Real Riemann Surfaces: Smooth and Discrete

TL;DR

The paper extends real Riemann surface theory to a discrete setting by formulating discrete real Riemann surfaces on bipartite quad-graphs with a linear discrete Cauchy–Riemann structure. It defines a discrete antiholomorphic involution, a discrete real-oval topology, and constructs a symplectic homology basis adapted to the involution, enabling a discrete period matrix with the canonical decomposition $\Pi = \frac{1}{2} H + i T$, where $H$ encodes topological data and $T$ is real. For discrete M-curves, the period matrix is purely imaginary, mirroring the smooth theory, and the work provides explicit constructions and algorithms to realize discrete real surfaces of arbitrary genus and type, along with convergence links to the continuous theory. This framework bridges combinatorial models and classical real algebraic geometry, with potential implications for numerical discrete complex analysis and integrable systems.

Abstract

This paper develops a discrete theory of real Riemann surfaces based on quadrilateral cellular decompositions (quad-graphs) and a linear discretization of the Cauchy-Riemann equations. We construct a discrete analogue of an antiholomorphic involution and classify the topological types of discrete real Riemann surfaces, recovering the classical results on the number of real ovals and the separation of the surface. Central to our approach is the construction of a symplectic homology basis adapted to the discrete involution. Using this basis, we prove that the discrete period matrix admits the same canonical decomposition $Π= \frac{1}{2} H + i T$ as in the smooth setting, where $H$ encodes the topological type and $T$ is purely imaginary. This structural result bridges the gap between combinatorial models and the classical theory of real algebraic curves.

Figures (10)

• Figure 1: A dividing real Riemann surface of genus $g=6$ with $k=3$ real ovals ($x_0, x_1, x_2$). The involution $\tau$ reflects across the plane containing the ovals, swapping the two components. The cycles $m_i, n_i, c_i$ indicate the symplectic basis elements derived in Proposition \ref{['prop:realhomology']}.
• Figure 2: Examples of non-dividing real Riemann surfaces using polygonal representations with identified edges. The dashed lines represent the fixed point set $\mathop{\mathrm{\mathrm{Fix}}}\nolimits(\tau)$. Both examples exhibit $k=1$ oval but remain connected after removing $\mathop{\mathrm{\mathrm{Fix}}}\nolimits(\tau)$.
• Figure 3: Fundamental $4g^{\prime}$-gon $F_{g^{\prime}}$ of the component $S_1$ (here $g^{\prime}=2$) bounded by real ovals $x_0, x_1, x_2$. The paths $y_i$ connect the base point $P_0$ on $x_0$ to $P_i$ on $x_i$.
• Figure 4: Construction of curves $c_i$ (blue) and $\tau(c_i)$ (green) for a non-dividing surface.
• Figure 5: Construction of the homology basis for the case $k = 0$.

Theorems & Definitions (77)

• Lemma 2.1
• proof
• proof
• Corollary 2.2
• proof
• Definition
• Remark : Discrete preview
• Lemma 2.3
• proof
• Definition