The Picard group of the moduli of \(G\)-bundles on a curve.

*(English)*Zbl 0976.14024Introduction: This paper is concerned with the moduli space of principal \(G\)-bundles on an algebraic curve of positive genus, for \(G\) a complex semi-simple group. While the case \(G=SL_r\), which corresponds to vector bundles, has been extensively studied in algebraic geometry, the general case has attracted much less attention until recently, when it became clear that these spaces play an important role in quantum field theory. In particular, if \(L\) is a holomorphic line bundle on the moduli space \(M_G\), the space \(H^0(M_G,L)\) is essentially independent of the curve \(X\), and can be naturally identified with what physicists call the space of conformal blocks associated to the most standard conformal field theory, the so-called WZW-model. This gives a strong motivation to determine the group \(\text{Pic} (M_G)\) of holomorphic line bundles on the moduli space.

Up to this point we have been rather vague about what we should call the moduli space of \(G\)-bundles on \(X\). Unfortunately there are two possible choices, and both are meaningful. Because \(G\)-bundles have usually nontrivial automorphisms, the natural solution to the moduli problem is not an algebraic variety, but a slightly more complicated object, the algebraic stack \({\mathcal M}_G\). This has all the good properties one expects from a moduli space; in particular, a line bundle on \({\mathcal M}_G\) is the functorial assignment, for every variety \(S\) and every \(G\)-bundle on \(X\times S\), of a line bundle on \(S\). There is also a more down-to-earth object, the coarse moduli space \(M_G\) of semi-stable \(G\)-bundles; the group \(\text{Pic}(M_G)\) is a subgroup of \(\text{Pic} ({\mathcal M}_G)\), but its geometric meaning is less clear.

We determine the groups \(\text{Pic} (M_G)\) and \(\text{Pic}({\mathcal M}_G)\) for essentially all classical semi-simple groups, i.e. of type \(A,B,C,D\) and \(G_2\). Since the simply-connected case was already treated by Y. Laszlo and C. Sorger [Ann. Sci. Ec. Norm. Supér., IV. Sér. 30, 499-525 (1997; Zbl 0918.14004)], we are mainly concerned with non simply-connected groups. One new difficulty appears: The moduli space is no longer connected, its connected components are naturally indexed by \(\pi_1(G)\). Let \(\widetilde G\) be the universal covering of \(G\); for each \(\delta\in \pi_1(G)\), we construct a natural ‘twisted’ moduli stack \({\mathcal M}^\delta_{\widetilde G}\) which dominates \({\mathcal M}_G^\delta\). (For instance if \(G=PGL_r\), it is the moduli stack of vector bundles on \(X\) of rank \(r\) and fixed determinant of degree \(d\), with \(e^{2\pi id /r}=\delta\).) This moduli stack carries in each case a natural line bundle \({\mathcal D}\), the determinant bundle associated to the standard representation of \(\widetilde G\). We can now state some of our results; for simplicity we only consider the adjoint groups.

Theorem. Put \(\varepsilon_G=1\) if the rank of \(G\) is even, 2 if it is odd. Let \(\delta\in \pi_1(G)\).

(a) The torsion subgroup of \(\text{Pic} ({\mathcal M}_G^\delta)\) is isomorphic to \(H^1(X,\pi_1 (G))\). The torsion-free quotient is infinite cyclic, generated by \({\mathcal D}^r\) if \(G=PGL_r\), by \({\mathcal D}^{\varepsilon_G}\) if \(G=PSp_{2l}\) or \(PSO_{2l}\).

(b) The group \(\text{Pic} (M_G^\delta)\) is infinite cyclic, generated by \({\mathcal D}^{r\varepsilon_G}\) if \(G=PGL_r\) by \({\mathcal D}^{2\varepsilon_G}\) if \(G=PSp_{2l}\) or \(PSO_{2l}\).

Unfortunately, though our method has some general features, it requires a case-by-case analysis; after our preprint appeared a uniform topological determination of \(\text{Pic} ({\mathcal M}_G)\) has been outlined by C. Teleman [“Borel-Weil-Bott theory on the moduli stack of \(G\)-bundles over a curve”, Invent. Math. 134, 1-57 (1998; Zbl 0980.14025)]. As a consequence of our analysis we prove that when \(G\) is of classical or \(G_2\) type, the moduli space \(M_G\) is locally factorial exactly when \(G\) is special in the sense of Serre [this is now also proved for exceptional groups: see S. Sorger, Ann. Sci. Éc. Norm. Supér., IV. Sér. 32, 127-133 (1999; Zbl 0969.14015)]. Nevertheless it is always a Gorenstein variety.

