Semi-finite vector bundles on complex tori

1. Problem Statement

The paper addresses the classification and structural decomposition of finite and semi-finite holomorphic vector bundles on complex tori.

• Context: In characteristic zero, Nori's theory establishes that finite vector bundles form a neutral Tannakian category, corresponding to the Nori fundamental group scheme $\pi_N(X, x)$. For complex manifolds, this coincides with the étale fundamental group scheme $\pi_{\text{\'et}}(X, x)$.
• Extension: Otabe introduced semi-finite bundles (bundles filtered by finite bundles) to define an extended Nori fundamental group scheme, denoted $\pi_{EN}(X, x)$.
• The Gap: While the structure of these bundles is understood for algebraic varieties (specifically abelian varieties) via different methods, a direct, explicit decomposition for complex tori (which may not be algebraic) was needed. The authors aim to explicitly describe these bundles and determine the structure of $\pi_{EN}$ for complex tori, specifically investigating whether it decomposes into the product of the étale and unipotent fundamental group schemes.

2. Methodology

The authors employ a combination of Tannakian duality, homological algebra, and complex analytic geometry.

• Tannakian Framework: They utilize the fact that the categories of finite ($C_N$), unipotent ($C_{\text{uni}}$), and semi-finite ($C_{EN}$) bundles form neutral Tannakian categories. By Tannakian duality, these categories correspond to affine group schemes ($\pi_N, \pi_{\text{uni}}, \pi_{EN}$).
• Deligne Tensor Product: A core methodological tool is the Deligne tensor product of abelian categories. The authors aim to show that the category of semi-finite bundles is equivalent to the Deligne tensor product of the categories of finite and unipotent bundles:
  $$C_{EN}(X) \simeq C_N(X) \otimes_D C_{\text{uni}}(X)$$
  This equivalence would imply the group scheme decomposition $\pi_{EN} \simeq \pi_N \times \pi_{\text{uni}}$.
• Inductive Filtration: To prove the structural decomposition of the bundles themselves, the authors use induction on the rank of the vector bundle, analyzing short exact sequences and extension groups ($\text{Ext}^1$).
• Cohomological Vanishing: They leverage specific cohomological properties of complex tori, particularly the vanishing of $H^1(X, L)$ for non-trivial torsion line bundles $L$, to prove that certain extensions split.

3. Key Contributions and Results

A. Decomposition of Finite Bundles (Lemma 3.1)

The authors prove that any finite holomorphic vector bundle $E$ on a complex torus $X$ decomposes into a direct sum of torsion line bundles:
$$E \simeq \bigoplus_{i=1}^n L_i$$
where each $L_i$ is a line bundle such that $L_i^{\otimes m} \cong \mathcal{O}_X$ for some integer $m$. This relies on the fact that the Nori fundamental group of a complex torus is isomorphic to the dual of the lattice ($\hat{\Lambda} \cong \widehat{\mathbb{Z}^{2g}}$), implying all indecomposable representations are rank 1.

B. Decomposition of Semi-Finite Bundles (Lemma 3.4)

The central result regarding the bundles themselves is that any semi-finite bundle $E$ on a complex torus decomposes as a direct sum of tensor products of unipotent bundles and torsion line bundles:
$$E \simeq \bigoplus_{i=1}^n (U_i \otimes L_i)$$
where:

• $U_i$ are unipotent bundles (iterated extensions of the trivial bundle).
• $L_i$ are torsion line bundles.
• Proof Strategy: The proof involves splitting the extension of a semi-finite bundle by a torsion line bundle. A critical step involves showing that $\text{Ext}^1(U \otimes L', \mathcal{O}_X) = 0$ for non-trivial torsion $L'$, which follows from the vanishing of $H^1(X, L')$ (Lemma 3.3).

C. Decomposition of the Extended Nori Fundamental Group Scheme (Theorem 3.6)

Using the Deligne tensor product and the bundle decomposition above, the authors prove the main theorem:
$$\pi_{EN}(X, 0) \simeq \pi_{\text{\'et}}(X, 0) \times \pi_{\text{uni}}(X, 0)$$
This establishes that the extended Nori fundamental group scheme of a complex torus is the direct product of its étale fundamental group scheme and its unipotent fundamental group scheme.

D. Explicit Structure of $\pi_{EN}$ (Corollary 3.9)

By analyzing the Lie algebra of the unipotent part, they provide an explicit isomorphism:
$$\pi_{EN}(X, 0) \simeq \widehat{\mathbb{Z}^{2g}} \times (\mathbb{G}_{a, \mathbb{C}})^g$$
where:

• $\widehat{\mathbb{Z}^{2g}}$ is the profinite completion of the lattice (representing the étale part).
• $(\mathbb{G}_{a, \mathbb{C}})^g$ is the additive group scheme of dimension $g$ (representing the unipotent part, derived from $H^1(X, \mathcal{O}_X) \cong \mathbb{C}^g$).

4. Significance and Implications

1. Analytic vs. Algebraic: While a similar result for abelian varieties was known (via [1]), that proof relied on the group scheme being abelian. The authors' proof for complex tori does not assume the group scheme is abelian a priori; instead, they derive the abelian nature of the unipotent part from the Künneth formula and the specific geometry of complex tori. This makes the result robust for non-algebraic complex tori.
2. Homogeneity: The paper connects semi-finite bundles to homogeneous bundles (translation-invariant bundles). It is shown that semi-finite bundles on complex tori are homogeneous, linking Tannakian categories to classical complex geometry results (Mukai, etc.).
3. Structural Clarity: The explicit decomposition $E \simeq \bigoplus (U_i \otimes L_i)$ provides a concrete classification tool for vector bundles in this category, separating the "topological" (torsion) and "geometric" (unipotent) components.
4. Tannakian Duality Application: The paper serves as a clear example of how the Deligne tensor product can be used to decompose fundamental group schemes by analyzing the underlying categories of vector bundles.

In summary, the paper successfully extends the structural theory of semi-finite bundles from algebraic varieties to the broader setting of complex tori, providing explicit decompositions for both the bundles and their associated fundamental group schemes.