Projectively Flat Holomorphic Vector Bundles

• Projectively flat holomorphic vector bundles are rank‑r bundles whose projectivizations admit a flat holomorphic connection, generalizing flat unitary bundles.
• They are constructed via pull-backs from universal quotient bundles over Grassmannians, leading to explicit curvature computations and applications in complex differential geometry.
• Their rigidity and classification connect these bundles to representation theory, moduli spaces, Yang–Mills theory, and integrable systems through standard holomorphic maps.

A projectively flat holomorphic vector bundle is a rank-$r$ holomorphic vector bundle whose projectivization admits a flat holomorphic connection. Such bundles generalize flat unitary bundles and play a central role in complex differential geometry, particularly in the paper of Kähler manifolds. The notion of projective flatness can arise from pull-backs of universal quotient bundles over Grassmannian manifolds under holomorphic maps, with a distinguished subclass—strongly projectively flat bundles—that are direct sums of projectively flat line bundles. These structures facilitate connections to representation theory, moduli spaces, Yang–Mills theory, and the rigidity of symmetric embeddings.

1. Definitions and Characterizations

Let $M$ be a compact Kähler manifold equipped with Kähler form $\omega$. Consider a holomorphic map $f: M \to Gr_p(\mathbb{C}^n)$ from $M$ into the complex Grassmannian of $p$-planes. Over $Gr_p(\mathbb{C}^n)$, the tautological exact sequence of holomorphic bundles

$$0 \to S \to \mathbb{C}^n \times Gr_p(\mathbb{C}^n) \to Q \to 0$$

defines the universal quotient bundle $Q$ of rank $q=n-p$, endowed with its Chern connection.

The pull-back $E = f^*Q \to M$ is a holomorphic vector bundle with induced Hermitian structure. The bundle $E$ is termed projectively flat if the curvature $F_E$ of its Chern connection satisfies

$F_E = \lambda \omega \otimes \operatorname{Id}_E,$

for some real function $\lambda$ on $M$, i.e., the curvature form is scalar-valued and proportional to the Kähler form. This ensures that the induced connection on the projectivization $PE$ is flat (Koga, 2015).

A map $f$ is strongly projectively flat if there exists a holomorphic Hermitian line bundle $L \to M$ such that

$E \simeq L^{\oplus q}$

as holomorphic Hermitian bundles. Here, each summand $L$ is itself projectively flat, with curvature proportional to $\omega$.

In the case $p = n-1$, $Gr_{n-1}(\mathbb{C}^n) = \mathbb{C}P^{n-1}$, and $Q \simeq \mathcal{O}(1)$, recovering the theory of projectively flat line bundles as the special case (Koga, 2015).

2. Curvature Computations and Structural Properties

Choose local coordinates so a given $p$-plane appears as the graph of a $q \times p$ matrix $Z$ over a standard chart of $Gr_p(\mathbb{C}^n)$. In these coordinates, $Q$ admits frames

$\sigma_j(Z) = e_{p+j} - \sum_{i=1}^p Z_{ji} e_i,$

with $j=1,\ldots,q$. The induced Hermitian metric is $H(Z) = I_q + ZZ^*$, and the Chern connection in this frame has one-form $A = H^{-1} \partial H$.

The curvature is then explicitly

$$F_A = -\partial \bar{\partial} \log \det(I + ZZ^*) \otimes \operatorname{Id}_q,$$

identifying $F_Q$ with the Fubini–Study Kähler form tensored by the identity. For a strongly projectively flat $f$, the pullback bundle satisfies $$F_E = f^*(\omega_{FS}) \otimes \operatorname{Id}_{L^{\oplus q}}$$, proportional to $\omega \otimes \operatorname{Id}_E$ (Koga, 2015).

