Complement Of Arrangement Research Articles

Cohomology of ℤ-Local Systems on Complex Hyperplane Arrangement Complements

Abstract We prove a Cohen-Dimca-Orlik-type theorem for rank one ${\mathbb{Z}}$-local systems on complex hyperplane arrangement complements. This settles a recent conjecture of S. Sugawara.

Milnor–Witt motivic cohomology of complements of hyperplane arrangements

Milnor–Witt motivic cohomology of complements of hyperplane arrangements

Integral homology groups of double coverings and rank one ℤ-local systems for a minimal CW complex

Given a finite CW complex X X , a nonzero cohomology class ω ∈ H 1 ( X , Z 2 ) \omega \in H^1(X,\mathbb {Z}_2) determines a double covering X ω X^\omega and a rank one Z \mathbb {Z} -local system L ω \mathcal {L}_\omega . We investigate the relations between the homology groups H ∗ ( X ω , Z ) H_*(X^{\omega },\mathbb {Z}) and H ∗ ( X , L ω ) H_*(X,\mathcal {L}_\omega ) , when X X is homotopy equivalent to a minimal CW complex. In particular, this settles a conjecture recently proposed by Ishibashi, Sugawara and Yoshinaga [Betti numbers and torsions in homology groups of double coverings, arxiv.org/abs/2209.02236, 2022, Conjecture 3.3], for a hyperplane arrangement complement.

ℤ-local system cohomology of hyperplane arrangements and a Cohen–Dimca–Orlik type theorem

Local system cohomology groups of the complements of hyperplane arrangements have played an important role in the theory of hypergeometric integrals, the topology of Milnor fibers and covering spaces. One of the important theorems is the vanishing theorem for generic [Formula: see text]-local systems which goes back to Aomoto’s work. Later, Cohen, Dimca, and Orlik proved a stronger version of the vanishing theorem. In this paper, we prove a Cohen–Dimca–Orlik type theorem for [Formula: see text]-local systems.

Cohomology of moduli spaces of Del Pezzo surfaces

AbstractWe compute the rational Betti cohomology groups of the coarse moduli spaces of geometrically marked Del Pezzo surfaces of degree 3 and 4 as representations of the Weyl groups of the corresponding root systems. The proof uses a blend of methods from point counting over finite fields and techniques from arrangement complements.

Gromov–Witten theory and invariants of matroids

We use techniques from Gromov–Witten theory to construct new invariants of matroids taking value in the Chow groups of spaces of rational curves in the permutohedral toric variety. When the matroid is realizable by a complex hyperplane arrangement, our invariants coincide with virtual fundamental classes used to define the logarithmic Gromov–Witten theory of wonderful models of arrangement complements, for any logarithmic structure supported on the wonderful boundary. When the boundary is empty, this implies that the quantum cohomology ring of a hyperplane arrangement’s wonderful model is a combinatorial invariant, i.e., it depends only on the matroid. When the boundary divisor is maximal, we use toric intersection theory to convert the virtual fundamental class into a balanced weighted fan in a vector space, having the expected dimension. We explain how the associated Gromov–Witten theory is completely encoded by intersections with this weighted fan. We include a number of questions whose positive answers would lead to a well-defined Gromov–Witten theory of non-realizable matroids.

The loop-linking number of line arrangements

In his Ph.D. thesis, Cadegan-Schlieper constructs an invariant of the embedded topology of a line arrangement which generalizes the \(\mathcal {I}\)-invariant introduced by Artal, Florens and the author. This new invariant is called the loop-linking number in the present paper. We refine the result of Cadegan-Schlieper by proving that the loop-linking number is an invariant of the homeomorphism type of the arrangement complement. We give two effective methods to compute this invariant, both are based on the braid monodromy. As an application, we detect an arithmetic Zariski pair of arrangements with 11 lines whose coefficients are in the 5th cyclotomic field. Furthermore, we also prove that the fundamental groups of their complements are not isomorphic; it is the Zariski pair with the fewest number of lines which have this property. We also detect an arithmetic Zariski triple with 12 lines whose complements have non-isomorphic fundamental groups. In the appendix, we give 29 combinatorial types which lead to similar ordered arithmetic Zariski pairs of 11 lines. To conclude this paper, we give a additivity theorem for the union of arrangements. This first allows us to prove that the complements of Rybnikov’s arrangements are not homeomorphic, and then leads us to a generalization of Rybnikov’s result. Lastly, we use it to prove the existence of homotopy-equivalent lattice-isomorphic arrangements which have non-homeomorphic complements.

