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The Kodaira dimension and minimal models of π’œ6

  • Haowu Wang
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Abstract

We prove that the moduli space π’œ6 of principally polarized abelian sixfolds has Kodaira dimension zero, determining the remaining case in the Kodaira classification of the spaces π’œg. More precisely, every nonzero scalar Siegel modular form of genus six, weight k and positive boundary order b satisfies kβ‰₯ 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 β„š-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 π’œ6 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.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026kodaira,
  title = {{The Kodaira dimension and minimal models of $\mathcalA_6$}},
  author = {Haowu Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40136},
  eprint = {2610.40136},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40136},
}
RIS
TY  - UNPB
TI  - The Kodaira dimension and minimal models of π’œ_6
AU  - Haowu Wang
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40136
UR  - https://arxiv.org/abs/2610.40136
AB  - We prove that the moduli space π’œ_6 of principally polarized abelian sixfolds has Kodaira dimension zero, determining the remaining case in the Kodaira classification of the spaces π’œ_g. More precisely, every nonzero scalar Siegel modular form of genus six, weight k and positive boundary order b satisfies kβ‰₯ 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 β„š-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 π’œ_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.
ER  -