arXiv research

Proves a theorem for complex flat vector bundles using differential forms.

We give a direct proof that the Freed-Lott differential analytic index is well defined and a condensed proof of the differential Grothendieck-Riemann-Roch theorem. As a byproduct we also obtain a direct proof that the R/Z analytic index is well defined and a condensed proof of the R/Z Grothendieck-Riemann-Roch theorem.

Paper introduces Lie algebroid index theory and a generalized Riemann-Roch theorem.

In this paper we give a simplified proof of the flat Grothendieck-Riemann-Roch theorem. The proof makes use of the local family index theorem and basic computations of the Chern-Simons form. In particular, it does not involve any adiabatic limit computation of the reduced eta-invariant.

Study of orbifold Chern character using superconnections.

Develops equivariant Chern characters for coherent sheaves with group actions.

In this paper we give a proof of an index theorem by Bismut. As a consequence we obtain another proof of the Grothendieck-Riemann-Roch theorem in differential cohomology.

The purpose of this paper is to give a proof of the real part of the Riemann-Roch-Grothendieck theorem for complex flat vector bundles at the differential form level in the even dimensional fiber case. The proof is, roughly speaking, an application of the local family index theorem for a perturbed twisted spin Dirac op…

In the present paper we discuss an independent on the Grothendieck-Sato isomorphism approach to the Riemann-Roch-Hirzebruch formula for an arbitrary differential operator. Instead of the Grothendieck-Sato isomorphism, we use the Topological Quantum Mechanics (more or less equivalent to the well-known constructions with…

We prove the analogue of the Riemann-Roch formula for the noncommutative two torus $A_θ = C(\mathbb{T}_θ^2)$ equipped with an arbitrary translation invariant complex structure and a Weyl factor represented by a positive element $k\in C^{\infty}(\mathbb{T}_θ^2)$. We consider a topologically trivial line bundle equipped…

New topological Riemann-Roch theorem for circle fibrations.

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.

In this paper, we introduce six axioms for relative Bott-Chern secondary characteristic classes and prove the uniqueness and existence theorem for them. Such a work provides us a natural way to understand and hence to prove the arithmetic Grothendieck-Riemann-Roch theorem.

Harder's reduction theory provides filtrations of euclidean buildings that allow one to deduce cohomological and homological properties of S-arithmetic groups over global function fields. In this survey I will sketch the main points of Harder's reduction theory starting from Weil's geometry of numbers and the Riemann-R…

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 differenti…

The goal of this paper is to re-examine D-brane Ramond-Ramond field couplings in the presence of a B-field. We will argue that the generalised geometry induced on the world volume by the B-field results in an important but subtle change on the coupling. In order to explain this, we use the language of differential K-th…

Develops a new index theory for odd Z/kZ K-theory.

Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The ``quantization commutes with reduction'' theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M, provided the quo…

Formula calculates Riemann-Roch number for singular symplectic quotients.

The mathematical physicists Bershadsky-Cecotti-Ooguri-Vafa (BCOV) proposed, in a seminal article from '94, a conjecture extending genus zero mirror symmetry to higher genera. With a view towards a refined formulation of the Grothendieck-Riemann-Roch theorem, we offer a mathematical description of the BCOV conjecture at…

The classical integral localization formula for equivariantly closed forms (Theorem 7.11 in [BGV]) is well-known and requires the acting Lie group to be compact. It is restated here as Theorem 2. In this article we extend this result to NONcompact groups. The main result is Theorem 20. Then, using this generalization, …

We calculate the local Riemann-Roch numbers of the zero sections of $T^*S^n$ and $T^*\R P^n$, where the local Riemann-Roch numbers are defined by using the $S^1$-bundle structure on their complements associated to the geodesic flows.

This paper proves positivity of Riemann-Roch polynomials for hyperkähler manifolds.

We compute the sum of the positive Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow. The computation is based on the analytic Riemann-Roch Theorem and uses a comparison of determinants of flat and hyperbolic Laplacians when the underlying Riemann surface degenerates.

In this paper, we develop differential twisted K-theory and define a twisted Chern character on twisted K-theory which depends on a choice of connection and curving on the twisting gerbe. We also establish the general Riemann-Roch theorem in twisted K-theory and find some applications in the study of twisted K-theory o…

We consider a smooth groupoid of the form Σ\rtimesΓwhere Σis a Riemann surface and Γa discrete pseudogroup acting on Σby local conformal diffeomorphisms. After defining a K-cycle on the crossed product C_0(Σ)\rtimesΓgeneralising the classical Dolbeault complex, we compute its Chern character in cyclic cohomology, using…

The first author conjectured certain relations for Morita-Mumford classes and Newton classes in the integral cohomology of mapping class groups (integral Riemann-Roch formulae). In this paper, the conjecture is verified for cyclic subgroups of mapping class groups.

