Riemann-Roch Research Articles

Index theorem for homogeneous spaces of Lie groups

We study index theory for homogeneous spaces associated to an almost connected Lie group in terms of the topological and analytic aspects. For the topological aspect, we obtain a topological formula as a result of the Riemann–Roch formula for proper cocompact actions of the Lie group, inspired by the work of Paradan and Vergne. For the analytic aspect, we apply heat kernel methods to obtain a local index formula representing the higher indices of equivariant elliptic operators with respect to a proper, cocompact action of the Lie group.

Topological recursion for Kadomtsev–Petviashvili tau functions of hypergeometric type

AbstractWe study the ‐point differentials corresponding to Kadomtsev–Petviashvili (KP) tau functions of hypergeometric type (also known as Orlov–Scherbin partition functions), with an emphasis on their ‐deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov–Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families, we prove in addition the so‐called projection property and thus the full statement of the Chekhov–Eynard–Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of ‐deformations of the Orlov–Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper, one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers. By virtue of topological recursion and the Grothendieck–Riemann–Roch formula, this, in turn, gives new and uniform proofs of almost all Ekedahl–Lando–Shapiro–Vainshtein (ELSV)‐type formulas discussed in the literature.

The Euler Characteristic of Parabolic Sheaves

Objectives: The primary aim of this study is to explicitly determine the Euler characteristic of the parabolic sheaves with rank 2 on a smooth projective algebraic surface defined over complex numbers with the smooth irreducible parabolic divisor . Methods: The computation of the parabolic Hilbert polynomial involves the use of -filtered sheaves on a smooth projective surface , with weights corresponding to the points where the filtration jumps. The Riemann-Roch theorem and Chern class computation have also been used. Findings: The study provides explicit computations of the parabolic Hilbert polynomial as well as the parabolic Chern classes for parabolic rank 2 bundles. Novelty: This work contributes to the understanding of parabolic sheaves on smooth projective surfaces, bridging the gap between different constructions of stable bundles. The explicit computation of the parabolic Hilbert polynomial for rank 2 bundles adds valuable insights to the study of moduli spaces of parabolic bundles. Keywords: Euler characteristic, Hilbert polynomial, Chern class, Parabolic sheaves, Smooth projective algebraic surface

Riemann–Hurwitz theorem and Riemann–Roch theorem for hypermaps

Riemann–Hurwitz theorem and Riemann–Roch theorem for hypermaps

The Riemann–Roch theorem for the Adams operations

We prove the classical Riemann–Roch theorems for the Adams operations ψj on K-theory: a statement with coefficients on Z[j−1], that holds for arbitrary projective morphisms, as well as another statement with integral coefficients, that is valid for closed immersions. In presence of rational coefficients, we also analyze the relation between the corresponding Riemann–Roch formula for one Adams operation and the analogous formula for the Chern character. To do so, we complete the elementary exposition of the work of Panin–Smirnov that was initiated by the first author in a previous paper. Their notion of oriented cohomology theory on algebraic varieties allows to use classical arguments to prove general and neat statements, which imply all the aforementioned results as particular cases.

Equivariant algebraic K-theory, G-theory and derived completions

In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K-Theory similar to the well-known Atiyah-Segal completion theorem for equivariant topological K-theory, the late Robert Thomason found the strong finiteness conditions that are required in such theorems to be too restrictive. Then he made a conjecture on the existence of a completion theorem in the sense of Atiyah and Segal for equivariant algebraic G-theory, for actions of linear algebraic groups on schemes that holds without any of the strong finiteness conditions that are required in such theorems proven by him, and also appearing in the original Atiyah-Segal theorem. The main goal of the present paper is to provide a proof of this conjecture in as broad a context as possible, making use of the technique of derived completion, and to consider several of the applications.Our solution is broad enough to allow actions by all linear algebraic groups, irrespective of whether they are connected or not, and acting on any quasi-projective scheme of finite type over a field, irrespective of whether they are regular or projective. This allows us therefore to consider the equivariant algebraic G-Theory of large classes of varieties like all toric varieties (for the action of a torus) and all spherical varieties (for the action of a reductive group). Restricting to actions by split tori, we are also able to consider actions on algebraic spaces. Moreover, the restriction that the base scheme be a field is also not required often, but is put in mainly to simplify some of our exposition. These enable us to obtain a wide range of applications, some of which are briefly sketched and which we plan to explore in detail in the future. In fact, we discuss an extension of our results to equivariant homotopy K-theory along with various Riemann-Roch theorems in a sequel. A comparison of our results with previously known results, none of which made use of derived completions, shows that without the use of derived completions one can only obtain results which are indeed very restrictive.

