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A signature formula for hyperelliptic broken Lefschetz fibrations  [PDF]
Kenta Hayano,Masatoshi Sato
Mathematics , 2011,
Abstract: A hyperelliptic broken Lefschetz fibration is a generalization of a hyperelliptic Lefschetz fibration. We construct and compute a local signature of hyperelliptic directed broken Lefschetz fibrations by generalizing Endo's local signature of hyperelliptic Lefschetz fibrations. It is described by his local signature and a rational-valued homomorphism on the subgroup of the hyperelliptic mapping class group which preserves a simple closed curve setwise.
The mondromy of the Lagrange Top and the Picard-Lefschetz formula  [PDF]
O. Vivolo
Physics , 2001,
Abstract: The purpose of this paper is to show that the monodromy of action variables of the Lagrange top and its generalizations can be deduced from the monodromy of cycles on a suitable hyperelliptic curve (computed by the Picard-Lefschetz formula).
On genus-1 simplified broken Lefschetz fibrations  [PDF]
Kenta Hayano
Mathematics , 2010, DOI: 10.2140/agt.2011.11.1267
Abstract: Auroux, Donaldson and Katzarkov introduced broken Lefschetz fibrations as a generalization of Lefshcetz fibrations in order to describe near-symplectic 4-manifolds. We first study monodromy representations of higher sides of genus-1 simplified broken Lefschetz fibrations. We then completely classify diffeomorphism types of such fibrations with connected fibers and with less than six Lefschetz singularities. In these studies, we obtain several families of genus-1 simplified broken Lefschetz fibrations, which we conjecture contain all such fibrations, and determine the diffeomorphism types of the total spaces of these fibrations. Our results are generalizations of Kas' classification theorem of genus-1 Lefschetz fibrations, which states that the total space of a non-trivial genus-1 Lefschetz fibration over $S^2$ is diffeomorphic to an elliptic surface E(n), for some $n\geq 1$.
Minimality and fiber sum decompositions of Lefschetz fibrations  [PDF]
R. Inanc Baykur
Mathematics , 2014,
Abstract: We give a short proof of a conjecture of Stipsicz on the minimality of fiber sums of Lefschetz fibrations, which was proved earlier by Usher. We then construct the first examples of genus g > 1 Lefschetz fibrations on minimal symplectic 4-manifolds which, up to diffeomorphisms of the summands, admit unique decompositions as fiber sums.
Universal Lefschetz fibrations and Lefschetz cobordisms  [PDF]
Daniele Zuddas
Mathematics , 2014,
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.
Invariants of real Lefschetz fibrations  [PDF]
Nermin Salepci
Mathematics , 2011,
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.
Lefschetz fibrations with small slope  [PDF]
Naoyuki Monden
Mathematics , 2012,
Abstract: We construct Lefschetz fibrations over $S^2$ which do not satisfy the slope inequality. This gives a negative answer to a question of Hain.
Lefschetz fibrations and the Hodge bundle  [PDF]
Ivan Smith
Mathematics , 1999, DOI: 10.2140/gt.1999.3.211
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.
Lefschetz fibrations and Torelli groups  [PDF]
R. Inanc Baykur,Dan Margalit
Mathematics , 2012,
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.
Grothendieck--Lefschetz type theorems for the local Picard group  [PDF]
János Kollár
Mathematics , 2012,
Abstract: We propose a strengthening of the Grothendieck--Lefschetz hyperplane theorem for the local Picard group, prove some special cases and derive several consequences to the deformation theory of log canonical singularities. Version 2: Main conjecture changed, following recent results of Bhatt and de Jong. Small changes to the proofs.
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