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Iitaka Conjecture


The Iitaka conjecture is the subadditivity assertion

 kappa(X)>=kappa(F)+kappa(Y)

for a surjective morphism with connected fibers f:X->Y between smooth projective varieties over an algebraically closed field of characteristic zero, where F is the geometric generic fiber. Here kappa is the Kodaira dimension, with (-infty)+a=-infty.

The logarithmic version uses reduced simple normal crossing divisors D_X and D_Y satisfying Supp(f^*D_Y) subset= Supp(D_X) and compares

 kappa(X,K_X+D_X)>=kappa(F,K_F+D_X|_F)+kappa(Y,K_Y+D_Y).

See also

Birational Morphism, Canonical Divisor, Iitaka Dimension, Kaehler Manifold, Kodaira Dimension, Logarithmic Kodaira Dimension, Simple Normal Crossing Divisor

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References

Fujino, O. "On Subadditivity of the Logarithmic Kodaira Dimension." J. Math. Soc. Japan 69, 1565-1581, 2017. https://doi.org/10.2969/jmsj/06941565.

Cite this as:

Weisstein, Eric W. "Iitaka Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/IitakaConjecture.html

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