9
$\begingroup$

The Hirzebruch Riemann-Roch Theorem (HRR) expresses an analytic/algebraic invariant, namely the Euler-Poincaré characteristic of a vector bundle $V$ over a compact complex/algebraic manifold $X$, as the evaluation of a cohomological expression. It has different manifestations in the analytic and algebraic categories; the general form in the analytic category is something like $$ \chi(X,V) = T(X,V), $$ where $X$ is a compact complex manifold, $V$ a holomorphic vector bundle, $\chi(M,V)$ $=$ $\sum_{i \ge 0} (-1)^i\dim_{\mathbb{C}}H^i(X,\mathcal{O}(V))$ the holomorphic Euler-Poincaré characteristic, and $T(X,V)$ a particular cohomological expression (see [1], § 21; [2]).

Subsequently, there were two important generalizations of this result:

  • The Grothendieck Rieman-Roch Theorem (GRR). This shifts the focus from objects $(X,V)$ to morphisms $f:X \rightarrow Y$, and to coherent sheaves, which are amenable to being pushed forward, and so its algebraic manifestion appears as a commutative diagram $\require{AMScd}$ \begin{CD}\tag{1} K_{\omega}(X) @>f_!>> K_{\omega}(Y)\\ @V\tau_XVV @VV\tau_YV\\ K_{\text{coh}}(X) @>>f_*> K_{\text{coh}}(Y) \end{CD} Here, $f:Y \rightarrow Y$ is a morphism of algebraic varieties, $K_{\omega} :=(K_{\omega}(-),(-)_*)$ an algebraic theory built from coherent sheaves, $K_{\text{coh}} := (K_{\text{coh}}(-), (-)_!)$ a cohomological theory, and $\tau:K_{\omega}\rightarrow K_{\text{coh}}$ a natural transformation of theories (see [3]; [4] Theorem 18.3; [5]; [6]). This generalizes (HRR) insofar as an appropriate instance of (HRR) is obtained from an appropriate instance of (GRR) by applying it to the morphism $X \rightarrow \text{pt}$.

  • The Atiyah-Singer Index Theorem (ASI). This shifts the focus from objects $(X,V)$ to (pseudo)elliptic operators $D:\mathcal{C}^{\infty}(X,E) \rightarrow \mathcal{C}^{\infty}(X;F)$ on real $\mathcal{C}^{\infty}$-vector bundles over a compact real $\mathcal{C}^{\infty}$-manifold $X$. It takes the form $$ \text{index}(D) = T(X;E,F) $$ where $$ \text{index}(D) := \dim \ker(D) - \dim \text{coker}(D) $$ is an analytical invariant and $T(X;E,F)$ a cohomological (topological) invariant (see [7], Theorem (6.8)). Then (HRR) arises by specializing $D$ to the Dolbeault operator $\overline{\partial}$ (see [8], Theorem (4.3).

Question:

Does there exist a generalization of (ASI) to a relative form, (GAS), say, in a similar fashion as (HRR) generalizes to (GRR), so that we have a square of generalizations
$\require{AMScd}$ \begin{CD}\tag{2} (HRR) @>>> (ASI)\\ @VVV @VVV\\ (GRR) @>>> (GAS) \end{CD}
so that (GAS) specializes, on one hand, to (ASI), and, on the other hand, to (GRR) (and, as the cherry on the cake, do everything equivariantly).

Addendum. Meanwhile, [9] has come to my attention, showing that I am with my question in very good company. Specifically, 6. Further remarks. (3) there refers to loc.cit. Theorem 1 (ASI) and loc.cit. Theorem 3 (HRR) as follows:

In view of Theorem 3 it is natural to hope that the Grothendieck-Riemann-Roch theorem for proper maps of complex manifolds will come out of a suitable generalization of Theorem 1.

Using the notation of loc.cit. one central issue might be, for the special case where the target theory is singular (co)homology, to give, for a proper map $f:X \rightarrow Y$ of real manifolds and an operator $D$ on $X$, an appropriate definition of $\text{ch}(f!D)$.

