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Media type:
E-Article
Title:
An algebraic proof for the symplectic structure of moduli space
Contributor:
Karshon, Yael
imprint:
American Mathematical Society (AMS), 1992
Published in:Proceedings of the American Mathematical Society
Language:
English
DOI:
10.1090/s0002-9939-1992-1112494-2
ISSN:
0002-9939;
1088-6826
Origination:
Footnote:
Description:
<p>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.</p>