Explicit matrices with low coherence based on algebraic geometric codes

A key element in the performance of a compressed sensing (CS) setup is the so called sensing matrix. It is known that the success of CS-based methods strongly rely on the properties of the employed sensing matrix. Random matrices are the widely-adopted choice due to their order-optimal performance and flexibility in size. In real world applications, however, random structures are rarely feasible. For this reason, the deterministic design of sensing matrices has been an ongoing research topic. In this paper, we introduce new classes of deterministic complex-valued sensing matrices based on algebraic curves. In particular, we design a number of algebraic-geometric codes with large minimum distances specifically for the construction of sensing matrices. Our approach is to find maximal curves in the Galois field Fpm‾, and transform them into Fp codes by a trace map. Invoking the Riemann-Roch theorem, we demonstrate that the resulting code has a large minimum distance compared to its length, which leads to a sensing matrix with small coherence value. For general m×n matrices, the Welch bound (≈1m) sets a universal lower-bound on the coherence value, and the bound is achievable only when n≤m2. In our designs, we are able to construct m×n matrices with n ranging from around 8m to values larger than O(m2) by tuning the parameters. Meanwhile, the coherence of the designed matrices differ from the Welch bound by only an O(logm) factor. Simulation results indicate that the performance of our matrices in recovering sparse vectors from compressed measurements is superior or equivalent to Gaussian random matrices.

Riemann-Roch for [formula omitted

We prove a Riemann-Roch theorem of an entirely novel nature for divisors on the Arakelov compactification of the algebraic spectrum of the integers. This result relies on the introduction of three key concepts: the cohomologies (attached to a divisor), their integer dimension, and Serre duality. These notions directly extend their classical counterparts for function fields. The Riemann-Roch formula equates the (integer valued) Euler characteristic of a divisor with a slight modification of the traditional expression in terms of the sum of the degree of the divisor and the logarithm of 2. Both the definitions of the cohomologies and of their dimensions rely on a universal arithmetic theory over the sphere spectrum that we had previously introduced using Segal's Gamma rings. By adopting this new perspective we can parallel Weil's adelic proof of the Riemann-Roch formula for function fields including the use of Pontryagin duality.

On singular generalizations of the Singer–Hopf conjecture

AbstractThe Singer–Hopf conjecture predicts the sign of the topological Euler characteristic of a closed aspherical manifold. In this note, we propose singular generalizations of the Singer–Hopf conjecture, formulated in terms of the Euler–Mather characteristic, intersection homology Euler characteristic and, resp., virtual Euler characteristic of a closed irreducible subvariety of an aspherical complex projective manifold. We prove these new conjectures under the assumption that the cotangent bundle of the ambient variety is numerically effective (nef), or, more generally, when the ambient manifold admits a finite morphism to a complex projective manifold with a nef cotangent bundle. The main ingredients in the proof are the semi‐positivity properties of nef vector bundles together with a topological version of the Riemann–Roch theorem, proved by Kashiwara.

Generalized Riemann Functions, Their Weights, and the Complete Graph

By a Riemann function we mean a function $f:\mathbb{Z}^n\to\mathbb{Z}$ such that $f(\mathbf{d})$ is equals $0$ for $d_1+\cdots+d_n$ sufficiently small, and equals $d_1+\cdots+d_n+C$ for a constant, $C$, for $d_1+\cdots+d_n$ sufficiently large. By adding $1$ to the Baker-Norine rank function of a graph, one gets an equivalent Riemann function, and similarly for related rank functions.
 To each Riemann function we associate a related function $W: \mathbb{Z}^n\to \mathbb{Z}$ via Möbius inversion that we call the weight of the Riemann function. We give evidence that the weight seems to organize the structure of a Riemann function in a simpler way: first, a Riemann function $f$ satisfies a Riemann-Roch formula iff its weight satisfies a simpler symmetry condition. Second, we will calculate the weight of the Baker-Norine rank for certain graphs and show that the weight function is quite simple to describe; we do this for graphs on two vertices and for complete graphs.
 For complete graphs, we build on the work of Cori and Le Borgne who gave a linear time method to compute the Baker-Norine rank of the complete graph. The associated weight function has a simple formula and is extremely sparse (i.e., mostly zero). Our computation of the weight function leads to a new linear time algorithm to compute the Baker-Norine rank, via a new formula likely related to one of Cori and Le Borgne, but seemingly simpler for general $\mathbf{d}\in \mathbb{Z}^n$, namely$$r_{{\rm BN},K_n}(\mathbf{d}) =-1+\biggl| \biggl\{ i=0,\ldots,\deg(\mathbf{d}) \ \Bigm|\ \sum_{j=1}^{n-2} \bigl( (d_j-d_{n-1}+i) \bmod n \bigr) \le \deg(\mathbf{d})-i\biggr\} \biggr|.$$However, the formula of Cori and Le Borgne, which requires $\mathbf{d}\in \mathbb{Z}^n$ to be a sorted parking function, is easier to evaluate for such $\mathbf{d}$.
 Our study of weight functions leads to a natural generalization of Riemann functions, with many of the same properties exhibited by Riemann functions.

