 
Summary: Singular Lefschetz pencils
D. Auroux
, S. K. Donaldson, L. Katzarkov
October 12, 2004
Abstract
We consider structures analogous to symplectic Lefschetz pencils in
the context of a closed 4manifold equipped with a "nearsymplectic"
structure (i.e., a closed 2form which is symplectic outside a union
of circles where it vanishes transversely). Our main result asserts
that, up to blowups, every nearsymplectic 4manifold (X, ) can be
decomposed into (a) two symplectic Lefschetz fibrations over discs,
and (b) a fibre bundle over S1 which relates the boundaries of the
Lefschetz fibrations to each other via a sequence of fibrewise handle
additions taking place in a neighbourhood of the zero set of the 2
form. Conversely, from such a decomposition one can recover a near
symplectic structure.
Contents
1 Introduction 2
2 Approximately holomorphic theory 7
3 Definition of the almostcomplex structure 12
