Shokurov's effective log adjunction conjecture

About 18 years old · traced to

Let (X,Δ)(X,\Delta) be a log klt pair and let f ⁣:X→Yf\colon X\to Y be a surjective morphism such that the Kodaira dimension of KX+ΔK_X+\Delta restricted to the general fibre is zero. Assume

KX+Δ=f∗(KY+BY+MY),K_X+\Delta=f^*(K_Y+B_Y+M_Y),

where BYB_Y is the discriminant divisor and MYM_Y is the moduli divisor. Shokurov's effective log adjunction conjecture. (1) There exists a birational contraction μ ⁣:Y′→Y\mu\colon Y'\to Y such that, after base change, the induced moduli divisor MY′M_{Y'} on Y′Y' is semiample. (2) If XηX_\eta is the generic fibre of ff, then

I(KXη+ΔXη)∼0,I(K_{X_\eta}+\Delta_{X_\eta})\sim 0,

where II depends only on dim⁡Xη\dim X_\eta and the horizontal multiplicities of Δ\Delta. (3) There exists a positive integer II depending only on the dimension of XX and the horizontal multiplicities of Δ\Delta such that IMY′IM_{Y'} is base point free on some model Y′/YY'/Y. These assertions concern semiampleness and effective bounds for the moduli part in the canonical bundle formula; the cited source does not provide evidence resolving them.

References

Primary source

Gueorgui Todorov, “Effective log Iitaka fibrations for surfaces and threefolds”, arXiv:0805.3494 (2008).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.