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We consider subtorus actions on complex toric varieties. A natural candidate for a categorical quotient of such an action is the so-called toric quotient, a universal object constructed in the toric category. We prove that if the toric quotient is weakly proper and if in addition the quotient variety is of expected dimension then the toric quotient is a categorical quotient in the category of algebraic varieties. For example, weak properness always holds for the toric quotient of a subtorus action...
We show that in the category of complex algebraic varieties, the Eilenberg–Moore
spectral sequence can be endowed with a weight filtration. This implies that it
degenerates if all spaces involved have pure cohomology. As application, we compute the
rational cohomology of an algebraic -variety ( being a connected algebraic
group) in terms of its equivariant cohomology provided that is pure. This is
the case, for example, if is smooth and has only finitely many orbits. We work in the
category...
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