By Michael Spivak
Booklet through Michael Spivak, Spivak, Michael
Read Online or Download A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd Edition PDF
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Extra info for A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd Edition
As a piece of notation, a positive symplectic divisor is a union of symplectic surfaces D = ai Σi with ai > 0, with the Σi pairwise having only isolated positive transverse intersections, and no triple intersections. 3 If a symplectic divisor satisﬁes D · Σj ≥ 0 for all j, then it can be symplectically smoothed. Proof (Sketch). Suppose ﬁrst that all the Σi are complex curves in a K¨ ahler surface. The hypothesis says that the line bundle O(D) has non-negative degree on each component. Then we can ﬁnd a smooth section γ of O(D) which is holomorphic near each intersection point and near its (transverse) zero set.
Note that a Lefschetz pencil is a family of symplectic surfaces of arbitrarily large volume and complexity (depending on k), so somewhat diﬀerent techniques are needed to ﬁnd symplectic surfaces in a prescribed homology class. 1 Symmetric Products Let (X, ωX ) be a symplectic four-manifold with integral [ωX ], and ﬁx a ˆ → S 2 on the blowLefschetz pencil, giving rise to a Lefschetz ﬁbration f : X ˆ with the symplectic form ωC = p∗ ωX + C f ∗ ωS 2 up. We will always equip X for some large C > 0.
Generic hyperplane sections of this projective embedding are smooth hypersurfaces in M , and a pencil of hyperplanes through a generic codimension 2 linear subspace deﬁnes a Lefschetz pencil. When the manifold M is only symplectic, the lack of integrability of J prevents the existence of holomorphic sections. Nonetheless, it is possible to ﬁnd an approximately holomorphic local model: a neighborhood of a point 18 D. Auroux and I. Smith x ∈ M , equipped with the symplectic form ω and the almost-complex structure J, can be identiﬁed with a neighborhood of the origin in Cn equipped with the standard symplectic form ω0 and an almost-complex structure of the form i + O(|z|).