Grothendeick Embedding of Real Numbers within Congruency Transforms of Energy Numbers: A Disproof of Grothendieck–Riemann–Roch theorem

Grothendeick Embedding of Real Numbers within Congruency Transforms of Energy Numbers: A Disproof of Grothendieck–Riemann–Roch theorem

Grothendeick Embedding of Real Numbers within Congruency Transforms of Energy Numbers: A Disproof of Grothendieck–Riemann–Roch theorem

Grothendeick Embedding of Real Numbers within Congruency Transforms of Energy Numbers: A Disproof of Grothendieck–Riemann–Roch theorem

A Survey of Riemann-Roch-type Theorems

The Riemann-Roch theorem is a classical result which forms a beautiful algebraic connection between complex analysis on a compact Riemann surface and a global topological property of that surface (the genus). We present a survey of the theorem and its many variants and generalizations. We also provide an alternative elementary proof of the Riemann-Roch formula for algebraic curves and outline some applications of the formula.

Topological K-theory for discrete groups and index theory

For any countable discrete group Γ (without any further assumptions on it) we first construct an explicit morphism from the Left-Hand side of the Baum-Connes assembly map, Ktop⁎(Γ), to the periodic cyclic homology of the group algebra, HP⁎(CΓ). This morphism, called the Chern-Baum-Connes assembly map, allows in particular to give a proper and explicit formulation for a Chern-Connes pairingKtop⁎(Γ)×HP⁎(CΓ)⟶C.Several theorems are needed to formulate the Chern-Baum-Connes assembly map. In particular we establish a delocalised Riemann-Roch theorem, the wrong way functoriality for periodic delocalised cohomology for Γ-proper actions, the construction of a Chern morphism between the Left-Hand side of Baum-Connes and a delocalised cohomology group associated to Γ which is an isomorphism once tensoring with C, and the construction of an explicit cohomological assembly map between the delocalised cohomology group associated to Γ and the homology group H⁎(Γ,FΓ) (where FΓ is the complex vector space freely generated by the set of elliptic elements in Γ). This last group H⁎(Γ,FΓ) identifies, by the work of Burghelea, as a direct factor of the cyclic periodic homology of the group algebra.Moreover, in this paper Ktop⁎(Γ) stands for the topological K-theory for discrete groups as originally proposed by Baum and Connes in their first paper, where wrong way functoriality in equivariant K-theory was used for its construction. As part of our results we prove that this model is indeed isomorphic to the analytic model for the left-hand side of the assembly map.Our main final theorem gives an explicit geometric index theoretical formula for the above mentioned pairing (without any further assumptions on Γ) in terms of pairings of invariant forms, associated to geometric cycles and given in terms of delocalized Chern and Todd classes, and currents naturally associated to group cocycles using Burghelea's computation. This gives a complete solution, for discrete countable groups, to the problem of defining and computing a geometric pairing between the left hand side of the Baum-Connes assembly map, given in terms of geometric cycles associated to proper actions of the discrete group on manifolds, and cyclic periodic cohomology of the group algebra.

Riemann–Roch theorems in monoidal 2-categories

Abstract Smooth and proper dg-algebras have an Euler class valued in the Hochschild homology of the algebra. This Euler class is worthy of this name since it satisfies many familiar properties including compatibility with the pairing on the Hochschild homology of the algebra and that of its opposite. This compatibility is the Riemann–Roch theorems of [21, 14]. In this paper, we prove a broad generalization of these Riemann–Roch theorems. We generalize from the bicategory of dg-algebras and their bimodules to symmetric monoidal bicategories and from the Euler class to traces of non-identity maps. Our generalization also implies spectral Riemann–Roch theorems. We regard this result as an instantiation of a 2-dimensional generalized cobordism hypothesis. This perspective draws the result close to many others that generalize results about Euler characteristics and classes to bicategorical traces.

