##
**Lagrangian matching invariants for fibred four-manifolds. I.**
*(English)*
Zbl 1143.53079

The paper under review is the first part of the author’s thesis, and is to study the symplectic feature of singular Lefschetz fibrations. Note that for symplectic 4-manifolds, Taubes proved that Seiberg-Witten invariants can be computed as Gromov-Witten invariants from counting \(J\)-holomorphic curves and their unramified coverings. On the other hand, Donaldson proved that every closed symplectic 4-manifold admits a symplectic Lefschetz pencil. Usher proved that Donaldson-Smith invariant (of gauge theory and symplectic topology) of the symplectic Lefschetz pencil can be evaluated as the Gromov invariant from adiabatic limiting of \(J\)-holomorphic curves.

Auroux, Donaldson and Katzarkov proved that every closed smooth oriented 4-manifold \(X\) with \(b_2^+ (X) > 0\) admits a broken Lefschetz pencil structure by using analytic methods. Recently, some preprints show that every closed smooth oriented 4-manifold admits a broken Lefschetz fibration [see Baykur, Lekili, Akbulut and Karakurt’s preprints on ArXiv]. To extend the Donaldson-Smith invariant DS from the Lefschetz pencil to the singular one, is the main goal of the paper and its sequel by Perutz. The invariant DS counts holomorphic sections of a relative Hilbert scheme that is constructed from the fibration. The extension of DS still counts \(J\)-holomorphic sections of a Hilbert scheme with certain Lagrangian boundary condition and matching the Lagrangian boundary condition related to a symplectic Floer homology group \(Y\), where \(Y\) is the boundary of a broken fibration.

The singular fibration arises from a one-submanifold \(Z \subset Int(X)\) and a discrete set \(D\) such that \(Z\cap D = \emptyset\) and \(\pi(Z) \cap \pi (D) = \emptyset\) for the smooth map \(\pi: X \to S\) and the local model around \(Z\) is given by \((t, x_1, x_2, x_3) \mapsto (t, \pm (x_1^2+x_2^2-x_3^2))\). When \(Z = \emptyset\), this reduces to the symplectic Lefschetz fibration. If \(Y\) is a closed oriented 3-manifold and \(f: Y \to S^1\) is a Morse function with index 1 or 2, then \(\pi : Y^1\times S^1 \to S^1 \times S^1 (\pi = f\times id_{S^1})\) is a broken fibration with \(D=\emptyset\).

The starting point of the paper is based on a observation in I. Smith’s paper [Topology, 42, 931–979 (2003; Zbl 1030.57038)] that the critical submanifold of the relative Hilbert scheme of \(n\) points \(\text{Hilb}_{\Delta}^n(E) \to \Delta \) (the closed unit disc) is naturally biholomorphic to \(\text{Sym}^{n-1}(\tilde{E}_0)\). The Lagrangian correspondence \(\hat{V}_L\) between \(\text{Sym}^n(\Sigma)\) and \(\text{Sym}^{n-1}(\overline{\Sigma})\) is defined as the graph of symplectic parallel transport over the ray \([0, 1]\subset \Delta\) into the critical set \(\text{Sym}^{n-1}(\tilde{E}_0)\), where \(\overline{\Sigma}\) is a Riemannian surface obtained by surgering out a circle \(L \subset \Sigma\).

Section 2 exploits the geometry of symplectic Morse-Bott fibration. In section 3, with a suitable choice of complex structures on \(\Sigma\) and \(\overline{\Sigma}\) and any pair of Kähler forms on \(\text{Sym}^n(\Sigma)\) and \(\text{Sym}^{n-1}(\overline{\Sigma})\), there exists a Lagrangian submanifold \(\hat{V}_L\) such that \(\pi_1: \hat{V}_L (\subset \text{Sym}^n(\Sigma) \times \text{Sym}^{n-1}(\overline{\Sigma})) \to \text{Sym}^n(\Sigma)\) is an embedding and \(\pi_2: \hat{V}_L \to \text{Sym}^{n-1}(\overline{\Sigma})\) is an \(S^1\)-bundle projection. This is Theorem A of the paper, and it is proved under general vanishing cycle construction structure of the singular locus of the Hilbert scheme of \(n\)-points on a nodal curve.

For each curve \(L \subset \Sigma\), there is an association of \(\hat{V}_L\); for a pair of \(L_1, L_2\), there is a symplectic Floer homology \(HF(\hat{V}_{L_1}, \hat{V}_{L_2})\) as a morphism in the TQFT setup. It may provide a category for the singular Lefschetz fibration. The last section 4 proves the parametrized version of Theorem A (Theorem B) still holds, and its proof follows: the \(S^1\)-parametrized construction of Theorem A for moving surfaces. In appendix A, the cohomology of \(\text{Hilb}_{\Delta}^n(E)\) is given.

