Local and global heights on weighted projective varieties and Vojta's conjecture

We develop the theory of local and global weighted heights a-la Weil for weighted projective spaces ${\mathbb P}_{{\mathfrak w}, k}^n$ via Cartier divisors by extending the definition of weighted heights for weighted projective varieties and their closed subvarieties, and weighted log pairs. We state Vojta's conjecture for smooth weighted projective varieties, weighted multiplier ideal sheaves, and weighted log pairs and prove that all three versions of the conjecture are equivalent. Furthermore, we introduce generalized weighted greatest common divisors and express them as heights of the weighted projective spaces blownup at a point, relative to an exceptional divisor. We show that a point ${\mathbf x} \in {\mathbb P}_{{\mathfrak w}, k}^n $ is smooth if and only if its generalized logarithmic weighted greatest common divisor $\log {h_{wgcd}} {\mathbf x} >0$. In the last part we prove that assuming Vojta's conjecture for weighted projective varieties one can bound the $\log {h_{wgcd}}$ for any subvariety of codimension $\geq 2$ and a finite set of places $S$. An analogue result is proved for weighted homogenous polynomials with integer coefficients.