Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On general type varieties admitting global holomorphic forms

Meng Chen, Zhi Jiang

TL;DR

The paper proves Noether-type inequalities for general type n-folds, establishing constants a_{n,k} and b_{n,k} so that vol(V) ≥ a_{n,k} h^0(Ω_V^k) − b_{n,k} and explores consequences for canonical stability indices and lifting principles. It develops a lifting framework driven by global 2-forms to bound the canonical stability index r_s in terms of irregularity, including sharp results for 3-folds with large h^{2,0} and a strong lifting principle when q > dim. It further derives Severi-type inequalities via Albanese and generic vanishing, and applies these to the minimal volume v_3, establishing v_3 = 1/420 with a precise equality case. The proofs blend divisor techniques, restriction theorems, multiplier ideals, and vanishing theorems to connect global sections, volumes, and log canonical centers, and extend the results to higher dimensions by leveraging fibration structures and extension theorems. Overall, the work provides explicit volume–Hodge-number bounds, refines the understanding of canonical stability indices, and introduces robust lifting principles for varieties of general type with abundant 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: $$\text{vol}(V)\geq a_{n,k}h^0(Ω_V^k)-b_{n,k}$$ 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 $χ({\mathcal O})\neq 2,3$ 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 $q>\dim(V)$.

On general type varieties admitting global holomorphic forms

TL;DR

The paper proves Noether-type inequalities for general type n-folds, establishing constants a_{n,k} and b_{n,k} so that vol(V) ≥ a_{n,k} h^0(Ω_V^k) − b_{n,k} and explores consequences for canonical stability indices and lifting principles. It develops a lifting framework driven by global 2-forms to bound the canonical stability index r_s in terms of irregularity, including sharp results for 3-folds with large h^{2,0} and a strong lifting principle when q > dim. It further derives Severi-type inequalities via Albanese and generic vanishing, and applies these to the minimal volume v_3, establishing v_3 = 1/420 with a precise equality case. The proofs blend divisor techniques, restriction theorems, multiplier ideals, and vanishing theorems to connect global sections, volumes, and log canonical centers, and extend the results to higher dimensions by leveraging fibration structures and extension theorems. Overall, the work provides explicit volume–Hodge-number bounds, refines the understanding of canonical stability indices, and introduces robust lifting principles for varieties of general type with abundant holomorphic forms.

Abstract

For all nonsingular projective -folds of general type, we prove the existence of Noether type inequalities in the following form: where , and are positive constants only depending on and . As applications, we prove the minimal volume conjecture for -folds of general type with 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 of general type with .
Paper Structure (16 sections, 23 theorems, 92 equations)

This paper contains 16 sections, 23 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Noether type inequalities in terms of $h^{k,0}$
  3. A Lifting principle of $\{r_n\}$ due to global $2$-forms
  4. Notions and notations
  5. Noether type inequalities between $\text{\rm vol}{}$ and $h^{k,0}$
  6. A general inequality
  7. A Severi type inequality between $\text{\rm vol}$ and $h^{1,0}$
  8. The minimal volume conjecture for $v_3$
  9. Proof of Theorem \ref{['3K']}
  10. The case with $\dim V_1=1$
  11. The case with $\dim V_1=2$ and $\text{\rm vol}(V_1)\geq 128$
  12. The case with $\dim V_1=2$ and $\mathrm{vol}(V_1)\leq 127$
  13. The subcase with $\deg \pi=m\geq 2$
  14. The subcase with $\pi$ being birational
  15. The proof of Theorem \ref{['5K']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

(=Theorem inequality1) Fix two integers $n$ and $k$ with $n>0$ and $0< k\leq n$. There exist positive numbers $a_{n, k}$ and $b_{n, k}$ such that the inequality holds for every smooth projective $n$-fold $V$ of general type.

Theorems & Definitions (47)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Example 1.5
  • Theorem 1.6
  • Example 1.7
  • Example 1.8
  • Example 1.9
  • Theorem 1.10
  • ...and 37 more