Equivariant Riemann–Roch theorems for curves over perfect fields

We prove an equivariant Riemann–Roch formula for divisors on algebraic curves over perfect fields. By reduction to the known case of curves over algebraically closed fields, we first show a preliminary formula with coefficients in . We then prove and shed some further light on a divisibility result that yields a formula with integral coefficients. Moreover, we give variants of the main theorem for equivariant locally free sheaves of higher rank.     

The Equivariant Hirzebruch–Riemann–Roch Theorem and the Geometry of Derived Loop Spaces

Mathematical Notes -     

Riemann–Roch theory for weighted graphs and tropical curves

We define a divisor theory for graphs and tropical curves endowed with a weight function on the vertices; we prove that the Riemann–Roch theorem holds in both cases. We extend Baker’s Specialization Lemma to weighted graphs.     

Monomials, binomials and Riemann–Roch

The Riemann–Roch theorem on a graph G is related to Alexander duality in combinatorial commutative algebra. We study the lattice ideal given by chip firing on G and the initial ideal whose standard monomials are the G-parking functions. When G is a saturated graph, these ideals are generic and the Scarf complex is a minimal free resolution. Otherwise, syzygies are obtained by degeneration. We also develop a self-contained Riemann–Roch theory for Artinian monomial ideals.     

The Riemann–Roch theorem for the Dynnikov–Novikov discrete complex analysis

We prove an analog of the Riemann–Roch Theorem for the Dynnikov–Novikov discrete complex analysis.     

A generalized Hirzebruch Riemann–Roch theorem

This short Note proves a generalization of the Hirzebruch Riemann–Roch theorem equivalent to the Cardy condition. This is done using an earlier result that explicitly describes what the Mukai pairing on Hochschild homology descends to in Hodge cohomology via the Hochschild–Kostant–Rosenberg map twisted by the root Todd genus. To cite this article: A.C. Ramadoss, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Un isomorphisme de type Deligne–Riemann–Roch

We prove an Adams–Riemann–Roch theorem for projective morphisms between regular schemes, in the sense of the program of P. Deligne on the functorial Riemann–Roch theorem and we deduce some geometric consequences. To cite this article: D. Eriksson, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

A Riemann–Roch theorem in tropical geometry

Recently, Baker and Norine have proven a Riemann–Roch theorem for finite graphs. We extend their results to metric graphs and thus establish a Riemann–Roch theorem for divisors on (abstract) tropical curves.     

On the Adams–Riemann–Roch theorem in positive characteristic

Let p > 0 be a prime number. We give a short proof of the Adams–Riemann–Roch theorem for the p-th Adams operation, when the involved schemes live in characteristic p. We also answer a question of B. Köck.     

Harder–Narasimhan theory for linear codes (with an appendix on Riemann–Roch theory)

In this text we develop some aspects of Harder–Narasimhan theory, slopes, semistability and canonical filtration, in the setting of combinatorial lattices. Of noticeable importance is the Harder–Narasimhan structure associated to a Galois connection between two lattices. It applies, in particular, to matroids.We then specialize this to linear codes. This could be done from at least three different approaches: using the sphere-packing analogy, or the geometric view, or the Galois connection construction just introduced. A remarkable fact is that these all lead to the same notion of semistability and canonical filtration. Relations to previous propositions toward a classification of codes, and to Wei's generalized Hamming weight hierarchy, are also discussed.Last, we study the two important questions of the preservation of semistability (or more generally the behavior of slopes) under duality, and under tensor product. The former essentially follows from Wei's duality theorem for higher weights—and its matroid version—which we revisit in an appendix, developing analogues of the Riemann–Roch, Serre duality, Clifford, and gap and gonality sequence theorems. Likewise the latter is closely related to the bound on higher weights of a tensor product, conjectured by Wei and Yang, and proved by Schaathun in the geometric language, which we reformulate directly in terms of codes. From this material we then derive semistability of tensor product.     

The Riemann–Roch theorem without denominators in motivic homotopy theory

The Riemann–Roch theorem without denominators for the Chern class maps on higher algebraic K-groups with values in motivic cohomology groups in the context of motivic homotopy theory is proved.     

Borel–Moore homology,Riemann–Roch transformations,and local terms

We develop various properties of étale Borel–Moore homology and study its relationship with intersection theory. Using Gabber's localized cycle classes we define étale homological Gysin morphisms and show that they are compatible with the cycle class map and Gysin morphisms in intersection theory. We also study étale versions of bivariant operations, and establish their compatibility with Riemann–Roch transformations and Fulton–MacPherson bivariant operations. As an application of these techniques we show that in certain situations local terms for correspondences acting on étale cohomology are given by cycle classes.     

Chip-firing based methods in the Riemann–Roch theory of directed graphs

Baker and Norine proved a Riemann–Roch theorem for divisors on undirected graphs. The notions of graph divisor theory are in duality with the notions of the chip-firing game of Björner, Lovász and Shor. We use this connection to prove Riemann–Roch-type results on directed graphs. We give a simple proof for a Riemann–Roch inequality on Eulerian directed graphs, improving a result of Amini and Manjunath. We also study possibilities and impossibilities of Riemann–Roch-type equalities in strongly connected digraphs and give examples. We intend to make the connections of this theory to graph theoretic notions more explicit via using the chip-firing framework.     

A Remark on an Algebraic Riemann—Roch Formula for Flat Bundles

Let f:X S be a smooth projective morphism over an algebraically closed field, with X and S regular. When E, ) is a flat bundle over X, then its Gauss–Manin bundles on S have a flat connection and one may ask for a Riemann–Roch formula relating the algebraic Chern–Simons and Cheeger–Simons invariants. We give an answer for X = Y × S, f = projection. The method of proof is inspired by the work of Hitchin and Simpson.     

Bases of Riemann–Roch spaces from Kummer extensions and algebraic geometry codes

We construct explicit bases of Riemann–Roch spaces from Kummer extensions and algebraic geometry codes with good parameters. This correspondence is a generalization of a work of Maharaj, Matthews, and Pirsic.     

Théorèmes de Riemann–Roch pour les champs de Deligne–Mumford

We develop a cohomology theory for Deligne–Mumford stacks, adapted to Hirzebruch–Riemann–Roch formulas. For this, we define the cohomology with coefficients in the representations and a Chern character, and we prove a Grothendieck–Riemann–Roch formula for the associated Riemann–Roch transformation.     

Single periodic Riemann problems for null-solutions to polynomially Cauchy–Riemann equations

In this paper, we focus on single periodic Riemann problems for a class of meta-analytic functions, i.e. null-solutions to polynomially Cauchy–Riemann equation. We first establish decomposition theorems for single periodic meta-analytic functions. Then, we give a series expansion of single periodic meta-analytic functions, and derive generalised Liouville theorems for them. Next, we introduce a definition of order for single periodic meta-analytic functions at infinity, and characterise their growth at infinity. Finally, applying the decomposition theorem for single periodic meta-analytic functions, we get explicit expressions of solutions and condition of solvability to Riemann problems for single periodic meta-analytic functions with a finite order at infinity.