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Lagrangian topology and enumerative geometry. (English) Zbl 1253.53079

Let \(L^n\) be a closed connected Lagrangian submanifold of a symplectic manifold \((M^{2n},\omega)\), \(J\) an almost complex structure compatible with \(\omega\), and let \(P\), \(Q\), \(R\) be three distinct points of \(L\). Then, the number \(n_{PQR}= n_{PQR}(L,J)\) of disks \(u: (D^2,\partial D^2)\to(M, L)\) that are \(J\)-holomorphic and that go through \(P\), \(Q\), \(R\), in this order, is asked for.
In this paper, as an answer to this question, if the Floer homology is \(HF(L,L)\neq 0\), the expression \[ \Delta= 4n_{PQR}+ n^2_P+ n^2_Q+ n^2_R-2n_P n_Q- 2n_Q n_R- 2nQn_R- 2n_R n_P, \] where \(n_P\) is the number of \(J\)-holomorphic disks of Maslov index 2 that go through \(P\) and cross transversally the edge \(\vec{QR}\), is shown to be independent of the triangle \(P\), \(Q\), \(R\) as well as of \(J\). This is proved in order to be able to show that \(\Delta\) coincides with the discriminant of a certain quadratic form that can be read off from the quantum homology product of \(L\) (Theorem 6.2.2). In fact, as a consequence of the classification of polynomial invariants associated to quadratic forms of Hilbert, \(\Delta\) is the unique symmetric polynomial, enumerative invariant that can be extracted from the quantum product (Remark 5.2.5).
Main properties of Lagrangian quantum homology \(QH(L)\) are reviewed in §2 following the authors’ paper [Geom. Topol. 13, No. 5, 2881–2989 (2009; Zbl 1180.53078)]. The product in \(QH(L)\) is denoted by \(*\). Delicate discussions on orientation in quantum homology are given in the appendix. §3 considers the representation variety \({\mathcal R}ep(L)= \{\rho:\pi_2(M, L)\to\mathbb{C}^*\}\). \(L\) is said to be \({\mathcal R}\)-wide if \(QH(L;{\mathcal R})= H(L;{\mathcal R})\). The wide varieties \({\mathcal W}_i\), \(i= 1,2\), associated to \(L\) are defined by \[ \begin{aligned}{\mathcal W}_2 &= \{\rho\in\operatorname{Hom}_0(H^D_2,\mathbb{C}^*)\mid L\text{ is }\Lambda^p\text{-wide}\},\\ {\mathcal W}_1 &= \{\rho'\in\hom(H_1, \mathbb{C}^*)\mid \rho'\circ\partial\in{\mathcal W}_2\}.\end{aligned} \] They are algebraic varieties and \(QH(L;{\mathcal O}({\mathcal W}_i)\otimes\Lambda^+)\cong H(L;{\mathcal O}({\mathcal W}_i)\otimes \Lambda^+)\), \(i= 1,2\) (§3.2). \(QH(L;{\mathcal O}({\mathcal W}_i)\otimes \Lambda^+)\), etc., are denoted by \(Q^+H(L;{\mathcal W}_i)\), etc. Then, the formula \[ C_i* C_j+ C_j* C_i= (-1)^n z_i z_j{\partial^2{\mathcal P}\over\partial z_i\partial z_j}\,Lt, \] where \([L]\in H_n(L;\mathbb{C})\subset QH_n(L;{\mathcal W}_1)\) is the unity, is derived (Proposition 3.3.4). Here, \({\mathcal P}: \operatorname{Hom}_0(H_1;\mathbb{C}^*)\to\mathbb{C}\) is the (Landau-Ginzburg) superpotential. The authors remark that this superpotential and the superpotential \({\mathcal P}'\) in [K. Fukaya et al., Duke Math. J. 151, No. 1, 23–175 (2010; Zbl 1190.53078)] are slightly different, and, as for \({\mathcal P}'\), \(C_i* C_j+ C_j* C_i= (-1)^n {\partial^2{\mathcal P}'\over\partial x_i\partial x_j}\) holds.
Quadratic forms associated to the quantum product with coefficients in \({\mathcal O}({\mathcal W})\) and their associated discriminants are defined in §4. In §5, the quantum product is investigated as the deformation of the intersection product. Then, invariant polynomials in the structural constants of the quantum product are discussed and the characterization of \(\Delta\) is remarked. After these preparations, the discriminant of \(\rho\in\operatorname{Hom}_0(H^D_2,\mathbb{C}^*)\) is computed and \(\Delta\) is shown to be independent of \(P\), \(Q\), \(R\) and \(J\). It is also shown that, if \(L\) is a Lagrangian torus and the trivial representation \(\rho=1\) belongs to \({\mathcal W}_2\), and \(N_L= 2\), then \(\Delta\equiv 0\pmod 2\) and \(\Delta\) admits only the value 0 or 1 modulo 4 (Theorem 6.3.1).
In §7, these results and definitions are applied to toric fibers. Then, the authors give an explanation of the translation of the structure of a Frobenius algebra of \(QH(M,\Lambda)\) to the Jacobian ring \({\mathcal O}({\mathcal W}_1)\otimes\Lambda\) established in [K. Fukaya et al., Sel. Math., New Ser. 17, No. 3, 609–711 (2011; Zbl 1234.53023)], using the arguments of this paper. As examples, the cases \(M=\mathbb{C} P^n\), \(S^2\times S^2\) and blow-ups of \(\mathbb{C} P^2\) are studied in §8, the last section.

MSC:

53D12 Lagrangian submanifolds; Maslov index
53D40 Symplectic aspects of Floer homology and cohomology
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References:

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