Complex-Valued Symplectic Lefschetz Fibration

• Complex-valued symplectic Lefschetz fibrations are symplectic structures defined on manifolds with local models based on complex Morse singularities that dictate critical loci and vanishing cycles.
• They enable the computation of global symplectic invariants via spectral sequences and Floer homology by leveraging precise monodromy data from isolated critical points.
• These fibrations bridge local analytic behavior with global topology, fostering constructions of exotic 4-manifolds and generalizations to higher-dimensional symplectic settings.

A complex-valued symplectic Lefschetz fibration is a symplectic geometric structure admitting a fibration morphism locally modeled on complex-valued Morse singularities, typically of the form $(z_1, z_2) \mapsto z_1^2 + z_2^2$, defined on a symplectic manifold and compatible with the symplectic form. The map need not be globally holomorphic, but must retain essential analytic features—especially pertaining to the structure of critical loci and vanishing cycles—permitting a rich interplay between symplectic topology, monodromy, Floer theory, and topological invariants.

1. Foundational Structure and Local Models

A complex-valued symplectic Lefschetz fibration consists of a tuple $(E, \pi, S, \omega)$, where $E$ is a symplectic manifold, $S$ is a complex 1–manifold (typically $\mathbb{C}$, $\mathbb{D}$, or %%%%6%%%%), $\pi: E \to S$ is a smooth map with isolated critical points, and $\omega$ a symplectic form satisfying certain compatibility near the critical locus. Around any critical point $q$, there exist Darboux charts with coordinates $(z_1, z_2)$ such that $\pi(z_1, z_2) = z_1^2 + z_2^2$.

Globally, the manifold $E$ is often either a completion ($\widehat{E}$) of a Liouville domain or, in the four-dimensional case, a closed or semi-positive symplectic manifold. In higher dimensions, the base and fiber may both have additional structure, as in symplectic bifibrations (Hayano, 8 Jan 2025). The symplectic structure is typically required to interact compatibly with the fibration in the sense that fibers are symplectic outside the critical set, and the singularities conform to Lefschetz–type local models.

2. Monodromy, Floer Homology, and Spectral Sequences

The central computational framework for complex-valued symplectic Lefschetz fibrations is the tight connection between the global symplectic topology of $E$ and the symplectic (Floer-theoretic) dynamics of the fiber monodromy.

Given a fibration $\pi: E \to S$ with cylindrical end modeled on $[1, \infty) \times M_\phi$, where $M_\phi$ is the mapping torus of the monodromy symplectomorphism $\phi$ of the completed fiber, there exists a spectral sequence converging to symplectic homology $SH_*(\widehat{E})$ (McLean, 2010). The filtration is induced by the winding number of periodic orbits around a base point in $S$, yielding a sequence of subcomplexes $F_k$. The associated graded pieces are

$H_*(F_k/F_{k-1}) \cong HF_{*+2k-1}(\phi, k),$

where $HF_*(\phi, k)$ denotes the Floer homology of the $k$-th iterate of the fiber monodromy.

The $E^1$-page of the spectral sequence is explicitly

$$E^1_{p, q} = \begin{cases} H^{n - q}(E) & \text{if } p = 0, \ HF_{q-p+1}(\phi, p) & \text{if } p > 0, \ 0 & \text{if } p < 0. \end{cases}$$

This filtration yields strong computational and structural consequences, including criteria for the existence of fixed points of $\phi^k$ for infinitely many $k$, linking fiber cohomology and the Floer theory of the monodromy. If the cohomology of the fiber has odd rank under certain stabilizations, then $HF^*(\phi^k) \neq 0$ for infinitely many $k$, ensuring infinitely many nontrivial dynamical fixed points (McLean, 2010).

3. Generalized Complex Structures and Type Change

Complex-valued Lefschetz fibrations in four dimensions naturally interface with generalized complex geometry, specifically with stable generalized complex structures, which interpolate between complex and symplectic structure (Goto et al., 2013, Cavalcanti et al., 2017). In these cases, the total space or its modifications admit structures where the canonical bundle section vanishes transversally along a codimension-2 type change locus.

Surgical operations such as $C^\infty$-logarithmic transformations along symplectic 2–tori are used to produce twisted generalized complex structures with arbitrarily many type changing loci, realized on genus 1 Lefschetz fibrations as well as their broken versions. Each such transformation increases the count of connected components of the type change locus by one. The local and global behavior, as well as the induced fibration structure, are controlled via explicit gluing dictated by SL(3, $\mathbb{Z}$) matrices (in the case of logarithmic transforms), and by leveraging the structure of the elliptic tangent bundle in the context of stable generalized complex geometry (Cavalcanti et al., 2017).

