Examples and counter‐examples of log‐symplectic manifolds
Examples and counter‐examples of log‐symplectic manifolds
复制标题
对数辛流形的例子和反例
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发表时间：
2013
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通讯作者：
G. Cavalcanti
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作者：
G. Cavalcanti
We study topological properties of log‐symplectic structures and produce examples of compact manifolds with such structures. Notably, we show that several symplectic manifolds do not admit bona fide log‐symplectic structures and several bona fide log‐symplectic manifolds do not admit symplectic structures; for example, #mCP2#nCP2¯ has bona fide log‐symplectic structures if and only if m,n>0 , while they only have symplectic structures for m=1 . We introduce surgeries that produce log‐symplectic manifolds out of symplectic manifolds and show that any compact oriented log‐symplectic 4‐manifold can be transformed into a collection of symplectic manifolds by reversing these surgeries. Finally, we show that if a compact manifold admits an achiral Lefschetz fibration with homologically essential fibres, then the manifold admits a log‐symplectic structure. Then, using results of Etnyre and Fuller (Int. Math. Res. Not. (2006), art. ID 70272), we conclude that if M is a compact, simply connected 4‐manifold then M#(S2×S2) and M#CP2#CP¯2 have log‐symplectic structures.