The construction of the paper is interesting, and it would be better to have some examples which cannot be distinguished by other invariants, or to see this construction explicitly for \(S^4\) and \(\mathbb C\mathbb P^2\).

Auroux, Donaldson and Katzarkov proved that every closed smooth oriented 4-manifold \(X\) with \(b_2^+ (X) > 0\) admits a broken Lefschetz pencil structure by using analytic methods. Recently, some preprints show that every closed smooth oriented 4-manifold admits a broken Lefschetz fibration [see Baykur, Lekili, Akbulut and Karakurt’s preprints on ArXiv]. To extend the Donaldson-Smith invariant DS from the Lefschetz pencil to the singular one, is the main goal of the paper and its sequel by Perutz. The invariant DS counts holomorphic sections of a relative Hilbert scheme that is constructed from the fibration. The extension of DS still counts \(J\)-holomorphic sections of a Hilbert scheme with certain Lagrangian boundary condition and matching the Lagrangian boundary condition related to a symplectic Floer homology group \(Y\), where \(Y\) is the boundary of a broken fibration.

The singular fibration arises from a one-submanifold \(Z \subset Int(X)\) and a discrete set \(D\) such that \(Z\cap D = \emptyset\) and \(\pi(Z) \cap \pi (D) = \emptyset\) for the smooth map \(\pi: X \to S\) and the local model around \(Z\) is given by \((t, x_1, x_2, x_3) \mapsto (t, \pm (x_1^2+x_2^2-x_3^2))\). When \(Z = \emptyset\), this reduces to the symplectic Lefschetz fibration. If \(Y\) is a closed oriented 3-manifold and \(f: Y \to S^1\) is a Morse function with index 1 or 2, then \(\pi : Y^1\times S^1 \to S^1 \times S^1 (\pi = f\times id_{S^1})\) is a broken fibration with \(D=\emptyset\).

The starting point of the paper is based on a observation in I. Smith’s paper [Topology, 42, 931–979 (2003; Zbl 1030.57038)] that the critical submanifold of the relative Hilbert scheme of \(n\) points \(\text{Hilb}_{\Delta}^n(E) \to \Delta \) (the closed unit disc) is naturally biholomorphic to \(\text{Sym}^{n-1}(\tilde{E}_0)\). The Lagrangian correspondence \(\hat{V}_L\) between \(\text{Sym}^n(\Sigma)\) and \(\text{Sym}^{n-1}(\overline{\Sigma})\) is defined as the graph of symplectic parallel transport over the ray \([0, 1]\subset \Delta\) into the critical set \(\text{Sym}^{n-1}(\tilde{E}_0)\), where \(\overline{\Sigma}\) is a Riemannian surface obtained by surgering out a circle \(L \subset \Sigma\).

Section 2 exploits the geometry of symplectic Morse-Bott fibration. In section 3, with a suitable choice of complex structures on \(\Sigma\) and \(\overline{\Sigma}\) and any pair of Kähler forms on \(\text{Sym}^n(\Sigma)\) and \(\text{Sym}^{n-1}(\overline{\Sigma})\), there exists a Lagrangian submanifold \(\hat{V}_L\) such that \(\pi_1: \hat{V}_L (\subset \text{Sym}^n(\Sigma) \times \text{Sym}^{n-1}(\overline{\Sigma})) \to \text{Sym}^n(\Sigma)\) is an embedding and \(\pi_2: \hat{V}_L \to \text{Sym}^{n-1}(\overline{\Sigma})\) is an \(S^1\)-bundle projection. This is Theorem A of the paper, and it is proved under general vanishing cycle construction structure of the singular locus of the Hilbert scheme of \(n\)-points on a nodal curve.

For each curve \(L \subset \Sigma\), there is an association of \(\hat{V}_L\); for a pair of \(L_1, L_2\), there is a symplectic Floer homology \(HF(\hat{V}_{L_1}, \hat{V}_{L_2})\) as a morphism in the TQFT setup. It may provide a category for the singular Lefschetz fibration. The last section 4 proves the parametrized version of Theorem A (Theorem B) still holds, and its proof follows: the \(S^1\)-parametrized construction of Theorem A for moving surfaces. In appendix A, the cohomology of \(\text{Hilb}_{\Delta}^n(E)\) is given.

The construction of the paper is interesting, and it would be better to have some examples which cannot be distinguished by other invariants, or to see this construction explicitly for \(S^4\) and \(\mathbb C\mathbb P^2\).

Reviewer: Weiping Li (Stillwater)

### MSC:

53D40 | Symplectic aspects of Floer homology and cohomology |

57R57 | Applications of global analysis to structures on manifolds |

57R15 | Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) |

### Keywords:

four manifolds; Lefschetz fibration; Seiberg-Witten invariant; pseudo-holomorphic curve; Lagrangian submanifold; Hilbert scheme### Citations:

Zbl 1030.57038### References:

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