Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations
Alexander Polishchuk, Arkady Vaintrob
备注：
v1: 45 pages, v2: added the generalized HRR theorem (Cardy condition) and new examples with the boundary-bulk maps, v3: 54 pages, to appear in Duke Math. J
We study the category of matrix factorizations for an isolated hypersurface singularity. We compute the canonical bilinear form on the Hochschild homology of this category. We find explicit expressions for the Chern character and the boundary-bulk maps and derive an analog of the Hirzebruch-Riemann-Roch formula for the Euler characteristic of the Hom-space between a pair of matrix factorizations. We also establish G-equivariant versions of these results.
We give a simple proof of the Hirzebruch-Riemann-Roch theorem for smooth complete toric varieties, based on Ishida's result that the Todd genus of a smooth complete toric variety is one.
We prove a Hirzebruch-Riemann-Roch type formula for global matrix factorizations. This is established by an explicit realization of the abstract Hirzebruch-Riemann-Roch type formula of Shklarov. We also show a Grothendieck-Riemann-Roch type theorem.
We introduce K-theoretic Gromov-Witten invariants of algebraic orbifold target spaces. Using the methods developed by Givental-Tonita we characterize Giventals Lagrangian cone of quantum K theory of orbifolds in terms of the cohomological cone.
We introduce the most general to date version of the permutation-equivariant quantum K-theory, and express its total descendant potential in terms of cohomological Gromov-Witten invariants. This is the higher-genus analogue of adelic characterization from the paper [7] by Givental-Tonita, and is based on the application of Kawasaki's Riemann-Roch formula to moduli spaces of stable maps.
V. Lunts has recently established Lefschetz fixed point theorems for Fourier-Mukai functors and dg algebras. In the same vein, D. Shklyarov introduced the noncommutative analogue of the Hirzebruch-Riemann-Roch theorem. In this short article, we see how these constructions and computations formally stem from their motivic counterparts.
This short note proves a generalization of the Hirzebruch Riemann-Roch theorem equivalent to the Cardy condition described in [1]. This is done using an earlier result [4] that explicitly describes what the Mukai pairing in [1] descends to in Hodge cohomology via the Hochschild-Kostant-Rosenberg map twisted by the root Todd genus.
Let D be a holomorphic differential operator acting on sections of a holomorphic vector bundle on an n-dimensional compact complex manifold. We prove a formula, conjectured by Feigin and Shoikhet, for the Lefschetz number of D as the integral over the manifold of a differential form. The class of this differential form is obtained via formal differential geometry from the canonical generator of the Hochschild cohomology of the algebra of differential operators in a formal neighbourhood of a point. If D is the identity, the formula reduces to the Riemann--Roch--Hirzebruch formula.
In this article we investigate Hirzebruch-Riemann-Roch formulae for line bundles on irreducible symplectic Kähler manifolds. As Huybrechts has shown, for every irreducible complex Kähler manifold $X$ of dimension $2n$, there are numbers $a_0, a_2, ..., a_{2n}$ such that $χ(L) = \sum_{k = 0}^n a_{2k}/(2k)! q_X(c_1(L))^k$ for the Euler characteristic of a line bundle $L$, where $q_X: H^2(X, \mathbbm C) \to \mathbbm C$ is the Beauville-Bogomolov quadratic form of $X$. Using Rozansky-Witten classes similar to Hitchin and Sawon, we obtain a formula expressing the $a_{2k}$ in terms of Chern numbers …
For an arbitrary proper DG algebra A (i.e. DG algebra with finite dimensional total cohomology) we introduce a pairing on the Hochschild homology of A and present an explicit formula for a Chern-type character of an arbitrary perfect A-module (the Chern characters take values in the Hochschild homology of A). The Hirzebruch-Riemann-Roch formula in this context expresses the Euler characteristic of the Hom-complex between two perfect A-modules in terms of the pairing of their Chern characters. We mention two examples of proper DG algebras and the HRR formulas for them. The first example is Ring…
We extract genus $0$ consequences of the all genera quantum HRR formula proved in Part IX. This includes re-proving and generalizing the adelic characterization of genus $0$ quantum K-theory found in [Givental A., Tonita V., in Symplectic, Poisson, and Noncommutative Geometry, <I>Math. Sci. Res. Inst. Publ.</I>, Vol. 62, Cambridge University Press, New York, 2014, 43-91]. Extending some results of Part VIII, we derive the invariance of a certain variety (the "big J-function"), constructed from the genus $0$ descendant potential of permutation-equivariant quantum K-theory, under the action of c…
We completely characterize genus-0 K-theoretic Gromov-Witten invariants of a compact complex algebraic manifold in terms of cohomological Gromov-Witten invariants of this manifold. This is done by applying (a virtual version of) the Kawasaki-Hirzebruch-Riemann-Roch formula for expressing holomorphic Euler characteristics of orbibundles on moduli spaces of genus-0 stable maps, analyzing the sophisticated combinatorial structure of inertia stacks of such moduli spaces, and employing various quantum Riemann--Roch formulas from "fake" (i.e. orbifold-ignorant) quantum K-theory of manifold and orbif…
Using different forms of the arithmetic Riemann-Roch theorem and the computations of Bott-Chern secondary classes, we compute the analytic torsion and the height of Hirzebruch surfaces.
