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Lagrangian Floer theory on compact toric manifolds. II: Bulk deformations. (English) Zbl 1234.53023
This is the second part of the authors’ series on Lagrangian Floer theory on toric manifolds (for Part I, see [the authors, Duke Math. J. 151, No. 1, 23–175 (2010; Zbl 1190.53078)]). Subsequent papers are [“Lagrangian Floer theory and mirror symmetry on compact toric manifolds”, arXiv:1009.1648, “Spectral invariants with bulk, quasimorphisms and Lagrangian Floer theory ”, arXiv:1105.5123]. The present paper mainly studies and applies bulk deformation of Lagrangian Floer theory (see [the authors, Lagrangian intersection Floer theory. Anomaly and obstruction. Part II. AMS/IP Studies in Advanced Mathematics 46,2. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press, 397–805 (2009; Zbl 1181.53003)]) and proves the following:
Theorem 1.1. Let $$X_k$$ be the $$k$$-point blow up of $$\mathbb{C} P^2$$ with $$k\geq 2$$. Then there exists a toric Kähler structure on $$X_k$$ such that there exists a continuum of non-displaceable Lagrangian fibers $$L(u)$$. Moreover, if $$\psi:X\to X$$ is a Hamiltonian isotopy such that $$\psi(L_u)$$ is transversal to $$L(u)$$, then $\sharp(\psi(L(u))\cap L(u))\geq 4.$ Here, a Lagrangian submanifold $$L$$ is called non-displacable if $$\psi(L)\cap L\neq\emptyset$$ for any Hamiltonian diffeomorphism $$\psi: X\to X$$.
As for general toric maniolds, the following theorem is proved:
Theorem 1.3. Let $$X$$ be a compact toric manifold and $$L(u)$$ its Lagrangian fiber such that the leading term equation of $$L(u)$$ has a solution in $$(\mathbb{C}\setminus\{0\})^n$$. Then there exist $${\mathfrak b}\in H^2(X;\Lambda_+)$$ and $${\mathfrak x}\in H(L(u);\Lambda_0)$$ satisfying $HF((L(u),({\mathfrak b},{\mathfrak x})), (L(u),({\mathfrak b},{\mathfrak x})); \Lambda_0)\cong H(T^n; \Lambda_0).$ Here, $$\Lambda_0$$ and $$\Lambda_+$$ are the universal Novikov ring and its maximal ideal, respectively. The leading term equation for Lagrangian fibers $$L(u)$$ of a toric manifold was introduced in Part I.
As a corollary (Corollary 1.4), if $$L(u)$$ is non-displaceable and if $$\psi: X\to X$$ is a Hamiltonian isotopy such that $$\psi(L(u))$$ is transversal to $$L(u)$$, it is proved that $\sharp(\psi(L))\cap L(u))\geq 2^n,$ where $$n= \dim L(u)$$.
The bulk deformation is derived from a family of operators ${\mathfrak q}_{\beta,\ell,k}:E_\ell(H[2])\otimes B_k(H^*(L;R)[1])\to H^*(L; R)[1].$ (Explanations of the notations used in this formula and its properties are given in §2.)
Then the potential function $${\mathfrak P}{\mathfrak O}$$ is a map from the set $$\widehat{\mathcal M}_{\text{weak,def}}(L; \Lambda^+_{0,\text{nov}})$$ of pairs $$({\mathfrak b},b)$$ to $$\Lambda_+$$, where $${\mathfrak b}$$ is a degree-2 element of $$H\otimes\Lambda^+_{0,\text{nov}}$$ and $$b$$ a weak bounding cochain of $$(H(L; \Lambda_{0,\text{nov}}), \{{\mathfrak m}^{\mathfrak b}_k\})$$. It is used to define the Floer cohomology $$HF((L,{\mathfrak b}_1),(L,{\mathfrak b}_0); \Lambda_{0,\text{nov}})$$.
