Todd Class Research Articles

On Chow Rings of Quiver Moduli

Abstract We describe the point class and Todd class in the Chow ring of a moduli space of quiver representations, building on a result of Ellingsrud–Strømme. This, together with the presentation of the Chow ring by the second author, makes it possible to compute integrals on quiver moduli. To do so, we construct a canonical morphism of universal representations in great generality, and along the way point out its relation to the Kodaira–Spencer morphism. We illustrate the results by computing some invariants of some “small” Kronecker moduli spaces. We also prove that the first non-trivial (6-dimensional) Kronecker moduli space is isomorphic to the zero locus of a general section of $\mathcal{Q}^{\vee }(1)$ on $\textrm{Gr}(2,8)$.

Cogroupoid structures on the circle and the Hodge degeneration

Abstract We exhibit the Hodge degeneration from nonabelian Hodge theory as a $2$ -fold delooping of the filtered loop space $E_2$ -groupoid in formal moduli problems. This is an iterated groupoid object which in degree $1$ recovers the filtered circle $S^1_{fil}$ of [MRT22]. This exploits a hitherto unstudied additional piece of structure on the topological circle, that of an $E_2$ -cogroupoid object in the $\infty $ -category of spaces. We relate this cogroupoid structure with the more commonly studied ‘pinch map’ on $S^1$ , as well as the Todd class of the Lie algebroid $\mathbb {T}_{X}$ ; this is an invariant of a smooth and proper scheme X that arises, for example, in the Grothendieck-Riemann-Roch theorem. In particular, we relate the existence of nontrivial Todd classes for schemes to the failure of the pinch map to be formal in the sense of rational homotopy theory. Finally, we record some consequences of this bit of structure at the level of Hochschild cohomology.

Atiyah Classes and Todd Classes of Pullback dg Lie Algebroids Associated with Lie Pairs

Atiyah Classes and Todd Classes of Pullback dg Lie Algebroids Associated with Lie Pairs

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.

Atiyah and Todd classes of regular Lie algebroids

Atiyah and Todd classes of regular Lie algebroids

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.

Derived categories of hyper-Kähler manifolds: extended Mukai vector and integral structure

We introduce a linearised form of the square root of the Todd class inside the Verbitsky component of a hyper-Kähler manifold using the extended Mukai lattice. This enables us to define a Mukai vector for certain objects in the derived category taking values inside the extended Mukai lattice which is functorial for derived equivalences. As applications, we obtain a structure theorem for derived equivalences between hyper-Kähler manifolds as well as an integral lattice associated to the derived category of hyper-Kähler manifolds deformation equivalent to the Hilbert scheme of a K3 surface mimicking the surface case.

An algebraic construction of sum-integral interpolators

This paper presents an algebraic construction of Euler-Maclaurin formulas for polytopes. The formulas obtained generalize and unite the previous lattice point formulas of Morelli and Pommersheim-Thomas, and the Euler-Maclaurin formulas of Berline-Vergne While the approach of this paper originates in the theory of toric varieties, and recovers previous results about characteristic classes of toric varieties, the present paper is self-contained and does not rely on results from toric geometry. We aim in particular to exhibit in a combinatorial way ingredients such as such Todd classes and cycle-level intersections in Chow rings, that first entered the theory of polytopes from algebraic geometry.

An exposition of the topological half of the Grothendieck–Hirzebruch–Riemann–Roch theorem in the fancy language of spectra

We give an informal exposition of pushforwards and orientations in generalized cohomology theories in the language of spectra. The whole note can be seen as an attempt at convincing the reader that Todd classes in Grothendieck–Hirzebruch–Riemann–Roch type formulas are not Devil’s appearances but rather that things just go in the most natural possible way. [Display omitted]

On the Todd class of the permutohedral variety

In the special case of braid fans, we give a combinatorial formula for the Berline-Vergne's construction for an Euler-Maclaurin type formula that computes number of lattice points in polytopes. Our formula is obtained by computing a symmetric expression for the Todd class of the permutohedral variety. By showing that this formula does not always have positive values, we prove that the Todd class of the permutohedral variety $X_d$ is not effective for $d\geq 24$. Additionally, we prove that the linear coefficient in the Ehrhart polynomial of any lattice generalized permutohedron is positive.

