Wronskian Line Bundle in Holomorphic Geometry

• The Wronskian line bundle is a holomorphic line bundle constructed via the Wronskian determinant of meromorphic sections of a projectively flat vector bundle over a Riemann surface.
• It encodes intrinsic geometric information by linking local differential properties to global divisor theory, including the computation of the first Chern class.
• Concrete examples, such as trivial and rank 2 bundles, illustrate its role in unifying classical Wronskian techniques with modern holomorphic geometry.

A Wronskian line bundle is a holomorphic line bundle canonically associated to a projectively flat holomorphic vector bundle over a compact Riemann surface. Its construction generalizes classical Wronskian techniques from the paper of scalar linear differential equations to non-abelian settings, providing a bridge between the local differential geometry of meromorphic sections and global divisor theory. The main idea is to use the transformation properties of the Wronskian determinant under transition maps of the projectively flat bundle to define a well-determined cocycle, yielding a line bundle whose divisor class encodes intrinsic geometric information, including the first Chern class and connections to Abel’s identity (Ajoodanian, 15 Nov 2025).

1. Construction of the Wronskian Line Bundle

Let $X$ be a compact Riemann surface and $\mathcal V \to X$ a holomorphic vector bundle of rank $n$ that is projectively flat. This means that with respect to some atlas $\{U_i\}$, the transition functions have the form: $$\Phi_{ij} = \lambda_{ij}\,T_{ij}, \qquad \lambda_{ij} : U_i \cap U_j \to \mathbb{C}^\times,\ T_{ij}\in GL_n(\mathbb{C})\ \text{constant}.$$ Each meromorphic section $A$ of $\mathcal V$ yields on $U_i$ a local representative $A_i$. If $A$ is generic (i.e., the Wronskian determinant does not vanish identically), the Wronskian is defined locally as: $$w(A_i) = \det \begin{pmatrix}A_i & A_i' & \dots & A_i^{(n-1)}\end{pmatrix}.$$ On overlaps, the transformation property is: $w(A_j) = \lambda_{ij}^n\,\det(T_{ij})\,w(A_i),$ so the collection $\{w(A_i)\}$ defines transition functions for a holomorphic line bundle on $X$. The divisor $\mathrm{div}(w(A))$ depends only on the isomorphism class of the bundle, not the choice of $A$.

The Wronskian line bundle $w(\mathcal V)$ is thus defined as the unique line bundle with transition functions: $$\left\{\frac{w(A_j)}{w(A_i)}\right\}_{i,j} = \{\lambda_{ij}^n\det(T_{ij})\}_{i,j},$$ or equivalently, $w(\mathcal V) = \mathcal O_X(\mathrm{div}(A))$ for generic $A$.

2. Division of Meromorphic Sections and Canonical Section

Given two meromorphic sections $A, B$ of $\mathcal V$ with $B$ generic, their Wronskian quotient is defined locally as: $$\frac{A}{B} = W(B)^{-1}W(A) \in \mathrm{End}(\mathbb{C}^n),$$ where $W(\cdot)$ denotes the matrix whose rows are derivatives up to order $n-1$. This quotient is independent of the projectively flat choice of trivialization, up to conjugation, and its determinant satisfies: $\det(A/B) = \frac{w(A)}{w(B)},$ which is a well-defined meromorphic function on $X$. The set of local functions $\{w(B_i)\}$ arising from a fixed generic $B$ provides a canonical meromorphic section of $w(\mathcal V)$, defined up to scaling.

3. Local Wronskian Determinants and Global Patching

Given a set of $n$ local meromorphic sections $s_1, \dots, s_n$ forming a frame on $U$, the Wronskian matrix is: $$W(s_1, \dots, s_n) = \det \begin{pmatrix} s_1 & s_2 & \cdots & s_n \ s_1' & s_2' & \cdots & s_n' \ \vdots & \vdots & \ddots & \vdots \ s_1^{(n-1)} & s_2^{(n-1)} & \cdots & s_n^{(n-1)} \ \end{pmatrix}.$$ If the projectively flat trivialization changes by $\Phi_{ij} = \lambda_{ij}T_{ij}$, the Wronskian transforms by $\lambda_{ij}^n\det(T_{ij})$, aligning with the cocycle for $w(\mathcal V)$. Thus, patched over the cover, the Wronskian determinants yield a global meromorphic section of the Wronskian line bundle.

