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For essential reduced hyperplane arrangements of 4 variables, we show thatthe pole order spectral sequence degenerates almost at $E_2$, and completely at$E_3$, generalizing the 3 variable case where the complete $E_2$-degenerationis known. These degenerations are useful to determine the roots ofBernstein-Sato polynomials supported at the origin. For the proof we improve anestimate of the Castelnuovo-Mumford regularity of logarithmic vector fieldswhich was studied by H. Derksen and J. Sidman.

Degeneration of pole order spectral sequences for hyperplane arrangements of 4 variables