Characteristic Classes of Singular Toric Varieties

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.

We obtain refined generating series formulae for equivariant characteristic classes of external and symmetric products of singular complex quasi-projective varieties. More concretely, we study equivariant versions of Todd, Chern and Hirzebruch classes for singular spaces, with values in delocalized Borel-Moore homology of external and symmetric products. As a byproduct, we recover our previous characteristic class formulae for symmetric products, and obtain new equivariant generalizations of these results, in particular also in the context of twisting by representations of the symmetric group.

We classify smooth toric Fano varieties of dimension $n\\geq 3$ containing a toric divisor isomorphic to the $(n-1)$-dimensional projective space. As a consequence of this classification, we show that any smooth complete toric variety $X$ of dimension $n\\geq 3$ with a fixed point $x\\in X$ such that the blow-up $B_x(X)$ of $X$ at $x$ is Fano is isomorphic either to the $n$-dimensional projective space or to the blow-up of the $n$-dimensional projective space along an invariant linear codimension two subspace. As expected, such results are proved\nusing toric Mori theory due to Reid.

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 $X$ 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 describe a practical and effective method for reconstructing the\ndeformation class of a Fano manifold X from a Laurent polynomial f that\ncorresponds to X under Mirror Symmetry. We explore connections to nef\npartitions, the smoothing of singular toric varieties, and the construction of\nembeddings of one (possibly-singular) toric variety in another. In particular,\nwe construct degenerations from Fano manifolds to singular toric varieties; in\nthe toric complete intersection case, these degenerations were constructed\npreviously by Doran--Harder. We use our method to find models of orbifold del\nPezzo surfaces as complete intersections and degeneracy loci, and to construct\na new four-dimensional Fano manifold.\n

The aim of this thesis is to study discrete structures associated with group- based models. Group-based models are statistical models on phylogenetic trees that can be parametrized by polynomial maps. The algebraic varieties given by these polynomial maps are toric. By the well-known correspondence between toric varieties and polyhedral fans, we can associate discrete structures, such as lattice polytopes and affine semigroups, with group-based models. We follow three main lines in this dissertation. In Chapters 2 and 3, we study the Hilbert polynomials of group-based models. This is motivated by a result of Buczyńska and Wiśniewski stating that the Hilbert polynomial of the Jukes- Cantor binary model on a trivalent tree does not depend on the shape of the tree. In Chapter 2, we give a simple combinatorial proof to this statement, and in Chapter 3, we show that the analogous statement does not hold for the Kimura 3-parameter model. In Chapters 4 and 5, we study the phylogenetic semigroups on graphs that generalize the Jukes-Cantor binary model on trees. In Chapter 4, we study the maximal degrees of the minimal generators of these semigroups. In Chapter 5, we investigate the minimal generators of the phylogenetic semigroups on graphs with a few holes, extending the work of Buczyńska. Finally, in Chapter 6, we establish a connection between Berenstein-Zelevinsky triangles from representation theory and group-based models. This is motivated by the recent work of Sturmfels, Xu, and Manon related to conformal block algebras.

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 compute the cohomological Brauer group of a normal toric variety whose singular locus has codimension less than or equal to $2$ everywhere.

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

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.

We describe how local toric singularities, including the Toric Lego construction, can be embedded in compact Calabi-Yau manifolds. We study in detail the addition of D-branes, including non-compact flavor branes as typically used in semi-realistic model building. The global geometry provides constraints on allowable local models. As an illustration of our discussion we focus on D3 and D7-branes on (the partially resolved) (dP 0)3 singularity, its embedding in a specific Calabi-Yau manifold as a hypersurface in a toric variety, the related type IIB orientifold compactification, as well as the corresponding F-theory uplift. Our techniques generalize naturally to complete intersections, and to a large class of F-theory backgrounds with singularities.

Using the algebraic geometry method of Berenstein and Leigh (BL) (hep-th/0009209 and hep-th/0105229), and considering singular toric varieties Vd+1 with NC irrational torus fibration, we construct NC extensions Mdnc of complex d dimension Calabi–Yau (CY) manifolds embedded in Vd+1nc. We give realizations of the NC C∗r toric group, derive the constraint equations for NC Calabi–Yau (NCCY) manifolds Mncd embedded in Vd+1nc and work out solutions for their generators. We study fractional D branes at singularities and show that, due to the complete reducibility property of C∗r group representations, there is an infinite number of noncompact fractional branes at fixed points of the NC toric group.