The Ambro-Kawamata effective nonvanishing conjecture predicts the existence of nonzero global sections under positivity assumptions on a Cartier
divisor and its difference from a canonical
divisor. In its smooth boundary-free case, 
is a smooth complex projective algebraic
variety, 
is a canonical divisor,
is a Cartier divisor, and
the assumptions are that 
and 
are nef divisors and that
is big. The conclusion is 
.
Ajit (2026) reported a smooth projective four-dimensional counterexample 
with 
semiample and big, 
basepoint-free and big, and 
. The construction starts with a quotient 
with terminal singularities
of the fourth power of an elliptic curve by an
action of order 3. On 
, both 
and 
are ample, 
, and 
. Moreover,
 |
for every integer 
. In particular, the first multiple has no section even
though higher multiples do. The reported counterexample has not been independently
verified here.
