Let X be a normal projective variety and let S+B be an effective Q-divisor satisfying (X,S+B) is a dlt pair,qquad lfloor S+B rfloor=S, K_X+S+B is nef, and K_X+S+B is Q-linearly equivalent to an effective divisor D such that S⊆operatornameSuppD⊆operatornameSupp(S+B). Extension conjecture for dlt pairs. The restriction map H^0 bigl(X, O_X(m(K_X+S+B))bigr)longrightarrow H^0 bigl(S, O_S(m(K_X+S+B))bigr) is surjective for sufficiently divisible integers m≥ 2. This conjecture concerns the extension of pluri-canonical sections from the reduced boundary S to a dlt pair. It is formulated in the cited work as part of the program toward abundance; the source gives no resolution status.

Extension conjecture for dlt pairs

Let $X$ be a normal projective variety and let $S+B$ be an effective $\mathbb{Q}$ -divisor satisfying

(X,S+B)\text{ is a dlt pair},\qquad \lfloor S+B\rfloor=S,

$K_X+S+B$ is nef, and $K_X+S+B$ is $\mathbb{Q}$ -linearly equivalent to an effective divisor $D$ such that

S\subseteq\operatorname{Supp}D\subseteq\operatorname{Supp}(S+B).

Extension conjecture for dlt pairs. The restriction map

H^0\bigl(X,\mathcal O_X(m(K_X+S+B))\bigr)\longrightarrow H^0\bigl(S,\mathcal O_S(m(K_X+S+B))\bigr)

is surjective for sufficiently divisible integers $m\geq 2$ .

This conjecture concerns the extension of pluri-canonical sections from the reduced boundary $S$ to a dlt pair. It is formulated in the cited work as part of the program toward abundance; the source gives no resolution status.

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.

The extension conjecture for dlt pairs
Let $(X,S+B)$ be an $n$ -dimensional projective dlt pair such that $B$ is an effective $\mathbb Q$ -divisor, and write $\lfloor S+B\rfloor=S$ . Assume that $K_X+S+B$ is nef and
$K_X+S+B\sim_{\mathbb Q}D\geq 0,$
where $S\subset\operatorname{Supp}D$ . Then, for all sufficiently divisible integers $m\geq 2$ , the restriction map
$H^0\bigl(X,\mathcal O_X(m(K_X+S+B))\bigr)\to H^0\bigl(S,\mathcal O_S(m(K_X+S+B))\bigr)$
is surjective.

This is the dlt-pair version of the extension problem for pluricanonical sections. The paper treats it as a conjectural component in the relationship between extension results and abundance, and gives no general resolution.
source: Osamu Fujino and Yoshinori Gongyo, “Log pluricanonical representations and abundance conjecture”, arXiv:1104.0361 (2012).

Sources & referencesView supporting material

Primary source

Shin-ichi Matsumura, “Injectivity theorems with multiplier ideal sheaves and their applications”, arXiv:1511.04226 (2015).

Additional references

3 papers in this index state this conjecture (2010–2015). The statement above is taken from the most recent of them; the others are arXiv:1406.6132, arXiv:1012.0493.

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Extension conjecture for dlt pairs

Equivalent formulations 1

The extension conjecture for dlt pairs

Primary source

Additional references

Progress summary

Solutions 0