1 Introduction

In [2] Borho and MacPherson proved that the nilpotent cone is a rational homology manifold. The proof relies on the celebrated Decomposition Theorem by Beilinson, Bernstein, Deligne and Gabber [1] and on the Springer’s theory of Weyl group representations (see [2] and the references therein).

The aim of this paper is to present a new proof, in our opinion conceptually very simple, based on the bivariant theory founded by Fulton and MacPherson in [4]. Actually, our approach enables us to prove a slightly more general statement (see Remark 2.4 below). By bivariant theory we intend the topological bivariant homology theory with coefficients in a Noetherian commutative ring with identity \({\mathbb {A}}\) [4, pp. 32, 83 and p. 86, Corollary 7.3.4].

That the nilpotent cone is a rational homology manifold can be seen as an easy consequence of a characterization of homology manifolds we recently proved in [3, Theorem 6.1]: given a resolution of singularities \(\pi :{\widetilde{{\mathcal {N}}}}\rightarrow {\mathcal {N}}\) of a quasi-projective variety \({\mathcal {N}}\) , then \({\mathcal {N}}\) is a homology manifold if and only if there exists a bivariant class of degree one for \(\pi \). A bivariant class of degree one for \(\pi \) is an element \(\eta \in H^0(\mathcal {\widetilde{N}}{\mathop {\rightarrow }\limits ^{\pi }}{\mathcal {N}})\) such that the induced Gysin homomorphism \(\eta _0:H^0({\widetilde{{\mathcal {N}}}})\rightarrow H^0({\mathcal {N}})\) sends \(1_{{\widetilde{{\mathcal {N}}}}}\) to \(1_{{\mathcal {N}}}\).

2 The Main Result

Theorem 2.1

Let \(\pi ':{\mathbf {{{\widetilde{g}}}}}\rightarrow {{\textbf{g}}}\) be a projective morphism between complex quasi-projective nonsingular varieties of the same dimension. Assume that \(\pi '\) is generically finite, of degree \(\delta \). Let \({\mathcal {N}}\subset {{\textbf{g}}}\) be a closed irreducible subvariety. Consider the induced fibre square diagram:

where \({\widetilde{{\mathcal {N}}}}:={\mathcal {N}}\times _{{\textbf{g}}}{\mathbf {{{\widetilde{g}}}}}\). If \({\widetilde{{\mathcal {N}}}}\) is irreducible and nonsingular and \(\pi \) is birational, then \({\mathcal {N}}\) is an \({\mathbb {A}}\)-homology manifold for every Noetherian commutative ring with identity \({\mathbb {A}}\) for which \(\delta \) is a unit.

Proof

Since \(\pi ':{\mathbf {{{\widetilde{g}}}}}\rightarrow {{\textbf{g}}}\) is a projective morphism between complex quasi-projective nonsingular varieties of the same dimension, it is a local complete intersection morphism of relative codimension 0 [4, p. 130]. Let

$$\begin{aligned} \theta ' \in H^0({\mathbf {{{\widetilde{g}}}}}{\mathop {\rightarrow }\limits ^{\pi '}}{\textbf{g}})\cong Hom_{D^{b}_{c}({{\textbf{g}}})}(R{\pi '}_*\mathbb A_{{\mathbf {{{\widetilde{g}}}}}}, {\mathbb {A}}_{{{\textbf{g}}}}) \end{aligned}$$

be the orientation class of \(\pi '\) [4, p. 131]. Let \(\theta '_0:H^0({\mathbf {{{\widetilde{g}}}}})\rightarrow H^0({{\textbf{g}}})\) be the induced Gysin map. It is clear that \(\theta '_0(1_{\mathbf {{{\widetilde{g}}}}})=\delta \cdot 1_{{\textbf{g}}}\in H^0({{\textbf{g}}})\), where \(\delta \) is the degree of \(\pi '\). Therefore, if we denote by

$$\begin{aligned} \theta :=i^*\theta '\in H^0({\widetilde{{\mathcal {N}}}}{\mathop {\rightarrow }\limits ^{\pi }}{\mathcal {N}})\cong Hom_{D^{b}_{c}({\mathcal {N}})}(R{\pi }_*{\mathbb {A}}_{\mathcal {\widetilde{N}}}, {\mathbb {A}}_{{\mathcal {N}}}) \end{aligned}$$

the pull-back of \(\theta '\), then \(\delta ^{-1}\cdot \theta \) is a bivariant class of degree one for \(\pi \) [3, 2. Notations, (ii)]. At this point, our claim follows by [3, Theorem 6.1]. For the Reader’s convenience, let us briefly summarize the argument.