Graded rings of Hermitian modular forms with singularities

We study graded rings of meromorphic Hermitian modular forms of degree two whose poles are supported on an arrangement of Heegner divisors. For the group mathrm {SU}_{2,2}({mathcal {O}}_K) where K is the imaginary-quadratic number field of discriminant -d, d in {4, 7,8,11,15,19,20,24} we obtain a polynomial algebra without relations. In particular the Looijenga compactifications of the arrangement complements are weighted projective spaces.

Cohomology of complements of toric arrangements associated with root systems

We develop an algorithm for computing the cohomology of complements of toric arrangements. In the case a finite group Gamma is acting on the arrangement, the algorithm determines the cohomology groups as representations of Gamma . As an important application, we determine the cohomology groups of the complements of the toric arrangements associated with root systems of exceptional type as representations of the corresponding Weyl groups.

Factorization statistics and bug-eyed configuration spaces

A recent theorem of Hyde proves that the factorizations statistics of a random polynomial over a finite field are governed by the action of the symmetric group on the configuration space of $n$ distinct ordered points in $\mathbb R^3$. Hyde asked whether this result could be explained geometrically. We give a geometric proof of Hyde's theorem as an instance of the Grothendieck--Lefschetz trace formula applied to an interesting, highly nonseparated algebraic space. An advantage of our method is that it generalizes uniformly to an arbitrary Weyl group. In the process we study certain non-Hausdorff models for complements of hyperplane arrangements, first introduced by Proudfoot.

A Leray Model for the Orlik–Solomon Algebra

Abstract We construct a combinatorial generalization of the Leray models for hyperplane arrangement complements. Given a matroid and some combinatorial blow-up data, we give a presentation for a bigraded (commutative) differential graded algebra. If the matroid is realizable over $\mathbb {C}$, this is the familiar Morgan model for a hyperplane arrangement complement, embedded in a blowup of projective space. In general, we obtain a cdga that interpolates between the Chow ring of a matroid and the Orlik–Solomon algebra. Our construction can also be expressed in terms of sheaves on combinatorial blowups of geometric lattices. As a key technical device, we construct a monomial basis via a Gröbner basis for the ideal of relations. Combining these ingredients, we show that our algebra is quasi-isomorphic to the classical Orlik–Solomon algebra of the matroid.

The homotopy type of elliptic arrangements

We give combinatorial models for the homotopy type of complements of elliptic arrangements (i.e., certain sets of abelian subvarieties in a product of elliptic curves). We give a presentation of the fundamental group of such spaces and, as an application, we treat the case of ordered configuration spaces of elliptic curves. Our models are finite polyhedral CW complexes, and our combinatorial tools of choice are acyclic categories (small categories without loops). As a stepping stone, we give a characterization of which acyclic categories arise as face categories of polyhedral CW complexes.

Bordifications of hyperplane arrangements and their curve complexes

The complement of an arrangement of hyperplanes in $\mathbb C^n$ has a natural bordification to a manifold with corners formed by removing (or "blowing up") tubular neighborhoods of the hyperplanes and certain of their intersections. When the arrangement is the complexification of a real simplicial arrangement, the bordification closely resembles Harvey's bordification of moduli space. We prove that the faces of the universal cover of the bordification are parameterized by the simplices of a simplicial complex $\mathcal{C}$, the vertices of which are the irreducible "parabolic subgroups" of the fundamental group of the arrangement complement. So, the complex $\mathcal{C}$ plays a similar role for an arrangement complement as the curve complex does for moduli space. Also, in analogy with curve complexes and with spherical buildings, we prove that $\mathcal{C}$ has the homotopy type of a wedge of spheres.

Twisted Alexander modules of hyperplane arrangement complements

We study torsion properties of the twisted Alexander modules of the affine complement M of a complex essential hyperplane arrangement, as well as those of punctured stratified tubular neighborhoods of complex essential hyperplane arrangements. We investigate divisibility properties between the twisted Alexander polynomials of the two spaces, compute the (first) twisted Alexander polynomial of a punctured stratified tubular neighborhood of an essential line arrangement, and study the possible roots of the twisted Alexander polynomials of both the complement and the punctured stratified tubular neighborhood of an essential hyperplane arrangement in higher dimensions. We apply our results to distinguish non-homeomorphic homotopy equivalent arrangement complements. We also relate the twisted Alexander polynomials of M with the corresponding twisted homology jump loci.

