The equivariant cohomology ring of regular varieties (Q1885574)

Let \(\mathfrak{B}\) be the upper-triangular subgroup of \(\text{SL} _{2}\left( \mathbb{C}\right) ,\) \(\mathfrak{T}\) its diagonal torus and \( \mathfrak{U}\) its unipotent radical. Let \(X\) be a smooth complex projective algebraic variety which is acted upon by \(\mathfrak{B}\) such that \(X^{ \mathfrak{U}}\) is a single point \(\mathfrak{o}\). This variety \(X\) is said to be regular. Let \(H_{\mathfrak{T}}^{\ast }\left( X\right) \) denote the equivariant cohomology ring of \(X_{\mathfrak{T}},\) where \(X_{\mathfrak{T} }=\left( X\times \mathcal{E}\right) /\mathfrak{T}\), \(\mathcal{E}\) a contractible space with a free action of \(\mathfrak{T.}\) This paper is a study of this cohomology ring. Let \(\mathcal{A}\) be the vector field on \(X\times \mathbb{A}^{1}\) given by \( \mathcal{A}_{\left( x,v\right) }=2\mathcal{V}_{x}-v\mathcal{W}_{x},\) where \( \mathcal{V}\) and \(\mathcal{W}\) are certain generators of \(\mathfrak{U}\) and \( \mathfrak{T}\) respectively, and let \(\mathcal{Z}\) be the zero scheme of \( \mathcal{A}.\) It is shown that there is an isomorphism of graded algebras \( H_{\mathfrak{T}}^{\ast }\left( X\right) \rightarrow \mathbb{C}\left[ \mathcal{Z}\right] .\) More generally, for \(Y\) a closed \(\mathfrak{B}\)-stable subvariety of \(X\) for which the restriction map \(H^{\ast }\left( X\right) \rightarrow H^{\ast }\left( Y\right) \) is surjective there is a graded isomorphism \(H_{\mathfrak{T}}^{\ast }\left( Y\right) \rightarrow \mathbb{C} \left[ \mathcal{Z}_{Y}\right] ,\) where \(\mathcal{Z}_{Y}\) is the union of the components of \(\mathcal{Z}\) lying in \(Y_{\mathfrak{o}}\cap \mathbb{A}^{1}\), \( Y_{\mathfrak{o}}=\left\{ y\in Y\,| \,\lim_{t\rightarrow \infty }\left( \begin{matrix} t & 0 \\ 0 & t^{-1} \end{matrix} \right) \cdot y=\mathfrak{0}\right\} .\) This isomorphism commutes with the restrictions \(H_{\mathfrak{T}}^{\ast }\left( X\right) \rightarrow H_{ \mathfrak{T}}^{\ast }\left( Y\right) \) and \(\mathbb{C}\left[ \mathcal{Z} \right] \rightarrow \mathbb{C}\left[ \mathcal{Z}_{Y}\right] .\) The authors compute the equivariant cohomology of the Peterson variety. Let \( G\) be a complex semi-simple linear algebraic group with Lie algebra \( \mathfrak{g}\) and \(B\) a Borel subgroup with Lie algebra \(\mathfrak{b}\). For \( M\) a \(B\)-submodule of \(\mathfrak{g}\) containing \(\mathfrak{b}\) we define for each \(x\in \mathfrak{g}\) \(Y_{M}\left( x\right) =\left\{ gB\in G/B\,| \,g^{-1}x\in M\right\} .\) Then the restriction map \(H^{\ast }\left( G/B\right) \rightarrow H^{\ast }\left( Y_{M}\left( e\right) \right) \) is surjective and hence \(H_{\mathfrak{T}}^{\ast }\left( Y_{M}\left( e\right) \right) \cong \mathbb{C}\left[ \mathcal{Z}_{Y_{M}\left( e\right) }\right] .\) This cohomology ring satisfies Poincaré duality, and its Poincaré polynomial is given. Finally, the equivariant push-forward map \(\int_{X}:H_{\mathfrak{T}}^{\ast }\left( X\right) \rightarrow \mathbb{C}\left[ z\right] \) corresponding to the map \(X\rightarrow \)(some point) is given.