10
$\begingroup$

The degree of the Todd class of the tangent bundle of a variety $Y$ gives the holomorphic genus of $Y$, which is a birational invariant. We also know that the $\hat{A}$-roof of a smooth variety $Y$ is closely related to its Todd class:

$$ (*)\quad Td(Y)= e^{\frac{1}{2}c_1}\ \hat{A}(Y), $$ where $c_1$ is the first Chern class of $Y$. I will call the degree of the $\hat{A}$-roof the $\hat{A}$-genus. On a spin manifold, the $\hat{A}$-genus gives the index of the Dirac operator by the Atiyah-Singer theorem.

I would like to know what is the invariance of the $\hat{A}$-genus. Here are few facts:

-Because the $\hat{A}$-genus can be expressed by Pontryagin numbers, the $\hat{A}$-genus is an oriented cobordism invariant and an oriented diffeomorphic invariant.

-The $\hat{A}$-genus is even an oriented homeomorphic invariant since Novikov proved in the 1960s that Pontryagin numbers are oriented homeomorphic invariants.

-The $\hat{A}$-genus of two birationally equivalent Calabi-Yau varieties are the same. This is a direct consequence of (*) and the birational invariance of the Todd class. Here, by Calabi-Yau, I only mean a variety with a trivial canonical class.

But is the $\hat{A}$-genus always a birational invariant as the Todd-genus?

In this generality, the answer will be negative by looking at the case of complex surfaces. For surfaces, the $\hat{A}$-genus is $(-c_1^2+2c_2)/{24}$ while the holomorphic Euler characteristic is $(c_1^2+c_2)/{12}$. Thus, if both were birational invariant, so would be $c_1^2$ and $c_2$, which is not the case by considering their behavior under the blowup of a smooth point.

But if we restrict to crepant birational maps, I suspect that we might get an invariant. (A well-known example of invariants of crepant birational maps between non-singular varieties are the Hodge numbers.) So my question is:

Is the $\hat{A}$-genus preserved under a crepant birational map between two smooth algebraic varieties?

(I work over the complex numbers and if it makes a difference, I am particularly interested in the case of projective fourfolds.)

Added: A crepant birational map is a birational map $f:X \dashrightarrow Y$ such that the canonical class is preserved $K_X=f^* K_Y$. For example,a flop between two crepant resolutions of the same underlying singular variety in a crepant birational map.

$\endgroup$
  • 2
    $\begingroup$ Did you look at work of Totaro and McTague on elliptic genera? I don't remember the details but it feels potentially relevant. $\endgroup$ – Neil Strickland Jul 18 '18 at 15:32
  • 1
    $\begingroup$ Could you explain what you call a crepant birational map? I suppose you are aware that a crepant birational morphism between smooth projective varieties is an isomorphism. $\endgroup$ – abx Jul 18 '18 at 17:22
  • $\begingroup$ abx: I added a note to answer your question. A crepant birational map is a birational map that preserves the canonical class. This is not necessarily an isomorphism. For example, you can have flops between smooth varieties that are only isomorphic in codimension one. $\endgroup$ – JME Jul 18 '18 at 18:11
  • 1
    $\begingroup$ The Pontryagin numbers are oriented cobordism invariants, but I don't know any reason to expect that they're homotopy invariants; a priori they depend on the isomorphism class of the tangent bundle. $\endgroup$ – Qiaochu Yuan Jul 20 '18 at 21:31
  • $\begingroup$ @QiaochuYuan. Thank you. You are right, I corrected. The only linear combination of Chern numbers that are oriented homotopy invariants are rational multiples of the L-genus. (That is a result of Khan in his paper "Chern numbers and oriented homotopy type"). $\endgroup$ – JME Jul 21 '18 at 9:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.