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On the isotropic subspace theorems

In this paper we explore the Isotropic Subspace Theorems of Catanese and Bauer, establishing relations between isotropic subspaces in the 1cohomology of a quasi-projective variety M and certain irrational pencils f : M ! C, from the point of view of the Tangent Cone Theorem due to Papadima, Suciu and the author. In the proper case the picture is completely clear, and is described in section 3. For the quasi-projective case and the associated logarithmic pencils, the results are satisfactory only under the additional technical restriction that M is 1-formal, see section 4. The example of the configuration space of n distinct labeled points on an elliptic curve, see Example 2.11, and that of the algebraic link of an isolated C ¤ -surface singularity, see subsection (4.10), illustrate well the difficulties in the general case.