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Remarks on the Direct Image Sheaf of Logarithmic Pluricanonical Bundles and the Non-Vanishing Conjecture

Houari Benammar Ammar

TL;DR

This work addresses non-vanishing and positivity questions for log pluricanonical bundles on both irregular and non-irregular varieties. By combining the Albanese map with the Chen–Jiang decomposition, it proves non-vanishing for lc pairs when the lower-dimensional cases hold, and, via canonical bundle formula, identifies a CJ-decomposable subsheaf of pushforwards $f_{*}(ND)$ that yields sections of $N(K_X+\Delta)$ for large $N$. It extends the Catanese–Fujita–Kawamata decomposition to the klt setting, obtaining $f_{*}(ND)=\mathcal{A}_{N}\oplus\mathcal{U}_{N}$ with a flat part and a generically ample part, and deduces corollaries guaranteeing sections of $K_X+\Delta$ under suitable positivity (e.g., $f$-big, $\kappa(Y)\ge0$). The paper also employs Viehweg’s trick to transfer positivity to log settings, showing that pushforwards can be realized as direct summands of canonical pushforwards on suitable covers, thereby enabling the log-CFK decomposition in additional contexts. Collectively, these results advance non-vanishing and abundance-type questions in the log setting, providing structural decompositions and tools for constructing sections in broader MMP scenarios.

Abstract

By applying the Chen-Jiang decomposition, we prove that the non-vanishing conjecture holds for an lc pair \((X, Δ)\), where \(X\) is an irregular variety, provided it holds for lower-dimensional varieties. In the second part, we extend the Catanese-Fujita-Kawamata decomposition to the klt case \((X, Δ)\), which leads to the existence of sections of \(K_X + Δ\) in certain situations.

Remarks on the Direct Image Sheaf of Logarithmic Pluricanonical Bundles and the Non-Vanishing Conjecture

TL;DR

This work addresses non-vanishing and positivity questions for log pluricanonical bundles on both irregular and non-irregular varieties. By combining the Albanese map with the Chen–Jiang decomposition, it proves non-vanishing for lc pairs when the lower-dimensional cases hold, and, via canonical bundle formula, identifies a CJ-decomposable subsheaf of pushforwards that yields sections of for large . It extends the Catanese–Fujita–Kawamata decomposition to the klt setting, obtaining with a flat part and a generically ample part, and deduces corollaries guaranteeing sections of under suitable positivity (e.g., -big, ). The paper also employs Viehweg’s trick to transfer positivity to log settings, showing that pushforwards can be realized as direct summands of canonical pushforwards on suitable covers, thereby enabling the log-CFK decomposition in additional contexts. Collectively, these results advance non-vanishing and abundance-type questions in the log setting, providing structural decompositions and tools for constructing sections in broader MMP scenarios.

Abstract

By applying the Chen-Jiang decomposition, we prove that the non-vanishing conjecture holds for an lc pair \((X, Δ)\), where is an irregular variety, provided it holds for lower-dimensional varieties. In the second part, we extend the Catanese-Fujita-Kawamata decomposition to the klt case \((X, Δ)\), which leads to the existence of sections of in certain situations.
Paper Structure (5 sections, 13 theorems, 57 equations)

This paper contains 5 sections, 13 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Non-vanishing and Chen-Jiang decomposition
  4. Catanese-Fujita-Kawamata decomposition
  5. Viehweg's trick machinery

Key Result

Theorem 1.2

Let $(X, \Delta)$ be a klt pair such that $D \sim_{\mathbb{Q}} m(K_X + \Delta)$ is a Cartier pseudo-effective divisor, with $q(X) > 0$. If the non-vanishing conjecture nonvanishingconjecture holds for lower-dimensional klt pairs, then it also holds for $(X, \Delta)$.

Theorems & Definitions (30)

  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Proposition 1.9
  • Definition 2.1: Greenlazarsfeld
  • ...and 20 more