Pseudonorms and theorems of Torelli type

Abstract

We show that for every positive integer n, there exists an explicit positive integer cn that only depends on n such that if $M$ and $M^{\prime}$ are canonically polarized complex projective manifolds of dimension $n$ and if $H^0 (M, mK_M)$ and $H^0(M^{\prime}, mK_{M^{\prime}})$ are linearly isometric with respect to the pseudonorm $\langle\!\langle \rangle\!\rangle$ for some $m \geqslant c_n$, then $M$ and $M^{\prime}$ are isomorphic. This generalizes a result of Royden for compact Riemann surfaces of genus greater than or equal to $2$. The same approach is used to prove similar and weaker results for projective manifolds with nonnegative Kodaira dimension. We also introduce a kind of singular index for singularities of pairs that refines the traditional log canonical threshold.