PomeProposition 19.9.23.5C0C7B20BA7A For every fixed holomorphic vector bundle $E$, the Euler characteristic $\chi (X,L^ p\otimes E)$ is a polynomial in $p$ of degree at most $r$.
Proposition 19.9.235C0C7B20BA7A
Proof. The analytic Hirzebruch–Riemann–Roch theorem gives
\[ \chi (X,L^ p\otimes E) =\int _ X\mathrm e^{p\beta }\operatorname {ch}(E) \operatorname {td}(T^{1,0}X) =\sum _{j=0}^{r}\frac{p^ j}{j!} \int _ X\beta ^ j\wedge \operatorname {ch}(E) \wedge \operatorname {td}(T^{1,0}X). \]
Here the integral selects real degree $2n$, and the Chern character and Todd forms are computed with fixed Hermitian metrics. The curvature normalization represents $c_1(L)$, and $\beta ^{r+1}=0$ pointwise by its rank bound. Every coefficient is independent of $p$. $\square$
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