PomeProposition 23.2.14.E5F423FF7707 On the interval where $\eta _ t=\omega +t\kappa _\lambda >0$, the polynomial $V(t)=\int _ X\eta _ t^ n$ has degree at most $n-1$ and
\[ V'(t)=n(n-1)\int _ XQ\wedge \omega \wedge \eta _ t^{n-2}>0. \]
Proposition 23.2.14E5F423FF7707
Proof. Since $\mathrm d\eta _ t=\lambda \wedge \omega $ and $\mathrm d\lambda ^{\mathrm c}=-\kappa _\lambda $, Stokes' theorem gives
\[ 0=\int _ X\mathrm d(\lambda ^{\mathrm c}\wedge \eta _ t^{n-1}) =-\int _ X\kappa _\lambda \wedge \eta _ t^{n-1} +(n-1)\int _ XQ\wedge \omega \wedge \eta _ t^{n-2}. \]
Differentiating the wedge polynomial proves the formula. The integrand on the right is positive wherever $\lambda \ne 0$, including when $n=2$. Its integral is therefore strictly positive. Finally, $\kappa _\lambda $ is closed and de Rham exact, so
\[ \int _ X\kappa _\lambda ^ n =-\int _ X\mathrm d(\lambda ^{\mathrm c}\wedge \kappa _\lambda ^{n-1})=0. \]
This removes the coefficient of degree $n$. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.