The Todd genus is the genus of the formal power series
with B2k as before, Bernoulli numbers. The first few values are
- Td0 = 1
- Td1 = c1/2
- Td2 = (c2 + c12)/12.
- Td3 = (c1c2)/24.
- Td4 = (-c14 + 4 c2c12+3c22+c3c1-c4)/720.
The Todd genus has the particular property that it assigns the value 1 to all complex projective spaces (i.e. ), and this suffices to show that the Todd genus agrees with the arithmetic genus for algebraic varieties as the arithmetic genus is also 1 for complex projective spaces. This observation is a consequence of the Hirzebruch–Riemann–Roch theorem, and in fact was one of the key developments that led to the formulation of that theorem.
Read more about this topic: Genus Of A Multiplicative Sequence
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