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Singular Lefschetz pencils , S. K. Donaldson, L. Katzarkov

Summary: Singular Lefschetz pencils
D. Auroux
, S. K. Donaldson, L. Katzarkov
October 12, 2004
We consider structures analogous to symplectic Lefschetz pencils in
the context of a closed 4-manifold equipped with a "near-symplectic"
structure (i.e., a closed 2-form which is symplectic outside a union
of circles where it vanishes transversely). Our main result asserts
that, up to blowups, every near-symplectic 4-manifold (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.
1 Introduction 2
2 Approximately holomorphic theory 7
3 Definition of the almost-complex structure 12


Source: Auroux, Denis - Department of Mathematics, Massachusetts Institute of Technology (MIT)


Collections: Mathematics