Quantization for Semipositive Adjoint Line Bundles

Let $L$ be a big and semipositive line bundle on a complex projective manifold $X$, and let $\theta\in c_1(L)$ be a smooth semipositive representative. In the adjoint setting $H^0(X,L^k\otimes K_X)$, we prove that Donaldson's quantized Monge--Amp\`ere energy converges to the Monge--Amp\`ere energy for every bounded $\theta$-plurisubharmonic function. This extends the quantization picture from the ample case to the big and semipositive setting, where smooth positive representatives are no longer available and non-pluripolar Monge--Amp\`ere theory is required. The main new input is a comparison theorem between adjoint Bergman kernels and their small ample twists. As a consequence, we prove that the normalized adjoint Bergman measures converge weakly to the corresponding non-pluripolar Monge--Amp\`ere measures. Our result partially answers a question of Berman--Freixas i Montplet concerning the convergence of quantized Monge--Amp\`ere energies in the semipositive setting.