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Counterexamples to the Ambro--Kawamata effective non-vanishing conjecture
Abstract
Let k be an algebraically closed field of characteristic $0$ or p≠3. We construct a terminal projective 4-fold X and an ample Cartier divisor D such that 3KX∼0, D−KX is ample, H0(X,D)=0. In char 0, this disproves Ambro--Kawamata effective non-vanishing conjecture. From this example we construct a smooth projective 4-fold Y with a semiample and big Cartier divisor DY such that DY−KY is basepoint-free and big, and H0(Y,DY)=0. We also construct a Gorenstein terminal 6-fold with K∼0, other geometric avatars, and a terminal order-5 Jacobian quotient.
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BibTeX
@misc{ajit2026counterexamples,
title = {{Counterexamples to the Ambro--Kawamata effective non-vanishing conjecture}},
author = {Rahul Ajit},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09270},
eprint = {2610.09270},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09270},
}
RIS
TY - UNPB
TI - Counterexamples to the Ambro--Kawamata effective non-vanishing conjecture
AU - Rahul Ajit
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.09270
UR - https://arxiv.org/abs/2610.09270
AB - Let k be an algebraically closed field of characteristic $0$ or p≠3. We construct a terminal projective 4-fold X and an ample Cartier divisor D such that 3K_X∼0, D−K_X is ample, H^0(X,D)=0. In char 0, this disproves Ambro--Kawamata effective non-vanishing conjecture. From this example we construct a smooth projective 4-fold Y with a semiample and big Cartier divisor D_Y such that D_Y−K_Y is basepoint-free and big, and H^0(Y,D_Y)=0. We also construct a Gorenstein terminal 6-fold with K∼0, other geometric avatars, and a terminal order-5 Jacobian quotient.
ER -