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Alan Stapledon

Publications and source records attributed to Alan Stapledon.

At least 19 recordsLinked to original sources
arXiv2025

Subdivisions of lower Eulerian posets

There is a natural notion of a subdivision of a lower Eulerian poset called a strong formal subdivision, which abstracts the notion of a polyhedral subdivision of a polytope, or a proper, surjective morphism of fans. We show that there is a canonical bijection between strong formal subdivisions and triples consisting of a lower Eulerian poset, a corresponding rank function, and a non-minimal element such that the join with any other element exists. The bijection uses the non-Hausdorff mapping cylinder construction introduced by Barmak and Minian. A corresponding bijection for $CW$-posets is given, as well as an application to computing the $cd$-index of an Eulerian poset. A companion paper explores applications to Kazhdan-Lusztig-Stanley theory.

math.CO↗
arXiv2025

Subdivisions of lower Eulerian posets and KLS theory

In a companion paper, a canonical bijection was established between strong formal subdivisions of lower Eulerian posets and triples consisting of a lower Eulerian poset, a corresponding rank function, and a non-minimal element such that the join with any other element exists. The main goal of this paper is to relate the local $h$-polynomials of a strong formal subdivision to the Kazhdan-Lusztig-Stanley (KLS) invariants associated to its corresponding lower Eulerian poset under this bijection. As an application, we show that Braden and MacPherson's relative $g$-polynomials are alternative encodings of corresponding local $h$-polynomials. We also further develop equivariant KLS theory and give equivariant generalizations of our main results, as well as an application to equivariant Ehrhart theory.

math.CO↗
arXiv2025

Complementary vectors of simplicial complexes

We classify the complementary vectors of doubly Cohen-Macaulay complexes. This proves a conjecture of Swartz, negatively answers a question of Athanasiadis and Tzanaki, and gives new bounds on the number of independent sets in a matroid. Our technique works more generally for certain level quotients of Stanley-Reisner rings, giving new bounds on the face numbers of Buchsbaum* complexes.

math.CO↗
arXiv2025

Lefschetz properties of local face modules

Local face modules are modules over face rings whose Hilbert function is the local $h$-vector of a triangulation of a simplex. We study when Lefschetz properties hold for local face modules. We prove new inequalities for local $h$-vectors of vertex-induced triangulations by proving Lefschetz properties for local face modules of these triangulations. We show that, even for regular triangulations, Lefschetz properties can fail for local face modules in positive characteristic.

math.CO↗
arXiv2024

Differential operators, anisotropy, and simplicial spheres

We find identities involving differential operators in the generic artinian reduction of the Stanley-Reisner ring of a simplicial sphere in any positive characteristic. These identities generalize the characteristic 2 identities used by Papadakis and Petrotou to give a proof of the algebraic g-conjecture. We show that these identities are a shadow of an identity on the degree map, and we use them to prove the anisotropy of certain forms on the generic artinian reduction of the Stanley--Reisner ring and to prove weak Lefschetz results.

math.CO↗
arXiv2024

Determinants of Hodge-Riemann forms

We calculate the determinant of the bilinear form in middle degree of the generic artinian reduction of the Stanley-Reisner ring of an odd-dimensional simplicial sphere. This proves the odd multiplicity conjecture of Papadakis and Petrotou and implies that this determinant is a complete invariant of the simplicial sphere. We extend this result to odd-dimensional connected oriented simplicial homology manifolds. In characteristic 2, we prove a generalization to the Hodge-Riemann forms of any connected simplicial homology manifold. To prove the latter theorem we establish the strong Lefschetz property for certain quotients of the Stanley-Reisner rings of connected simplicial pseudomanifolds.

math.AC↗
arXiv2023

Equivariant Ehrhart theory, commutative algebra and invariant triangulations of polytopes

Ehrhart theory is the study of the enumeration of lattice points in lattice polytopes. Equivariant Ehrhart theory is a generalization of Ehrhart theory that takes into account the action of a finite group acting via affine transformations on the underlying lattice and preserving the polytope. We further develop equivariant Ehrhart theory in part by establishing connections with commutative algebra as well as the question of when there exists an invariant lattice triangulation of a lattice polytope.