4. Symplectic and Holomorphic Fillings; Rational Blowdown and Surgery

Complex-valued symplectic Lefschetz fibrations are instrumental in classifying symplectic fillings, especially for contact 3–manifolds appearing as links of isolated complex surface singularities. Every minimal weak symplectic filling of the canonical contact structure on a lens space is represented by a positive allowable Lefschetz fibration (PALF) over the disk (Bhupal et al., 2013). These can be algorithmically described via explicit monodromy factorizations encoding both the fibration and the handlebody topology.

Rational blowdown and monodromy substitution techniques extend the set of manifolds realized by such fibrations. Monodromy factorizations admit local substitutions, such as lantern and chain relations, that correspond geometrically to removing negative-definite configurations (e.g., linear plumbings or spheres of square $-4$) and replacing them with rational homology balls. Changes in the monodromy directly translate to new symplectic structures and allow for constructions of exotic 4–manifolds, minimal fillings with prescribed invariants, and the population of previously unreachable regions in the symplectic geography plane (Akhmedov et al., 2018, Baykur et al., 2022).

5. Almost Complex, Multisection, and Monodromy Structures

The mapping class group framework is central for the encoding and manipulation of the monodromy of these fibrations. Positive factorizations of the identity correspond to Lefschetz fibrations; their decorated versions (framed mapping class groups) control multisection and section behavior (Baykur et al., 2013). The monodromy data fully determines invariants such as the genus and self-intersection of multisections via explicit combinatorial formulas, e.g.,

$$g(S) = \frac{k + r}{2} - n + 1, \quad m = -\left( \sum a_i \right) + 2k + r,$$

where $n$ is the degree, $k,r$ count branch points, and the $a_i$ the powers of boundary Dehn twists.

Monodromy substitutions (for example, via the generalized lantern relation) allow for the construction of fiber-sum–indecomposable fibrations without low-degree section classes, which provides counterexamples to conjectures on indecomposability and establishes the existence of exotic Lefschetz pencils.

6. Global Invariants, Geography, and Exotic Structures

The precise manipulation of monodromy and the surgery techniques above allow for constructions of symplectic 4–manifolds spanning large portions of the geography plane. For genus 2 Lefschetz fibrations, strong inequalities analogous to Noether and BMY constraints arise: $2\chi_h(X) - 6 \leq c_1^2(X) \leq 6\chi_h(X) - 3,$ with equality occurring in extremal instances (Nakamura, 2018). Full ranges of admissible invariants are realized, including for spin structures and prescribed fundamental groups (Arabadji et al., 2023).

Explicit monodromy factorizations and substitutions (including breeding methods) have been used to construct Lefschetz fibrations with arbitrary signature—including those with positive and zero signature, thus settling longstanding conjectures (Baykur et al., 2020). Accordingly, exotic smooth structures on 4-manifolds homeomorphic but not diffeomorphic to connected sums of $S^2 \times S^2$ or standard complex rational surfaces can be produced.

7. Higher-Dimensional Generalizations

Higher-dimensional analogues—such as bifibrations or Lefschetz–Bott fibrations—extend the theory to symplectic 6–manifolds and beyond (Hayano, 8 Jan 2025, Oba, 2019, Oba, 2023). In these cases, compatible pairs comprising mapping class group data and braid group relations encode the bifibration structure. Global invariants, including Chern numbers and counts of cusp singularities, are computable via Thom polynomial techniques and monodromy data. The resulting symplectic topology features an expanded landscape, with flexibility surpassing what is possible in the strict holomorphic (complex) case. For example, there exist symplectic 6-manifolds carrying infinitely many codimension-2 symplectic submanifolds that are homologous but not homotopic, in sharp contrast to the uniqueness for complex hypersurfaces in projective spaces (Oba, 2023).

---

Complex-valued symplectic Lefschetz fibrations thus serve as a central object in the paper of symplectic topology, encoding both local complex-analytic structure and global symplectic and topological invariants. The interplay of their monodromy, spectral invariants, and surgery moves underpins a broad range of constructions—ranging from the enumeration of symplectic fillings, and the realization of exotic smooth structures, to advances in the classification and geography of 4–manifolds, and generalizations to higher dimensions. The formalism bridges symplectic Floer theory, mapping class group combinatorics, and modern approaches to generalized complex geometry, offering a unified and systematic method to probe the symplectic and topological complexity of manifolds modeled locally on complex-valued Lefschetz singularities.