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• W1972812033 abstract "Goldman has constructed a symplectic form on the moduli space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper H o m left-parenthesis pi comma upper G right-parenthesis slash upper G> <mml:semantics> <mml:mrow> <mml:mi>Hom</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>π<!-- π --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>G</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>operatorname {Hom} (pi ,G)/G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, of flat <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-bundles over a Riemann surface <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=application/x-tex>S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose fundamental group is <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding=application/x-tex>pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The construction is in terms of the group cohomology of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding=application/x-tex>pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The proof that the form is closed, though, uses de Rham cohomology of the surface <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding=application/x-tex>S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, with local coefficients. This symplectic form is shown here to be the restriction of a tensor, that is defined on the infinite product space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G Superscript pi> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>G</mml:mi> <mml:mi>π<!-- π --></mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>{G^pi }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This point of view leads to a direct proof of the closedness of the form, within the language of group cohomology. The result applies to all finitely generated groups <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding=application/x-tex>pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose cohomology satisfies certain conditions. Among these are the fundamental groups of compact Kähler manifolds." @default.
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• W1972812033 date "1992-01-01" @default.
• W1972812033 modified "2023-10-13" @default.
• W1972812033 title "An algebraic proof for the symplectic structure of moduli space" @default.
• W1972812033 cites W1983033166 @default.
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• W1972812033 doi "https://doi.org/10.1090/s0002-9939-1992-1112494-2" @default.
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