We survey the Hirzebruch signature theorem as a special case of the Atiyah-Singer index theorem. The family version of the Atiyah-Singer index theorem in the form of the Riemann-Roch-Grothendieck-Quillen (RRGQ) formula is then applied to the complexified signature operators varying along the universal family of ellipti…

We consider a proper flat fibration with real base and complex fibers. First we construct odd characteristic classes for such fibrations by a method that generalizes constructions of Bismut-Lott. Then we consider the direct image of a fiberwise holomorphic vector bundle, which is a flat vector bundle on the base. We gi…

These notes are the first chapter of a monograph, dedicated to a detailed proof of the equivariant index theorem for transversally elliptic operators. In this preliminary chapter, we prove a certain number of natural relations in equivariant cohomology. These relations include the Thom isomorphism in equivariant cohomo…

This note announces a general construction of characteristic currents for singular connections on a vector bundle. It develops, in particular, a Chern-Weil-Simons theory for smooth bundle maps $α: E \rightarrow F$ which, for smooth connections on $E$ and $F$, establishes formulas of the type $$ φ\ = \ \text{\rm Res}_φΣ…

We generalize work of Deligne and Gillet-Soulé on a Riemann-Roch type isometry, to the case of the trivial sheaf on cusp compactifications of Riemann surfaces $Γ\backslash\mathbb{H}$, for $Γ\subset PSL_{2}(\mathbb{R})$ a fuchsian group of the first kind, equipped with the Poincaré metric. This metric is singular at cus…

We present a new proof, as well as a ${\bf C/Q}$ extension, of the Riemann-Roch-Grothendieck theorem of Bismut-Lott for flat vector bundles. The main techniques used are the computations of the adiabatic limits of $η$-invariants associated to the so-called sub-signature operators. We further show that the Bismut-Lott a…

This note is based on a talk given at the 2019 ISAAC Congress in Aveiro, Portugal. We give an expository account of joint work with Daniele Alessandrini and Gye-Seon Lee on Hitchin components for orbifold groups (arXiv:1811.05366), recasting part of it in the language of analytic orbi-curves. This reduces the computati…

Paper constructs Chern character for coherent sheaves.

Novel threefold partitioning of $L^2$-spinor space on cone links.

The paper establishes inequalities for Chern classes and Riemann-Roch type inequalities for projective manifolds.

We study algebro-geometric consequences of the quantised extremal Kähler metrics, introduced in the previous work of the author. We prove that the existence of quantised extremal metrics implies weak relative Chow polystability. As a consequence, we obtain asymptotic weak relative Chow polystability and $K$-semistabili…

In this paper, we extend Deligne's functorial Riemann-Roch isomorphism for hermitian holomorphic line bundles on Riemann surfaces to the case of flat, not necessarily unitary connections. The Quillen metric and star-product of Gillet-Soule are replaced with complex valued logarithms. On the determinant of cohomology si…

Generalized differential cohomology theories, in particular differential K-theory (often called "smooth K-theory"), are becoming an important tool in differential geometry and in mathematical physics. In this survey, we describe the developments of the recent decades in this area. In particular, we discuss axiomatic ch…

Formula derived for Bott-Chern classes in complex blow-ups.

We define a local Riemann-Roch number for an open symplectic manifold when a complete integrable system without Bohr-Sommerfeld fiber is provided on its end. In particular when a structure of a singular Lagrangian fibration is given on a closed symplectic manifold, its Riemann-Roch number is described as the sum of the…

Paper extends variational formula for Bismut-Cheeger eta form, proving key theorems in K-theory.

This is the sequel of the first part math.DG/0611281. Here, the procedure of transgressing the families index theorem (the so-called $η$-form) is adapted to take in account the case of Dirac type operators with kernels of varying dimension. The constructed form is then used to define the direct image under proper subme…

Researchers express spectral determinants on hyperbolic cones.

If E is a C^\infty complex vector bundle on an oriented C^\infty manifold Σ, diffeomorphic to a circle, then the space of sections of E has a canonical polarization in the sense of Pressley and Segal and so one has its determinantal gerbe with lien C^*, the group of nonzero complex numbers. If q:Σ-->B is a smooth famil…

Let $Δ\subset \mathbb{R}^n$ be an $n$-dimensional integral Delzant polytope. It is well-known that there exist the $n$-dimensional compact toric manifold $X_Δ$ and the very ample $(\mathbb{C}^\times)^n$-equivariant line bundle $L_Δ$ on $X_Δ$ associated with $Δ$. In the present paper, we give a necessary and sufficient …

Study quantizes topological numbers on degenerating Einstein manifolds.