Resonance Varieties Overview

• Resonance varieties are cohomology jump loci arising from graded algebras and spaces, defined via the Aomoto complex that detects jumps in cohomology dimensions.
• They are studied in diverse settings such as hyperplane arrangements and matroids, linking algebraic structures to topological and geometric phenomena.
• Advanced investigations extend their analysis through determinantal structures, duality, and representation theory, offering insights into group formality and Chen ranks.

Resonance varieties are cohomology jump loci attached to a graded-commutative algebra, a space, or a commutative differential graded algebra. In their classical rank-one form, they are defined from the Aomoto complex obtained by multiplication by a degree-one class, and they record those parameters for which the cohomology in a fixed degree jumps. They are homogeneous algebraic sets, and they are studied in a variety of topological, combinatorial, and geometric contexts, including hyperplane arrangements, matroids, finitely generated groups, Lie algebra representations, and vector bundles (Denham et al., 2015, Aprodu et al., 2023).

1. Definitions and basic constructions

For a graded, graded-commutative connected algebra $A=\bigoplus_{i\ge 0}A^i$ over $\mathbb{C}$, the resonance varieties are defined by

$$\mathscr{R}_d^i(A)=\left\{ a\in A^1 \;\Big|\; \dim H^i(A,a)\ge d\right\},$$

where $(A,a)$ is the Aomoto complex with differential $d_a(u)=a\cdot u$. For a group $G$ with a finite-type classifying space, one sets $$\mathscr{R}_d^i(G):=\mathscr{R}_d^i(H^*(G,\mathbb{C}))$$ (Suciu et al., 2016). For a space $X$, with $A=H^*(X,k)$, one also writes

$$\mathcal{R}_d^i(X,k)=\{a\in H^1(X,k)\mid \dim_k H^i(A,a)\ge d\},$$

and frequently abbreviates $\mathbb{C}$0 (Denham et al., 2015).

In degree $\mathbb{C}$1, a widely used equivalent description for a group $\mathbb{C}$2 is

$\mathbb{C}$3

For hyperplane arrangement complements, higher-degree and higher-depth loci are defined by

$\mathbb{C}$4

where $\mathbb{C}$5 is the cohomology of the Orlik-Solomon algebra with differential $\mathbb{C}$6 (Suciu et al., 2018, Budur, 2011).

The formalism extends to non-abelian coefficients. Given a cdga $\mathbb{C}$7, a finite-dimensional Lie algebra $\mathbb{C}$8, and a representation $\mathbb{C}$9, one considers the space of flat connections

$$\mathscr{R}_d^i(A)=\left\{ a\in A^1 \;\Big|\; \dim H^i(A,a)\ge d\right\},$$0

and the associated Aomoto complex $$\mathscr{R}_d^i(A)=\left\{ a\in A^1 \;\Big|\; \dim H^i(A,a)\ge d\right\},$$1. The higher-rank resonance varieties are then

$$\mathscr{R}_d^i(A)=\left\{ a\in A^1 \;\Big|\; \dim H^i(A,a)\ge d\right\},$$2

(Papadima et al., 2013).

Characteristic $$\mathscr{R}_d^i(A)=\left\{ a\in A^1 \;\Big|\; \dim H^i(A,a)\ge d\right\},$$3 requires a different differential. If $$\mathscr{R}_d^i(A)=\left\{ a\in A^1 \;\Big|\; \dim H^i(A,a)\ge d\right\},$$4 and $$\mathscr{R}_d^i(A)=\left\{ a\in A^1 \;\Big|\; \dim H^i(A,a)\ge d\right\},$$5 is the Bockstein homomorphism, then $$\mathscr{R}_d^i(A)=\left\{ a\in A^1 \;\Big|\; \dim H^i(A,a)\ge d\right\},$$6, and for $$\mathscr{R}_d^i(A)=\left\{ a\in A^1 \;\Big|\; \dim H^i(A,a)\ge d\right\},$$7 one defines the Aomoto-Bockstein complex by

$$\mathscr{R}_d^i(A)=\left\{ a\in A^1 \;\Big|\; \dim H^i(A,a)\ge d\right\},$$8

The corresponding resonance varieties are

$$\mathscr{R}_d^i(A)=\left\{ a\in A^1 \;\Big|\; \dim H^i(A,a)\ge d\right\},$$9

(Suciu, 2022).

