Generalised abundance conjecture for klt pairs with nef adjoint divisors

Let (X,B)(X,B) be a projective klt pair over a field kk. Assume that KX+BK_X+B is pseudo-effective, and let MM be a nef Q\mathbb{Q}-Cartier Q\mathbb{Q}-divisor on XX. A divisor is num-semiample if it is numerically equivalent to a semiample Q\mathbb{Q}-Cartier Q\mathbb{Q}-divisor.

Generalised abundance conjecture. If KX+B+MK_X+B+M is nef, then KX+B+MK_X+B+M is num-semiample.

This extends abundance and the semiampleness conjecture to adjoint divisors involving a nef summand. The supplied text gives no resolution of the general statement, so it remains open.

Sources & referencesView supporting material

Primary source

Fabio Bernasconi and Liam Stigant, “Semiampleness for Calabi–Yau surfaces in positive and mixed characteristic”, arXiv:2207.03002 (2022).

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