Abstract
Logarithmic foliations
Highlights
Recall that a logarithmic form on a complex manifold M is a meromorphic q-form η on M such that the pole divisors of η and dη are reduced
This statement were generalized by Deligne in the context of logarithmic forms as follows: Theorem 1.1. — Let η be a logarithmic q-form on a compact Kähler manifold M
Assume that the pole divisor (η)∞ of η is an hypersurface with normal crossing singularities
Summary
Recall that a logarithmic form on a complex manifold M is a meromorphic q-form η on M such that the pole divisors of η and dη are reduced. These results give necessary conditions for a codimension p foliation F to be a local or global product in terms of the codimension of the singular set of its intersection with a (p + 1)plane: if there is a (p + 1)-plane Σ such that cod(Sing(F|Σ)) 3 F = F ∗(F |Σ), where F : Pn− → Pp+1 is induced by a linear map of maximal rank f : Cn+1 → Cp+2 (Theorem 5.1). — Let F be the logarithmic foliation on Pn defined in homogeneous coordinates by an integrable p-form η on Cn+1 as below: η=. We would like to observe that his proof of the stability of logarithmic 2-forms is purely algebraic: he computes the Zariski tangent space at a generic point
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