Chapter 4 Algebro-geometric interpretation

In algebraic geometry the classical (abelian) Torelli is a theorem and an algorithm that reconstructs a closed Riemann surface (smooth projective algebraic curve) \(C\) from its Jacobian \(J(C)\) equipped with a principal polarization by \(\Theta\)-divisor. Usually “Torelli package” also has a description of the automorphism group of \((J(C),\Theta)\) in terms of automorphisms of \(C\), or more generally classifies all isomorphisms between \((J(C),\Theta)\) and \((J(C'),\Theta')\).

The Jacobian \(J(C)\) has an interpretation as (a connected component of) the Picard variety, that parametrizes isomorphism classes of line bundles of fixed numerical degree. Similarly the moduli space \(U_r(C)\) parametrizes (\(S\)-equivalence classes of semi-)stable vector bundles of rank \(r\). Let \(SU_r(C,L)\) be the fiber of the determinantal fibration \(\det: U_r(C) \to U_1(C)\) over a line bundle \(L\in U_1(C)\). Any isomorphism of curves \(f : C \to C'\) induces an isomorphism \(SU_r(C,L) \to SU_r(C',f_* L)\), and any line bundle \(M \in U_1(C)\) induces an isomorphism \(SU_r(C,L) \to SU_r(C,L\otimes M^{\otimes r})\) given by \(E \mapsto E \otimes M\). These isomorphisms generate a groupoid with objects being the pairs \((C,L)\) of an algebraic curve \(C\) equipped with a line bundle \(L\) and morphisms from \((C,L)\) to \((C',L')\) given by triples \((f,M';\iota)\) of an isomorphism of curves \(f: C \to C'\), a line bundle \(M'/C'\) an an isomorphism of line bundles \(\iota: f_* L \to L' \otimes M'^{\otimes r}\), so the association \((C,L) \mapsto SU_r(C,L)\) is promoted to a functor from this groupoid to the groupoid of algebraic varieties and isomorphisms.

Unlike Jacobians the spaces \(SU_r(C,L)\) have (anti-)canonical polarizations, and a non-abelian Torelli theorem for smooth curves was formulated by Andrey Tyurin following the foundational theorems of Narasimhan-Ramanan and Newstead. Kouvidakis and Pantev (1995) use Higgs bundles to reconstruct the curve \(C\) from the respective moduli space \(SU_r(C,L)\) and also describe the group of automorphisms of these moduli spaces as an extension of the automorphisms of the curve by \(r\)-torsion subgroup in Jacobian, effectively proving that under mild assumptions \(SU_r\) considered as a functor between groupoids is full for any \(r>1\).

In the combinatorial versions of these theorems trivalent graphs are avatars of Riemann surfaces, colorings are avatars of line bundles, polytopes are avatars of moduli spaces, and the functor \(P\) is an avatar of the functor \(SU_2\).

The analogy between curves and trivalent graphs is standard and can be explained by the following underlying geometry. With a graph one can associate a graph curve, the union of projective lines enumerated by vertices of the graph with some common points (nodes) corresponding to the edges. Leaves (if there are any) correspond to additional marked smooth points. These curves form a finite set of the deepest corners of the Deligne-Mumford moduli space of stable marked curves. A path between the corner and the interior of the Deligne-Mumford space corresponds to a degeneration of a smooth algebraic curve to the graph curve, and the respective vanishing cycles give the Thurston cut system of the Riemann surface into trinions (i.e. spheres with three holes or pairs-of-pants) encoded by the same graph.

Moduli spaces such as Jacobians and \(SU_r(C,L)\) vary together with the smooth curve \(C\), and it is natural to look for their limits as \(C\) tends to a graph curve \(C(G)\).