By Casim Abbas
This publication presents an advent to symplectic box thought, a brand new and significant topic that's at present being constructed. the place to begin of this thought are compactness effects for holomorphic curves confirmed within the final decade. the writer provides a scientific advent supplying loads of heritage fabric, a lot of that is scattered through the literature. because the content material grew out of lectures given by means of the writer, the most objective is to supply an access aspect into symplectic box thought for non-specialists and for graduate scholars. Extensions of convinced compactness effects, that are believed to be precise by means of the experts yet haven't but been released within the literature intimately, refill the scope of this monograph.
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Extra resources for An Introduction to Compactness Results in Symplectic Field Theory
With this in mind we consider what can be said about moduli spaces of holo morphic and pseudo-holomoxphic curves. In the integrable case one can of cours apply a great deal of existing algebro-geometric theory. First, in a general wa~ the moduli spaces of holomorphic curves will be quasi-projective varieties - an' the coresponding hyperplane class has a simple toploogical description, much as i ' the Yang-Mills case. While it is a difficult problem to find holomoxphic curves i general there is one class of examples which are easy to find and describe - th" curves given by complete intersections of hypersurlaces in a complex manifold.
Nn sum up in the following theorem 'aUI:ouEM 13. ,,,1 k he even with 4k > 6 + 3b+(X). (~ €i contains terms of degree at most (3/2)(1 + b+(Xi)) in H 2 (Xi). ioual cohomology: the latter are killed by connected sums, since the glueing 14(1(' ttl. er is rationally trivial, but the torsion classes can detect the gluing parame- ,.... nud p;ive potentially non-trivial invariants. irold does not split off an 8 2 x S2 summand. olIlorphism group of X realises all symmetries of the intersection form.
In "' HiInilar vein, one can show that for large k the sections Hk generate k , so they cl,~fille a smooth map: j" : V --+ P(HZ). e It. l(unorphic" map. REFERENCES I. Jones, Topological GSpects of Yang-Mills Theory, Commun. Math. Phys. 61 (1978), 97 -118. '/.. Peters and A. Van de Ven, "Compact Complex Surfaces," Springer, Berlin, 1984. :I. Bismut, Demailly's Asymptotic Morse Inequalities: A Heat equation proof, Jour. of I'unctional Analysis 72 (1987), 263-278. 4. Chern, "Complex manifolds without potential theory," Springer, Berlin, 1979.