We show that the Nakai--Moishezon ampleness criterion holds for real line bundles on complete schemes. As applications, we treat the relative Nakai--Moishezon ampleness criterion for real line bundles and the Nakai--Moishezon ampleness criterion for real line bundles on complete algebraic spaces. The main ingredient of this paper is Birkars characterization of augmented base loci of real divisors on projective schemes.
In this note, we prove effective semi-ampleness conjecture due to Prokhorov and Shokurov for a special case, more concretely, for Q-Gorenstein klt-trivial fibrations over smooth projective curves whose fibers are all klt log Calabi-Yau pairs of Fano type.
The $J$-equation proposed by Donaldson is a complex Hessian quotient equation on Kahler manifolds. The solvability of the $J$-equation is proved by Song-Weinkove to be equivalent to the existence of a subsolution. It is also conjectured by Lejmi-Szekelyhidi to be equivalent to a stability condition in terms of holomorphic intersection numbers as an analogue of the Nakai-Moishezon criterion in algebraic geometry. The conjecture is recently proved by Chen under a stronger uniform stability condition. In this paper, we establish a Nakai-Moishezon type criterion for pairs of Kahler classes on analytic Kahler varieties. As a consequence, we prove Lejmi-Szekelyhidis original conjecture for the $J$-equation. We also apply such a criterion to obtain a family of constant scalar curvature Kahler metrics on smooth minimal models.
For any two nef line bundles F and G on a toric variety X represented by lattice polyhedra P respectively Q, we present the universal equivariant extension of G by F under use of the connected components of the set theoretic difference of Q and P.
For a projective variety $X$ defined over a non-Archimedean complete non-trivially valued field $k$, and a semipositive metrized line bundle $(L, phi)$ over it, we establish a metric extension result for sections of $L^{otimes n}$ from a sub-variety $Y$ to $X$. We form normed section algebras from $(L, phi)$ and study their Berkovich spectra. To compare the supremum algebra norm and the quotient algebra norm on the restricted section algebra $V(L_{X|Y})$, two different methods are used: one exploits the holomorphic convexity of the spectrum, following an argument of Grauert; another relies on finiteness properties of affinoid algebra norms.
In this paper we count the number of isomorphism classes of geometrically indecomposable quasi-parabolic structures of a given type on a given vector bundle on the projective line over a finite field. We give a conjectural cohomological interpretation for this counting using the moduli space of Higgs fields on the given vector bundle over the complex projective line with prescribed residues. We prove a certain number of results which bring evidences to the main conjecture. We detail the case of rank 2 vector bundles.