A Local Approach to the Deligne-Riemann-Roch Theorem

The classical Riemann-Roch theorem on compact Riemann surfaces was discovered in the XIX-th century (in 1857 and 1865) and describes the dimensions of the spaces of sections of holomorphic line bundles on these surfaces. Since then, this theorem has been greatly generalized. At the end of the XIX-th century, it was generalized by M. Noether to projective algebraic surfaces. In the middle and second half of the XX-th century, it was generalized first by F. Hirzebruch to projective complex algebraic varieties, and then A. Grothendieck generalized this theorem to the relative situation, that is, to some morphisms between algebraic varieties. Finally, in the mid-80s of the XX-th century, P. Deligne obtained a functorial version of this theorem, which is related to the moduli space of pairs: an algebraic curve and a line bundle on it, and gives an isomorphism of some line bundles on the moduli space.
I will describe a local approach to this Deligne-Riemann-Roch theorem. It consists in calculating the class of a central extension of some infinite-dimensional group. The tangent Lie algebra of this group is the Lie algebra of differential operators of order at most 1 acting on the formal punctured disk. In this case, calculating the class of a central extension of a group is reduced to calculating the corresponding class of a central extension of the Lie algebra. The mentioned group acts on the moduli space of the following data (quintets): a projective algebraic curve, a line bundle on this curve, a smooth point on this curve, a local parameter at this point, a trivialization of the line bundle in a formal neighborhood of this point. This action induces surjective maps from the Lie algebra to the tangent spaces of this moduli space. Besides, the moduli space of these quintets is naturally mapped to the moduli space of the pairs mentioned above.