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Todd Class

The Todd class of a complex vector bundle with formal Chern roots x_j is

td(E)=product_(j)(x_j)/(1-e^(-x_j)).

The expression is interpreted as a power series in rational cohomology. Its first terms are 1+c_1/2+(c_1^2+c_2)/12+c_1c_2/24. For a complex manifold X, td(X) means the Todd class of its tangent bundle.

For a smooth projective threefold with trivial canonical bundle, c_1(X)=0, so the square root with constant term 1 is sqrt(td(X))=1+c_2(X)/24. This factor occurs in a large-volume central charge (Bayer 2009).

See also

Canonical Bundle, Chern Character, Chern Class, Tangent Bundle

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References

Bayer, A. "Polynomial Bridgeland Stability Conditions and the Large Volume Limit." Geom. Topol. 13, 2389-2425, 2009. https://doi.org/10.2140/gt.2009.13.2389.The Stacks Project Authors. "Todd Classes." The Stacks Project, Tag 02UN. https://stacks.math.columbia.edu/tag/02UN.

Cite this as:

Weisstein, Eric W. "Todd Class." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ToddClass.html

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