Logarithmic cohomology of the complement of a plane curve.

*(English)*Zbl 1010.32016The main object of study in this paper is the de Rham complex of holomorphic differential forms with logarithmic poles along a divisor in a complex manifold. It had been showed earlier by the last three authors that if the divisor is a locally quasi-homogeneous free divisor then the above mentioned complex calculates the cohomology of the complement of the divisor (“the logarithmic comparison theorem holds”).

In the present paper the authors study a partial converse to this statement, i.e. they study plane curve germs (which are always free divisors) and find that the mentioned complex calculates the cohomology of the complement only if the curve germ is quasi-homogeneous. The main argument of the proof shows that if the condition holds then there is a vector field germ \(\chi\) such that \(\chi\cdot h=h\), where \(h\) is the local equation of the curve. Then they can use Saito’s result proving that \(h\) is quasi-homogeneous in certain coordinates. In higher dimensions Saito’s result can not be applied, so the authors only conjecture the existence of the the vector field \(\chi\).

In the present paper the authors study a partial converse to this statement, i.e. they study plane curve germs (which are always free divisors) and find that the mentioned complex calculates the cohomology of the complement only if the curve germ is quasi-homogeneous. The main argument of the proof shows that if the condition holds then there is a vector field germ \(\chi\) such that \(\chi\cdot h=h\), where \(h\) is the local equation of the curve. Then they can use Saito’s result proving that \(h\) is quasi-homogeneous in certain coordinates. In higher dimensions Saito’s result can not be applied, so the authors only conjecture the existence of the the vector field \(\chi\).

Reviewer: Richárd Rimányi (Ohio)