On general type varieties admitting global holomorphic forms

TL;DR

This paper establishes new inequalities relating volume and holomorphic forms for general type varieties, proves the minimal volume conjecture in certain cases, and introduces lifting principles for canonical stability indices.

Contribution

It introduces generalized Noether type inequalities for all nonsingular projective varieties of general type, advancing understanding of their geometric properties.

Findings

Proved Noether type inequalities for all nonsingular projective varieties of general type.

Confirmed the minimal volume conjecture for certain 3-folds of general type.

Disclosed new lifting principles for canonical stability indices.

Abstract

For all nonsingular projective $n$-folds $V$ of general type, we prove the existence of Noether type inequalities in the following form: $$\text{vol}(V)\geq a_{n,k}h^0(\Omega_V^k)-b_{n,k}$$ where $0< k\leq n$, $a_{n,k}$ 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 $\chi({\mathcal O})\neq 2,3$ 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 $q>\dim(V)$.