Non-vanishing theorem for generalized log canonical pairs with a polarization (Q2674691)

Generalized polarized pairs were introduced by \textit{C. Birkar} and \textit{D.-Q. Zhang} [Publ. Math., Inst. Hautes Étud. Sci. 123, 283--331 (2016; Zbl 1348.14038)]. They play an important role in the development of birational geometry in recent years. Generalized polarized pairs share many properties of usual log pairs especially when they have klt singularities and certain bigness condition is imposed. However, not all the properties/conjectures of log pairs are expected to hold for generalized polarized pairs. The paper under review established the non-vanishing theorem for generalized lc pairs with a polarization: {Theorem.} Let \((X, B+M)\) be a projective NQC generalized lc pair. Let \(A\) be an ample \(\mathbb R\)-divisor on \(X\). Suppose that \(K_X+B+A+M\) is pseudo-effective. Then there exsits an effective \(\mathbb R\)-divisor \(D\) on \(X\) such that \[ K_X+B+A+M\sim_{\mathbb R} D. \] When \(M\) is \(\mathbb R\)-Cartier, the paper obtains a stronger non-vanishing theorem. Because the generalized polarized pair is assumed to be generalized lc, there are no simple modifications to reduce \((X,B+A+M)\) to an lc log pair. The proof of the theorem is an induction by dimensions which uses the full strength of the minimal model program and deep theorems in birational geometry. Part of the argument is similar to the paper [\textit{K. Hashizume} and \textit{Z.-Y. Hu}, J. Reine Angew. Math. 767, 109--159 (2020; Zbl 1476.14038)]. The paper also contains interesting examples that the Kodaira type vanishing theorem does not hold for generalized polarized pairs.