Schnell's fiber-space conjecture is the assertion that a morphism with connected fibers 
between smooth connected projective complex varieties
with geometric generic fiber 
of Kodaira dimension
zero satisfies 
if 
is pseudo-effective for some positive integer 
and ample Cartier divisor 
on 
. The ample comparison is essential. For example, a product
of an elliptic curve with 
has fibers of Kodaira dimension
zero over 
,
but total Kodaira dimension 
and fails that numerical hypothesis.
Schnell Fiber Space Conjecture
See also
Canonical Divisor, Iitaka Conjecture, Iitaka Dimension, Kodaira Dimension, Pseudo-Effective DivisorExplore with Wolfram|Alpha
References
Kim, H. "Canonical Bundle Formula and a Conjecture on Certain Algebraic Fiber Spaces by Schnell." 27 Dec 2024. https://arxiv.org/abs/2412.19769.Schnell, C. "Singular Metrics and a Conjecture by Campana and Peternell." Pure Appl. Math. Quart. 21, 1269-1281, 2025. https://doi.org/10.4310/PAMQ.250116023222.Cite this as:
Weisstein, Eric W. "Schnell Fiber Space Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SchnellFiberSpaceConjecture.html