Notes on Logarithmic Vector Fields, Logarithmic Differential Forms and Free Divisors David Mond November 12, 2012 1 Introduction Let D = {h =0}⊂ C n , where h is a polynomial or a germ of holomorphic function. Then Der(− log D) := {χ ∈ Der C n : χ is tangent to D reg } (1.1) = {χ ∈ Der C n : χ · h ∈ (h)} (1.2) Ω p (log D) := {ω ∈ Ω p C n (∗D): hω and hdω are both regular } (1.3) := {ω ∈ Ω p C n (∗D): hω and dh ∧ ω are both regular } (1.4) These definitions really refer to sheaves of germs of vector fields and differential forms, but in these notes I will not be careful to indicate this. D is a free divisor if Der(− log D) is a locally free O C n -module. Outside D, Der(− log D) coincides with Der C n , and therefore has rank n. So if D is a free divisor, it follows that at each point there is a (local) basis for Der(− log D) consisting of exactly n logarithmic vector fields. The definitions and theorems in this section and the next three are due to Kyoji Saito, [9]. For background on commutative algebra I recommend the book of Matsumura, [7]. 2 Singular free divisors are singular in codimension 1 Obviously, smooth hypersurfaces are free divisors. But suppose that D is a free divisor and that x 0 is a singular point of D. From the exact sequence 0  Der(− log D) i  θ C n dh  J h O D  0, (2.1) where i is inclusion, together with the Depth Lemma 1 we deduce that D is free ⇔ depth J h O D = n − 1, (2.2) where J h is the Jacobian ideal of h. The Depth Lemma, applied now to the short exact sequence 0 → J h O D →O D →O D /J h O D → 0, 1 If 0 → A → B → C → 0 is an exact sequence of R-modules then depth B ≥ min{depth A, depth B}, and if the inequality is strict then depth A = depth B +1. 1