Flat connections and resonance varieties: From rank one to higher ranks

Proc. London Math. Soc.

2017

Self Cite

A central question in arrangement theory is to determine whether the characteristic polynomial Δq of the algebraic monodromy acting on the homology group Hqfalse(F(A),double-struckCfalse) of the Milnor fiber of a complex hyperplane arrangement scriptA is determined by the intersection lattice L(A). Under simple combinatorial conditions, we show that the multiplicities of the factors of Δ1 corresponding to certain eigenvalues of order a power of a prime p are equal to the Aomoto–Betti numbers βpfalse(scriptAfalse), which in turn are extracted from L(A). When scriptA defines an arrangement of projective lines with only double and triple points, this leads to a combinatorial formula for the algebraic monodromy. To obtain these results, we relate nets on the underlying matroid of scriptA to resonance varieties in positive characteristic. Using modular invariants of nets, we find a new realizability obstruction (over double-struckC) for matroids, and estimate the number of essential components in the first complex resonance variety of scriptA. Our approach also reveals a rather unexpected connection of modular resonance with the geometry of prefixSL2false(double-struckCfalse)‐representation varieties, which are governed by the Maurer–Cartan equation.

“…In Proposition , we associate to every special cocycle

τ \in Z_{double-struckk}^{'} false(scriptA false) ∖ B_{double-struckk} false(scriptA false)

an embedding

{prefixev}_{τ} : H^{double-struckk} false(frakturg false) ↪ F false(A (A) \otimes double-struckC, frakturg false)

, which preserves the regular parts. Building on recent work from , we then exploit this construction in two ways, for

g = {fraktursl}_{2} false(double-struckC false)

. On one hand, as noted in Remark , the construction gives the inverse of the map

λ_{k}

from Theorem (i).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…As before, let

frakturg

be a finite‐dimensional Lie algebra. As noted in , the canonical isomorphism

ι : A^{1} \otimes g \overset{≃}{\to} {prefixHom}_{double-struckC} false(A_{1}, frakturg false)

restricts to a functorial isomorphism

ι : F false(A, frakturg false) \overset{≃}{\to} {prefixHom}_{prefixLie} false(frakturh (A), frakturg false) .

Under this isomorphism, the subset

F^{(1)} false(A, frakturg false)

corresponds to the set

{prefixHom}_{prefixLie}^{1} false(frakturh (A), frakturg false)

of Lie algebra morphisms whose image is at most one‐dimensional.…”

Section: Flat Connections and Holonomy Lie Algebrasmentioning

confidence: 94%

g = {fraktursl}_{2}

is crucial.…”

Section: Evaluation Maps and Multinetsmentioning

confidence: 99%

“…Noteworthy is the case when

g = {fraktursl}_{2}

, a case studied in a more general context in . In this setting, an

{sl}_{2}

‐valued 1‐form

ω = {(ω_{u})}_{u \in M}

is a solution to the system of equations if and only if, for each

X \in L_{2} false(scriptM false)

either \sum_{v \in X} ω_{v} = 0, or dim span {ω_{v}}_{v \in X} ⩽ 1 .

…”

Section: Flat Connections and Holonomy Lie Algebrasmentioning

confidence: 99%

“…Here, the assumption that

g = {fraktursl}_{2}

is crucial. In the proof of [, Proposition 5.3], it is also shown that

P \otimes g \subseteq F (A, g)

, when

P

is isotropic. Claim (4) follows then from .

□

…”

Section: Evaluation Maps and Multinetsmentioning

confidence: 99%

See 3 more Smart Citations

The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy

Papadima

Proc. London Math. Soc.

2017

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Non-Abelian Resonance: Product and Coproduct Formulas

Papadima¹,

Bridging Algebra, Geometry, and Topology

2014

Self Cite

Around the Tangent Cone Theorem

2016

Springer INdAM Series

Self Cite

A cornerstone of the theory of cohomology jump loci is the Tangent Cone theorem, which relates the behavior around the origin of the characteristic and resonance varieties of a space. We revisit this theorem, in both the algebraic setting provided by cdga models, and in the topological setting provided by fundamental groups and cohomology rings. The general theory is illustrated with several classes of examples from geometry and topology: smooth quasi-projective varieties, complex hyperplane arrangements and their Milnor fibers, configuration spaces, and elliptic arrangements.