1
$\begingroup$

Let $X$ be an algebraic surface, $E$ be a rank $r$ vector bundle over $X$, then the Riemann-Roch formula $$\chi(E)=r\chi(\mathscr O_X)+\frac12\left(c^2_1(E)-c_1(E)c_1(K_X)\right)-c_2(E),$$ holds.

Q: I know how to show it using index theory.

But I do not know how to show it in sheaf language, i.e. by inductive method, construct a sheaf for rank two case.

Could any one help me ?

  • 0
    For line bundles this can be found in Hartshorne, Algebraic Geometry, V Theorem 1.6 . – 2017-02-19
  • 1
    And the general case follows from ibid, Appendix A, Theorem 4.1 (Hirzebruch-Riemann-Roch). – 2017-02-19
  • 0
    @Niels For the line bundle case, this is the Riemann-Roch for the embedded curve, and by induction of rank, we need to show the rank $2$ case. And what is ibid? – 2017-02-19
  • 0
    For higher rank case just use exact sequence to make the vector bundle to be an extension of two vector bundles of lower ranks, and then use induction – 2017-02-19
  • 0
    @DLIN http://lmgtfy.com/?q=ibid – 2017-02-19

0 Answers 0