Section 4.2: Projective varieties and Lorentzian polynomials (continued)

Recall the main theorem: If \(H_1, \dots, H_n\) are nef divisors on \(Y\), then \(\text{vol}_H(w)=(w_1H_1 + \dots + w_nH_n)^d\) is a Lorentzian polynomial. To prove this, last week we clarified Kleiman's theorem.

The core strategy of the proof relies on reducing the problem to the Hodge index theorem on an algebraic surface via partial derivatives. By Kleiman's theorem, we can assume the nef divisors are very ample by taking limits. According to Theorem 2.25, it then suffices to show that the \((d-2)\)-th partial derivatives of \(\text{vol}_H(w)\) are Lorentzian, which means their corresponding quadratic forms have exactly one positive eigenvalue. By Bertini's theorem, intersecting \((d-2)\) very ample divisors yields an irreducible surface \(S \subseteq Y\). Finally, applying the Hodge index theorem to \(S\) (or its resolution of singularities combined with the projection formula) guarantees the existence of exactly one positive eigenvalue, which completes the proof.