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Chern classes and Hirzebruch-Riemann-Roch theorem for coherent sheaves on complex-projective orbifolds with isolated singularities. (English) Zbl 0949.14006

From the introduction: Let \(X\) be a complex-projective manifold and let \({\mathcal F}\) be a locally free sheaf on \(X\); set \(n:= \dim_{\mathbb{C}} X\). Then the construction and the properties of the Chern classes \(c_i({\mathcal F})\in H^{2i} (X,\mathbb{Z})\) \((i=0,\dots, n)\) are well known; moreover it is well known that the holomorphic Euler characteristic \(\chi(X,{\mathcal F})\) can be computed from the Chern class \(c_i({\mathcal F})\) and \(c_j({\mathcal T}_X)\) using the Hirzebruch-Riemann-Roch (HRR) theorem \(\chi(X,{\mathcal F})= \int_X \text{td} ({\mathcal T}_X)\cdot \text{ch} ({\mathcal F})\), where \({\mathcal T}_X\) is the holomorphic tangent sheaf of \(X\).
Since 1954, when F. Hirzebruch annouced his RR theorem, this theory has been generalized by many people into various directions. However, one generalization of the theory of Chern classes and of the RR theorem is missing and may even be possible only to a limited extension: This is the theory of the Chern classes and of the RR theorem for pairs \((X,{\mathcal F})\), where \(X\) and \({\mathcal F}\) are allowed to be singular at the same time; i.e. e.g. \(X\) is an arbitrary normal complex-projective variety and \({\mathcal F}\) is an arbitrary coherent sheaf on \(X\). The main purpose of this paper is to solve this problem in case \(X\) has only isolated quotient singularities; the main results are theorems 3.5, 3.17, 5.9 and 6.1.
A first remark is on terminology: Following I. Satake, we always speak of “V-manifolds” instead of “orbifolds”; moreover, for the sake of brevity, we speak of “Vi-manifolds” when we mean “V-manifolds with isolated singularities”. In section 1, we study thoroughly the de Rham cohomology of \(V\)-manifolds and in particular the connection between the de Rham cohomology of a Vi-manifold \(X\) to that of its resolution \(\widetilde{X}\). In section 2, we study locally V-free sheaves on V-manifolds and their Chern classes, following I. Satake, W. Baily, R. Kobayashi, and Y. Kawamata.
Definition of the holomorphic orbifold Euler characteristic of a locally V-free sheaf \({\mathcal F}\) on a V-manifold \(X: \chi_{\text{orb}} (X,{\mathcal F}):= \int_X \text{ch}({\mathcal F})\cdot \text{td}(X)\). Then, for a Vi-manifold \(X\), the HRR theorem 3.5 states that \[ \chi(X,{\mathcal F})= \chi_{\text{orb}} (X,{\mathcal F})+ \sum_{x\in \operatorname {Sing}X} \mu_{X,x} ({\mathcal F}); \] the correction terms \(\mu_{X,x}\) are group homomorphisms from the Grothendieck groups \(K_{VV} (X,x)\) to the rational numbers \(\mathbb{Q}\), where \(K_{VV}(X,x) \cong R(G_x)\), the representation ring of the local fundamental group of the V-manifold \(X\) at the point \(x\). It is an important observation that \(\mu_{X,x}\) does not descend to the “usual” Grothendieck group \(K(X,x)\) on the singularity \((X,x)\). The proof of theorem 3.5 uses resolution of singularities \(\widetilde{X}@> \sigma>>X\) of the given V-manifold \(X\); moreover, local Chern classes play an important role in the proof of theorem 3.5. This also leads to the local HRR theorem 3.16: \[ \chi(x, \widetilde{\mathcal F})= -\chi_{\text{orb}} (x,\widetilde{\mathcal F})+ \mu_{X,x} ({\mathcal F}), \] where \({\mathcal F}= (\sigma_*\widetilde{\mathcal F})^{\vee\vee}\), and where \(\chi(x, \widetilde{\mathcal F})\), respectively \(\chi_{\text{orb}} (x,\widetilde{\mathcal F})\) are local characteristics of the sheaf \(\widetilde{\mathcal F}\) with respect to the resolution \(\sigma\). In theorem 3.17, we work out the purely group theoretical Atiyah-Bott expression \[ \mu_{X,x} ({\mathcal F})= \frac{1}{\# G}\cdot \sum_{g\neq \text{id}} \frac{\text{trace} (\rho(g))} {\det (1_n-g)}, \] where \(\rho\) is the representation associated to \({\mathcal F}\).
Paragraph 4 is devoted to the study of asymptotic RR theorems and contains a comparison of our local Chern numbers to a recent notion of local Chern number due to J. Wahl. In section 5, we construct the Chern classes of coherent sheaves of compact Vi-manifolds, see theorem 5.9. The local construction is carried out in theorem 5.8. In the last paragraph, we prove a HRR theorem for coherent sheaves \(\mathcal S\) on Vi-manifolds which uses the Chern classes constructed in section 5. We choose liftings \(\widehat{\mathcal S}|_{(Y,y)}\) of \({\mathcal S}|_{(X,x)}\) to every local smoothing covering \((Y,y)\to(X,x)\) of \(X\), and then associate the Chern classes \(c_i\) to the pair \(({\mathcal S},\{{\mathcal S}|_{(Y,y)}|\) \(x\in \text{Sing} X\}).\)
Similar to theorem 3.5, it states that \[ \chi(X,{\mathcal S})= \chi_{\text{orb}} (X,({\mathcal S},\{\widehat{\mathcal S}_x))+ \sum_{x\in \text{Sing }X} \mu_X,x (({\mathcal S},\widehat{\mathcal S}_x)) \] where we now have to consider the pairs \(({\mathcal S},\{\widehat{\mathcal S}_x\})\) rather than the sheaf \({\mathcal S}\) alone, cf. theorem 6.1.

MSC:

14C40 Riemann-Roch theorems
32S20 Global theory of complex singularities; cohomological properties
14B05 Singularities in algebraic geometry
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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References:

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