Equivariant Grothendieck–Riemann–Roch theorem via formal deformation theory

We use the formalism of traces in higher categories to prove a common generalization of the holomorphic Atiyah-Bott fixed point formula and the Grothendieck-Riemann-Roch theorem. The proof is quite different from the original one proposed by Grothendieck et al.: it relies on the interplay between self dualities of quasi- and ind- coherent sheaves on $X$ and formal deformation theory of Gaitsgory-Rozenblyum. In particular, we give a description of the Todd class in terms of the difference of two formal group structures on the derived loop scheme $\mathcal LX$. The equivariant case is reduced to the non-equivariant one by a variant of the Atiyah-Bott localization theorem.

The positivity of local equivariant Hirzebruch class for toric varieties

The central object of investigation of this paper is the Hirzebruch class, a deformation of the Todd class, given by Hirzebruch (for smooth varieties) in his celebrated book "Topological Methods in Algebraic Geometry". The generalization for singular varieties is due to Brasselet-Sch\"urmann-Yokura. Following the work of Weber, we investigate its equivariant version for (possibly singular) toric varieties. The local decomposition of the Hirzebruch class to the fixed points of the torus action and a formula for the local class in terms of the defining fan are mentioned. After this review part, we prove the positivity of local Hirzebruch classes for all toric varieties, thus proving false the alleged counterexample given by Weber.

Remarks on Kawamata’s effective non-vanishing conjecture for manifolds with trivial first Chern classes

Kawamata proposed a conjecture predicting that every nef and big line bundle on a smooth projective variety with trivial first Chern class has nontrivial global sections. We verify this conjecture for several cases, including (i) all hyperk\"{a}hler varieties of dimension $\leq 6$; (ii) all known hyperk\"{a}hler varieties except for O'Grady's 10-dimensional example; (iii) general complete intersection Calabi-Yau varieties in certain Fano manifolds (e.g. toric ones). Moreover, we investigate the effectivity of Todd classes of hyperk\"{a}hler varieties and Calabi-Yau varieties. We prove that the fourth Todd classes are "fakely effective" for all hyperk\"{a}hler varieties and general complete intersection Calabi-Yau varieties in products of projective spaces.

On Analytic Todd Classes of Singular Varieties

AbstractLet $(X,h)$ be a compact and irreducible Hermitian complex space. This paper is devoted to various questions concerning the analytic K-homology of $(X,h)$. In the 1st part, assuming either $\dim (\operatorname{sing}(X))=0$ or $\dim (X)=2$, we show that the rolled-up operator of the minimal $L^2$-$\overline{\partial }$ complex, denoted here $\overline{\eth }_{\textrm{rel}}$, induces a class in $K_0 (X)\equiv KK_0(C(X),\mathbb{C})$. A similar result, assuming $\dim (\operatorname{sing}(X))=0$, is proved also for $\overline{\eth }_{\textrm{abs}}$, the rolled-up operator of the maximal $L^2$-$\overline{\partial }$ complex. We then show that when $\dim (\operatorname{sing}(X))=0$ we have $[\overline{\eth }_{\textrm{rel}}]=\pi _*[\overline{\eth }_M]$ with $\pi :M\rightarrow X$ an arbitrary resolution and with $[\overline{\eth }_M]\in K_0 (M)$ the analytic K-homology class induced by $\overline{\partial }+\overline{\partial }^t$ on $M$. In the 2nd part of the paper we focus on complex projective varieties $(V,h)$ endowed with the Fubini–Study metric. First, assuming $\dim (V)\leq 2$, we compare the Baum–Fulton–MacPherson K-homology class of $V$ with the class defined analytically through the rolled-up operator of any $L^2$-$\overline{\partial }$ complex. We show that there is no $L^2$-$\overline{\partial }$ complex on $(\operatorname{reg}(V),h)$ whose rolled-up operator induces a K-homology class that equals the Baum–Fulton–MacPherson class. Finally in the last part of the paper we prove that under suitable assumptions on $V$ the push-forward of $[\overline{\eth }_{\textrm{rel}}]$ in the K-homology of the classifying space of the fundamental group of $V$ is a birational invariant.

