Abstract:
We construct universal Lefschetz fibrations, that are defined in analogy with the classical universal bundles. We also introduce the cobordism groups of Lefschetz fibrations, and we see how these groups are quotient of the singular bordism groups via the universal Lefschetz fibrations.

Abstract:
In this note we introduce certain invariants of real Lefschetz fibrations. We call these invariants {\em real Lefschetz chains}. We prove that if the fiber genus is greater than 1, then the real Lefschetz chains are complete invariants of real Lefschetz fibrations with only real critical values. If however the fiber genus is 1, real Lefschetz chains are not sufficient to distinguish real Lefschetz fibrations. We show that by adding a certain binary decoration to real Lefschetz chains, we get a complete invariant.

Abstract:
Integral symplectic 4-manifolds may be described in terms of Lefschetz fibrations. In this note we give a formula for the signature of any Lefschetz fibration in terms of the second cohomology of the moduli space of stable curves. As a consequence we see that the sphere in moduli space defined by any (not necessarily holomorphic) Lefschetz fibration has positive "symplectic volume"; it evaluates positively with the Kahler class. Some other applications of the signature formula and some more general results for genus two fibrations are discussed.

Abstract:
For each g > 2 and h > 1, we explicitly construct (1) fiber sum indecomposable relatively minimal genus g Lefschetz fibrations over genus h surfaces whose monodromies lie in the Torelli group, (2) fiber sum indecomposable genus g surface bundles over genus h surfaces whose monodromies are in the Torelli group (provided g > 3), and (3) infinitely many genus g Lefschetz fibrations over genus h surfaces that are not fiber sums of holomorphic ones.

Abstract:
In this note, we verify that the complex Kodaira dimension $\kappa^h$ equals the symplectic Kodaira dimension $\kappa^s$ for smooth 4-manifolds with complex and symplectic structures. We also calculate the Kodaira dimension for many Lefschetz fibrations.

Abstract:
We prove that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We then describe the topology of the regular and singular fibres, in particular calculating their middle Betti numbers. For the example of sl(2,C) we compute the Fukaya--Seidel category of Lagrangian vanishing cycles.

Abstract:
We employ a certain labeled finite graph, called a chart, in a closed oriented surface for describing the monodromy of a(n achiral) Lefschetz fibration over the surface. Applying charts and their moves with respect to Wajnryb's presentation of mapping class groups, we first generalize a signature formula for Lefschetz fibrations over the 2-sphere obtained by Endo and Nagami to that for Lefschetz fibrations over arbitrary closed oriented surface. We then show two theorems on stabilization of Lefschetz fibrations under fiber summing with copies of a typical Lefschetz fibration as generalizations of a theorem of Auroux.

Abstract:
The existence of a positive allowable Lefschetz fibration on a compact Stein surface with boundary was established by Loi and Piergallini by using branched covering techniques. Here we give an alternative simple proof of this fact and construct explicitly the vanishing cycles of the Lefschetz fibration, obtaining a direct identification of the set of compact Stein manifolds with positive allowable Lefschetz fibrations over a 2-disk. In the process we associate to every compact Stein manifold infinitely many nonequivalent such Lefschetz fibrations.

Abstract:
In this article, we characterize isomorphism classes of Lefschetz fibrations with multisections via their monodromy factorizations. We prove that two Lefschetz fibrations with multisections are isomorphic if and only if their monodromy factorizations in the relevant mapping class groups are related to each other by a finite collection of modifications, which extend the well-known Hurwitz equivalence. This in particular characterizes isomorphism classes of Lefschetz pencils. We then show that, from simple relations in the mapping class groups, one can derive new (and old) examples of Lefschetz fibrations which cannot be written as fiber sums of blown-up pencils.