Curves on algebraic surfaces satistying $K^2_{X}\cdot C^2\leq K_XC\cdot K_XC$.

Question

Let $X$ be a minimal algebraic surface of general type and $C\subset X$ an ample irreducible curve. I am reading a text in which the following inequality is used sometimes: $$ K^2_{X}\cdot C^2\leq K_XC\cdot K_XC. $$ Moreover, if I am not wrong, it is claimed to be a consequence of Hodge Index Theorem. How could we prove that inequality? Is it indeed related to the Hodge Index Theorem? Do we need more assumptions on $C$ or $K_X$?

Answer 1

Yes, it's Hodge Index. Since $C$ is ample, some multiple of it is very ample, and so $C$ is a basis for the positive definite part of $NS(X)$. Now let $n=C.C, m=C.K, \ell = K.K$; since $C.(mC-nK)=0$, $(mC-nK)^2\le 0$ -- with equality, in fact, if and only if $mC=nK$. This gives us $m^2n+n^2\ell -2m^2n \le 0$, which rearranges to what you want.