Since \(\delta ^{-1}\cdot \theta \) is a bivariant class of degree one for \(\pi \), it follows that \(\left( \delta ^{-1}\cdot \theta \right) \circ \pi ^*={\text {id}}_{{\mathbb {A}}_{\mathcal {N}}}\) in \({D^{b}_{c}({\mathcal {N}})}\), i.e. that \(\delta ^{-1}\cdot \theta \) is a section of the pull-back \(\pi ^*:{\mathbb {A}}_{{\mathcal {N}}}\rightarrow R\pi _*{\mathbb {A}}_{{\widetilde{{\mathcal {N}}}}}\) [3, Remark 2.1, (i)]. Hence, \({\mathbb {A}}_{{\mathcal {N}}}\) is a direct summand of \(Rf_*{\mathbb {A}}_{{\widetilde{{\mathcal {N}}}}} \) in \({D^{b}_{c}(\mathcal N)}\) [3, Lemma 3.2] and so we have a decomposition

$$\begin{aligned} Rf_*{\mathbb {A}}_{{\widetilde{{\mathcal {N}}}}} \cong {\mathbb {A}}_{\mathcal { N}} \oplus {\mathcal {K}}. \end{aligned}$$
(1)

Now, set \(\nu =\dim {\mathcal {\widetilde{N}}}=\dim {{\mathcal {N}}}\) and let \([{{\widetilde{{\mathcal {N}}}}}]\in H_{2\nu }({{\widetilde{{\mathcal {N}}}}})\) be the fundamental class of \({{\widetilde{{\mathcal {N}}}}}\). We have:

$$\begin{aligned} {[}{{\widetilde{{\mathcal {N}}}}}]\in H_{2\nu }({\mathcal {\widetilde{N}}})\cong H^{-2\nu }({{\widetilde{{\mathcal {N}}}}}{\mathop {\rightarrow }\limits ^{}}pt.)\cong Hom_{D^{b}_{c}({{\widetilde{{\mathcal {N}}}}})}(\mathbb A_{{\widetilde{{\mathcal {N}}}}}[\nu ],D\left( \mathbb A_{{\widetilde{{\mathcal {N}}}}}[\nu ]\right) ), \end{aligned}$$

where D denotes Verdier dual. Therefore, \([{\mathcal {\widetilde{N}}}]\) corresponds to a morphism

$$\begin{aligned} {\mathbb {A}}_{{\widetilde{{\mathcal {N}}}}}[\nu ]\rightarrow D\left( \mathbb A_{{\widetilde{{\mathcal {N}}}}}[\nu ]\right) , \end{aligned}$$
(2)

whose induced map in hypercohomology is nothing but the duality morphism

$$\begin{aligned} {\mathcal {D}}_{{\widetilde{{\mathcal {N}}}}}: x\in H^{\bullet }({{\widetilde{{\mathcal {N}}}}})\rightarrow x\cap [{\mathcal {\widetilde{N}}}]\in H_{2\nu -\bullet }({{\widetilde{{\mathcal {N}}}}}). \end{aligned}$$
(3)

If we assume that \({{\widetilde{{\mathcal {N}}}}}\) is nonsingular (actually it suffices that \({{\widetilde{{\mathcal {N}}}}}\) is an \(\mathbb A\)-homology manifold), the morphisms (2) and (3) are isomorphisms. The first one induces an isomorphism

$$\begin{aligned} R\pi _*{\mathbb {A}}_{{\widetilde{{\mathcal {N}}}}}[\nu ]\rightarrow D\left( R\pi _*{\mathbb {A}}_{{\widetilde{{\mathcal {N}}}}}[\nu ]\right) , \end{aligned}$$

which in turn, via decomposition (1), induces two projections

$$\begin{aligned} {\mathbb {A}}_{{\mathcal {N}}}[\nu ] \rightarrow D\left( \mathbb A_{{\mathcal {N}}}[\nu ]\right) , \quad {\mathcal {K}}[\nu ]\rightarrow D\left( {\mathcal {K}}[\nu ]\right) . \end{aligned}$$
(4)

