On the logarithmic comparison theorem for integrable logarithmic connections

On the logarithmic comparison theorem for integrable logarithmic connections

LetX be a complex analytic manifold, D ⊂ X a Koszul free divisor with jacobian ideal of linear type (e.g. a locally quasi-homogeneous free divisor), j : U = X −D →֒ X the corresponding open inclusion, E an integrable logarithmic connection with respect to D and L the local system of the horizontal sections of E on U . In this paper we prove that the canonical morphisms ΩX(logD)(E(kD)) −→ Rj∗L, j...

On the Logarithmic Connections over Curves

We study two different actions on the moduli spaces of logarithmic connections over smooth complex projective curves. Firstly, we establish a dictionary between logarithmic orbifold connections and parabolic logarithmic connections over the quotient curve. Secondly, we prove that fixed points on the moduli space of connections under the action of finite order line bundles are exactly the push-f...

Testing the Logarithmic Comparison Theorem for Free Divisors

Moduli of Semistable Logarithmic Connections

Under the Riemann Hilbert correspondence, which is an equivalence of categories, local systems on a nonsingular complex projective variety X correspond to pairs E = (g' , V) where g' is a locally free sheaf on X , and V: g' -t n ~ 0g' is an integrable connection on g'. Generalizing this correspondence to noncomplete quasi-projective varieties, Oeligne (see [0]) showed that the correct algebro-g...

Logarithmic Comparison Theorem and Euler Homogeneity for Free Divisors

We prove that if the Logarithmic Comparison Theorem holds for a free divisor in a complex manifold then this divisor is Euler homogeneous. F.J. Calderón–Moreno et al. have conjectured this statement and have proved it for reduced plane curves.