In the note, we obtain the estimates for the rate of convergence for a sequence of q-Bernstein polynomials {B n,q (f )} for 0 < q < 1 by the modulus of continuity of f, and the estimates are sharp with respect to the order for Lipschitz continuous functions. We also get the exact orders of convergence for a family of functions f (x) = x , > 0, = 1, and the orders do not depend on , unlike the classical case.
We prove the Nonvanishing conjecture for uniruled projective log canonical pairs of dimension n, assuming the Nonvanishing conjecture for smooth projective varieties in dimension n−1. We also show that the existence of good minimal models for non-uniruled projective klt pairs in dimension n implies the existence of good minimal models for projective log canonical pairs in dimension n.
We prove a superadditivity result for the Kodaira dimension of algebraic fiber spaces over abelian varieties, an additivity result for (almost) smooth morphisms, as well as a subadditivity result for log pairs.Key words and phrases. Kodaira dimension; abelian varieties; positivity. MP was partially supported by the NSF grant DMS-2040378.
We prove some vanishing and torsion-freeness results for higher direct images of adjoint pairs satisfying relative abundance and nefness conditions. These are applied to generic vanishing and weak positivity.
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