Let (X,B) be a projective klt pair and let L be a nef line bundle on X such that L-(K_X+B) is ample. Ambro--Kawamata effective non-vanishing conjecture. Then H^0(X,L)≠ 0. This conjecture is the effective non-vanishing input used in the paper to prove the Mukai-type conjecture for the total index. Its general validity is not established in the source.

Ambro--Kawamata effective non-vanishing conjecture

About 3 years old · traced to 2023

Let $(X,B)$ be a projective klt pair and let $L$ be a nef line bundle on $X$ such that

L-(K_X+B)

is ample.

Ambro--Kawamata effective non-vanishing conjecture. Then

H^0(X,L)\ne 0.

This conjecture is the effective non-vanishing input used in the paper to prove the Mukai-type conjecture for the total index. Its general validity is not established in the source.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Ambro–Kawamata effective non-vanishing conjecture
Let $(X,\Delta)$ be a klt pair, and let $H$ be an ample Cartier divisor such that $H-K_X-\Delta$ is ample.

Ambro–Kawamata conjecture. One has
$H^0(X,H) \neq 0.$
This is an effective non-vanishing statement for adjoint divisors. The paper proves it for the weighted complete intersections covered by its main theorem, while the general klt-pair statement remains open.
source: Alessandro Passantino, “Effective non-vanishing for weighted complete intersections of low codimension”, arXiv:2501.13267 (2025).

References

Primary source

Yoshinori Gongyo, “Effective non-vanishing and Mukai type conjecture”, arXiv:2306.08841 (2023).

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Ambro–Kawamata effective non-vanishing conjecture

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