Lagrangian Floer theory on compact toric manifolds. II: Bulk deformations.

*(English)*Zbl 1234.53023This 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 \({\mathfrak b}\in{\mathcal A}(\Lambda_0)\).

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 \({\mathfrak b}\in{\mathcal A}(\Lambda_0)\).

Reviewer: Akira Asada (Takarazuka)

##### 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 |

##### Keywords:

toric manifolds; Floer cohomology; weakly unobstructed Lagrangian submanifolds; potential function; Jacobian ring; bulk deformations; bulk-balanced Lagrangian submanifolds; open-closed Gromov-Witten invariant##### References:

[1] | Berkovich, B.G.: Spectral theory and analytic geometry over non-archimedean fields. Mathematical Surveys and Monographs 33. American Mathematical Society, Providence, RI (1990) · Zbl 0715.14013 |

[2] | Bosch, S., Güntzer, U., Remmert, R.: Non-Archimedean Analysis. A Systematic Approach to Rigid Analytic Geometry, Grundlehren der Mathematischen Wissenschaften 261. Springer-Verlag, Berline (1984) · Zbl 0539.14017 |

[3] | Cho C.-H.: Non-displaceable Lagrangian submanifolds and Floer cohomology with non-unitary line bundle. J. Geomet. Phys. 58, 1465–1476 (2008) arXiv:0710.5454 · Zbl 1161.53076 |

[4] | Cho C.-H., Oh Y.-G.: Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds. Asian J. Math. 10, 773–814 (2006) · Zbl 1130.53055 |

[5] | Dubrovin, B.: Geometry of 2D topological field theories, Lecture Notes in Mathematics 1620. 120–348 (1996) · Zbl 0841.58065 |

[6] | Duistermaat J.J.: On global action-angle coordinates. Commun. Pure Appl. Math. 33(6), 687–706 (1980) · Zbl 0451.58016 |

[7] | Entov M., Polterovich L.: Calabi quasimorphism and quantum homology. Int. Math. Res. Not. 2003(30), 1635–1676 (2003) · Zbl 1047.53055 |

[8] | Floer A.: Morse theory for Lagrangian intersections. J. Differen. Geomet. 28, 513–547 (1988) · Zbl 0674.57027 |

[9] | Fukaya K.: Mirror symmetry of abelian variety and multi theta function. J. Algebra Geomet. 11, 393–512 (2002) · Zbl 1002.14014 |

[10] | Fukaya, K.: Application of Floer homology of Lagrangian submanifolds to symplectic topology. In: Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, NATO Sci. Ser. II Math. Phys. Chem. 217, pp. 231–276. Springer, Dordrecht (2006) · Zbl 1089.53064 |

[11] | Fukaya, K.: Differentiable operad, Kuranishi correspondence, and Foundation of topological field theories based on pseudo-holomorphic curve. In: Arithmetic and Geometry around Quantization, Progr. Math. Birkäuser 279, pp. 123–200 (2009) · Zbl 1214.53066 |

[12] | Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory-anomaly and obstruction. Kyoto University preprint (2000) · Zbl 1181.53003 |

[13] | Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory-anomaly and obstruction. Expanded version of [FOOO1] (2006 & 2007) · Zbl 1181.53003 |

[14] | Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian intersection Floer theory-anomaly and obstruction, Part I, & Part II. AMS/IP Studies in Advanced Mathematics, vol. 46.1, & 46.2. Amer. Math. Soc./International Press (2009) · Zbl 1181.53002 |

[15] | Fukaya K., Oh Y.-G., Ohta H., Ono K.: Lagrangian Floer theory on compact toric manifolds I. Duke Math. J. 151(1), 23–174 (2010) arXiv:0802.1703 · Zbl 1190.53078 |

[16] | Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Lagrangian Floer theory and mirror symmetry on compact toric manifolds, preprint (2010). arXiv:1009.1648 · Zbl 1190.53078 |

[17] | Fukaya, K., Oh, Y.-G., Ohta, H., Ono, K.: Spectral invariants with bulk, quasimorphisms and Lagrangian Floer theory (in preparation) |

[18] | Fukaya K., Ono K.: Arnold conjecture and Gromov-Witten invariant. Topology 38(5), 933–1048 (1999) · Zbl 0946.53047 |

[19] | Guillemin V.: Kähler structures on toric varieties. J. Differen. Geom. 43, 285–309 (1994) · Zbl 0813.53042 |

[20] | Fulton W.: Introduction to Toric Varieties, Annals of Math. Studies, 131. Princeton University Press, Princeton (1993) · Zbl 0813.14039 |

[21] | Hofer H.: On the topological properties of symplectic maps. Proc. R. Soc. Edinb. 115, 25–38 (1990) · Zbl 0713.58004 |

[22] | Hori, K., Vafa, C.: Mirror symmetry. preprint (2000). hep-th/0002222 |

[23] | Katz S., Liu C.-C.: Enumerative geometry of stable maps with Lagrangian boundary conditions and multiple covers of the disc. Adv. Theor. Math. Phys. 5, 1–49 (2001) · Zbl 1026.32028 |

[24] | Kontsevich, M., Soibelman, Y.: Affine structures and non-archimedean analytic spaces. In: Etingof, P., Retakh, V., Singer, I.M. (eds.) The Unity of Mathematics, pp. 321–385. Progr. Math. 244, Birkhäuser (2006) · Zbl 1114.14027 |

[25] | Mather, J.: Stratifications and mappings. Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971) 41, pp. 195–232. Academic Press, New York (1973) |

[26] | Ostrover Y., Tyomkin I.: On the quantum homology algebra of toric Fano manifolds. Selecta Math. 15, 121–149 (2009) arXiv:0804.0270 · Zbl 1189.53083 |

[27] | Ruan Y.: Virtual neighborhood and pseudoholomorphic curve. Turkish J. Math. 23, 161–231 (1999) · Zbl 0967.53055 |

[28] | Saito K.: Period mapping associated to a primitive form. Publ. R.I.M.S. 19, 1231–1261 (1983) · Zbl 0539.58003 |

[29] | Takahashi, A.: Primitive forms, Topological LG model coupled with gravity, and mirror symmetry, preprint. arXiv:9802059 |

[30] | Weinstein A.: Symplectic manifolds and their Lagrangian submanifolds. Adv. Math. 6, 329–346 (1971) · Zbl 0213.48203 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.