3. Rigidity and Classification of Equivariant Maps

A central result is the rigidity of strongly projectively flat, equivariant holomorphic maps from a homogeneous Kähler manifold $M = G/K$ into Grassmannians. Given a full, $G$-equivariant, strongly projectively flat map $f: M \to Gr_p(\mathbb{C}^n)$, there exists an irreducible $G$-module $W$ and a standard $G$-equivariant embedding $f_0: M \to Gr_{N-1}(W)$, with $\mathbb{C}^n \simeq W^{\oplus q}$, such that $f$ is, up to unitary equivalence, the composite

$$M \to Gr_{N-1}(W)^{q} \to Gr_p(\mathbb{C}^n),\ x \mapsto (f_0(x), \ldots, f_0(x)).$$

This rigidity shows that equivariant, strongly projectively flat maps are uniquely determined by the globally generated line bundle $L$ on $M$ and its space of sections $W = H^0(M, L)$. The only freedom is given by equivariant isomorphisms, forced to be scalar multiples by Schur’s lemma (Koga, 2015).

As a result, the classification of such maps reduces to identifying $G$-invariant line bundles $L \to M$, establishing global generation, and analyzing the associated standard maps.

4. Examples and Special Geometries

Specific cases of projectively flat and strongly projectively flat bundles include:

• Projective Spaces: For $M = \mathbb{C}P^m$, $G = SU(m+1)$, the hyperplane bundle $\mathcal{O}(1)$ is projectively flat. Strongly projectively flat, $G$-equivariant immersions into higher projective spaces correspond to standard linear systems, such as the Veronese embedding (Koga, 2015).
• Quadrics and Hermitian Symmetric Spaces: The spinor bundle on the hyperquadric $Q^m$ (the Hermitian symmetric space $SO(m+2)/S(O(2) \times O(m))$) and its $q$-fold direct sum realize the universal quotient of $Gr_2(\mathbb{C}^{m+2})$. Compact irreducible Hermitian symmetric spaces of tube type admit strongly projectively flat embeddings into suitable Grassmannians via their minimal representation (Koga, 2015).
• Riemann Surfaces with Projective Structure: On a compact connected Riemann surface $X$ of genus $g \geq 2$, the moduli space $\mathcal{B}_g(r)$ classifies triples $(X, K_X^{1/2}, F)$ with $F$ a stable holomorphic vector bundle. Projectively flat connections here correspond to second-order matrix differential operators with oper normalization, encoded as cotangent torsors $\mathcal{H}_g(r)$ over $\mathcal{B}_g(r)$ (Biswas et al., 2021).

5. Connections with Moduli Spaces and Symplectic Geometry

On Riemann surfaces, $\mathcal{H}_g(r)$ parametrizes pairs $(X, K_X^{1/2}, F, D)$, where $D$ is a "projectively flat" extension of $F$, equivalently a second-order holomorphic differential operator $L$ with symbol $\operatorname{Id}$ from $F \otimes K_X^{-1/2}$ to $F \otimes K_X^{1/2}$, modulo lower-order gauge. This space inherits the structure of a $T^*\mathcal{B}_g(r)$-torsor, modeled by $H^1(X, \operatorname{At}(F))^*$. The tautological $1$-form induces a holomorphic symplectic structure, canonically isomorphic to the structure on the space of holomorphic connections on the determinant theta line bundle over $\mathcal{B}_g(r)$ (Biswas et al., 2021).

The identification with spaces of "matrix opers" ties projectively flat holomorphic bundles to classical objects in integrable systems and geometric representation theory.

6. Implications, Moduli, and Open Directions

The paper of projectively flat holomorphic bundles informs the structure of stable vector bundles with constant scalar curvature, and connects to the theory of Yang–Mills connections. Strongly projectively flat bundles inherit advantageous curvature properties, and can be explicitly constructed via pull-backs from Grassmannians, enabling geometric and representation-theoretic classification (Koga, 2015).

Absent $G$-equivariance, the isomorphism class of the projectively flat bundle depends on the semi-positive Hermitian endomorphism $T$ up to unitary conjugacy, leading to moduli problems concerning such operators.

Open questions, as identified in recent research, include:

• Characterization of projectively flat bundles on non-homogeneous Kähler manifolds via holomorphic maps into infinite Grassmannians.
• Extension of rigidity results to flag manifolds, where the geometry of universal bundles is more intricate.
• Investigation into the links between projectively flat bundles, harmonic maps, and the Hitchin–Kobayashi correspondence (Koga, 2015).

A plausible implication is that further developments in the understanding of projectively flat vector bundles could unify geometric representation theory, complex differential geometry, and the theory of integrable systems, especially via their realization as "opers" on higher genus Riemann surfaces and higher-dimensional analogues.