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PreprintCC BY 4.0arXiv

Log Canonical Counterexamples to the Chen--Jiang Decomposition

  • Houari Benammar Ammar
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Abstract

We construct a projective log-smooth surface pair (X,Δ) and a morphism f:X→ E to an elliptic curve such that f*OOX(KX+Δ) is Atiyah's indecomposable rank-two bundle F2 of degree zero. This gives a negative answer to Question 3.4 of Jiang \cite{Jiang}. We also construct a log canonical pair (X,Δ) and a morphism f to an abelian surface such that f*OOX(m(KX+Δ)) does not admit a Chen--Jiang decomposition for any integer m≥2. Products yield counterexamples in every higher direct image degree and over abelian bases of arbitrary positive dimension.

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Cite

BibTeXRIS
BibTeX
@misc{ammar2026canonical,
  title = {{Log Canonical Counterexamples to the Chen--Jiang Decomposition}},
  author = {Houari Benammar Ammar},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.13623},
  eprint = {2609.13623},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.13623},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Log Canonical Counterexamples to the Chen--Jiang Decomposition
AU  - Houari Benammar Ammar
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.13623
UR  - https://arxiv.org/abs/2609.13623
AB  - We construct a projective log-smooth surface pair (X,Δ) and a morphism f:X→ E to an elliptic curve such that f_*OO_X(K_X+Δ) is Atiyah's indecomposable rank-two bundle F_2 of degree zero. This gives a negative answer to Question 3.4 of Jiang \cite{Jiang}. We also construct a log canonical pair (X,Δ) and a morphism f to an abelian surface such that f_*OO_X(m(K_X+Δ)) does not admit a Chen--Jiang decomposition for any integer m≥2. Products yield counterexamples in every higher direct image degree and over abelian bases of arbitrary positive dimension.
ER  -