Formality and Kontsevich–Duflo type theorems for Lie pairs

Kontsevich's formality theorem states that there exists an L∞ quasi-isomorphism from the dgla Tpoly•(M) of polyvector fields on a smooth manifold M to the dgla Dpoly•(M) of polydifferential operators on M, which extends the classical Hochschild–Kostant–Rosenberg map. In this paper, we extend Kontsevich's formality theorem to Lie pairs, a framework which includes a range of diverse geometric contexts such as complex manifolds, foliations, and g-manifolds (that is, manifolds endowed with an action of a Lie algebra g). The spaces tot(Γ(Λ•A∨)⊗RTpoly•) and tot(Γ(Λ•A∨)⊗RDpoly•) associated with a Lie pair (L,A) each carry an L∞ algebra structure canonical up to L∞ isomorphism. These two spaces serve as replacements for the spaces of polyvector fields and polydifferential operators, respectively. Their corresponding cohomology groups HCE•(A,Tpoly•) and HCE•(A,Dpoly•) admit canonical Gerstenhaber algebra structures. We establish the following formality theorem for Lie pairs: there exists an L∞ quasi-isomorphism from tot(Γ(Λ•A∨)⊗RTpoly•) to tot(Γ(Λ•A∨)⊗RDpoly•) whose first Taylor coefficient is equal to hkr∘(tdL/A∇)12. Here the cocycle (tdL/A∇)12 acts on tot(Γ(Λ•A∨)⊗RTpoly•) by contraction. Furthermore, we prove a Kontsevich–Duflo type theorem for Lie pairs: the Hochschild–Kostant–Rosenberg map twisted by the square root of the Todd class of the Lie pair (L,A) is an isomorphism of Gerstenhaber algebras from HCE•(A,Tpoly•) to HCE•(A,Dpoly•). As applications, we establish formality and Kontsevich–Duflo type theorems for complex manifolds, foliations, and g-manifolds. In the case of complex manifolds, we recover the Kontsevich–Duflo theorem of complex geometry.

Todd class via homotopy perturbation theory

We compute the quantized cycle class of a closed embedding of complex manifolds defined by Grivaux using homotopy perturbation theory. In the case of a diagonal embedding, our approach provides a novel perspective of the usual Todd class of a complex manifold.

Atiyah and Todd classes arising from integrable distributions

In this paper, we study the Atiyah class and the Todd class of the DG manifold (F[1],dF) corresponding to an integrable distribution F⊂TKM=TM⊗RK, where K=R or ℂ. We show that these two classes are canonically identical to those of the Lie pair (TKM,F). As a consequence, the Atiyah class of a complex manifold X is isomorphic to the Atiyah class of the corresponding DG manifold (TX0,1[1],∂̄). Moreover, if X is a compact Kähler manifold, then the Todd class of X is also isomorphic to the Todd class of the corresponding DG manifold (TX0,1[1],∂̄).

Spectral Hirzebruch–Milnor classes of singular hypersurfaces

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 hyperplane section is not 1. These formulas using hyperplane sections instead of hypersurface ones are easier to calculate in certain cases. Here we use the Thom–Sebastiani theorem for the underlying filtered D-modules of vanishing cycles, from which we can deduce the Thom–Sebastiani type theorem for spectral Hirzebruch–Milnor classes. For the Chern classes after specializing to $$y=-\,1$$, we can give a relatively simple formula for the localized Milnor classes, which implies a new formula for the Euler numbers of projective hypersurfaces, using iterated hyperplane sections. Applications to log canonical thresholds and Du Bois singularities are also explained; for instance, the latter can be detected by using Hirzebruch–Milnor classes in the projective hypersurface case.

Twisted Hilbert modular surfaces, arithmetic intersections and the Jacquet–Langlands correspondence

We study arithmetic intersections on quaternionic Hilbert modular surfaces and Shimura curves over a real quadratic field. Our first main result is the determination of the degree of the top arithmetic Todd class of an arithmetic twisted Hilbert modular surface. This quantity is then related to the arithmetic volume of a Shimura curve, via the arithmetic Grothendieck–Riemann–Roch theorem and the Jacquet–Langlands correspondence.

Equivariant characteristic classes of external and symmetric products of varieties

We obtain refined generating series formulae for equivariant characteristic classes of external and symmetric products of singular complex quasi-projective varieties. More concretely, we study equivariant versions of Todd, Chern and Hirzebruch classes for singular spaces, with values in delocalized Borel-Moore homology of external and symmetric products. As a byproduct, we recover our previous characteristic class formulae for symmetric products, and obtain new equivariant generalizations of these results, in particular also in the context of twisting by representations of the symmetric group.