In this paper the authors consider a possible generalization of the following important conjecture in the minimal model program:
Conjecture 1.1 (Nonvanishing Conjecture). Let \( (X, B)\) be a projective log canonical pair such that \( K_X + B\) is pseudo-effective. Then \( K_X + B \sim_{\mathbb{R}} D\) for some effective \({\mathbb{R}}\)-Cartier \({\mathbb{R}}\)-divisor \(D\).
A question raised by C. Birkar and Z. Hu [Nagoya Math. J. 215, 203–224 (2014; Zbl 1314.14028)] which is a modification of the above conjecture is:
Conjecture 1.2 (Numerical Nonvanishing for Generalized Polarized Pairs). Let \((X, B + M)\) be a projective generalized log canonical pair. Suppose that
(i) \(K_X + B + M_X\) is pseudo-effective,
(ii) \(M =\sum_j \mu_jM_j\), where \(\mu_j \in {\mathbb{R}}_{>0} \) and \( M_j\) are nef Cartier \(b\)-divisors.
Then \(K_X + B + M_X \equiv D\) for some effective \({\mathbb{R}}\)-Cartier \({\mathbb{R}}\)-divisor \(D\).
In this paper, the authors
1. prove that Conjecture 1.2 is true in dimension two;
2. confirm Conjecture 1.2 in higher dimensions if \(K_X +M_X\) is not pseudo-effective;
3. prove the numerical nonvanishing for projective generalized \( lc\) threefolds with rational singularities by scaling the nef part:
Theorem. Let \( (X, B +M)\) be a projective generalized lc threefold with rational singularities such that \( M\) is an \({\mathbb{R}}_{>0}\)-linear combination of nef Cartier b-divisors. If \(K_X + B + M_X\) is pseudo-effective and \(M_X\) is \({\mathbb{R}}\) Cartier, then there exists a \(0 \leq t \leq 1\) such that \( K_X + B + tM_X\) is numerically equivalent to an effective \({\mathbb{R}}\)-Cartier \({\mathbb{R}}\)-divisor.
To prove that Conjecture 1.2 is true conditionally, the key technique (due to J.-P. Demailly et al. [Acta Math. 210, No. 2, 203–259 (2013; Zbl 1278.14022)]) is to construct a Mori fibre space \( X\dashrightarrow Y \rightarrow Z\), run appropriate MMPs over \( Y\) or \(Z\) to reach a generalized \(lc\)-trivial fibration, then apply an induction on dimension.