2. Hyperplane arrangements and matroids

In the arrangement-theoretic setting, resonance is built from the Orlik-Solomon algebra. For a simple matroid $(A,a)$0 on a ground set $(A,a)$1 of $(A,a)$2 elements, its Orlik-Solomon algebra is

$(A,a)$3

where $(A,a)$4 and $(A,a)$5 is generated by the circuit relations. For $(A,a)$6 and $(A,a)$7, the resonance varieties are

$(A,a)$8

(Suciu, 28 Sep 2025). For a central, essential, indecomposable hyperplane arrangement $(A,a)$9 of degree $d_a(u)=a\cdot u$0 in $d_a(u)=a\cdot u$1, the complement $d_a(u)=a\cdot u$2 has Betti numbers $d_a(u)=a\cdot u$3, and these are, up to sign, the Whitney numbers of the first kind for the associated matroid. The resonance loci are tied to the cohomology of local systems, to information about the lower central series of the arrangement’s fundamental group, and to the study of critical points of master functions (Budur, 2011).

Degree $d_a(u)=a\cdot u$4 resonance is especially rigid. For realizable matroids over $d_a(u)=a\cdot u$5, the positive-dimensional components of $d_a(u)=a\cdot u$6 correspond bijectively to multinets on submatroids, and one has

$d_a(u)=a\cdot u$7

where $d_a(u)=a\cdot u$8 is the linear space associated to a multinet $d_a(u)=a\cdot u$9. The survey literature also records that, in degree $G$0, the structure as unions of linear spaces persists for non-realizable matroids by the same combinatorial argument, whereas for higher degree $G$1 it remains open whether the irreducible components are always unions of rational linear subspaces (Suciu, 28 Sep 2025).

Higher-degree resonance reveals phenomena not visible in degree $G$2. For a matroid $G$3 of rank $G$4, the propagation chain

$G$5

holds, and for complex hyperplane arrangements $G$6 is a union of linear subspaces for all $G$7. At the same time, the higher-degree theory is more delicate: components may arise from submatroids, duals, and matroid operations, and deletion need not preserve components of higher resonance (Denham, 2015).

The first resonance variety of an arrangement also admits effective geometric realizations. One description interprets $G$8 as the locus of decomposable two-tensors in the quadratic part of the Orlik-Solomon ideal, giving

$G$9

where $$\mathscr{R}_d^i(G):=\mathscr{R}_d^i(H^*(G,\mathbb{C}))$$0 is the Grassmannian of decomposable tensors and $$\mathscr{R}_d^i(G):=\mathscr{R}_d^i(H^*(G,\mathbb{C}))$$1 is the linear space determined by the quadratic generators. This method is described as much faster than previous alternatives based on Fitting ideals or Ext modules (Lima-Filho et al., 2008). In a related direction, non-local components of $$\mathscr{R}_d^i(G):=\mathscr{R}_d^i(H^*(G,\mathbb{C}))$$2 give rise to determinantal syzygies of the Orlik-Terao algebra $$\mathscr{R}_d^i(G):=\mathscr{R}_d^i(H^*(G,\mathbb{C}))$$3, and $$\mathscr{R}_d^i(G):=\mathscr{R}_d^i(H^*(G,\mathbb{C}))$$4 lies on a scroll (Schenck, 2010).

