A generalized Second Main Theorem for closed subschemes
Volume 129 / 2022
Abstract
Let $Y_{1}, \ldots , Y_{q}$ be closed subschemes in $\ell $-subgeneral position with index $\kappa $ in a complex projective variety $X$ of dimension $n.$ Let $A$ be an ample Cartier divisor on $X.$ We show that if a holomorphic curve $f:\mathbb C \to X$ has Zariski-dense image, then for every $\epsilon \gt 0,$ \[ \sum ^{q}_{j=1}\epsilon _{Y_{j}}(A)m_{f}(r,Y_{j})\leq _{\rm exc} \biggl (\frac {(\ell -n+\kappa )(n+1)}{\kappa }+\epsilon \bigg )T_{f,A}(r). \] This generalizes the Second Main Theorems for the general position case due to Heier–Levin [Amer. J. Math. 143 (2021), 213–226] and for the subgeneral position case due to He–Ru [J. Number Theory 229 (2021), 125–141]. In particular, whenever all the $Y_j$ are reduced to Cartier divisors, we also give a Second Main Theorem with the distributive constant. The corresponding Schmidt subspace theorem for closed subschemes in Diophantine approximation is also given.