We establish a Hirzebruch-Riemann-Roch type theorem and Grothendieck-Riemann-Roch type theorem for matrix factorizations on quotient Deligne-Mumford stacks. For this we first construct a Hochschild-Kostant-Rosenberg type isomorphism explicit enough to yield a categorical Chern character formula. We next find an expression of the canonical pairing of Shklyarov under the isomorphism.
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.
We prove a new formula for the Hirzebruch-Milnor classes of global complete intersections with arbitrary singularities describing the difference between the Hirzebruch classes and the virtual ones. This generalizes a formula for the Chern-Milnor classes in the hypersurface case that was conjectured by S. Yokura and was proved by A. Parusinski and P. Pragacz. It also generalizes a formula of J. Seade and T. Suwa for the Chern-Milnor classes of complete intersections with isolated singularities.
In this paper, we show that the cohomology of a general stable bundle on a Hirzebruch surface is determined by the Euler characteristic provided that the first Chern class satisfies necessary intersection conditions. More generally, we compute the Betti numbers of a general stable bundle. We also show that a general stable bundle on a Hirzebruch surface has a special resolution generalizing the Gaeta resolution on the projective plane. As a consequence of these results, we classify Chern characters such that the general stable bundle is globally generated.
In this paper, we formulate axioms of certain graded cohomology theory for which Chern class maps from higher K-theory are defined, following the method of Gillet [Gi1]. We will not include homotopy invariance nor purity in our axioms. It will turn out that the Grothendieck-Riemann-Roch theorem and the Riemann-Roch theorem without denominators hold for our higher Chern classes. We will also give two applications of our Riemann-Roch results.
Using the arithmetic Schottky uniformization theory, we show the arithmeticity of $PSL_{2}({\mathbb C})$ Chern-Simons invariant. In terms of this invariant, we give an explicit formula of the Riemann-Roch isomorphism as Zograf-Mcintyre-Takhtajan's infinite product for families of algebraic curves. By this formula, we determine the unknown constant which appears in the holomorphic factorization formula of determinant of Laplacians on Riemann surfaces via the classical Liouville action. As an application, we show the rationality of Ruelle zeta values for Schottky uniformized $3$-manifolds.
Let $f: X \to S$ be flat morphism over an algebraically closed field $k$ with a relative normal crossings divisor $Y\subset X$, $(E, \nabla)$ be a bundle with a connection with log poles along $Y$ and curvature with values in $f^*Ω^2_{k(S)}$. Then the Gauß-Manin sheaf $R^if_*(Ω^*_{X/S}({\rm log} Y)\otimes E)$ carries a Gauß-Manin connection $GM^i(\nabla)$. We establish a Riemann-Roch formula relating the algebraic Chern-Simons invariants of $\nabla$, $GM^i(\nabla)$ and the top Chern class of $Ω^1_{X/S}({\rm log}Y)$.
We give a general formula for the defect appearing in the Verdier-type Riemann-Roch formula for Chern-Schwartz-MacPherson classes in the case of a regular embedding. Our proof of this formula uses the constructible function version of Verdier's specialization functor, together with a specialization property of Chern-Schwartz-MacPherson classes and the corresponding Riemann-Roch theorem for smooth morphisms. As a very special case we get a formula for the Milnor-class of a local complete intersection in a smooth manifold, which in the case of a hypersurface gives back a result of Parusinski-Pra…
For a proper scheme X with a fixed 1-perfect obstruction theory, we define virtual versions of holomorphic Euler characteristic, chi y-genus, and elliptic genus; they are deformation invariant, and extend the usual definition in the smooth case. We prove virtual versions of the Grothendieck-Riemann-Roch and Hirzebruch-Riemann-Roch theorems. We show that the virtual chi y-genus is a polynomial, and use this to define a virtual topological Euler characteristic. We prove that the virtual elliptic genus satisfies a Jacobi modularity property; we state and prove a localization theorem in the toric …
Motivic Chern and Hirzebruch classes are polynomials with K-theory and homology classes as coefficients, which specialize to Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes. We provide formulas to compute the motivic Chern and Hirzebruch classes of Grassmannian and vexillary degeneracy loci. We apply our results to obtain the Hirzebruch $χ_y$-genus of classical and one-pointed Brill-Noether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature.
