On finite generation of the section ring of the determinant of cohomology line bundle. (English) Zbl 1433.14027
Let \(C\) be a stable (projective) curve of arithmetic genus \(g \ge 2\).over complex numbers. Let \(Bun_G(C)\) denote the stack of principal \(G\)-bundles on \(C\), \(G\) being a simple, simply connected, complex linear algebraic group. Let \(D = D(V)\) denote the determinant of cohomology line bundle on \(\mathrm{Bun}_G(C)\) associated to a linear representation \(G \to \mathrm{GL}(V)\), for \(G = \mathrm{SL}(r)\) one takes the standard representation.
Main Theorem 1. For \(G = \mathrm{SL}(r)\), the section ring \[A^C_{\bullet} := \bigoplus_{m \in {\mathbb Z}\ge 0} H^0(\mathrm{Bun}_{\mathrm{SL}(r)}(C), D^{\otimes m})\] is finitely generated. As an application, the authors prove the following theorem.
Theorem 2. Let \(\overline{\mathcal M}_g\) denote the Deligne-Mumford compactification of the moduli space \(\mathcal{M}_g\) of smooth curves of genus \(g\). There is a flat family \(p: \chi \rightarrow \overline{\mathcal M}_g\), with \(\chi\) relatively projective over \(\overline{\mathcal M}_g\) such that \(\chi_{\mathbb{C}} \cong \mathrm{Proj}(A^C_{\bullet})\) is integral, normal and irreducible for \([C] \in \overline{\mathcal M}_g\), and \(\chi_{\mathbb C} \cong \mathrm{SU}_C(r)\) for \([C] \in \mathcal M_g\) where \(\mathrm{SU}_C(r)\) is the moduli space parametrising semistable vector bundles of rank \(r\) and degree \(0\) on \(C\).
It is well known that for a smooth curve \(C\), \(H^0(\mathrm{Bun}_{\mathrm{SL}(r)}(C), D^{\otimes m}) \cong H^0(\mathrm{SU}_C(r), \theta^{\otimes m})\),
Let g denote the Lie algebra of \(G\). For an integer \(\ell\), let \(\overline{\lambda}= (\lambda_1, \cdots, \lambda_n)\) be an \(n\)-tuple of dominant integral weights of \(\textbf{g}\) at level \(\ell\). Associated to this data, there is a vector bundle of conformal blocks, denoted by \(\mathbb{V}(\textbf{g}, \overline{\lambda}, \ell)\). It is a vector bundle over \(\overline{\mathcal{M}_{g,n}}\), the moduli space of \(n\)-pointed semistable curves. The authors show that the fibres of this bundle may be interpreted as the space of sections of a line bundle \(\mathcal{L}_G(C, \overline{p}, \overline{\lambda})\) on the stack parametrising generalised parabolic \(G\)-bundles on \(C\).
As a second application of Theorem 1, they show that under certain assumptions there are geometric interpretations for fibres of \(\mathbb{V}(sl_r, \ell)\vert^*_C\) at singular stable curves \([C] \in \overline{\mathcal{M}}_g\), as global sections of a line bundle on a projective variety.
Main Theorem 1. For \(G = \mathrm{SL}(r)\), the section ring \[A^C_{\bullet} := \bigoplus_{m \in {\mathbb Z}\ge 0} H^0(\mathrm{Bun}_{\mathrm{SL}(r)}(C), D^{\otimes m})\] is finitely generated. As an application, the authors prove the following theorem.
Theorem 2. Let \(\overline{\mathcal M}_g\) denote the Deligne-Mumford compactification of the moduli space \(\mathcal{M}_g\) of smooth curves of genus \(g\). There is a flat family \(p: \chi \rightarrow \overline{\mathcal M}_g\), with \(\chi\) relatively projective over \(\overline{\mathcal M}_g\) such that \(\chi_{\mathbb{C}} \cong \mathrm{Proj}(A^C_{\bullet})\) is integral, normal and irreducible for \([C] \in \overline{\mathcal M}_g\), and \(\chi_{\mathbb C} \cong \mathrm{SU}_C(r)\) for \([C] \in \mathcal M_g\) where \(\mathrm{SU}_C(r)\) is the moduli space parametrising semistable vector bundles of rank \(r\) and degree \(0\) on \(C\).
It is well known that for a smooth curve \(C\), \(H^0(\mathrm{Bun}_{\mathrm{SL}(r)}(C), D^{\otimes m}) \cong H^0(\mathrm{SU}_C(r), \theta^{\otimes m})\),
Let g denote the Lie algebra of \(G\). For an integer \(\ell\), let \(\overline{\lambda}= (\lambda_1, \cdots, \lambda_n)\) be an \(n\)-tuple of dominant integral weights of \(\textbf{g}\) at level \(\ell\). Associated to this data, there is a vector bundle of conformal blocks, denoted by \(\mathbb{V}(\textbf{g}, \overline{\lambda}, \ell)\). It is a vector bundle over \(\overline{\mathcal{M}_{g,n}}\), the moduli space of \(n\)-pointed semistable curves. The authors show that the fibres of this bundle may be interpreted as the space of sections of a line bundle \(\mathcal{L}_G(C, \overline{p}, \overline{\lambda})\) on the stack parametrising generalised parabolic \(G\)-bundles on \(C\).
As a second application of Theorem 1, they show that under certain assumptions there are geometric interpretations for fibres of \(\mathbb{V}(sl_r, \ell)\vert^*_C\) at singular stable curves \([C] \in \overline{\mathcal{M}}_g\), as global sections of a line bundle on a projective variety.
Reviewer: Usha N. Bhosle (Bangalore)
MSC:
| 14H60 | Vector bundles on curves and their moduli |
| 14D20 | Algebraic moduli problems, moduli of vector bundles |
| 14L24 | Geometric invariant theory |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
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