3. Determinantal structure, propagation, and duality

A major structural result for arrangement complements is that all higher resonance varieties are determinantal. For the singular module $$\mathscr{R}_d^i(G):=\mathscr{R}_d^i(H^*(G,\mathbb{C}))$$5, the Eisenbud-Popescu-Yuzvinsky resolution yields matrices $$\mathscr{R}_d^i(G):=\mathscr{R}_d^i(H^*(G,\mathbb{C}))$$6 of linear forms, and for $$\mathscr{R}_d^i(G):=\mathscr{R}_d^i(H^*(G,\mathbb{C}))$$7,

$$\mathscr{R}_d^i(G):=\mathscr{R}_d^i(H^*(G,\mathbb{C}))$$8

with equality away from lower-degree resonance loci under additional hypotheses. One consequence is the codimension estimate

$$\mathscr{R}_d^i(G):=\mathscr{R}_d^i(H^*(G,\mathbb{C}))$$9

together with connectedness statements and propagation inclusions such as

$X$0

(Budur, 2011).

Propagation also occurs in a broader homological setting. If a space $X$1 has the EPY property, then resonance varieties propagate: $X$2 For a complex hyperplane arrangement $X$3 in $X$4, the projective complement $X$5 has the EPY property, and hence both resonance and characteristic varieties propagate (Denham et al., 2015).

Poincaré duality imposes additional symmetry. For a PD$X$6 algebra $X$7,

$X$8

for all $X$9 and $A=H^*(X,k)$0. In the special case of PD$A=H^*(X,k)$1 algebras, the resonance varieties are controlled by the associated alternating $A=H^*(X,k)$2-form

$A=H^*(X,k)$3

and $A=H^*(X,k)$4 may be described as a degeneracy locus of a skew-symmetric matrix, equivalently by Pfaffians (Suciu, 2018). Over $A=H^*(X,k)$5, if $A=H^*(X,k)$6 is a PD-$A=H^*(X,k)$7 cdga, then the same duality $A=H^*(X,k)$8 holds, and for a closed orientable manifold $A=H^*(X,k)$9, the top-degree resonance $$\mathcal{R}_d^i(X,k)=\{a\in H^1(X,k)\mid \dim_k H^i(A,a)\ge d\},$$0 characterizes orientability (Suciu, 2022).

4. Scheme structure, irreducibility, and reducedness

Beyond their set-theoretic support, resonance varieties carry a scheme structure. Given a vector space $$\mathcal{R}_d^i(X,k)=\{a\in H^1(X,k)\mid \dim_k H^i(A,a)\ge d\},$$1 and a subspace $$\mathcal{R}_d^i(X,k)=\{a\in H^1(X,k)\mid \dim_k H^i(A,a)\ge d\},$$2, the Koszul module $$\mathcal{R}_d^i(X,k)=\{a\in H^1(X,k)\mid \dim_k H^i(A,a)\ge d\},$$3 defines the resonance scheme

$$\mathcal{R}_d^i(X,k)=\{a\in H^1(X,k)\mid \dim_k H^i(A,a)\ge d\},$$4

with projectivization

$$\mathcal{R}_d^i(X,k)=\{a\in H^1(X,k)\mid \dim_k H^i(A,a)\ge d\},$$5

(Aprodu et al., 2023). This viewpoint introduces conditions such as linear resonance, isotropicity, separability, and strong isotropicity. If all components of $$\mathcal{R}_d^i(X,k)=\{a\in H^1(X,k)\mid \dim_k H^i(A,a)\ge d\},$$6 are linear subspaces of $$\mathcal{R}_d^i(X,k)=\{a\in H^1(X,k)\mid \dim_k H^i(A,a)\ge d\},$$7, then separability implies that the projectivized resonance scheme is reduced and its components are disjoint; conversely, if $$\mathcal{R}_d^i(X,k)=\{a\in H^1(X,k)\mid \dim_k H^i(A,a)\ge d\},$$8 is reduced and isotropic, then it is separable. For an isotropic component $$\mathcal{R}_d^i(X,k)=\{a\in H^1(X,k)\mid \dim_k H^i(A,a)\ge d\},$$9, reducedness, generic reducedness, and strong isotropicity are equivalent (Aprodu et al., 2023).