Hyperplane arrangement complement with top degree Betti number being small

Hyperplane arrangement complement with top degree Betti number being small

Chow rings of heavy/light Hassett spaces via tropical geometry

We compute the Chow ring of an arbitrary heavy/light Hassett spaceM‾0,w. These spaces are moduli spaces of weighted pointed stable rational curves, where the associated weight vector w consists of only heavy and light weights. Work of Cavalieri et al. [3] exhibits these spaces as tropical compactifications of hyperplane arrangement complements. The computation of the Chow ring then reduces to intersection theory on the toric variety of the Bergman fan of a graphic matroid. Keel [16] has calculated the Chow ring A⁎(M‾0,n) of the moduli space M‾0,n of stable nodal n-marked rational curves; his presentation is in terms of divisor classes of stable trees of P1's having one nodal singularity. Our presentation of the ideal of relations for the Chow ring A⁎(M‾0,w) is analogous. We show that pulling back under Hassett's birational reduction morphism ρw:M‾0,n→M‾0,w identifies the Chow ring A⁎(M‾0,w) with the subring of A⁎(M‾0,n) generated by divisors of w-stable trees, which are those trees which remain stable in M‾0,w.

The Dunkl weight function for rational Cherednik algebras

In this paper we prove the existence of the Dunkl weight function $$K_{c, \lambda }$$ for any irreducible representation $$\lambda $$ of any finite Coxeter group W, generalizing previous results of Dunkl. In particular, $$K_{c, \lambda }$$ is a family of tempered distributions on the real reflection representation of W taking values in $$\text {End}_\mathbb {C}(\lambda )$$, with holomorphic dependence on the complex multi-parameter c. When the parameter c is real, the distribution $$K_{c, \lambda }$$ provides an integral formula for Cherednik’s Gaussian inner product $$\gamma _{c, \lambda }$$ on the Verma module $$\Delta _c(\lambda )$$ for the rational Cherednik algebra $$H_c(W, \mathfrak {h})$$.queryPlease check and confirm the inserted city name ‘Stanford’ for the affiliation is correct. In this case, the restriction of $$K_{c, \lambda }$$ to the hyperplane arrangement complement $$\mathfrak {h}_{\mathbb {R}, reg}$$ is given by integration against an analytic function whose values can be interpreted as braid group invariant Hermitian forms on $$KZ(\Delta _c(\lambda ))$$, where KZ denotes the Knizhnik–Zamolodchikov functor introduced by Ginzburg–Guay–Opdam–Rouquier. This provides a concrete connection between invariant Hermitian forms on representations of rational Cherednik algebras and invariant Hermitian forms on representations of Iwahori–Hecke algebras, and we exploit this connection to show that the KZ functor preserves signatures, and in particular unitarizability, in an appropriate sense.

Hyperbolicity of the complement of arrangements of non complex lines

The goal of this paper is twofold. We study holomorphic curves f:C ? C3 avoiding four complex hyperplanes and a real subspace of real dimension five in C3 where we study the cases where the projection of f into the complex projective space CP2 is constant. On the other hand, we investigate the kobayashi hyperbolicity of the complement of five perturbed lines in CP2.

Double coverings of arrangement complements and 2-torsion in Milnor fiber homology

We prove that the mod $2$ Betti numbers of double coverings of a complex hyperplane arrangement complement are combinatorially determined. The proof is based on a relation between the mod $2$ Aomoto complex and the transfer long exact sequence. Applying the above result to the icosidodecahedral arrangement ($16$ planes in the three dimensional space related to the icosidodecahedron), we conclude that the first homology of the Milnor fiber has non-trivial $2$-torsion.

Hyperplane Arrangements and Tensor Product Invariants

In the first part of this paper, we consider, in the context of an arbitrary weighted hyperplane arrangement, the map from compactly supported cohomology to the usual cohomology of a local system. We obtain a formula (i.e., an explicit algebraic de Rham representative) for a generalized version of this map. In the second part, we apply these results to invariant theory: Schechtman and Varchenko connect invariant theoretic objects to the cohomology of local systems on complements of hyperplane arrangements. The first part of this paper is then used, following and completing arguments of Looijenga, to determine the image of invariants in cohomology. In suitable cases (e.g., corresponding to positive integral levels) the space of invariants acquires a mixed Hodge structure over a cyclotomic field. We investigate the Hodge filtration on the space of invariants and characterize the subspace of conformal blocks in Hodge theoretic terms.