$Td_p$ notation of Kotschick

Question

In this paper, notation $Td_p$ is used without explicit definition (it is stated that it is a certain combination of Chern numbers). It is claimed that HRR theorem implies $$ Td_p(M)=\sum_{q}(-1)^q h^{p, q}(M) $$ for any closed complex manifold $M$.

Am I correct assuming that $Td_p$ stands for the pairing of $Td(M)Ch(\Omega^p)$ with the fundamental class of $M$? Here $\Omega$ is the canonical bundle.

Answer 1

The HRR-Theorem asserts $$\int Td(M)ch(E)=\chi(M,E)=\sum_q(-1)^q\dim H^q(M,E)$$ for every vector bundle $E$. With $E=\Omega^p$ the sheaf of holomorphic $p$-forms you get $$\int Td(M)ch(\Omega^p)=\sum_q(-1)^q\dim H^q(M,\Omega^p)=\sum_q(-1)^q h^{p,q}.$$ So your guess is right.