In this paper we study some new theories of characteristic homology classes for singular complex algebraic varieties. First we introduce a natural transformation T_{y}: K_{0}(var/X) -> H_{*}(X,Q)[y] commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. Here K_{0}(var/X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration. T_{y} is a homology class version of the motivic measure corresponding to a suitable specialization of the well known Hodge poly…
In this paper we prove a categorification of the Grothendieck-Riemann-Roch theorem. Our result implies in particular a Grothendieck-Riemann-Roch theorem for Toën and Vezzosi's secondary Chern character. As a main application, we establish a comparison between the Toën-Vezzosi Chern character and the classical Chern character, and show that the categorified Chern character recovers the classical de Rham realization.
In this paper we study some new theories of characteristic homology classes for singular complex algebraic (or compactifiable analytic) spaces. We introduce a motivic Chern class transformation mC_{*}: K_{0}(var/X)-> G_{0}(X)[y], which generalizes the total λ-class of the cotangent bundle to singular spaces. Here K_{0}(var/X) is the relative Grothendieck group of complex algebraic varieties over X as introduced and studied by Looijenga and Bittner in relation to motivic integration, and G_{0}(X) is the Grothendieck group of coherent sheaves of O_{X}-modules. We define a natural transformation …
We study the singular Bott-Chern classes introduced by Bismut, Gillet and Soule. Singular Bott-Chern classes are the main ingredient to define direct images for closed immersions in arithmetic K-theory. In this paper we give an axiomatic definition of a theory of singular Bott-Chern classes, study their properties, and classify all possible theories of this kind. We identify the theory defined by Bismut, Gillet and Soule as the only one that satisfies the additional condition of being homogeneous. We include a proof of the arithmetic Grothendieck-Riemann-Roch theorem for closed immersions that…
We describe the pushforward of a matrix factorisation along a ring morphism in terms of an idempotent defined using relative Atiyah classes, and use this construction to study the convolution of kernels defining integral functors between categories of matrix factorisations. We give an elementary proof of a formula for the Chern character of the convolution generalising the Hirzebruch-Riemann-Roch formula of Polishchuk and Vaintrob.
We present a solution of the generalized Hirzebruch problem on the relations between the Chern numbers of a stably almost complex manifold and the Chern numbers of its virtual Chern submanifolds.
In the paper, by the singular Riemann-Roch theorem, it is proved that the class of the e-th Frobenius power can be described using the class of the canonical module for a normal local ring of positive characteristic. As a corollary, we prove that the coefficient of the second term of the Hilbert-Kunz function of a finitely generated A-module M vanishes if A is a Q-Gorenstein ring and M is of finite projective dimension. For a normal algebraic variety X over a perfect field of positive characteristic, it is proved that the first Chern class of the direct image of the structure sheaf via e-th Fr…
In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the complete case fit nicely with a generalized Hirzebruch-Riemann-Roch theorem. As special cases, we obtain new (or recover well-known) formulae for the Baum-Fulton-MacPherson Todd (or MacPherson-Chern) classes of toric varieties, as well as for the Thom-Milnor L-classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of …
The Milnor class is a generalization of the Milnor number, defined as the difference (up to sign) of Chern--Schwartz--MacPherson's class and Fulton--Johnson's canonical Chern class of a local complete intersection variety in a smooth variety. In this paper we introduce a "motivic" Grothendieck group $K^{\mathcal Prop}_{\ell.c.i}(\mathcal V/X \to S)$ and natural transformations from this Grothendieck group to the homology theory. We capture the Milnor class, more generally Hirzebruch--Milnor class, as a special value of a distinguished element under these natural transformations. We also show a…
In 1954 Hirzebruch asked which linear combinations of Chern numbers are topological invariants of smooth complex projective varieties. We give a complete answer to this question in small dimensions, and also prove partial results without restrictions on the dimension.
We deduce the Riemann-Roch type formula expressing the microlocal Euler class of a perfect complex of D-modules in terms of the Chern character of the associated symbol complex and the Todd class of the manifold from the Riemann-Roch type theorem for periodic cyclic cocycles of a symplectic deformation quantization. The proof of the latter is contained in the sequel to this paper.
We show that, in characteristic zero, the obvious integral version of the Grothendieck-Riemann-Roch formula obtained by clearing the denominators of the Todd and Chern characters is true (without having to divide the Chow groups by their torsion subgroups). The proof introduces an alternative to Grothendieck's strategy: we use resolution of singularities and the weak factorization theorem for birational maps.