_

[1] Hirzebruch, F. -- Topological Methods in Algebraic Geometry (Reprint of the 1978 Edition). Springer 1978.

[2] O'Brian, N.R. et al. -- Hirzebruch-Riemann-Roch for coherent sheaves, Amer. J. Math. 103, (1981), 253-271.

[3] Borel, A, & Serre, J.-P. -- Le théorème de Riemann-Roch (d'après Grothendieck), Bull. Soc. Math. France 86 (1958), 97 - 136.

[4] Fulton, W. -- Intersection Theory (2nd Ed.), Springer 1998. (Reprint of the 1978 Edition). Springer 1978.

[5] Baum, P. et al. -- Riemann-Roch and topological K-theory for singular varieties, Acta Math. 43 (1979),155-192.

[6] O'Brian, N.R. et al. -- A Grothendieck-Riemann-Roch formula for maps of complex manifolds, Math. Ann. 271, (1985), 493-526.

[7] Atiyah, M.F & Singer, I.M. -- The index of elliptic operators I. Ann. Math. 87 (1968), 484-530.

[8] Atiyah, M.F & Singer, I.M. -- The index of elliptic operators III. Ann. Math. 87 (1968), 546-604.

[9] Atiyah, M.F & Singer, I.M. -- The index of elliptic operators on compact manifolds. Bull. Amer. Math. Soc. 69 (1963), no. 3, 422--433.

Improve this question
$\endgroup$
10
  • 8
    $\begingroup$ Isn’t this just the families index theorem, proved in Atiyah-Singer IV? $\endgroup$
    – Aaron Bergman
    Commented Aug 7, 2020 at 1:14
  • 2
    $\begingroup$ I guess the families index theorem corresponds to the special case of GRR where the map is smooth. $\endgroup$
    – Dustin Clausen
    Commented Aug 7, 2020 at 8:51
  • $\begingroup$ @AaronBergmann Rather not, I think, for else, (GRR) would arise just by specializing some parameters in its formulation, just as (HRR) arises from (GRR) by specializing the morphism and from (ASI) by specializing the operator, and I know of no way to formulate the Index Theorem for Families in such a way that specializing some parameters ieads to (GRR). By the way, this would make the appearance of elaborate papers like [6] 14 years later look somewhat strange. $\endgroup$
    – MathCrawler
    Commented Aug 8, 2020 at 23:50
  • $\begingroup$ @MathCrawler It seems to me that multiple generalizations are going on here. GRR both generalizes HRR to the relative case and to the case of coherent sheaves. The families index theorem, I believe, gives you the relative generalization you are looking for. However, for coherent sheaves you have to be a bit more careful because they can be trickier on analytic manifolds than on algebraic varieties, even in the smooth case. Cont’d. $\endgroup$
    – Aaron Bergman
    Commented Aug 9, 2020 at 1:57
  • 1
    $\begingroup$ To amplify Dustin's comment, I believe the Families Index Theorem applies to maps that are "smooth" in the sense of algebraic geometry, i.e., submersions, while GRR applies to general proper maps, such as resolutions of singularities -- i.e. the families allowed in algebraic geometry are typically much less constrained than in differential topology. Then again maybe all that's missing (given general functoriality) is to know a version of Atiyah-Singer for closed embeddings, and maybe that's well known? $\endgroup$
    – David Ben-Zvi
    Commented Aug 10, 2020 at 0:07

1 Answer 1

Reset to default
8
$\begingroup$

I'm sorry for the self-citation. But your question is largely answered in the monograph Coherent Sheaves, Superconnections, and Riemann-Roch-Grothendieck, or the arxiv version, joint work of Jean-Michel Bismut, Shu Shen, and me.

In the above work we proved the Grothendieck-Riemann-Roch theorem for coherent sheaves on compact complex manifolds. The target theory is the Bott-Chern cohomology.

We use antiholomorphic flat superconnections as a tool to study coherent sheaves on complex manifolds, on which we can build Chern-Weil theory. We then proved the Grothendieck-Riemann-Roch theorem as a version of the family index theorem.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .