Complete subvarieties in the projectivized strata of holomorphic differentials

TL;DR

The paper proves that for any holomorphic differential stratum $\Omega\mathcal{M}_{g}(\mu)$, the projectivized stratum $\mathbb P\Omega\mathcal{M}_{g}(\mu)$ contains no positive-dimensional complete subvarieties, resolving Harris's subcanonical-points question in the minimal case $\mathbb P\Omega\mathcal{M}_{g}(2g-2)$. The authors refine Gendron's maximum modulus method by establishing rigidity of cylinder moduli and of period ratios on any such subvariety, using period coordinates and the pluriharmonic behavior of relevant holomorphic ratios. They show that a hypothetical complete family would force the existence of flexible saddle connections or cylinder-pairs, which is incompatible with the rigidity results, thereby ruling out nontrivial deformations; the argument extends to $k$-differentials with pole orders bounded by $k-1$ via cyclic coverings. This advances the affine and algebraic geometry of differential strata and has implications for canonical-divisor geometry and the theory of subcanonical points.

Abstract

We show that the projectivized strata of holomorphic differentials with prescribed zero orders contain no positive-dimensional complete subvarieties. In the case of the minimal strata, this resolves a question of Harris concerning the existence of complete families of subcanonical points. Our proof relies on the geometry of flat cylinders.

Theorems & Definitions (23)

• Theorem 1.1
• Corollary 1.2
• Proposition 2.1
• Proposition 2.2
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Corollary 3.3
• proof