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Lagrangian embeddings into subcritical Stein manifolds. (English) Zbl 1165.53378

For a pair \((W, L)\) with \(W\) geometrically bounded symplectic manifold and \(L\) a compact Lagrangian embedded submanifold allowing disjunction by a Hamiltonian isotopy, the reviewer [Int. Math. Res. Not. 1996, No. 7, 305–346 (1996; Zbl 0858.58017)] and Yu. V. Chekanov [Duke Math. J. 95, No. 1, 213–226 (1998; Zbl 0977.53077)] studied some symplectic topological obstructions of the Lagrangian embedding \(L\) using the Floer homology theory. In the paper [Math. Res. Lett. 4, No. 6, 895–905 (1997; Zbl 0899.58020)], the reviewer simplified their arguments by studying the moduli space of a two-parameter family of Floer’s perturbed Cauchy-Riemann equations obtained by connecting zero, a Hamiltonian \(H\) whose time one map realizes the disjunction, and then back to zero. One interesting particular class of such pairs that the authors studied is the ones either when \(W=V\) is subcritical Stein, or when \(W = V \times X\) of subcritical Stein \(V\) and compact symplectic \(X\) which they call subcritical split: a Stein manifold/domain \((V, J, \phi)\) is called subcritical [Y. Eliashberg, Int. J. Math. 1, No. 1, 29–46 (1990; Zbl 0699.58002)] if all the critical points of the plurisubharmonic Morse function \(\varphi\) have Morse indices less than \(\frac{1}{2} \dim V\). The authors extract some topological restrictions on compact Lagrangian embeddings into subcritical Stein manifolds by translating the three main theorems in the paper which are summarized below.
The most notable result from the paper under review, among other things, is Theorem 1.1, the cohomological sphericality of the Lagrangian embeddings \(L\) with \(H_1(L;\mathbb Z) = 0\) into subcritical split manifold \(W = V \times \mathbb C P^n\) with \(c_1(V)| _{\pi_2(V)} = 0\) with \(2 \leq \dim V \leq 2(n+1)\). This is an immediate corollary, which is an interesting observation of the authors, of two results that the reviewer previously proved: the first is that under the assumption that the minimal Maslov number \(\Sigma_{(W,L)}\) satisfying \(\Sigma_{(W,L)} \geq 2\), the Floer cohomology \(HF^*(L;\mathbb Z_2)\) of such a Lagrangian embedding is defined and invariant under the Hamiltonian isotopy [the reviewer, Commun. Pure Appl. Math. 46, No. 7, 949–994 (1993; Zbl 0795.58019) and ibid. 48, No. 11, 1299–1302 (1995; Zbl 0847.58036)], and the second is that under the assumption that \(\Sigma_{(W,L)}\geq \dim L+1\), \(HF^k(L,\mathbb Z_2) \cong H^k(L,\mathbb Z_2)\) for \(2 \leq k \leq \dim L - 1 \,(\text{mod} \,\, \Sigma_{(W,L)})\) [Theorem II in: loc. cit., Zbl 0858.58017]. It should be noted that the pair \((W,L)\) that the authors looked at in Theorem 1.1 is monotone with the minimal Maslov number \(\Sigma_{(W,L)}=2(n+1)\) and \(2 \leq \dim L \leq 2n+1\) (and hence \(\Sigma_{(W,L)} \geq \dim L+1\)) and allows a disjunction of \(L\) by a compactly supported Hamiltonian isotopy, this theorem immediately follows by combining the above mentioned two results of the reviewer. For such a disjunction obviously implies vanishing of the Floer cohomology \(HF^*(L, \mathbb Z_2)\) by definition, once it is defined and invariant under the Hamiltonian isotopy. This point is somewhat obscure in authors’ presentation of the paper in review and would have resulted in much simpler and clearer presentation of Theorem 1.1 and its proof under the presence of the above mentioned results of the reviewer which have been relatively well-known among the experts in the area.
Theorem 2.1 in the paper says that for any such pair \((W,L)\) that the authors look at, there must exist either a (pseudo)-holomorphic sphere \(S \subset W\) with \(2c_1(S) \leq \dim L +1\) or a disc \((D,\partial D) \subset (W,L)\) with its Maslov index \(\mu(D) \leq \dim L + 1\). This result itself indeed holds whenever the above mentioned disjunction is possible, not just in the context of the theorem, for a subcritical split manifold. Its proof is based on the standard dimension counting argument that is similar to the one used by L. Polterovich [Math. Z. 207, No. 2, 217–222 (1991; Zbl 0703.58018)] in the setting of the above mentioned moduli space [loc. cit., Zbl 0899.58020].
Theorem 2.2 is a slight refinement of the inequality, by (1) in Theorem 2.1, for monotone Lagrangian embeddings which is a re-statement, in the case of subcritical split manifolds, of one of reviewer’s results [Theorem II in: loc. cit., Zbl 0858.58017] for the monotone Lagrangian embeddings with minimal Maslov number greater than or equal to 2 and of \(\dim L \geq 2\).

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
32Q28 Stein manifolds
53D12 Lagrangian submanifolds; Maslov index
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
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