Let $X$ be a compact Kähler manifold and $\omega$ be a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${ℰ}_{\chi }(X,\omega )$ of $\omega$-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $\chi$ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class ${ℰ}_{\chi }(X,\omega )$. This is done by establishing...

We generalize and give an elementary proof of Kelly’s refinement [9] of Shoemaker’s result [11] on the birationality of certain BHK-mirrors. Our approach uses a construction that is equivalent to the Krawitz generalization [10] of the duality in Berglund-Hübsch [2].