Making explicit the isomorphism induced in cohomology and homology by (1), one may prove [3, Corollary 5.1] that \(\mathcal D_{{\widetilde{{\mathcal {N}}}}}\) is the direct sum of \(P_1\) and \(P_2\), where

$$\begin{aligned} P_1: H^{\bullet }({{\mathcal {N}}})\rightarrow H_{2\nu -\bullet }({{\mathcal {N}}}) \quad {\text {and}}\quad P_2:{\mathbb {H}}({\mathcal {K}}[\nu ])\rightarrow \mathbb H(D\left( {\mathcal {K}}[\nu ]\right) ) \end{aligned}$$

are the maps induced in hypercohomology by the projections (4). It follows that \(P_1\) is an isomorphism, because so is \({\mathcal {D}}_{{\widetilde{{\mathcal {N}}}}}\), and this holds true when restricting to every open subset U of \({\mathcal {N}}\). For instance (see also [3, Corollary 5.1]), if \({{\widetilde{U}}} = \pi ^{-1}(U)\), the vanishing of the morphism \({\mathbb {H}}^{\bullet }({\mathcal {K}}_U[\nu ]) \rightarrow H_{2\nu -\bullet }(U)\) derives from projection formula [4, p. 26, G4, (ii)]:

$$\begin{aligned} \pi _*([{{\widetilde{U}}}]\cap \lambda _* w)=\pi _*(\delta ^{-1}\theta ^*[U]\cap \lambda _* w)=\delta ^{-1}(\theta _*\lambda _* w)\cap [U]=0, \quad \forall w \in {\mathbb {H}}^{\bullet }({\mathcal {K}}_U[\nu ]), \end{aligned}$$

where \(\lambda _*\) is the morphism induced in hypercohomology by \({\mathcal {K}}_U[\nu ]\rightarrow R\pi _*{\mathbb {A}}_{{{\widetilde{U}}}}[\nu ]\).

Therefore, we have \({\mathbb {A}}_{{\mathcal {N}}}[\nu ] \cong D\left( {\mathbb {A}}_{{\mathcal {N}}}[\nu ]\right) \), which is equivalent to say that \({\mathcal {N}}\) is an \({\mathbb {A}}\)-homology manifold. \(\square \)

Remark 2.2

Observe that, as a scheme, \({\widetilde{{\mathcal {N}}}}\) could also be nonreduced, but what matters is that, for the usual topology, it is a nonsingular variety [4, p. 32, 3.1.1].

Corollary 2.3

The nilpotent cone is a rational homology manifold.

Proof

Let \(\pi : {\widetilde{{\mathcal {N}}}}\rightarrow {\mathcal {N}}\) be the Springer resolution of the nilpotent cone \({\mathcal {N}}\). It extends to a generically finite projective morphism \(\pi ':{\mathbf {{{\widetilde{g}}}}}\rightarrow {{\textbf{g}}}\), known as the Grothendieck simultaneous resolution, between complex quasi-projective nonsingular varieties of the same dimension [2, p. 49]. Therefore, Theorem 2.1 applies. \(\square \)

Remark 2.4

If the Grothendieck simultaneous resolution \(\pi ': {\mathbf {{\widetilde{g}}}}\rightarrow {{\textbf{g}}}\) has degree \(\delta \), by Theorem 2.1 we deduce that the nilpotent cone \({\mathcal {N}}\) is an \({\mathbb {A}}\)-homology manifold for every Noetherian commutative ring with identity \({\mathbb {A}}\) for which \(\delta \) is a unit. For instance, for the variety \({\mathcal {N}}\) of nilpotent matrices in \({\text {GL}}(n,{\mathbb {C}})\), we have \(\delta =n!\). Therefore, in this case, \({\mathcal {N}}\) is also a \({\mathbb {Z}}_h\)-homology manifold for every integer h relatively prime with n! in \({\mathbb {Z}}\).