1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

LECTURE 2. THE GEOMETRY OF THE MOMENT MAP 307 which has the polytope P as its moment polytope; if P has d faces of codimension 1 ...

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Lecture 3. Equivariant Cohomology and the Cartan Model Let M be a manifold equipped with the action of a (compact) group G. If t ...

310 L. C. JEFFREY, HAMILTONIAN GROUP ACTIONS 3.1. The Cartan model There is a De Rham description of equivariant cohomology, whi ...

LECTURE 3. EQUIVARIANT COHOMOLOGY AND THE CARTAN MODEL 311 Example 3.4. Equivariant characteristic classes in the Cartan model. ...

312 L. C. JEFFREY, HAMILTONIAN GROUP ACTIONS Since c 1 (vF,j)/f3F,j(X) is nilpotent, we find that we may define the inverse of e ...

LECTURE 3. EQUIVARIANT COHOMOLOGY AND THE CARTAN MODEL 313 that i} ( i F) * = e F is multiplication by the equivariant Euler cla ...

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LECTURE 4 The Duistermaat-Heckman Theorem and Applications to the Cohomology of Symplectic Quotients Let M be a symplectic manif ...

316 L. C. JEFFREY, HAMILTONIAN GROUP ACTIONS The lemma of stationary phase ([21], Section 33) asserts that the oscillatory integ ...

LECTURE 4. THE DUISTERMAAT-HECKMAN THEOREM 317 Suppose a symplectic manifold M is equipped with a complex structure com- patible ...

318 L. C. JEFFREY, HAMILTONIAN GROUP ACTIONS 4.4.1. The residue formula A related result is the residue formula, Theorem 8.1 of ...

LECTURE 4. THE DUISTERMAAT-HECKMAN THEOREM 319 Thus we need to prove the result only for torus actions. A sketch of the proof wh ...

320 L. C. JEFFREY, HAMILTONIAN GROUP ACTIONS The sum in Theorem 4.9 is over those Ti and Mi for which a (generic) ray in t* from ...

Lecture 5. Moduli Spaces of Vector Bundles over Riemann Surfaces 5.1. Prototype: The Jacobian Let :E be a compact Riemann surfac ...

322 L. C. JEFFREY, HAMILTONIAN GROUP ACTIONS There is an alternative description of t he symplectic structure via de Rham cohomo ...

LECTURE 5. MODULI SPACES OF VECTOR BUNDLES 323 as a moduli space of gauge equivalence classes of fl.at connections on E ; ...

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Exercises Hamiltonian group actions and symplectic reduction: Exercise 1 (Lectures 1 and 2) Let f: C ___.,IR be the moment map ...

326 L. C. JEFFREY, HAMILTONIAN GROUP ACTIONS are the vector fields X on T*JR.^3 generated by the action of X E JR.^3 = g on T*JR ...