The Kodaira dimension and minimal models of $\mathcal{A}_6$

Source abstract

We prove that the moduli space $\mathcal{A}_6$ of principally polarized abelian sixfolds has Kodaira dimension zero, determining the remaining case in the Kodaira classification of the spaces $\mathcal{A}_g$. More precisely, every nonzero scalar Siegel modular form of genus six, weight $k$ and positive boundary order $b$ satisfies $k\geq 7b$, with equality precisely for powers of the weight-fourteen harmonic theta series of Dittmann, Salvati Manni and Scheithauer. The canonical ring is therefore a polynomial ring on one generator of degree two. We also prove numerical dimension zero and construct the unique projective $\mathbb{Q}$-factorial terminal minimal model. Its Picard number is one and its canonical divisor has order two. The perfect cone compactification is a Mori dream space, and the birational automorphism group of $\mathcal{A}_6$ is trivial. The proof combines normal vanishing on the Prym locus, polar intersections on abelian boundary fibers, and an exact theta calculation. A discrepancy estimate recovers the modular open subset from the minimal model.