The upper McCool groups provide a prominent non-reduced example. For $\mathbb{C}$00,

$\mathbb{C}$01

where each $\mathbb{C}$02 is a $\mathbb{C}$03-dimensional linear subspace of $\mathbb{C}$04 defined by explicit linear equations, and the components are mutually projectively disjoint. For $\mathbb{C}$05, however, the resonance scheme is not reduced: in addition to the isolated components $\mathbb{C}$06, there are embedded $\mathbb{C}$07-dimensional linear subspaces $\mathbb{C}$08, and $\mathbb{C}$09 is not weakly reduced (Suciu et al., 2018). This example also shows that projective disjointness does not by itself force reducedness.

Irreducibility questions have a parallel algebraic formulation over exterior algebras. For a stable monomial ideal $\mathbb{C}$10, the first resonance variety $\mathbb{C}$11 is irreducible. If $\mathbb{C}$12 is the Orlik-Solomon ideal of an essential central hyperplane arrangement, then $\mathbb{C}$13 is irreducible if and only if the subideal $\mathbb{C}$14 generated by all degree-$\mathbb{C}$15 elements has a $\mathbb{C}$16-linear resolution. For arrangements of rank $\mathbb{C}$17, componentwise linearity of $\mathbb{C}$18 is equivalent to irreducibility of $\mathbb{C}$19 (Thieu, 2011).

A common simplification is to identify resonance with its support only set-theoretically. The scheme-theoretic results show that multiplicities, embedded components, and reducedness questions are intrinsic to the subject rather than peripheral refinements. A plausible implication is that asymptotic formulas derived from Koszul modules depend not merely on the support, but on how the support sits scheme-theoretically.

5. Formality, Chen ranks, and other jump loci

Formality governs the relation between resonance and characteristic varieties. For a finitely generated, $\mathbb{C}$20-formal group $\mathbb{C}$21, the analytic germs at the origin of the relative resonance varieties and the relative characteristic varieties are analytically isomorphic. In particular, the tangent cone formula

$\mathbb{C}$22

holds, and in the rank-one case all irreducible components of $\mathbb{C}$23 are linear subspaces of $\mathbb{C}$24, defined over $\mathbb{C}$25 (Dimca et al., 2009).

This yields sharp obstructions to $\mathbb{C}$26-formality. For the pure virtual braid groups $\mathbb{C}$27 and $\mathbb{C}$28, the degree-$\mathbb{C}$29, depth-$\mathbb{C}$30 resonance varieties are not unions of linear subspaces when $\mathbb{C}$31; for instance, $\mathbb{C}$32 is an irreducible $\mathbb{C}$33-dimensional, degree $\mathbb{C}$34 subvariety in $\mathbb{C}$35. Consequently,

$\mathbb{C}$36

(Suciu et al., 2016). By contrast, the upper McCool groups $\mathbb{C}$37 are $\mathbb{C}$38-formal, so their first resonance variety is set-theoretically a union of rational linear subspaces, but for $\mathbb{C}$39 the non-reduced resonance scheme obstructs the Chen ranks conjecture (Suciu et al., 2018).

The Chen ranks are closely tied to resonance through Koszul modules. If $\mathbb{C}$40 is $\mathbb{C}$41-formal, then

$\mathbb{C}$42

and if $\mathbb{C}$43 is strongly isotropic, with disjoint linear components $\mathbb{C}$44, then for $\mathbb{C}$45,

$\mathbb{C}$46

This asymptotic formula is presented as a Chen rank formula under reducedness and strong isotropicity hypotheses (Aprodu et al., 2023).

Resonance also controls Dwyer-Fried invariants for straight spaces. If $\mathbb{C}$47 is $\mathbb{C}$48-straight, then

$\mathbb{C}$49

where

$\mathbb{C}$50

In general, translated components of characteristic varieties may force the corresponding inclusion to be strict, so resonance alone does not always determine $\mathbb{C}$51-invariants (Suciu, 2011).

