Combinatorics and topology of proper toric maps

Cataldo, Mark Andrea de; Migliorini, Luca; Mustaţă, Mircea

doi:10.1515/crelle-2015-0104

Cited by 23 publications

(30 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This approach is explained in detail in Section 6, and we use it to prove the results above for the mixed h-polynomial h P (S; u, v). We note that this extends ideas of Stanley [43] and also overlaps with recent work of de Cataldo, Migliorini, and Mustata [20].…”

Section: Introductionsupporting

confidence: 86%

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

Katz¹,

Stapledon²

2016

Advances in Mathematics

View full text Add to dashboard Cite

Abstract. 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.

show abstract

Section: Introductionsupporting

confidence: 86%

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

Katz¹,

Stapledon²

2016

Advances in Mathematics

View full text Add to dashboard Cite

show abstract

“…This lemma is proved in [DeLo,Lemma 6.5], in the case when E = IC S is the intersection complex (automatically G m -equivariant); this seems to be rooted in [KL,Lemma 4.5.(a)]. The proof of the above simple generalization to the direct image under a proper map of a weakly equivariant G m -equivariant complex is contained in the proof of [dMM,Lemma 4.2]. We also draw the reader's attention to [Spr84,Cor.…”

Section: A Semisimplicity Conjecturementioning

confidence: 93%

“…In what follows, we are going to use freely the notion of incidence algebra of the poset associated with the B-orbits in G/B as summarized in [dMM], §6. In particular, (2.1) is the analogue of [dMM,Theorem 7.3] in the context of the map p below. The reader is warned that in [dMM], the poset of orbits in the toric variety has the order opposite to the one employed below in G/B, i.e.…”

Section: Corollary 223 (Goodness For Convolution Products)mentioning

confidence: 99%

Frobenius semisimplicity for convolution morphisms

Cataldo

Haines

2017

Math. Z.

Self Cite

View full text Add to dashboard Cite

Abstract. This article concerns properties of mixed ℓ-adic complexes on varieties over finite fields, related to the action of the Frobenius automorphism. We establish a fiberwise criterion for the semisimplicity and Frobenius semisimplicity of the direct image complex under a proper morphism of varieties over a finite field. We conjecture that the direct image of the intersection complex on the domain is always semisimple and Frobenius semisimple; this conjecture would imply that a strong form of the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber is valid over finite fields. We prove our conjecture for (generalized) convolution morphisms associated with partial affine flag varieties for split connected reductive groups over finite fields. As a crucial tool, we develop a new schematic theory of big cells for loop groups. With suitable reformulations, the main results are valid over any algebraically closed ground field.

show abstract

“…Another application with surprising consequences of arithmetic flavor is contained in [40] where a variation of the method is used to get estimates on the Betti numbers of the fibres of a map. The paper [14] contains a complete characterization of the supports of toric maps between toric varieties in terms of the combinatorics of the fans associated to two varieties.…”

Section: Higher Discriminantsmentioning

confidence: 99%