Riemann–Roch coefficients for Kleinian orbisurfaces

Suppose mathcal {S} is a smooth, proper, and tame Deligne–Mumford stack. Toën’s Grothendieck–Riemann–Roch theorem requires correction terms, involving components of the inertia stack, to the standard formula for schemes. We give a brief overview of Toën’s Grothendieck–Riemann–Roch theorem, and explicitly compute the correction terms in the case of an orbifold surface with stabilizers of types ADE.

Formal Bott–Thurston Cocycle and Part of a Formal Riemann–Roch Theorem

Formal Bott–Thurston Cocycle and Part of a Formal Riemann–Roch Theorem

Formal Bott-Thurston Cocycle and Part of a Formal Riemann-Roch Theorem

Коцикл Ботта-Тeрстона - это $2$-коцикл на группе сохраняющих ориентацию диффеоморфизмов окружности. Вводится и изучается формальный аналог коцикла Ботта-Тeрстона. Формальный коцикл Ботта-Тeрстона - это $2$-коцикл на группе непрерывных $A$-автоморфизмов алгебры $A((t))$ рядов Лорана над коммутативным кольцом $A$ со значениями в группе $A^*$ обратимых элементов кольца $A$. Доказывается, что центральное расширение, заданное формальным коциклом Ботта-Тeрстона, эквивалентно 12-кратной сумме Бэра детерминантного центрального расширения, если $A$ является $\mathbb Q$-алгеброй. В качестве следствия этого результата доказывается часть новой формальной теоремы Римана-Роха. Эта теорема Римана-Роха применяется к окольцованному пространству на отделимой схеме $S$ над полем $\mathbb Q$, где структурный пучок окольцованного пространства локально на схеме $S$ изоморфен пучку $\mathcal O_S((t))$ и склеивающие автоморфизмы непрерывны. Локально на схеме $S$ это окольцованное пространство соответствует проколотой формальной окрестности сечения гладкого морфизма в $U$ относительной размерности $1$, где $U \subset S$ - открытое подмножество.

Virtual Riemann–Roch theorems for almost perfect obstruction theories

This is the third in a series of works devoted to constructing virtual structure sheaves and K-theoretic invariants in moduli theory. The central objects of study are almost perfect obstruction theories, introduced by Y.-H. Kiem and the author as the appropriate notion in order to define invariants in K-theory for many moduli stacks of interest, including generalized K-theoretic Donaldson–Thomas invariants. In this paper, we prove virtual Riemann–Roch theorems in the setting of almost perfect obstruction theory in both the non-equivariant and equivariant cases, including cosection localized versions. These generalize and remove technical assumptions from the virtual Riemann–Roch theorems of Fantechi–Göttsche and Ravi–Sreedhar. The main technical ingredients are a treatment of the equivariant K-theory and equivariant Gysin map of sheaf stacks and a formula for the virtual Todd class.

Seiberg–Witten and Gromov invariants for self-dual harmonic 2–forms

Author(s): Gerig, Chris | Advisor(s): Hutchings, Michael | Abstract: For a closed oriented smooth 4-manifold X with $b^2_+(X)g0$, the Seiberg-Witten invariants are well-defined. Taubes' SW=Gr theorem asserts that if X carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes' Gromov invariants. In the absence of a symplectic form there are still nontrivial closed self-dual 2-forms which vanish along a disjoint union of circles and are symplectic elsewhere. This thesis describes well-defined counts of pseudoholomorphic curves in the complement of the zero set of such 2-forms, and it is shown that they recover the Seiberg-Witten invariants (modulo 2). This is an extension of Taubes' SW=Gr theorem to non-symplectic 4-manifolds.The main results are the following. Given a suitable form w and tubular neighborhood N of its zero set, there are well-defined counts of pseudoholomorphic curves in a completion of the symplectic cobordism (X-N, w) which are asymptotic to certain Reeb orbits on the ends. They can be packaged together to form near-symplectic Gromov invariants as a map on the set of spin-c structures of X. They are furthermore equal to the Seiberg-Witten invariants with mod 2 coefficients, where w determines the chamber for defining the latter invariants when $b^2_+(X)=1$.In the final chapter, as a non sequitur, a new proof of the Fredholm index formula for punctured pseudoholomorphic curves is sketched. This generalizes Taubes' proof of the Riemann-Roch theorem for compact Riemann surfaces.