6. Higher-rank, representation-theoretic, and algebro-geometric extensions

Non-abelian resonance replaces the degree-one parameter $\mathbb{C}$52 by a flat $\mathbb{C}$53-valued connection $\mathbb{C}$54. Product and coproduct formulas describe how the corresponding jump loci behave under tensor products and wedge sums of cdga’s; for instance, when $\mathbb{C}$55 and $\mathbb{C}$56 have zero differentials and $\mathbb{C}$57 or $\mathbb{C}$58, exact product formulas are available (Papadima et al., 2013). A complementary structural result states that if the rank-one resonance decomposes as a finite union of linear subspaces, then for $\mathbb{C}$59 or its Borel subalgebra $\mathbb{C}$60, the space of flat connections and the higher-rank resonance varieties decompose accordingly; along essentially rank-one flat connections $\mathbb{C}$61, membership in higher-rank resonance is determined by the rank-one resonance through the eigenvalues of $\mathbb{C}$62 (Macinic et al., 2013).

Representation theory supplies a vanishing criterion. For an irreducible $\mathbb{C}$63-module $\mathbb{C}$64 and a submodule $\mathbb{C}$65, one defines

$\mathbb{C}$66

The associated Koszul module satisfies

$\mathbb{C}$67

and the paper gives a roots-and-weights criterion in terms of the weights $\mathbb{C}$68 for simple roots $\mathbb{C}$69 (Papadima et al., 2012).

Algebro-geometric versions of resonance arise from vector bundles. Given a smooth projective curve $\mathbb{C}$70 and a vector bundle $\mathbb{C}$71, with $\mathbb{C}$72 and determinant map

$\mathbb{C}$73

one sets $\mathbb{C}$74 and defines $\mathbb{C}$75. Its points correspond to classes of global sections $\mathbb{C}$76 for which there exists another section $\mathbb{C}$77 with $\mathbb{C}$78, meaning that their span generates a sub-line bundle or subpencil (Spiridon, 25 Apr 2026). The flattening stratification

$\mathbb{C}$79

organizes the resonance by the degrees of saturated sub-line bundles; each $\mathbb{C}$80 is locally closed, only finitely many are nonempty, and

$\mathbb{C}$81

(Spiridon, 25 Apr 2026).

For rank-two bundles over an elliptic curve, the geometry can be completely explicit. If $\mathbb{C}$82 is non-split, then

$\mathbb{C}$83

For split bundles, the behavior depends on the summands: for example, if $\mathbb{C}$84 with $\mathbb{C}$85, then $\mathbb{C}$86, whereas if $\mathbb{C}$87, then $\mathbb{C}$88 is the Segre quadric $\mathbb{C}$89 (Spiridon, 25 Apr 2026).

A further algebro-geometric realization uses restricted universal quotient bundles over linear sections of Grassmannians. For $\mathbb{C}$90 a positive-dimensional transversal linear section and $\mathbb{C}$91, the resonance of $\mathbb{C}$92 coincides with the resonance variety constructed from the orthogonal linear section $\mathbb{C}$93. In the case $\mathbb{C}$94, the analysis shows that any resonance variety in $\mathbb{C}$95 consisting of fourteen disjoint lines is the resonance of some bundle which appeared in the work of Mukai (Aprodu et al., 10 Oct 2025).

Across these settings, resonance varieties retain a common role as jump loci for Aomoto-type complexes, yet their geometry ranges from unions of rational linear subspaces to irreducible nonlinear varieties, from reduced determinantal loci to non-reduced schemes with embedded components, and from matroidal combinatorics to Quot schemes and linear sections of Grassmannians. A plausible implication is that resonance is best viewed not as a single rigid invariant, but as a family of closely related constructions whose behavior is controlled by the ambient homological, representation-theoretic, and geometric framework.