[Paper Review] On the equivariant cohomology of subvarieties of a B-regular variety

James B. Carrell, Kiumars Kaveh|ArXiv.org|Sep 6, 2008

Algebraic Geometry and Number Theory6 references12 citations

TL;DR

This paper classifies ${\mathfrak{B}}$-invariant subvarieties $Y$ of a ${\mathfrak{B}}$-regular variety $X$ for which the restriction map $H^*(X) \to H^*(Y)$ is surjective (called principal subvarieties). It establishes that such subvarieties correspond exactly to those for which the ${\mathfrak{T}}$-equivariant cohomology $H^*_{\mathfrak{T}}(Y)$ is isomorphic to the coordinate ring of the curve ${\mathcal{Z}}_Y \subset Y \times \mathbb{P}^1$, and proves that $Y$ is principal if and only if $H^*_{\mathfrak{T}}(Y) = {\mathcal{H}}_{\mathfrak{B}}^*(Y)$, where the latter is the subalgebra generated by Chern classes of ${\mathfrak{B}}$-equivariant vector bundles on $Y$. This provides a cohomological characterization of principal subvarieties in terms of equivariant geometry.

ABSTRACT

By a $B$-regular variety, we mean a smooth projective variety over $C$ admitting an algebraic action of the upper triangular Borel subgroup $B \subset SL_2(C)$ such that the unipotent radical in $B$ has a unique fixed point. A result of M. Brion and the first author describes the equivariant cohomology algebra (over $C$) of a $B$-regular variety $X$ as the coordinate ring of a remarkable affine curve in $X imes P^1$. The main result of this paper uses this fact to classify the $B$-invariant subvarieties $Y$ of a $B$-regular variety $X$ for which the restriction map $i_Y:H^*(X) o H^*(Y)$ is surjective.

Motivation & Objective

• To classify ${\mathfrak{B}}$-invariant subvarieties $Y$ of a ${\mathfrak{B}}$-regular variety $X$ for which the restriction map $H^*(X) \to H^*(Y)$ is surjective.
• To characterize such subvarieties using the structure of their ${\mathfrak{T}}$-equivariant cohomology algebras.
• To establish a precise correspondence between the cohomological surjectivity of restriction maps and the isomorphism $H^*_{\mathfrak{T}}(Y) \cong \mathbb{C}[{\mathcal{Z}}_Y]$, where ${\mathcal{Z}}_Y$ is a canonical curve in $Y \times \mathbb{P}^1$.
• To prove that a ${\mathfrak{B}}$-invariant subvariety $Y$ is principal if and only if its ${\mathfrak{T}}$-equivariant cohomology equals the subalgebra ${\mathcal{H}}_{\mathfrak{B}}^*(Y)$ generated by Chern classes of ${\mathfrak{B}}$-equivariant vector bundles on $Y$.

Proposed method

• The paper uses the known isomorphism $H^*_{\mathfrak{T}}(X) \cong \mathbb{C}[{\mathcal{Z}}_X]$, where ${\mathcal{Z}}_X$ is a ${\mathfrak{T}}$-stable affine curve in $X \times \mathbb{P}^1$, to analyze the restriction map $i_Y^*: H^*_{\mathfrak{T}}(X) \to H^*_{\mathfrak{T}}(Y)$ for ${\mathfrak{B}}$-invariant subvarieties $Y$.
• It defines ${\mathcal{Z}}_Y$ as the reduced intersection ${\mathcal{Z}}_X \cap (Y \times \mathbb{C})$, and constructs a ${\mathbb{C}}[v]$-algebra homomorphism $\rho_Y: {\mathcal{H}}_{\mathfrak{B}}^*(Y) \to \mathbb{C}[{\mathcal{Z}}_Y]$.
• The key technical tool is the construction of a regular section $s(v)$ of the endomorphism bundle $\textup{End}(\tilde{E}_{|{\mathcal{Z}}_Y})$, which allows the Chern classes $c_k^{{\mathfrak{T}}}(E)$ to be realized as regular functions on ${\mathcal{Z}}_Y$.
• The paper applies localization techniques and the structure of the ${\mathfrak{B}}$-action to show that $\rho_Y$ is an isomorphism when $Y$ has vanishing odd cohomology.
• It proves that the image of $H^*_{\mathfrak{T}}(X)$ under $i_Y^*$ is exactly $H^*_{\mathfrak{T}}(Y)$ if and only if $H^*_{\mathfrak{T}}(Y) = {\mathcal{H}}_{\mathfrak{B}}^*(Y)$, linking the cohomological condition to the algebraic structure of the equivariant cohomology.
• The proof relies on the commutativity of a diagram involving $\rho_X$, $\rho_Y$, and the restriction maps, establishing the isomorphism $\rho_Y \circ i_Y^* \circ \rho_X^{-1} = \text{id}$ on $\mathbb{C}[{\mathcal{Z}}_Y]$.

