The Kodaira Dimension of Lefschetz Fibrations

Abstract

In this note, we verify that the complex Kodaira dimension κ h equals the symplectic Kodaira dimension κ s for smooth 4−manifolds with complex and symplectic structures. We also calculate the Kodaira dimension for many Lefschetz fibrations.2.1. Complex Kodaira dimension. If the manifold M admits a complex structure J, then the Kodaira dimension is defined as follows (in the case dim R M = 4, see Def 2.3 for the 2 dimensional case): The n-th plurigenus P n (M, J) of a complex manifold is defined by P n (M, J) = h 0 (K ⊗n J ), with K J the canonical bundle of (M, J). We denote by c 1 = c 1 (X, J) the first Chern class of the complex manifold (X, J).Definition 2.1. The complex Kodaira dimension κ h (M, J) is defined as

Geometric structures, Gromov norm and Kodaira dimensions

“…Furthermore, when M is a surface bundle over circle, M 3 × S 1 is a surface bundle over torus. Thus by classification results on these manifolds in [9], we also have [κ t ] = κ h in this case.…”

“…The corresponding results of [15] for circle bundles and mapping tori are further discussed in [16] and [32] respectively. For a surface bundle over surface, when the base is a positive genus surface, the additivity is established in [9]. When the base is S 2 , the bundle is either a ruled surface or a Hopf surface; the latter case occurs when the fiber is T 2 and homologically trivial.…”

“…Moreover, it is shown in [31] that a symplectic manifold with κ s = 0 has the same homological invariants as one of the manifolds listed above. When κ s = 1 or 2, no classification is possible since symplectic manifolds in both categories could admit arbitrary finitely presented group as their fundamental group [19].In [9], the authors prove that complex and symplectic Kodaira dimensions are compatible with each other. More precisely, when a 4-manifold M admits at the same time both complex and symplectic structures (but the structures are not necessarily compatible with each other), then κ s (M ) = κ h (M, J).…”

A note on collapse, entropy, and vanishing of the Yamabe invariant of symplectic 4-manifolds

“…The symplectic Kodaira dimension of a symplectic manifold is defined as the Kodaira dimension of its minimal model, and in the presence of a holomorphic structure, the holomorphic and the symplectic Kodaira dimensions coincide [Li106,DZ13]. Theorem 1.…”

Virtual Betti numbers and the symplectic Kodaira dimension of fibered $4$-manifolds

“…A similar study for surface bundles and Lefschetz fibrations is carried out in [5], where the authors treat the trickiest case of surface bundles over T 2 by invoking the subadditivity of Kodaira dimensions result from [10]. A similar study for surface bundles and Lefschetz fibrations is carried out in [5], where the authors treat the trickiest case of surface bundles over T 2 by invoking the subadditivity of Kodaira dimensions result from [10].…”