math.CO↗
arXiv2022

The local motivic monodromy conjecture for simplicial nondegenerate singularities

We prove the local motivic monodromy conjecture for singularities that are nondegenerate with respect to a simplicial Newton polyhedron. It follows that all poles of the local topological zeta functions of such singularities correspond to eigenvalues of monodromy acting on the cohomology of the Milnor fiber of some nearby point, as do the poles of Igusa's local $p$-adic zeta functions for large primes $p$.

math.AG↗
arXiv2022

Resolutions of local face modules, functoriality, and vanishing of local $h$-vectors

We study the local face modules of triangulations of simplices, i.e., the modules over face rings whose Hilbert functions are local $h$-vectors. In particular, we give resolutions of these modules by subcomplexes of Koszul complexes as well as functorial maps between modules induced by inclusions of faces. As applications, we prove a new monotonicity result for local $h$-vectors and new results on the structure of faces in triangulations with vanishing local $h$-vectors.

math.CO↗
arXiv2019

Splitting groups with cubic Cayley graphs of connectivity two

A group $G$ splits over a subgroup $C$ if $G$ is either a free product with amalgamation $A \underset{C}{\ast} B$ or an HNN-extension $G=A \underset{C}{\ast} (t)$. We invoke Bass-Serre theory and classify all infinite groups which admit cubic Cayley graphs of connectivity two in terms of splittings over a subgroup.

math.CO↗
arXiv2016

Counting lattice points in free sums of polytopes

We show how to compute the Ehrhart polynomial of the free sum of two lattice polytopes containing the origin $P$ and $Q$ in terms of the enumerative combinatorics of $P$ and $Q$. This generalizes work of Beck, Jayawant, McAllister, and Braun, and follows from the observation that the weighted $h^*$-polynomial is multiplicative with respect to the free sum. We deduce that given a lattice polytope $P$ containing the origin, the problem of computing the number of lattice points in all rational dilates of $P$ is equivalent to the problem of computing the number of lattice points in all integer dilates of all free sums of $P$ with itself.

math.CO↗
arXiv2014

Local $h$-polynomials, invariants of subdivisions, and mixed Ehrhart theory

There are natural polynomial invariants of polytopes and lattice polytopes coming from enumerative combinatorics and Ehrhart theory, namely the $h$- and $h^*$-polynomials, respectively. In this paper, we study their generalization to subdivisions and lattice subdivisions of polytopes. By abstracting constructions in mixed Hodge theory, we introduce multivariable polynomials which specialize to the $h$-, $h^*$- polynomials. These polynomials, the mixed $h$-polynomial and the (refined) limit mixed $h^*$-polynomial have rich symmetry, non-negativity, and unimodality properties, which both refine known properties of the classical polynomials, and reveal new structure. For example, we prove a lower bound theorem for a related invariant called the local $h^*$-polynomial. We introduce our polynomials by developing a very general formalism for studying subdivisions of Eulerian posets that extends the work of Stanley, Brenti and Athanasiadis on local $h$-vectors. In particular, we prove a conjecture of Nill and Schepers, and answer a question of Athanasiadis.

math.CO↗
arXiv2014

Formulas for monodromy

Given a family $X$ of complex varieties degenerating over a punctured disc, one is interested in computing related invariants called the motivic nearby fiber and the refined limit mixed Hodge numbers, both of which contain information about the induced action of monodromy on the cohomology of a fiber of $X$. Our first main result is that the motivic nearby fiber of $X$ can be computed by first stratifying $X$ into locally closed subvarieties that are non-degenerate in the sense of Tevelev, and then applying an explicit formula on each piece of the stratification that involves tropical geometry. Our second main result is an explicit combinatorial formula for the refined limit mixed Hodge numbers in the case when $X$ is a family of non-degenerate hypersurfaces. As an application, given a complex polynomial, then, under appropriate conditions, we give a combinatorial formula for the Jordan block structure of the action of monodromy on the cohomology of the Milnor fiber, generalizing a famous formula of Varchenko for the associated eigenvalues. In addition, we give a formula for the Jordan block structure of the action of monodromy at infinity.

