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Lefschetz pencils on symplectic manifolds

Symplectic 4-Manifolds and Algebraic Surfaces pp Cite as. This set of lectures aims to give an overview of Donaldson's theory of linear systems on symplectic manifolds and the algebraic and geometric invariants to which they give rise.

After collecting some of the relevant background, we discuss topological, algebraic and symplectic viewpoints on Lefschetz pencils and branched covers of the projective plane. The later lectures discuss invariants obtained by combining this theory with pseudo-holomorphic curve methods. Unable to display preview. Download preview PDF. Skip to main content. This service is more advanced with JavaScript available. Advertisement Hide. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, log in to check access. Bogomolov, L. Katzarkov, T. Pantev, Symplectic Lefschetz fibrations with arbitrary fundamental groups, J. Amram, D. Goldberg, M. Teicher, U. Auroux, Asymptotically holomorphic families of symplectic submani-folds, Geom. Auroux, Symplectic 4-manifolds as branched coverings of CP 2Invent. Auroux, Symplectic maps to projective spaces and symplectic invariants, Turkish J.

Google Scholar. Symplectic Geom. Auroux, Fiber sums of genus 2 Lefschetz fibrations, Turkish J. Auroux, S. Donaldson, L. Katzarkov, Luttinger surgery along La-grangian tori and non-isotopy for singular symplectic plane curves, Math.

Katzarkov, Singular Lefschetz pencils, Geom. Katzarkov, M. Yotov, Fundamental groups of complements of plane curves and symplectic invariants, Topology 43Auroux, L.

Katzarkov, Branched coverings of CP 2 and invariants of symplectic 4-manifolds, Invent. Katzarkov, The degree doubling formula for braid mon-odromies and Lefschetz pencils, Pure Appl.

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Katzarkov, D. Orlov, Mirror symmetry for weighted projective planes and their noncommutative deformations, Ann. Auroux, V. Presas, Lagrangian submanifolds and Lefschetz pencils, J. MathSciNet Google Scholar. Biran, A stability property of symplectic packing in dimension 4, Invent.In mathematicsa Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetzused to analyse the algebraic topology of an algebraic variety V.

A pencil is a particular kind of linear system of divisors on Vnamely a one-parameter family, parametrised by the projective line. This means that in the case of a complex algebraic variety Va Lefschetz pencil is something like a fibration over the Riemann sphere ; but with two qualifications about singularity. The first point comes up if we assume that V is given as a projective varietyand the divisors on V are hyperplane sections.

There is a rational mapping. To make a well-defined mapping, some blowing up must be applied to V. The second point is that the fibers may themselves 'degenerate' and acquire singular points where Bertini's lemma applies, the general hyperplane section will be smooth.

A Lefschetz pencil restricts the nature of the acquired singularities, so that the topology may be analysed by the vanishing cycle method. The fibres with singularities are required to have a unique quadratic singularity, only. It has been shown that Lefschetz pencils exist in characteristic zero.

They apply in ways similar to, but more complicated than, Morse functions on smooth manifolds. Simon Donaldson has found a role for Lefschetz pencils in symplectic topologyleading to more recent research interest in them.

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From Wikipedia, the free encyclopedia. Categories : Geometry of divisors. Hidden categories: CS1: long volume value. Namespaces Article Talk. Views Read Edit View history. Help Learn to edit Community portal Recent changes Upload file.

lefschetz pencils on symplectic manifolds

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lefschetz pencils on symplectic manifolds

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Lefschetz Pencils, Branched Covers and Symplectic Invariants

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lefschetz pencils on symplectic manifolds

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Hard Lefschetz Theorem

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But Shigure was a legend even within the IJN, considered the good luck ship of the fleet. I keep hearing Shigure would be better but no one ever explains why. Best ship at Guadalcanal. The Littirio class is my absolute favorite ship from when i played Seapower (models on floors).

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With some premiums too ofc. An IJN or two besides Mutsu. Posts are automatically archived after 6 months. This is an archived post.