Experimental results

Research questions

• RQ1 Which ${\mathfrak{B}}$-invariant subvarieties $Y$ of a ${\mathfrak{B}}$-regular variety $X$ have surjective restriction map $H^*(X) \to H^*(Y)$?
• RQ2 How can the ${\mathfrak{T}}$-equivariant cohomology $H^*_{\mathfrak{T}}(Y)$ be described geometrically for such subvarieties?
• RQ3 What is the precise algebraic condition on $Y$ that ensures $H^*_{\mathfrak{T}}(Y) \cong \mathbb{C}[{\mathcal{Z}}_Y]$, where ${\mathcal{Z}}_Y$ is the curve in $Y \times \mathbb{P}^1$?
• RQ4 Under what conditions is the subalgebra ${\mathcal{H}}_{\mathfrak{B}}^*(Y)$ of $H^*_{\mathfrak{T}}(Y)$ equal to the full equivariant cohomology algebra?
• RQ5 When is the restriction map $i_Y^*: H^*_{\mathfrak{T}}(X) \to H^*_{\mathfrak{T}}(Y)$ surjective, and how does this relate to the Chern classes of ${\mathfrak{B}}$-equivariant vector bundles on $Y$?

Key findings

• A ${\mathfrak{B}}$-invariant subvariety $Y$ of a ${\mathfrak{B}}$-regular variety $X$ is principal (i.e., the restriction $H^*(X) \to H^*(Y)$ is surjective) if and only if $H^*_{\mathfrak{T}}(Y) = {\mathcal{H}}_{\mathfrak{B}}^*(Y)$, where ${\mathcal{H}}_{\mathfrak{B}}^*(Y)$ is the subalgebra generated by Chern classes of ${\mathfrak{B}}$-equivariant vector bundles on $Y$.
• The ${\mathfrak{T}}$-equivariant cohomology $H^*_{\mathfrak{T}}(Y)$ is isomorphic to the coordinate ring $\mathbb{C}[{\mathcal{Z}}_Y]$ of the curve ${\mathcal{Z}}_Y = {\mathcal{Z}}_X \cap (Y \times \mathbb{C})$, which is the reduced intersection of the canonical curve ${\mathcal{Z}}_X \subset X \times \mathbb{P}^1$ with $Y \times \mathbb{C}$.
• The restriction map $i_Y^*: H^*_{\mathfrak{T}}(X) \to H^*_{\mathfrak{T}}(Y)$ has image equal to $H^*_{\mathfrak{T}}(Y)$ if and only if $H^*_{\mathfrak{T}}(Y) = {\mathcal{H}}_{\mathfrak{B}}^*(Y)$, establishing a cohomological criterion for principality.
• The Chern classes $c_k^{{\mathfrak{T}}}(E)$ of any ${\mathfrak{B}}$-equivariant vector bundle $E$ on $Y$ map under $\rho_Y$ to regular functions on ${\mathcal{Z}}_Y$, which are given by the trace of the endomorphism $s(v)_y$ on $\bigwedge^k E_y$.
• The isomorphism $\rho_Y: H^*_{\mathfrak{T}}(Y) \to \mathbb{C}[{\mathcal{Z}}_Y]$ is a ${\mathbb{C}}[v]$-algebra isomorphism, and the diagram involving $\rho_X$, $\rho_Y$, and the restriction maps commutes.
• If $Y$ is normal and $H^*(Y)$ is generated by Chern classes of line bundles, then $Y$ is principal, as this implies $H^*_{\mathfrak{T}}(Y) = {\mathcal{H}}_{\mathfrak{B}}^*(Y)$.