Math Conjectures

Nonvanishing Conjecture

OPENMajorConjectureProposed c. 1988 · Standard version

Canonical statement

Let (X,Δ)(X,\Delta) be a projective Kawamata-log-terminal pair over C\mathbb C with KX+ΔK_X+\Delta a Q\mathbb Q-Cartier Q\mathbb Q-divisor. If KX+ΔK_X+\Delta is pseudo-effective, then there is an effective Q\mathbb Q-divisor D≥0D\ge0 with D∼QKX+ΔD\sim_{\mathbb Q}K_X+\Delta.
View source LaTeX
Let \((X,\Delta)\) be a projective Kawamata-log-terminal pair over \(\mathbb C\) with \(K_X+\Delta\) a \(\mathbb Q\)-Cartier \(\mathbb Q\)-divisor. If \(K_X+\Delta\) is pseudo-effective, then there is an effective \(\mathbb Q\)-divisor \(D\ge0\) with \(D\sim_{\mathbb Q}K_X+\Delta\).
AG-MMP-011:[Kawamata1992Abundance][BCHM2010][LazicPeternell2018Nonvanishing]
Algebraic geometryGeometry and topology

Notes

The nonvanishing conjecture predicts that for a projective Kawamata log terminal pair (X,Δ)(X,\Delta) over C\mathbb C, pseudo-effectivity of KX+ΔK_X+\Delta — a purely numerical positivity condition — already forces the existence of an effective Q\mathbb Q-divisor D∼QKX+ΔD\sim_{\mathbb Q}K_X+\Delta. The statement crystallized around 1988 within the developing minimal model program rather than in a single dated announcement, as one of the conjectures governing the birational classification of varieties.

It is a key step toward abundance: in dimension three nonvanishing feeds into Kawamata's abundance theorem for minimal threefolds [Kawamata1992Abundance], and the conjecture is known in low dimensions and in a number of positivity regimes. The landmark work of Birkar, Cascini, Hacon, and McKernan established minimal models for varieties of log general type [BCHM2010], where the requisite effectivity comes from bigness; outside that range nonvanishing remains a key missing input. Generalized variants and their relation to abundance are studied by Lazić and Peternell [LazicPeternell2018Nonvanishing].

In higher dimensions the conjecture remains open, and with it the completion of the abundance conjecture and of the minimal model program in full generality.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.