The Milnor-Hirzebruch class of a locally complete intersection X in an algebraic manifold M measures the difference between the (Poincare dual of the) Hirzebruch class of the virtual tangent bundle of X and, respectively, the Brasselet-Schuermann-Yokura (homology) Hirzebruch class of X. In this note, we calculate the Milnor-Hirzebruch class of a globally defined algebraic hypersurface X in terms of the corresponding Hirzebruch invariants of singular strata in a Whitney stratification of X. Our approach is based on Schuermann's specialization property for the motivic Hirzebruch class transforma…
Let $X$ be a nonsingular complex variety and $D$ a reduced effective divisor in $X$. In this paper we study the conditions under which the formula $c_{SM}(1_U)=c(\textup{Der}_X(-\log D))\cap [X]$ is true. We prove that this formula is equivalent to a Riemann-Roch type of formula. As a corollary, we show that over a surface, the formula is true if and only if the Milnor number equals the Tjurina number at each singularity of $D$. We also show the Rimann-Roch type of formula is true if the Jacobian scheme of $D$ is nonsingular or a complete intersection.
In this article, we show that the combination of the constructions done in SGA 6 and the A^1-homotopy theory naturally leads to results on higher algebraic K-theory. This applies to the operations on algebraic K-theory, Chern characters and Riemann-Roch theorems.
We introduce spectral Hirzebruch-Milnor classes for singular hypersurfaces. These can be identified with Steenbrink spectra in the isolated singularity case, and may be viewed as their global analogues in general. Their definition uses vanishing cycles of mixed Hodge modules and the Todd class transformation. These are compatible with the pushforward by proper morphisms, and the classes can be calculated by using resolutions of singularities. Formulas for Hirzebruch-Milnor classes of projective hypersurfaces in terms of these classes are given in the case where the multiplicity of a generic hy…
The Hirzebruch $td_y(X)$ class of a complex manifold X is a formal combination of Chern characters of the sheaves of differential forms multiplied by the Todd class. The related $χ_y$-genus admits a generalization for singular complex algebraic varieties. The equivariant version of the Hirzebruch class can be developed as well. The general theory applied in the situation when a torus acts on a singular variety allows to apply powerful tools as the Localization Theorem of Atiyah and Segal for equivariant K-theory and Berline-Vergne formula for equivariant cohomology. We obtain a meaningful inva…
This is the first article in an upcoming series of papers. They have arisen through an attempt to answer open questions of Deligne proposed in "Le determinant de la cohomologie", Contemp. Mathematics 67 (1987). It amounts to functorial and metrized versions of the Grothendieck-Riemann-Roch theorem, as well as a Lefschetz-Riemann-Roch formula in the sense of Thomason, cf. "Lefschetz-Riemann-Roch theorem and coherent trace formula" Ann. Sci. Ec. Norm. Sup. 18 (1985), Theorem 3.5. In this article we treat various preliminary results on virtual categories which are the categories where the Deligne…
In the present paper we extend the Riemann-Roch formalism to structure algebras of moment graphs. We introduce and study the Chern character and pushforwards for twisted fibrations of moment graphs. We prove an analogue of the Riemann-Roch theorem for moment graphs. As an application, we obtain the Riemann-Roch type theorem for equivariant $K$-theory of some Kac-Moody flag varieties.
We prove a conjecture of Shklyarov concerning the relationship between K. Saito's higher residue pairing and a certain pairing on the periodic cyclic homology of matrix factorization categories. Along the way, we give new proofs of a result of Shklyarov and Polishchuk-Vaintrob's Hirzebruch-Riemann-Roch formula for matrix factorizations.
We show that a version of dimensional interpolation for the Riemann--Roch--Hirzebruch formalism in the case of a grassmannian leads to an expression for the Euler characteristic of line bundles in terms of a Selberg integral. We propose a way to interpolate higher Bessel equations, their wedge powers, and monodromies thereof to non--integer orders, and link the result with the dimensional interpolation of the RRH formalism in the spirit of the gamma conjectures.
We construct a functor, from the category of schemes to the category of graded rings, that is an initial object for having a theory of Chern classes with an additive first Chern class. For any scheme $X$, the graded ring that our functor associates to $X$ is related to the associated graded ring of the $γ$-filtration on the Grothendieck ring of finite rank locally free sheaves on $X$ via a Grothendieck-Riemann-Roch type theorem.
We show a Riemann-Roch theorem for group ring bundles over an arithmetic surface; this is expressed using the higher adeles of Beilinson-Parshin and the tame symbol via a theory of adelic equivariant Chow groups and Chern classes. The theorem is obtained by combining a group ring coefficient version of the local Riemann-Roch formula as in Kapranov-Vasserot with results on K-groups of group rings and an explicit description of group ring bundles over P^1. Our set-up provides an extension of several aspects of the classical Fr"ohlich theory of the Galois module structure of rings of integers of …
