Riemann-Roch Research Articles

On a generalized Petrieʼs conjecture via index theorems

The aim of this paper is to show some results related to Petrieʼs conjecture for torus manifolds which are homotopy equivalent to a product ∏i=1mCPni of complex projective spaces CPni via the Atiyah–Singer index theorem and the Riemann–Roch formula.

Pushing forward matrix factorizations

We describe the pushforward of a matrix factorization along a ring morphism in terms of an idempotent defined using relative Atiyah classes, and we use this construction to study the convolution of kernels defining integral functors between categories of matrix factorizations. We give an elementary proof of a formula for the Chern character of the convolution generalizing the Hirzebruch–Riemann–Roch formula of Polishchuk and Vaintrob.

Higher Chow groups of varieties with group action

We give explicit descriptions of the higher Chow groups of toric bundles and flag bundles over schemes. We derive several consequences of these descriptions for the equivariant and ordinary higher Chow groups of schemes with group action. We prove a decomposition theorem for the equivariant higher Chow groups of a smooth scheme with action of a diagonalizable group. This theorem is applied to compute the equivariant and ordinary higher Chow groups of smooth toric varieties. The results of this paper play fundamental roles in the proof of the Riemann–Roch theorems for equivariant higher K-theory.

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.

Geometry of Vector Bundle Extensions and Applications to a Generalised Theta Divisor

Let $E$ and $F$ be vector bundles over a complex projective smooth curve $X$, and suppose that $0 \to E \to W \to F \to 0$ is a nontrivial extension. Let $G \subseteq F$ be a subbundle and $D$ an effective divisor on $X$. We give a criterion for the subsheaf $G(-D) \subset F$ to lift to $W$, in terms of the geometry of a scroll in the extension space ${\mathbf{P}} H^{1}(X, \mathrm{Hom}(F, E))$. We use this criterion to describe the tangent cone to the generalised theta divisor on the moduli space of semistable bundles of rank $r$ and slope $g-1$ over $X$, at a stable point. This gives a generalisation of a case of the Riemann-Kempf singularity theorem for line bundles over $X$. In the same vein, we generalise the geometric Riemann-Roch theorem to vector bundles of slope $g-1$ and arbitrary rank.

A NOTE ON THE GENUS OF GLOBAL FUNCTION FIELDS

AbstractTo give a relatively elementary proof of the Brumer–Stark conjecture in a function field context involving no algebraic geometry beyond the Riemann–Roch theorem for curves, Hayes Compos. Math., vol. 55, 1985, pp. 209–239) defined a normalizing field $H_\mathfrak{e}^*$ associated with a fixed sgn-normalized Drinfeld module and its extension field $K_\mathfrak{m}$, which is an analogue of cyclotomic function fields over a rational function field. We present explicitly in this note the formulae for the genus of the two fields and the maximal real subfield $H_\mathfrak{m}$ of $K_\mathfrak{m}$. In some sense, our results can be regarded as generalizations of formulae for the genus of classical cyclotomic function fields obtained by Hayes Trans. Amer. Math. Soc., vol. 189, 1974, pp. 77–91) and Kida and Murabayashi (Tokyo J. Math., vol. 14(1), 1991, pp. 45–56).

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

The goal of this paper is to prove Riemann-Roch type theorems for Deligne-Mumford algebraic stacks. To this end, we introduce a cohomology with coefficients in representations and a Chern character, and we prove a Grothendieck-Riemann-Roch theorem for the Riemann-Roch transformation it defines. As a corollary we obtain an Hirzebruch-Riemann-Roch formula for the Euler characteristic of a coherent sheaf, and some formulas for the different topological Euler characteristics of complex algebraic stacks.

The Witten–Reshetikhin–Turaev invariants of finite order mapping tori I

We formulate the Asymptotic Expansion Conjecture for the Witten–Reshetikhin–Turaev quantum invariants of closed oriented 3-manifolds. For of finite order mapping tori, we study these quantum invariants via the geometric gauge theory approach to the corresponding quantum representations and prove, using a version of the Lefschetz–Riemann–Roch theorem due to Baum, Fulton, MacPherson & Quart, that the quantum invariants can be expressed as a sum over the components of the moduli space of flat connections on the mapping torus. Moreover, we show that the term corresponding to a component is a polynomial in the level k, weighted by a complex phase, which is k times the Chern–Simons invariant corresponding to the component. We express the coefficients of these polynomials in terms of cohomological pairings on the fixed point set of the moduli space of flat connections on the surface. We explicitly describe the fixed point set in terms of moduli spaces of the quotient orbifold Riemann surface and for the smooth components we express the aforementioned coefficients in terms of the known generators of the cohomology ring. We provide an explicit formula in terms of the Seifert invariants of the mapping torus for the contributions from each of the smooth components. We further establish that the Asymptotic Expansion Conjecture for manifolds and we verify the Growth Rate Conjecture for these finite order mapping tori.

Ice cream and orbifold Riemann–Roch

We give an orbifold Riemann–Roch formula in closed form for the Hilbert series of a quasismooth polarized -fold , under the assumption that is projectively Gorenstein with only isolated orbifold points. Our formula is a sum of parts each of which is integral and Gorenstein symmetric of the same canonical weight; the orbifold parts are called ice cream functions. This form of the Hilbert series is particularly useful for computer algebra, and we illustrate it on examples of surfaces and Calabi–Yau 3-folds. These results apply also with higher dimensional orbifold strata (see [1] and [2]), although the precise statements are considerably trickier. We expect to return to this in future publications.

