By Frances Clare Kirwan

ISBN-10: 0691083703

ISBN-13: 9780691083704

These notes describe a normal approach for calculating the Betti numbers of the projective quotient kinds that geometric invariant thought affiliates to reductive staff activities on nonsingular complicated projective forms. those quotient kinds are fascinating particularly due to their relevance to moduli difficulties in algebraic geometry. the writer describes various techniques to the matter. One is solely algebraic, whereas the opposite makes use of the tools of symplectic geometry and Morse conception, and includes extending classical Morse thought to sure degenerate functions.

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**Extra info for Cohomology of quotients in symplectic and algebraic geometry**

**Example text**

L p (y) = r; (y) :;; MaXC~1(2hf (am) - h f (b m) ) ,0) . l :;; 'J , p(y) < implies v E Loo ' Rv(y) > 0 whieh implies in turn that Rv(y) ;: 1 (hence poo(y) = 0 00 (y) = 0 ). Let and be the greatest eommon denominator of the p > N a(y) < 00 co, and a m , bm's ; for n ... 4 \1/n ( den Nn/(bm)n) ~ nie /Stirling), see the appendix. The and former two estimates, together with the divergence of 'i'~ L p-1 ' show that p>N p(y) < 00 * ~ The latter two estimates show that v ~ 0(y) < 00 the first and the third estimates show that finiteness for p and 0 , * ~ ~ ~ v = \) .

When they satisfy condition (4) above, we shall say the M is normalized. 1. Let endowed with bases {m j } and be two a-modules over {m;} respectively. ) ~ Let G, resp. ' • J be the matrices which represent (resp. M') with recourse to check that A , {mi} a in M resp. {mi} . One readily satisfies the following differential system: GH-HG' G'X' One can pass from solutions of (1)': ax' to solutions of (1) by setting (17) X=HX'. Let us now assume that H is invertible in GL~(A) this means that H represents an isomorphism of a-modules.

Co ° (L~) / log (cf. Lemma 2g)); k Integration of any formal power series but the Hadamard produet xy y is nothing * L1 . 4 Generalized hypergeometrie funetions For a , a E al : = (al' ... ,x) LEMMA. := The three eonditions I: n;:O (~)n/(b) p(y) - n < 00, 0(y) < co u = and 'J are equivalent. l p (y) = r; (y) :;; MaXC~1(2hf (am) - h f (b m) ) ,0) . l :;; 'J , p(y) < implies v E Loo ' Rv(y) > 0 whieh implies in turn that Rv(y) ;: 1 (hence poo(y) = 0 00 (y) = 0 ). Let and be the greatest eommon denominator of the p > N a(y) < 00 co, and a m , bm's ; for n ...

### Cohomology of quotients in symplectic and algebraic geometry by Frances Clare Kirwan

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