Canonical Models of Foliations

Thirty years after its publication, it remains true that the only absolutely satisfactory theorem on families of curves on surfaces (so a fortiori on higher dimensional) varieties of general type is the conclusion of [B2] that on surfaces with many symmetric tensors curves of a given genus are bounded in moduli. A revealing re-interpretation of this result is provided by Gromov’s view of the isoperemetric inequality for (holomorphic) discs as not just a combinatorial consequence of, but as morally equivalent to, negative (holomorphic) sectional curvature. As such, we may re-phrase Bogomolov’s theorem as asserting that up to the exception of finitely many rational and elliptic curves the isoperemetric inequality holds for discs in the algebraic directions. Unfortunately, despite the fact that any holomorphic disc may be arbitrarily well approximated by those inside algebraic curves, the isoperemetric inequality that one obtains from Bogomolov’s theorem is far too dependent on the genus of the curves in question as to answer even a rather qualitative question such as whether such surfaces admit a Zariski dense