# Purity, formality, and arrangement complements

@article{Dupont2015PurityFA, title={Purity, formality, and arrangement complements}, author={Cl'ement Dupont}, journal={arXiv: Algebraic Geometry}, year={2015} }

We prove a "purity implies formality" statement in the context of the rational homotopy theory of smooth complex algebraic varieties, and apply it to complements of hypersurface arrangements. In particular, we prove that the complement of a toric arrangement is formal. This is analogous to the classical formality theorem for complements of hyperplane arrangements, due to Brieskorn, and generalizes a theorem of De Concini and Procesi.

## 15 Citations

### MIXED HODGE STRUCTURES AND FORMALITY OF SYMMETRIC

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- 2018

We use mixed Hodge theory to show that the functor of singular chains with rational coefficients is formal as a lax symmetric monoidal functor, when restricted to complex varieties whose weight…

### On the cohomology of arrangements of subtori

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- 2022

Given an arrangement of subtori of arbitrary codimension in a complex torus, we compute the cohomology groups of the complement. Then, by using the Leray spectral sequence, we describe the…

### Homotopy transfer and formality

- MathematicsAnnales de l'Institut Fourier
- 2021

In a recent paper, the second author and Joana Cirici proved a theorem that says that given appropriate hypotheses, $n$-formality of a differential graded algebraic structure is equivalent to the…

### A differential algebra and the homotopy type of the complement of a toric arrangement

- MathematicsRendiconti Lincei - Matematica e Applicazioni
- 2020

We show that the rational homotopy type of the complement of a toric arrangement is completely determined by two sets of combinatorial data. This is obtained by introducing a differential graded…

### Combinatorics of toric arrangements

- MathematicsRendiconti Lincei - Matematica e Applicazioni
- 2019

In this paper we build an Orlik-Solomon model for the canonical gradation of the cohomology algebra with integer coefficients of the complement of a toric arrangement. We give some results on the…

### Étale cohomology, purity and formality with torsion coefficients

- MathematicsJournal of Topology
- 2022

We use Galois group actions on étale cohomology to prove results of formality for dg‐operads and dg‐algebras with torsion coefficients. Our theory applies, among other related objects, to the…

### A basis for the cohomology of compact models of toric arrangements

- Mathematics
- 2022

In this paper we ﬁnd monomial bases for the integer cohomology rings of compact wonderful models of toric arrangements. In the description of the monomials various combinatorial objects come into…

### The Goresky-MacPherson formula for toric arrangements

- Mathematics
- 2018

A subspace arrangement is a finite collection of affine subspaces in $\mathbb{R}^n$. One of the main problems associated to arrangements asks up to what extent the topological invariants of the union…

### Corrigendum to “Orlik-Solomon-type presentations for the cohomology algebra of toric arrangements”

- MathematicsTransactions of the American Mathematical Society
- 2021

We give an explicit presentation for the integral cohomology ring of the complement of any arrangement of level sets of characters in a complex torus (alias "toric arrangement"). Our description…

### Ranks of homotopy and cohomology groups for rationally elliptic spaces and algebraic varieties

- MathematicsHomology, Homotopy and Applications
- 2022

We discuss inequalities between the values of homotopical and cohomological Poincaré polynomials of the self-products of rationally elliptic spaces. For rationally elliptic quasi-projective…

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