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Graded Lagrangian submanifolds. (English) Zbl 0992.53059
Floer theory assigns, in favourable circumstances, an abelian group \(HF(L_0,L_1)\) to a pair \((L_0,L_1)\) of Lagrangian submanifolds of a symplectic manifold \((M,\omega)\). Following A. Floer [A relative Morse index for symplectic action, Commun. Pure Appl. Math. 41, 393-407 (1988; Zbl 0693.58009)], one can equip \(HF(L_0,L_1)\) with a canonical \(\mathbb{Z}/N\)-grading, where \(1 \leq N \leq \infty\) is a number which depends on \((M,\omega)\), \(L_0\) and \(L_1\).
In the paper the author takes another approach and he considers Lagrangian submanifolds with certain extra structure and calls them graded Lagrangian submanifolds. This extra structure removes the ambiguity and defines an absolute \(\mathbb{Z}/N\)-grading on Floer cohomology.
He applies the theory to Lagrangian submanifolds of \(\mathbb{C}\mathbb{P}^n\), weighted homogeneous singularities and symplectically knotted Lagrangian spheres, which application seems to be most important from the point of view of the “graded” theory as, according to the author, the “graded” framework allows to state a basic geometric property of generalized Dehn twists which seems to be hard to encode in any other way.
In the paper the author improves on the results in [Lagrangian two-spheres can be symplectically knotted, J. Differ. Geom. 52, 145-171 (1999; Zbl 1032.53068)], there are examples of compact symplectic 4-manifolds \(M\) (with boundary) with the following property: there is a family of embedded Lagrangian 2-spheres \(L^{(k)} \subset M\), \(k \in \mathbb{Z}\) such that any two of them are isotopic as smooth submanifolds but no two are isotopic as Lagrangian submanifolds – we say that \(M\) contains infinitely many symplectically knotted Lagrangian 2-spheres. The construction and the proof works for all even numbers. In this paper the proof is much simplified and new similar examples of Lagrangian \(n\)-spheres are produced for all odd numbers greater or equal to 5.

MSC:
53D12 Lagrangian submanifolds; Maslov index
53D40 Symplectic aspects of Floer homology and cohomology
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
57R40 Embeddings in differential topology
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