A symplectic isotopy of a Dehn twist on ℂℙn × ℂℙn+1
2009
The complex manifold CPn �~ CPn+1 with symplectic form �Ð�E = �ÐCPn + �E�ÐCPn+1, where �ÐCPn
and �ÐCPn+1 are normalized Fubini.Study forms, n �¸ N and �E > 1 a real number, contains
a natural Lagrangian sphere L�E. We prove that the Dehn twist along L�E is symplectically
isotopic to the identity for all �E > 1. This isotopy can be chosen so that it pointwise fixes a
complex hypersurface in CPn �~ CPn+1 and lifts to the blow-up of CPn �~ CPn+1 along a complex
n-dimensional submanifold.
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