4. Divisor Theory and Independence

For any generic section $A$, define the divisor $\mathrm{div}(A) = \mathrm{div}(w(A))$. For any two generic sections $A, B$,

$$\mathrm{div}(A) - \mathrm{div}(B) = \mathrm{div}\left(\frac{w(A)}{w(B)}\right),$$

and as $\frac{w(A)}{w(B)}$ is a global meromorphic function, the difference is principal. Therefore, the divisor class $[\mathrm{div}(A)] \in \mathrm{Pic}(X)$ is independent of the section and depends only on the bundle.

It follows that

$w(\mathcal V) = \mathcal O_X(\mathrm{div}(A))$

for any generic section $A$.

5. Abel’s Identity and the First Chern Class

Locally, a generic section $A_i$ satisfies an $n$th order linear ODE: $$A_i^{(n)} = p_1^i\,A_i^{(n-1)} + \cdots + p_{n-1}^i\,A_i,$$ with the classical Abel’s identity relating the derivative of the Wronskian: $$\frac{d}{dz} (\det W(A_i)) = p_1^i\, \det W(A_i), \qquad \text{thus} \qquad d\log w(A_i) = p_1^i\,dz.$$ On overlaps, the difference

$d\log\frac{w(A_i)}{w(A_j)} = (p_1^i - p_1^j)dz$

captures the transition difference.

The first Chern class of $w(\mathcal V)$ is therefore given by the Čech–de Rham cocycle: $$c_1(w(\mathcal V)) = \left[\,\{\,d\log(w(A_i)/w(A_j))\,\}\,\right] = \left[\,\{\,p_1^i - p_1^j\,\}\,\right] \in H^1(X, \Omega^1),$$ identifying Abel’s identity with the cohomological data of $w(\mathcal V)$. Alternatively, equipping $w(\mathcal V)$ with a Hermitian metric gives curvature $\bar\partial\partial\log\|w(A)\|^2$, paralleling this interpretation. In the classical line bundle case, vanishing of $c_1(\mathcal L)$ corresponds exactly to the existence of a meromorphic section with trivial Wronskian (Ajoodanian, 15 Nov 2025).

6. Explicit Computations and Examples

Several constructions elucidate the geometric content of the Wronskian line bundle:

• Trivial Bundle $\mathcal V = \mathcal{O}_X^{\oplus n}$:

Choosing a nonconstant meromorphic function $f$ on $X$ and setting $A = (1, f, f^2, \dots, f^{n-1})$, the Wronskian is:

$w(A) = (1!)(2!)\cdots(n-1)! (f')^{n(n-1)/2}$

with $f'$ a section of the canonical bundle $K_X$. Thus,

$$w(\mathcal{O}_X^{\oplus n}) = \tfrac{n(n-1)}{2} K_X,$$

demonstrating the relation to the canonical bundle.

• Direct Sum of a Line Bundle:

For $\mathcal V = L^{\oplus n}$, it follows from tensor product properties:

$$w(\mathcal V) = w(\mathcal{O}_X^{\oplus n}) \otimes L^n = \left(\tfrac{n(n-1)}{2} K_X\right) \otimes L^n.$$

• Rank 2 Projectively Flat Bundles:

Let $\mathcal V$ have determinant line bundle $M$ and be projectively flat. An open question is whether

$w(\mathcal V) = M + K_X,$

always holds. For split cases $\mathcal V = L_1 \oplus L_2$,

$$w(\mathcal V) = 2L_1 + K_X = 2L_2 + K_X = L_1 + L_2 + K_X,$$

matching the naive $\det + K_X$ expectation.

These concrete cases illustrate the role of the Wronskian line bundle in encoding both the projectively flat structure and the global geometry of the underlying Riemann surface (Ajoodanian, 15 Nov 2025).