Terao and Yuzvinsky's algebraic logarithmic comparison conjecture for hyperplane arrangements

Let f=f1fdR=C[x1,,xn]f=f_1\cdots f_d\in R=\mathbb{C}[x_1,\dots,x_n] define a reduced hyperplane arrangement A\mathscr{A} in X=CnX=\mathbb{C}^n, and let U=XAU=X\setminus\mathscr{A}. Write ΩR(logA)\Omega_R^{\bullet}(\log\mathscr{A}) for the algebraic logarithmic de Rham complex and ΩR(A)\Omega_R^{\bullet}(\star\mathscr{A}) for the algebraic rational de Rham complex, allowing arbitrary-order poles along A\mathscr{A}. Terao and Yuzvinsky's conjecture. The natural inclusion is a quasi-isomorphism:

ΩR(logA,d)q.i.ΩR(A,d)H(U,CU).\Omega_R^{\bullet}(\log\mathscr{A},d)\xhookrightarrow{\operatorname{q.i.}}\Omega_R^{\bullet}(\star\mathscr{A},d)\simeq H^{\bullet}(U,\mathbb{C}_U).

This conjecture asserts the algebraic logarithmic comparison theorem for every reduced hyperplane arrangement. It was posed by Terao and Yuzvinsky and motivates the paper's study of logarithmic comparison theorems; the supplied text does not establish whether it has been resolved.

Sources & referencesView supporting material

Primary source

Daniel Bath, “Hyperplane Arrangements Satisfy (un)Twisted Logarithmic Comparison Theorems, Applications to D_X-modules”, arXiv:2202.01462 (2024).

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