The geometric correspondence for singular curves (Q618250)

In the article under review, Poirier generalizes the geometric Langlands correspondence over \(\mathbb C\) from smooth curves to curves with ordinary multiple point singularities. In the first section, the author works within the category of complex analytic spaces. For an integral projective singular curve \(X\) of positive genus, she considers the image \(\underline{\Omega}\) of the sheaf of holomorphic differentials on \(X\) within the direct image of the sheaf of holomorphic differentials of the normalization of \(X\). She defines a point \(P\) on \(X\) to be a mild singularity of \(X\) if the natural morphism \(\underline{d}_P:\mathcal O_{X,P}\rightarrow\underline{\Omega}_P\) satisfies \(\underline{d}_P\mathfrak m_P^s=\mathfrak m_P^{s-1}\underline{\Omega}_P\) for all \(s\geq 1\), where \(\mathfrak m_P\) denotes the maximal ideal of \(\mathcal O_{X,P}\). If \(P\) is mildly singular, then \(\underline{d}_P\) is surjective. Examples for mild singularities include special singularities, which are singular points \(P\) of \(X\) with the property that the conductor of \(\mathcal O_{X,P}\) equals \(\mathfrak m_P\); ordinary multiple points are special. The author gives an example of a curve \(X_{\text{not mild}}\) with a cusp singularity \(P\) where \(\underline{d}_P\) fails to be surjective. Poirier then defines connections \((\mathcal{M},\nabla)\) on general curves \(X\) as above, using the sheaf \(\underline{\Omega}\) instead of the sheaf of holomorphic differentials \(\Omega^1_{X/\mathbb C}\). For the curve \(X_{\text{not mild}}\), she gives an example of a connection whose kernel is not a local system, thereby showing that the equivalence of categories between connections and local systems, valid for smooth curves, fails for \(X_{\text{not mild}}\). She then proves the first main theorem of her paper, stating that if \(X\) has at most mildly singular points, then the functor \((\mathcal{M},\nabla)\mapsto\ker(\mathcal{M},\nabla)\) establishes an equivalence between the category of connections on \(X\) and the category of local systems on \(X\). Her proof consists in finding, locally at a singular point \(P\) and after a choice of basis for \(\mathcal{M}_P\), a fundamental matrix \(Y\) for \(\nabla_P\). Poirier first constructs a formal solution \(\hat{Y}\) to the problem, with entries in the formal completion \(\hat{\mathcal O}_{X,P}\) of \(\mathcal O_{X,P}\), using the mildness assumption; she then uses general results on differential equations to show that the entries of \(\hat{Y}\) are convergent. In the second section, Poirier continues to work in the above setting, albeit imposing the stronger assumption that the singularities of \(X\) be at most multiple ordinary points. A local system \(E\) on \(X\) yields, via pullback, a local system \(s^*E\) on the normalization \(s:\tilde{X}\rightarrow X\) of \(X\). To go the opposite way, level structures are required. Poirier defines notions of level structures for local systems and connections on \(\tilde{X}\): a level structure on a local system \(E\) on \(\tilde{X}\) is defined to consist, for each singular point \(P\) of \(X\), of a compatible collection of isomorphisms between the stalks \(E_{P'}\) in the points \(P'\) of the \(s\)-fiber of \(P\). Level structures for connections on \(\tilde{X}\) are defined similarly; here the definition can be extended to any curve with at most special singularities. Poirier observes that the categories of local systems with level structure on \(\tilde{X}\) and connections with level structure on \(\tilde{X}\) are equivalent, and she also establishes an equivalence with the categories of local systems respectively connections on \(X\). In section 3, Poirier works in the category of algebraic varieties over \(\mathbb C\); as before, she considers integral projective curves of positive genus. She first recalls the fact that given a smooth such curve together with a finite set of positive divisors \(D_1,\dots,D_r\) with disjoint supports, one can construct a curve \(X=Y_{D_1,\dots,D_r}\) with at most special singularities having \(Y\) as a normalization such the support of \(\sum_i D_i\) coincides with the preimage of the singular locus of \(X\). She then defines an equivalence relation on the group of divisors of \(X_{\text{smooth}}\) by declaring that two such divisors be equivalent if and only if they differ by a rational function that is constant modulo the \(D_i\); she defines \(J_{D_1,\dots,D_r}\) to be the resulting group of equivalence classes. Given a divisor \(M\) on \(X_{\text{smooth}}\), the author computes the space of positive divisors on \(X_{\text{smooth}}\) that are equivalent to \(M\) in the above sense, and she gives a criterion for this space being nonempty, using the generalized Riemann-Roch theorem and regular differential forms. Poirier then defines a notion of \((D_1,\dots,D_r)\)-level structure for invertible sheaves on \(\tilde{X}\), and she introduces an equivalence relation on the set of invertible sheaves on \(\tilde{X}\) with level structure. The resulting set \(F(D_1,\dots,D_r)\) of equivalence classes carries a natural group structure, and there is a canonical group homomorphism onto \(\text{Pic}(\tilde{X})\) whose kernel is computed explicitly. Poirier establishes a natural isomorphism \(J_{D_1,\dots,D_r}\overset{\sim}{\rightarrow}F(D_1,\dots,D_r)\); its surjectivity is based on the aforementioned criterion for the existence of certain positive divisors. The author then defines a functor \(\mathcal F(D_1,\dots,D_r)\) from the category of \(\mathbb C\)-schemes to the category of abelian groups whose group of \(\mathbb C\)-valued points is \(F(D_1,\dots,D_r)\); she exhibits a surjection of presheaves from \(\mathcal F(D_1,\dots,D_r)\) onto the relative Picard functor \(\text{Pic}_X\) of \(X\), and she computes its kernel. Using rigidified line bundels with level structure, Poirier shows that \(\mathcal F(D_1,\dots,D_r)\) is representable by a commutative group scheme of finite type \(J_{D_1,\dots,D_r}\) over \(\mathbb C\). The author then shows that \(J_{D_1,\dots,D_r}\) is canonically isomorphic to the Jacobian variety of \(X\); as a corollary, she gives a description of the Jacobian of an arbitrary integral projective curve over \(\mathbb C\) of positive genus, using birational approximation by a special singular curve. In the fourth and final section, Poirier works in the \(\mathbb C\)-analytic category; she considers integral projective curves over \(\mathbb C\) with positive genus and only ordinary multiple points as singularities, and she generalizes the geometric Hecke correspondence to curves \(X\) of this type. More specifically, she defines, using the explicit description of \(J_{D_1,\dots,D_r}\) given in section 3, a morphism \(H\) from \((\tilde{X}\setminus S)\times J_{D_1,\dots,D_r}\) to \(J_{D_1,\dots,D_r}\), where \(\tilde{X}\) is the normalization of \(X\) and where \(S\subset\tilde{X}\) is the union of the fibers above the singular points of \(X\). She then shows the final main result of the paper, that a rank one connection with level structure on \(\tilde{X}\) admits a `preimage' under \(H\). The proof again uses the explicit structure of \(J_{D_1,\dots,D_r}\) and its relation to the relative Picard functor. The paper is very well written; Poirier explains the arguments in detail, and her exposition is clear.