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We construct a terminal projective 4-fold $X$ with an ample Cartier divisor $D$ satisfying $3K_X \sim 0$, $D-K_X$ ample, and $H^0(X,D)=0$. This example disproves the Ambro--Kawamata effective non-vanishing conjecture in characteristic 0 and $p \neq 3$. Further constructions yield a smooth projective 4-fold $Y$ with a big and semiample divisor $D_Y$ such that $D_Y-K_Y$ is big and basepoint-free, yet $H^0(Y,D_Y)=0$.
A terminal 4-fold with $3K_X \sim 0$ and an ample divisor $D$ vanishing on sections disproves a fundamental conjecture in birational geometry.
Let $k$ be an algebraically closed field of characteristic $0$ or $p\neq3$. We construct a terminal projective 4-fold $X$ and an ample Cartier divisor $D$ such that \[ 3K_X\sim0,\qquad D-K_X\ \text{is ample},\qquad H^0(X,D)=0. \] In char 0 this disproves the 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 give geometric variants and a separate terminal order 5 Jacobian quotient.