mathematical setting. In this paper we take the first steps in this direction. We begin by giving a precise definition of positive geometries and canonical forms, and introduce two general methods for finding forms for more complicated positive geometries from simpler ones — via “triangulation” on the one hand, and “push-forward” maps between geometries on the other. We present numerous examples of positive geometries in projective spaces, Grassmannians, and toric, cluster and flag varieties, both for the simplest “simplex-like” geometries and the richer “polytope-like” ones. We also illustrate a number of strategies for computing canonical forms for large classes of positive geometries, ranging from a direct determination exploiting knowledge of zeros and poles, to the use of the general triangulation and push-forward methods, to the representation of the form as volume integrals over dual geometries and contour integrals over auxiliary spaces. These methods yield interesting representations for the canonical forms of wide classes of positive geometries, ranging from the simplest Amplituhedra to new expressions for the volume of arbitrary convex polytopes.

The paper is concerned with cohomology of the small quantum group at a root of unity, and of its upper triangular subalgebra, with coefficients in a tilting module. It turns out to be related to irreducible objects in the heart of a certain t-structure on the derived category of equivariant coherent sheaves on the Springer resolution, and to equivariant coherent IC sheaves on the nil-cone. The support of the cohomology is described in terms of cells in affine Weyl groups. The basis in the Grothendieck group provided by the cohomology modules is shown to coincide with the Kazhdan-Lusztig basis, as predicted by J. Humphreys and V. Ostrik. The proof is based on the results of [ABG ], [AB] and [B], which allow us to reduce the question to purity of IC sheaves on affine flag varieties.