Up to this point we have been rather vague about what we should call the moduli space of \(G\)-bundles on \(X\). Unfortunately there are two possible choices, and both are meaningful. Because \(G\)-bundles have usually nontrivial automorphisms, the natural solution to the moduli problem is not an algebraic variety, but a slightly more complicated object, the algebraic stack \({\mathcal M}_G\). This has all the good properties one expects from a moduli space; in particular, a line bundle on \({\mathcal M}_G\) is the functorial assignment, for every variety \(S\) and every \(G\)-bundle on \(X\times S\), of a line bundle on \(S\). There is also a more down-to-earth object, the coarse moduli space \(M_G\) of semi-stable \(G\)-bundles; the group \(\text{Pic}(M_G)\) is a subgroup of \(\text{Pic} ({\mathcal M}_G)\), but its geometric meaning is less clear.

We determine the groups \(\text{Pic} (M_G)\) and \(\text{Pic}({\mathcal M}_G)\) for essentially all classical semi-simple groups, i.e. of type \(A,B,C,D\) and \(G_2\). Since the simply-connected case was already treated by Y. Laszlo and C. Sorger [Ann. Sci. Ec. Norm. Supér., IV. Sér. 30, 499-525 (1997; Zbl 0918.14004)], we are mainly concerned with non simply-connected groups. One new difficulty appears: The moduli space is no longer connected, its connected components are naturally indexed by \(\pi_1(G)\). Let \(\widetilde G\) be the universal covering of \(G\); for each \(\delta\in \pi_1(G)\), we construct a natural ‘twisted’ moduli stack \({\mathcal M}^\delta_{\widetilde G}\) which dominates \({\mathcal M}_G^\delta\). (For instance if \(G=PGL_r\), it is the moduli stack of vector bundles on \(X\) of rank \(r\) and fixed determinant of degree \(d\), with \(e^{2\pi id /r}=\delta\).) This moduli stack carries in each case a natural line bundle \({\mathcal D}\), the determinant bundle associated to the standard representation of \(\widetilde G\). We can now state some of our results; for simplicity we only consider the adjoint groups.

Theorem. Put \(\varepsilon_G=1\) if the rank of \(G\) is even, 2 if it is odd. Let \(\delta\in \pi_1(G)\).

(a) The torsion subgroup of \(\text{Pic} ({\mathcal M}_G^\delta)\) is isomorphic to \(H^1(X,\pi_1 (G))\). The torsion-free quotient is infinite cyclic, generated by \({\mathcal D}^r\) if \(G=PGL_r\), by \({\mathcal D}^{\varepsilon_G}\) if \(G=PSp_{2l}\) or \(PSO_{2l}\).

(b) The group \(\text{Pic} (M_G^\delta)\) is infinite cyclic, generated by \({\mathcal D}^{r\varepsilon_G}\) if \(G=PGL_r\) by \({\mathcal D}^{2\varepsilon_G}\) if \(G=PSp_{2l}\) or \(PSO_{2l}\).

Unfortunately, though our method has some general features, it requires a case-by-case analysis; after our preprint appeared a uniform topological determination of \(\text{Pic} ({\mathcal M}_G)\) has been outlined by C. Teleman [“Borel-Weil-Bott theory on the moduli stack of \(G\)-bundles over a curve”, Invent. Math. 134, 1-57 (1998; Zbl 0980.14025)]. As a consequence of our analysis we prove that when \(G\) is of classical or \(G_2\) type, the moduli space \(M_G\) is locally factorial exactly when \(G\) is special in the sense of Serre [this is now also proved for exceptional groups: see S. Sorger, Ann. Sci. Éc. Norm. Supér., IV. Sér. 32, 127-133 (1999; Zbl 0969.14015)]. Nevertheless it is always a Gorenstein variety.

##### MSC:

14H60 | Vector bundles on curves and their moduli |

14C22 | Picard groups |

14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

14L30 | Group actions on varieties or schemes (quotients) |