Hirzebruch-Riemann-Roch theorem

The Hirzebruch-Riemann-Roch theorem is a theorem of algebraic geometry . It can be understood as a generalization of the Riemann-Roch theorem and is named after the mathematicians Friedrich Hirzebruch , Bernhard Riemann and Gustav Roch . Hirzebruch proved this theorem for projective complex manifolds. In the version formulated below, it applies generally to complex manifolds . The Hirzebruch-Riemann-Roch theorem itself can be understood as a special case of the Grothendieck-Riemann-Roch and Atiyah-Singer theorems .

sentence

Let be a holomorphic vector bundle over a compact complex manifold . Then applies ${\displaystyle E\to M}$

${\displaystyle \chi (M,E):=\sum _{i}(-1)^{i}\dim H^{i}(M,E)=\int _{M}Td(M)ch(E),}$

where the Todd class of the tangent bundle , the total Chern class of and the sheaf cohomology of the sheaf of cuts in is. ${\displaystyle Td(M)}$ ${\displaystyle ch(E)}$ ${\displaystyle E}$ ${\displaystyle H^{i}(M,E)}$ ${\displaystyle E}$

Riemann surfaces

For a divisor on a Riemann surface , the straight line bundle corresponding to the divisor is considered and obtained ${\displaystyle D}$ ${\displaystyle S}$

${\displaystyle \dim H^{0}({\mathcal {O}}(D))-\dim H^{1}({\mathcal {O}}(D))=c_{1}({\mathcal {O}}(D))+{\frac {1}{2}}c_{1}(TS),}$

what about the classic phrase by Riemann-Roch

${\displaystyle l(D)-l(K-D)=deg(D)+1-g}$

is equivalent.

FUNCTIONAL ACCESS

The set of Grothendieck Riemann-Roch is a generalization of the theorem for morphisms and has this functorial access a simpler proof. The Hirzebruch-Riemann-Roch theorem is the special case for . ${\displaystyle f\colon M\to N}$ ${\displaystyle N=\left\{{\text{Punkt}}\right\}}$

literature

• Friedrich Hirzebruch : New Topological Methods in Algebraic Geometry . Springer-Verlag 1956. ISBN 978-3-662-41083-7

Individual evidence

1. ^ Obituary Hirzebruch Friedrich - Bavarian Academy of Science