math.AG↗
arXiv2014

Tropical geometry, the motivic nearby fiber and limit mixed Hodge numbers of hypersurfaces

The motivic nearby fiber is an invariant obtained from degenerating a complex variety over a disc. It specializes to the Euler characteristic of the original variety but also contains information on the variation of Hodge structure associated to the degeneration which is encoded as a limit mixed Hodge structure. However, this invariant is difficult to compute in practice. Using the techniques of tropical geometry we present a new formula for the motivic nearby fiber. Moreover, since there is a range of available software implementing the main algorithms in tropical geometry, our formula can be computed in practice. We specialize to the case of families of sch\"{o}n complex hypersurfaces of tori where we provide explicit formulas describing the action of the unipotent part of monodromy on the graded pieces (with respect to the Deligne weight filtration) of the cohomology with compact supports. These families are described combinatorially by a polyhedral subdivision of the associated Newton polytope. We develop new mixed Hodge theory-inspired combinatorial invariants of such subdivisions, among them the `refined limit mixed $h^*$-polynomial'. These invariants are related to Stanley's combinatorial study of subdivisions: in a companion combinatorial paper whose results are applied here, we situate our invariants in Stanley's theory where they become multi-variable extensions of his invariants. Our results generalize work of Danilov and Khovanski{\u\i} and Batyrev and Borisov on the Hodge numbers of hypersurfaces. We also present analogous formulas describing the action of the unipotent part of monodromy on the intersection cohomology groups of a family of sch\"on hypersurfaces of a projective toric variety.

math.AG↗
arXiv2010

New mirror pairs of Calabi-Yau orbifolds

We prove a representation-theoretic version of Borisov-Batyrev mirror symmetry, and use it to construct infinitely many new pairs of orbifolds with mirror Hodge diamonds, with respect to the usual Hodge structure on singular complex cohomology. We conjecture that the corresponding orbifold Hodge diamonds are also mirror. When $X$ is the Fermat quintic in $\P^4$, and $\tilde{X}^*$ is a $\Sym_5$-equivariant, toric resolution of its mirror $X^*$, we deduce that for any subgroup $\Gamma$ of the alternating group $A_5$, the $\Gamma$-Hilbert schemes $\Gamma$-$\Hilb(X)$ and $\Gamma$-$\Hilb(\tilde{X}^*)$ are smooth Calabi-Yau threefolds with (explicitly computed) mirror Hodge diamonds.

math.AG↗
arXiv2010

Tropical Geometry and the Motivic Nearby Fiber

We construct motivic invariants of a subvariety of an algebraic torus from its tropicalization and initial degenerations. More specifically, we introduce an invariant of a compactification of such a variety called the "tropical motivic nearby fiber." This invariant specializes in the schon case to the Hodge-Deligne polynomial of the limit mixed Hodge structure of a corresponding degeneration. We give purely combinatorial expressions for this Hodge-Deligne polynomial in the cases of schon hypersurfaces and smooth tropical varieties. We also deduce a formula for the Euler characteristic of a general fiber of the degeneration.

math.AG↗
arXiv2010

Representations on the cohomology of hypersurfaces and mirror symmetry

We study the representation of a finite group acting on the cohomology of a non-degenerate, invariant hypersurface of a projective toric variety. We deduce an explicit description of the representation when the toric variety has at worst quotient singularities. As an application, we conjecture a representation-theoretic version of Batyrev and Borisov's mirror symmetry between pairs of Calabi-Yau hypersurfaces, and prove it when the hypersurfaces are both smooth or have dimension at most 3. An interesting consequence is the existence of pairs of Calabi-Yau orbifolds whose Hodge diamonds are mirror, with respect to the usual Hodge structure on singular cohomology.

math.RT↗