Complete subvarieties in the projectivized strata of meromorphic differentials

Technical Summary: Complete Subvarieties in the Projectivized Strata of Meromorphic Differentials

Problem Statement
The paper addresses the question of whether projectivized strata of strictly meromorphic differentials, denoted $P\Omega M_g(\mu)$, contain positive-dimensional complete algebraic subvarieties. Here, $\mu$ represents a tuple of integers summing to $2g-2$, prescribing the orders of zeros and poles. While it is known that strata of holomorphic differentials (where $\mu$ has non-negative entries) contain no such subvarieties, and that the unprojectivized strata of strictly meromorphic differentials also lack them (proven via algebro-geometric methods involving divisor classes), a flat-geometric proof for the projectivized strictly meromorphic case was previously unavailable. The authors aim to fill this gap by constructing a specific analytic function on these strata.

Methodology
The authors employ a flat-geometric approach, utilizing the translation surface structures induced by meromorphic differentials. The core methodology involves constructing a globally defined, strictly plurisubharmonic function on the projectivized strata $P\Omega M_g(\mu)$. This is achieved by combining two distinct functions:

1. The Length Function ($\ell^{-2}$): Introduced by Mondello, this function measures the total length of a basis of short saddle connections. Specifically, for a basis $B$ of relative homology, $\ell^{-2}_B = \sum_{\gamma \in B} |P_\gamma|^{-2}$, where $P_\gamma$ is the period along the saddle connection $\gamma$. The global function $\ell^{-2}$ is the supremum over all such bases.
2. The Polar Size Function ($S$): A new function introduced in this work. In the flat structure of a strictly meromorphic differential, the surface decomposes into a "core" (a finite-area convex hull of zeros) and "polar domains" (infinite-area regions surrounding each pole). The boundary of these polar domains consists of "peripheral saddle connections." The function $S(\omega)$ is defined as the maximum modulus of the sum of periods over all subsets of these boundary saddle connections:
   $$S(\omega) := \max_{E \subset D_\omega} \left| \sum_{\gamma \in E} P_\gamma \right|$$
   where $D_\omega$ is the union of boundary edges of all polar domains.

The authors analyze the deformation properties of these functions. They establish that while $S$ is homogeneous of degree 1 and $\ell^{-2}$ is homogeneous of degree -2, their combination $2 \log S + \log \ell^{-2}$ is well-defined on the projectivized stratum (where scaling $\omega$ by $\lambda \in \mathbb{C}^*$ leaves the value invariant).

Key Contributions and Results

• Construction of a Strictly Plurisubharmonic Function: The primary result (Theorem 1.3) proves that the function $F = 2 \log S + \log \ell^{-2}$ is strictly plurisubharmonic on any projectivized stratum $P\Omega M_g(\mu)$ of strictly meromorphic differentials.

  • The proof relies on showing that $\log \ell^{-2}$ is strictly plurisubharmonic on submanifolds transverse to the rays of the stratum (a result adapted from Mondello).
  • Crucially, the authors prove that $\log S$ is plurisubharmonic. This is achieved by demonstrating that the maximum defining $S$ is always attained by "coherent homology classes." These are specific sums of peripheral saddle connections that persist as coherent classes under small deformations, allowing $S$ to be locally expressed as the maximum of finitely many holomorphic modulus functions.

• Non-existence of Complete Subvarieties: As a direct consequence (Corollary 1.4), the authors deduce that no positive-dimensional complete subvariety exists within $P\Omega M_g(\mu)$. The argument follows from the maximum principle: a non-constant plurisubharmonic function cannot exist on a complete variety. Since $F$ is strictly plurisubharmonic, its restriction to any positive-dimensional subvariety would be non-constant, leading to a contradiction if that subvariety were complete.

• Extension to $k$-differentials: The result is extended to strata of $k$-differentials with poles of order at least $k$ (Corollary 1.5). This is justified by the fact that such differentials can be lifted via a canonical cyclic covering to the $k$-th power of a strictly meromorphic differential one-form, thereby inheriting the non-existence property.

Significance and Claims
The paper claims to provide the first flat-geometric proof that projectivized strata of strictly meromorphic differentials contain no positive-dimensional complete subvarieties. Prior proofs for the meromorphic case relied on the positivity of divisor classes in the algebro-geometric setting.

The authors highlight that the constructed polar size function $S$ and its plurisubharmonic nature serve as an "analytic incarnation" of the positivity of $\psi$ classes (associated with the $S^1$-bundle at poles) used in previous algebro-geometric proofs. They note that this interplay between metric geometry (flat structures) and cohomological classes ($\psi$ classes) parallels techniques used by Kontsevich and Mirzakhani in the context of Witten's conjecture. The work aims to bridge the gap between affine geometry perspectives and moduli theory, offering a new tool for studying the geometry of strata of differentials.