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Symplectic submanifold

Author: KongLingHui
Tutor: WangHongYu
School: Yangzhou University
Course: Applied Mathematics
Keywords: Submanifold Codimension Symplectic manifold Complex manifold Open subset Symplectic structure Cohomology Subspace Tangent space Existence proof
CLC: O189.33
Type: Master's thesis
Year: 2007
Downloads: 19
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Abstract


In this paper we develop a general procedure for constructing symplectic submannifold. The idea is from Donaldson about the proof of the existence of codimension-2 symplectic submanifolds. In general , questions about complex submanifolds of high codimension can be intractable, but one has a rather good grip on sunmanifolds of complex codimension 1, which can be studied through the familiar aparatus of line bundles, linear systems and cohomology-effectively linearising the problem. The idea of this paper is to extend these techniques in complex geometry to general symplectic manifolds. The main result that we prove is the following existence theorem.Theorem 1: Let (V ,ω) be a compact symplectic manifold of dimension 2n , and suppose that the de Rham cohomology class [ω/2π]∈H2 (V;R) lies in the integral lattice H 2 (V;Z)/Torsion. Let h∈H2 (V;Z) be a lift of [ω/2π] to an integral class . Then for sufficiently large integers k the Poincarédual of kh , in H2n-2(V;Z), can be realised by a symplectic submanifold W ? V.We will now explain some of the ideas involved in the proof in more detail, and formulate our results more precisely. We begin with a little linear algebra. Let C n have its standard metric and symplectic formω, and let G be the Grassmannian of oriented real 2n-2-planes in C n. Write G + ? G for the open set of“symplectic”(2n-2)-planesΠ, those for which the restriction ofωn?1 toΠis positive, relative to the orientation onΠ. Clearly G+ depends only on the symplectic structure on C n. Given the metric, and hence a volume formΩΠon each subspace, we can define a map-the“k?hler angle”-One can show, although we do not need this, thatθcompletely classifies the orbits of U(n) acting on G. The complex-linear subspaces are just those withθ(Π)=0, soθmeasures the amount by which a subspace fails to be complex-linear. Clearly the set G+ isθ-1 [0,π/2).Now suppose that a linar subspacesΠin C n is obtained as the kernel of an R-linear map A : Cn→C. We can write A as the sum a ’ + a" where a ’ is complex linear and a " is antilinear. We let |a’|,|a"| be the standard norms defined by the Hermitian metric. A little calculation shows that 1. A has (real) rank 2 unless (a"|-) = eiαa’ for some realα. 2. If A has rank 2 andΠ= ker( A), then One sees from this that , so the ratio eontrols the deviationof the kernel from being a complex linear subspaces.For key observation we need for our main results is the following: Proposition 2: If a ’ , a ": C n→C are respectively complex linear and anti-linear mapsand if |a"|<|a’|, then the subspace ker( a ’ + a ") ? Cn is symplectic. Of course it is easy to verify this directly, without introducing the functionθ. Now consider a symplectic manifold (V ,ω) with a compatible almost complex structure. If W (?)V is a Csubmanifold of real codimension 2, we can define at each point p of W a numberθp (W ), by applying the above discussion to the tangent space of W in V. This measures the extent to which W fails to be a pseudo-holomorphic submanifold. Suppose that L→V is a complex line bundles, and s is a smooth ofξ. The derivation▽s is well defined on the zero-set of s and can be split into the complex linear and antilinear parts (?)s ,(?)s . We see then that if |(?)s|<|(?)s| everywhere on the zero-set, then this zero set is a symplectic codimension 2 submanifold of V, with orientation compatible with symplectic structure. The homology class of the zero set is of course the Poincarédual of the first Chern class of L. This is the way in which we will construct symplectic submanifolds. However we will actually be able to manage rather more. Recall first that, given the hypotheses of Theorem 1, there is a line bundle L over V with c1 ( L )= h, an integral lift of [ω/2π]. We can endow L with a unitary connection having curvature form ? iω, which will play a fundamental role in the proof, although it is not actually involved in the statement of the main result, as follows:Theorem 3. Let L→V be a comlex line bundle over a compact symplectic manifold V with compatible almost-complex structure, and with c1 = [ω2π]. Then there is a constant C such that for all large k there is a section s of L?k with on the zero set of s.This Theorem, together with Proposition 2, implies Theorem 1.

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CLC: > Mathematical sciences and chemical > Mathematics > Geometry, topology > Topology ( the situation in geometry ) > Analytic topology > Differentiable manifolds
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