The Minimal Hirsch-Brown Model Via Classical Hodge Theory

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In our book on cohomological methods in transformation groups the minimal Hirsch-Brown model was used to good effect. The construction there, however, was rather abstract. Here, for smooth compact connected Lie group actions on smooth closed manifolds, we give a much more explicit construction of the minmal Hirsch-Brown model using operators from classical Hodge theory and the small Cartan model. As a simple example, we discuss circle actions on complex projective space, computing the deformation terms in the product by means of the moment map in two different ways.