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The Deligne-Riemann-Roch isomorphism

Dennis Eriksson, Gerard Freixas i Montplet

TL;DR

The paper develops a comprehensive geometric framework for Deligne’s refined Grothendieck–Riemann–Roch program, constructing a canonical Deligne–Riemann–Roch isomorphism for lci morphisms via a universal Chern-category and a relative intersection-theory of line distributions. It systematically builds DRR for regular closed immersions and projective bundles, then assembles these into a broad, functorial DRR isomorphism for general lci morphisms, with a detailed treatment of base change, projection, and Grothendieck duality. The authors derive far-reaching consequences, including explicit Knudsen–Mumford expansion terms, a determinant-of-de-Rham-cohomology formula, and BCOV-type isomorphisms for Calabi–Yau families, linking to mirror symmetry in genus one. The work unifies and extends prior results (Mumford, Moret-Bailly, Ducrot, Saito–Terasoma) within a robust categorical framework, enabling new applications to Arakelov geometry, higher CM-line bundles, and Calabi–Yau moduli. Overall, the DRR formalism provides a canonical, base-change compatible, and compositional approach to determinants of cohomology in higher dimensions, with clear geometric and arithmetic implications.

Abstract

This work establishes the geometric component of Deligne's longstanding program on refined Grothendieck-Riemann-Roch formulas expressed through determinants of cohomology. The approach relies on a newly developed universal category of Chern classes together with an associated relative intersection theory. As an example of the applications, we provide a structural description of the coefficients in the Knudsen-Mumford expansion and establish a fundamental Mumford-type isomorphism for the alternating product of Griffiths bundles.

The Deligne-Riemann-Roch isomorphism

TL;DR

The paper develops a comprehensive geometric framework for Deligne’s refined Grothendieck–Riemann–Roch program, constructing a canonical Deligne–Riemann–Roch isomorphism for lci morphisms via a universal Chern-category and a relative intersection-theory of line distributions. It systematically builds DRR for regular closed immersions and projective bundles, then assembles these into a broad, functorial DRR isomorphism for general lci morphisms, with a detailed treatment of base change, projection, and Grothendieck duality. The authors derive far-reaching consequences, including explicit Knudsen–Mumford expansion terms, a determinant-of-de-Rham-cohomology formula, and BCOV-type isomorphisms for Calabi–Yau families, linking to mirror symmetry in genus one. The work unifies and extends prior results (Mumford, Moret-Bailly, Ducrot, Saito–Terasoma) within a robust categorical framework, enabling new applications to Arakelov geometry, higher CM-line bundles, and Calabi–Yau moduli. Overall, the DRR formalism provides a canonical, base-change compatible, and compositional approach to determinants of cohomology in higher dimensions, with clear geometric and arithmetic implications.

Abstract

This work establishes the geometric component of Deligne's longstanding program on refined Grothendieck-Riemann-Roch formulas expressed through determinants of cohomology. The approach relies on a newly developed universal category of Chern classes together with an associated relative intersection theory. As an example of the applications, we provide a structural description of the coefficients in the Knudsen-Mumford expansion and establish a fundamental Mumford-type isomorphism for the alternating product of Griffiths bundles.

Paper Structure

This paper contains 68 sections, 64 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Origin of the Deligne program
  3. The program and the example of curves
  4. Interesection bundles
  5. Line distributions
  6. The main theorem
  7. Some geometric consequences
  8. Examples in low dimensions
  9. The Knudsen--Mumford expansion
  10. The determinant of the de Rham cohomology
  11. Griffiths bundles and the BCOV isomorphism
  12. Relations to other works
  13. Organization of the paper
  14. Preliminaries
  15. Notation and conventions
  16. ...and 53 more sections

Key Result

Theorem A

Let $f\colon X \to S$ be a flat surjective locally projective morphism of quasi-compact schemes, of local complete intersection and relative dimension $n\geq 0$. Let $E$ be a vector bundle on $X$. Then, there exists a canonical isomorphism of $\mathbb Q$-line bundles satisfying the above list of properties. The isomorphism is characterized by further requiring a compatibility with closed immersio

Theorems & Definitions (124)

  • Theorem A: Deligne--Riemann--Roch isomorphism
  • Proposition B
  • Proposition C
  • Proposition D
  • Proposition E
  • Proposition F: BCOV isomorphism
  • Definition : Condition $(C_n)$
  • Definition A: Divisorial scheme
  • Theorem B: DRR1
  • Lemma C
  • ...and 114 more