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Misha Verbitsky
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The Hodge * operator action on cohomology is generally speaking metric-dependent, hence * is not well-defined without fixing the metric. There are some caveats. On complex curves, for example, the Hodge * operator is complex rotation, which depends only on complex structure. This gives an example of a manifold when the action of * is metric-dependent: indeed, the action of complex rotation on $H^1$ determines the biholomorphism class of the complex curve (Torelli).

On compact 2n-dimensional manifolds, the *-operator in the middle cohomology is determined by the conformal structure. On compact Lie groups, you can identify harmonic forms and bi-invariant forms, hence the Hodge * is the same for all bi-invariant metrics.

Misha Verbitsky
  • 9.2k
  • 1
  • 28
  • 48