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

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 1

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).

Sources & referencesView supporting material

Primary source

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

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Equivalent formulations 1

Ambro–Kawamata effective non-vanishing conjecture

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