Abstract
We exhibit a new proof, relying on bivariant theory, that the nilpotent cone is rationally smooth. Our approach enables us to prove a slightly more general statement.
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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
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
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
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:
where D denotes Verdier dual. Therefore, \([{\mathcal {\widetilde{N}}}]\) corresponds to a morphism
whose induced map in hypercohomology is nothing but the duality morphism
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
which in turn, via decomposition (1), induces two projections
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
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)]:
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 \({\mathbb {Z}}\).
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References
Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981), Astérisque, 100, Soc. Math. France, Paris, , 5–171 (1982)
Borho, W., MacPherson, R.: Partial Resolutions of Nilpotent Varieties. Astérisque 101–102, 23–74 (1983)
Di Gennaro, V., Franco, D., Sessa, C.: Bivariant class of degree one. Results Math. 77(6),(2022). Paper No. 227
Fulton, W., MacPherson, R.: Categorical framework for the study of singular spaces. Mem. Am. Math. Soc. 31(243) (1981). pp. vi+165
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Di Gennaro, V., Franco, D. & Sessa, C. Nilpotent Cone and Bivariant Theory. Results Math 78, 223 (2023). https://doi.org/10.1007/s00025-023-01989-1
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DOI: https://doi.org/10.1007/s00025-023-01989-1
Keywords
- Nilpotent cone
- decomposition theorem
- Springer theory
- Weyl group
- bivariant theory
- Gysin homomorphism
- homology manifold
- resolution of singularities
- Grothendieck simultaneous resolution