Todd class

The Todd class is a construction from the algebraic topology of the characteristic classes . The Todd class of a vector bundle can be explained with the theory of the Chern classes and exists where they exist, especially in differential topology , the theory of complex manifolds, and in algebraic geometry . Roughly speaking, it acts like a reciprocal Chern class or is related to it like a bundle of normals to a bundle of normals. The Todd class plays a fundamental role in the generalization of the Riemann-Roch Theorem to higher dimensions, in the Hirzebruch-Riemann-Roch Theorem or the Grothendieck-Hirzebruch-Riemann-Roch Theorem .

It is named after the English mathematician John Arthur Todd , who introduced a special case into algebraic geometry in 1937, before the definition of the Chern classes. The geometric idea is sometimes called the Todd-Eger class , the general definition in higher dimensions comes from Friedrich Hirzebruch (in his book Topological Methods of Algebraic Geometry ).

definition

In order to define the Todd class for a complex -dimensional vector bundle on a topological space , it is usually possible to restrict oneself to a Whitney sum (i.e. direct sum) of line bundles using a general method from the theory of characteristic classes, the Chern roots . Look at ${\ displaystyle \ operatorname {td} (E)}$ ${\ displaystyle n}$ ${\ displaystyle E}$ ${\ displaystyle X}$

{\ displaystyle {\ begin {aligned} Q (x) = & {\ frac {x} {1-e ^ {- x}}} \\ = & \ sum _ {k = 0} ^ {\ infty} { \ frac {(-1) ^ {k} B_ {k}} {k!}} x ^ {k} = 1 + {\ frac {1} {2}} x + {\ frac {1} {12}} x ^ {2} + \ ldots \ end {aligned}}}

as a formal power series , where the coefficients are Bernoulli numbers . If that has as Chern roots, is ${\ displaystyle B_ {i}}$ ${\ displaystyle E}$ ${\ displaystyle \ alpha _ {i}}$

{\ displaystyle \ operatorname {td} (E) = \ prod _ {i = 1} ^ {n} Q (\ alpha _ {i}) = \ prod _ {i = 1} ^ {n} \ sum _ { k = 0} ^ {\ infty} {\ frac {(-1) ^ {k} B_ {k}} {k!}} \ alpha _ {i} ^ {k} \ ,,}

what is computed in the cohomology ring of (or in its completion, if one considers infinite-dimensional manifolds). ${\ displaystyle X}$

The explicit form of the Todd class as a formal power series in the Chern classes is:

{\ displaystyle \ operatorname {td} (E) = 1 + c_ {1} \ cdot {\ frac {1} {2}} + (c_ {1} ^ {2} + c_ {2}) \ cdot {\ frac {1} {12}} + c_ {1} c_ {2} \ cdot {\ frac {1} {24}} + \ ldots \ ,,}

where the cohomology classes are the Chern classes of and are in the cohomology group . If is finite dimensional, most of the terms vanish and is a polynomial in the Chern classes. ${\ displaystyle c_ {i}}$ ${\ displaystyle E}$ ${\ displaystyle H ^ {2i} (X)}$ ${\ displaystyle X}$ ${\ displaystyle \ operatorname {td} (E)}$

literature

J. Todd : The arithmetical theory of algebraic loci. In: Proceedings of the London Mathematical Society. 43, 1937, ISSN 0024-6115 , pp. 190-225.
Friedrich Hirzebruch : Topological methods in algebraic geometry ( Grundlehren der Mathematischen Wissenschaften 131). 2nd corrected printing of the 3rd edition. Springer, Berlin et al. 1978, ISBN 3-540-03525-7 .