Terao's conjecture on logarithmic comparison for hyperplane arrangements

Let A\mathscr{A} be a central hyperplane arrangement, let Ω(logA)\Omega^\bullet(\log\mathscr{A}) be its logarithmic differential-form complex, and let OS(A)\operatorname{OS}(\mathscr{A}) be the Orlik–Solomon algebra generated by the forms dαi/αi{\mathrm d}\alpha_i/\alpha_i. The natural inclusion is

OS(A)Ω(logA).\operatorname{OS}(\mathscr{A})\to\Omega^\bullet(\log\mathscr{A}).

Terao's conjecture. The inclusion

OS(A)Ω(logA)\operatorname{OS}(\mathscr{A})\to\Omega^\bullet(\log\mathscr{A})

is a quasi-isomorphism.

Brieskorn proved the analogous quasi-isomorphism from the Orlik–Solomon algebra to the meromorphic de Rham complex, and the source identifies this logarithmic comparison as the prominent open problem in the area. It gives no resolution evidence for the stated claim.

Sources & referencesView supporting material

Primary source

Anton Leykin and Uli Walther, “Survey on the D-module f^s”, arXiv:1504.07516 (2015).

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