Improved bounds on the number of holomorphic maps between compact Riemann surfaces

Technical Summary: Improved Bounds on the Number of Holomorphic Maps Between Compact Riemann Surfaces

Problem Statement
The paper addresses the problem of establishing upper bounds for the number of non-constant holomorphic maps originating from a fixed compact Riemann surface $X$ of genus $g > 1$. While the Theorem of de Franchis guarantees the finiteness of such maps for a fixed target $Y$, and further guarantees that only finitely many target surfaces $Y_i$ admit such maps, the specific challenge lies in quantifying this finiteness. The goal is to derive an upper bound for the total number of isomorphism classes of non-constant holomorphic maps, denoted as $\#I(X)$, that depends solely on the genus $g$ of the source surface $X$, independent of the specific complex structure of $X$ or the target surfaces.

Methodology
The author employs a synthesis of complex geometry, the theory of Jacobian varieties, and the geometry of numbers. The core strategy involves analyzing the pullbacks of holomorphic differentials and the associated homomorphisms between Jacobian varieties.

1. Jacobian Varieties and Endomorphisms: The paper utilizes the correspondence between holomorphic maps $f: X \to Y$ and homomorphisms between their Jacobian varieties, $f: J(X) \to J(Y)$. The study focuses on the dual Jacobian varieties $\hat{J}(X)$ and the induced dual maps. The rational representations of these homomorphisms are treated as integer matrices acting on the homology lattice $H_1(X, \mathbb{Z})$.
2. Rosati Involution and Norms: A key technical tool is the Rosati involution, which allows for the definition of an inner product and a norm on the underlying real vector space of the Jacobian. This structure is used to bound the growth of vectors under the action of the induced endomorphisms associated with the holomorphic maps.
3. Geometry of Numbers: The proof leverages the theory of successive minima of lattices. By considering the lattice of the dual Jacobian variety, the author estimates the number of possible "pullbacks" of elements in $H_1(X, \mathbb{Z})$. A packing argument is used to bound the number of distinct rational representations that satisfy specific orthogonality and congruence conditions.
4. Combinatorial Reduction: The problem is reduced to counting the number of isomorphism classes of maps that share specific properties regarding the pullback of holomorphic differentials. This leads to a combinatorial summation involving binomial coefficients, which is then bounded using Vandermonde's convolution identity and Stirling's approximation.

Key Contributions and Results
The primary contribution of the paper is the derivation of a new, tighter upper bound for $\#I(X)$ compared to previously established results by Kani (1986), Alzati and Pirola (1990), and the author's own previous work (2011).

• Main Theorem (Theorem 1.2): For a compact Riemann surface $X$ of genus $g > 1$, the number of isomorphism classes of non-constant holomorphic maps satisfies:
  $$\#I(X) \leq \left[ \sum_{k=4}^{2g} (2g-1)^k \right] \binom{6g-6}{2g-2}$$
  Alternatively, utilizing the asymptotic bound for the binomial coefficient, this is expressed as:
  $$\#I(X) \leq \left[ \sum_{k=4}^{2g} (2g-1)^k \right] \left( \frac{27}{4} \right)^{2g-2}$$

• Improvement over Previous Bounds: The author demonstrates that the ratio of this new bound (1.3) to the bound provided by Chamizo and Fuertes (2011) tends to 0 as $g \to \infty$. This indicates a significant asymptotic improvement.

• Refined Combinatorial Estimates: The paper improves upon the author's previous work [T2] by refining the estimate for the number of isomorphic classes of maps sharing a specific pullback of a differential. The term $(2g-1)^d$ in the previous bound is replaced by the tighter combinatorial term $\binom{4g-4}{d}$, which is subsequently summed and bounded using the identity $\sum \binom{2g-2}{k}\binom{4g-4}{k} \leq \binom{6g-6}{2g-2}$.

Significance
The paper claims significance by providing the sharpest known upper bound for the number of holomorphic maps between compact Riemann surfaces that depends only on the genus. The work advances the field by integrating the geometry of numbers (specifically lattice packing and successive minima) with the algebraic geometry of Jacobian varieties to refine the counting of these maps. The author notes that while Kani's construction of function fields shows that $M(g)$ (the maximum number of maps over all surfaces of genus $g$) cannot be bounded by any polynomial in $g$, this paper provides a concrete exponential bound that improves upon the state-of-the-art estimates. The proof relies on the finiteness of maps established by de Franchis and Severi, offering a quantitative refinement of these foundational finiteness theorems.