On general type varieties admitting global holomorphic forms

Abstract

For all nonsingular projective $n$-folds $V$ of general type, we prove the existence of Noether type inequalities in the following form: $$\text{vol}(V)\ge a_{n,k}{h}^0(\hat{I}{\text{c◯}}_V^k)-b_{n,k}$$ where $0<k\le n$, $a_{n,k}$ and $b_{n,k}$ are positive constants only depending on $n$ and $k$. As applications, we prove the minimal volume conjecture for $3$-folds of general type with $\ddot{I}‡(\mathcal{O})\ne 2,3$ and disclose a new type of lifting principles for the sequence of canonical stability indices for varieties of general type. Finally we prove a theorem about ``strong lifting principle'' on varieties $V$ of general type with $q>\dim (V)$.

Improved and simplified version. 28 pages