A Riemann-Roch theorem for dg algebras

— Given a smooth proper dg-algebra A, a perfect dg A-module M and an endomorphism f of M , we define the Hochschild class of the pair (M, f) with values in the Hochschild homology of the algebra A. Our main result is a Riemann-Roch type formula involving the convolution of two such Hochschild classes.

Brill–Noether theory of squarefree modules supported on a graph

We investigate the analogy between squarefree Cohen–Macaulay modules supported on a graph and line bundles on a curve. We prove a Riemann–Roch theorem, we study the Jacobian and gonality of a graph, and we prove Clifford’s theorem.

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.

Permutohedral spaces and the Cox ring of the moduli space of stable pointed rational curves

We study the Cox ring of the moduli space of stable pointed rational curves, \({\overline{M}_{0,n}}\), via the closely related permutohedral (or Losev-Manin) spaces \({\overline{L}_{n-2}}\). Our main result establishes \(\left(\begin{array}{ll} n 2 \end{array}\right)\) polynomial subrings of \({{\rm Cox}(\overline{M}_{0,n})}\), thus giving collections of boundary variables that intersect the ideal of relations of \({{\rm Cox}(\overline{M}_{0,n})}\) trivially. As applications, we give a combinatorial way to partially solve the Riemann-Roch problem for \({\overline{M}_{0,n}}\), and we show that all relations in degrees of \({{\rm Cox}(\overline{M}_{0,6})}\) arising from certain pull-backs from projective spaces are generated by the Plucker relations.

The differential analytic index in Simons–Sullivan differential K-theory

We define the Simons–Sullivan differential analytic index by translating the Freed–Lott differential analytic index via explicit ring isomorphisms between Freed–Lott differential K-theory and Simons–Sullivan differential K-theory. We prove the differential Grothendieck–Riemann–Roch theorem in Simons–Sullivan differential K-theory using a theorem of Bismut.

The J-invariant and Tits algebras for groups of inner type E 6

A connection between the indices of the Tits algebras of a split linear algebraic group G and the degree one parameters of its motivic J-invariant was introduced by Queguiner-Mathieu, Semenov and Zainoulline through use of the second Chern class map in the Riemann-Roch theorem without denominators. In this paper we extend their result to higher Chern class maps and provide applications to groups of inner type E 6.

A Riemann–Roch theorem for hypermaps

Starting from the Riemann–Roch theorem for graphs of Baker and Norine, we give a version of such a theorem for the divisors on the points of algebraic hypermaps.

On the Birational Invariance of the BCOV Torsion of Calabi-Yau Threefolds

Fang et al. (J. Diff. Geom. 80(2):175–259, 2008, Sect. 4, Conj. 4.17) conjecture that a certain spectral string-theoretic invariant of Calabi-Yau threefolds (the BCOV torsion) is a birational invariant. We prove a weak form of this conjecture. The proof combines the arithmetic Riemann-Roch theorem in Arakelov geometry with some inputs from motivic integration theory.

An arithmetic Hilbert–Samuel theorem for pointed stable curves

Let (\mathcal{O},∑,F_∞) be an arithmetic ring of Krull dimension at most 1, \mathcal S_ = \mathrm{Spec} \mathcal O and (\mathcal X → \mathcal S; σ_1, ..., σ_n) a pointed stable curve. Write \mathcal U =\mathcal Χ \backslash \bigcup_j σ_j(\mathcal S) . For every integer k\geq 0 , the invertible sheaf ω^{k+1}_{Χ/{\mathcal S}}(kσ_1 + ... + kσ_n) inherits a singular hermitian structure from the hyperbolic metric on the Riemann surface \mathcal U_∞ . In this article we define a Quillen type metric ‖ ∙ ‖_Q on the determinant line λ_{k+1} = λ(ω^{k+1}_{Χ/{\mathcal S}}(kσ_1 + ... + kσ_n)) and compute the arithmetic degree of (λ_{k+1}, ‖ ∙ ‖_Q) by means of an analogue of the Riemann–Roch theorem in Arakelov geometry. As a byproduct, we obtain an arithmetic Hilbert–Samuel formula: the arithmetic degree of (λ_{k+1}, ‖ ∙ ‖_{L^2}) admits an asymptotic expansion in k , whose leading coefficient is given by the arithmetic self-intersection of (ω_{Χ/{\mathcal S}}(σ_1 + ... + σ_n), ‖ ∙ ‖_{\mathrm{hyp}}) . Here ‖ ∙ ‖_{L^2} and ‖ ∙ ‖_{\mathrm{hyp}} denote the L^2 metric and the dual of the hyperbolic metric, respectively. Examples of application are given for pointed stable curves of genus 0.

THE JACQUET–LANGLANDS CORRESPONDENCE AND THE ARITHMETIC RIEMANN–ROCH THEOREM FOR POINTED CURVES

We show how the Jacquet–Langlands correspondence and the arithmetic Riemann–Roch theorem for pointed curves, relate the arithmetic self-intersection numbers of the sheaves of modular forms — with their Petersson norms — on modular and Shimura curves: these are equal modulo ∑l∈Sℚ log l, where S is a controlled set of primes. These quantities were previously considered by Bost and Kühn (modular curve case) and Kudla–Rapoport–Yang and Maillot–Roessler (Shimura curve case). By the work of Maillot and Roessler, our result settles a question raised by Soulé.

On a conjucture of Kashiwara relating Chern and Euler classes of $\mathcal{O}$-modules

In this note we prove a conjecture of Kashiwara, which states that the Euler class of a coherent analytic sheaf $\mathcal{F}$ on a complex manifold $X$ is the product of the Chern character of $\mathcal{F}$ with the Todd class of $X$. As a corollary, we obtain a functorial proof of the Grothendieck–Riemann–Roch theorem in Hodge cohomology for complex manifolds.