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Testing the logarithmic comparison theorem for free divisors. (English) Zbl 1071.14024
Summary: We propose in this work a computational criterion to test if a free divisor \(D\subset\mathbb{C}^n\) verifies the logarithmic comparison theorem (LCT); that is, whether the complex of logarithmic differential forms computes the cohomology of the complement of \(D\) in \(\mathbb{C}^n\). For Spencer free divisors \(D\equiv(f=0)\), we solve a conjecture about the generators of the annihilating ideal of \(1/f\) and make a conjecture on the nature of Euler homogeneous free divisors which verify LCT. In addition, we provide examples of free divisors defined by weighted homogeneous polynomials that are not locally quasi-homogeneous.

MSC:
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14F40 de Rham cohomology and algebraic geometry
32C38 Sheaves of differential operators and their modules, \(D\)-modules
68W30 Symbolic computation and algebraic computation
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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