§3 specializes these constructions to toric fibers. In this case, denote the free abelian group generated by toric divisors of $$X$$ by $${\mathcal A}$$. There is a canonical inclusion ${\mathcal A}(\Lambda_+)\times H^1(L(u); \Lambda_+)\to \widehat{\mathcal M}_{\text{weak,def}}(L(u)),$ where $${\mathcal A}(\Lambda_+)={\mathcal A}\times_{\mathbb{Z}}\Lambda_+$$. By using this fact, the potential function with bulk ${\mathfrak P}{\mathfrak O}^u_{\mathfrak b}={\mathfrak P}{\mathfrak O}^u({\mathfrak b},\cdot): H^1(L(u); \Lambda_+)\to \Lambda_+$ is introduced. $${\mathfrak P}{\mathfrak O}^u$$ allows for a Fourier expansion in $$y_i= e^{x_i}$$, where $$x_i$$ are the coordinates of $$H^1(L(u);\Lambda_+)$$. By using a non-Archimedean topology of $$\Lambda$$, the notion of strict convergence of such a Fourier series is introduced (cf. [S. Bosch, U. Güntzer and R. Remmert, Non-Archimedean analysis. A systematic approach to rigid analytic geometry. Grundlehren der Mathematischen Wissenschaften, 261. Berlin etc.: Springer Verlag (1984; Zbl 0539.14017)]). Convergence of the expansion of $${\mathfrak P}{\mathfrak O}^u$$ is shown.
The authors note that the $$\mathbb{C}$$-reduction in $${\mathfrak P}{\mathfrak O}$$ corresponds to the precise form of the physical Landau-Ginzburg potential function associated to toric manifolds (cf. [K. Hori et al., Mirror symmetry. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1044.14018)]). Then, introducing the notion of bulk balanced Lagrangian fiber, the statements of Corollary 1.4 (mentioned above) are derived for bulk balanced Lagrangian fibers (Proposition 3.19). The authors ask if the converse of Proposition 3.19 holds.
Leading terms of the expansion of $${\mathfrak P}{\mathfrak O}^u$$ are studied in §4. They are eliminated by solving the leading term equations. When $$X$$ is the 2-point blow up of $$\mathbb{C} P^2$$, the leading term equation takes the form $1- y^{-2}_2= 0,\quad 1+ y_2= 0.$ Starting from this fact, Theorem 1.1 is proved in §5.
The rest of the paper (§§6–12) deals with technical details of statements of §§2–4.
§6 and §7 study the moduli space of holomorphic disks and its effects on the operator $${\mathfrak q}$$ and on the potential function $${\mathfrak P}{\mathfrak O}$$. The authors state that the results of §6 are basically due to C.-H. Cho and Y.-G. Oh [Asian J. Math. 10, No. 4, 773–814 (2006; Zbl 1130.53055)]. §8 discusses applications of Floer cohomology and the potential function to the study of the non-displacement property of Lagrangian submanifolds. In this way, bulk deformation in the study of non-displacement of Lagrangian submanifolds was already described in [Zbl 1181.53003] by using singular cohomology. In this section, an alternative method using de Rham cohomology is presented. The techniques of using a continuous family of multisections and integrating along the fiber on their zero sets so that the smooth correspondence by spaces with Kuranishi structure induces a map between de Rham complexes are reviewed in §12 (cf. [Y. Ruan, Turk. J. Math. 23, No. 1, 161–231 (1999; Zbl 0967.53055)]). §9 proves convergence of the potential function using an explicit description of $${\mathfrak P}{\mathfrak O}^u(w_1,\dots, w_B;{\mathfrak x})$$. This formula is also used to derive the equality ${\mathfrak E}({\mathfrak P}{\mathfrak O}^u)={\mathfrak P}{\mathfrak O}^u,$ where $${\mathfrak E}$$ is the Euler vector field defined by ${\mathfrak E}= \sum^B_{i=m+1} \Biggl(1-{d_i\over 2}\Biggr)\,w_i{\partial\over\partial w_i}+ \sum^m_{i=1} {\partial\over\partial{\mathfrak w}_i}.$ In §4 and §8, bulk deformation of Lagrangian Floer cohomology by divisor cycles $${\mathfrak b}\in{\mathcal A}(\Lambda_+)$$ was used; in §11, the case of $${\mathfrak b}\in{\mathcal A}(\Lambda_0)$$ is studied and it is shown that Theorem 3.16 and Proposition 3.19 hold for $${\mathfrak b}\in{\mathcal A}(\Lambda_0)$$.

##### MSC:
 53D12 Lagrangian submanifolds; Maslov index 53D40 Symplectic aspects of Floer homology and cohomology 53D35 Global theory of symplectic and contact manifolds 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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