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Affine holomorphic bundles over $\mathbb{P}^1_\mathbb{C}$ and apolar ideals

Naoufal Bouchareb

TL;DR

The paper develops a general classification framework for affine holomorphic bundles by translating the problem into extensions and the cohomology invariant $h_A\in H^1(X,\mathcal{L}(A))$, yielding a bijection between isomorphism classes with fixed linearisation and the Aut$(E)$-orbit of $H^1(X,\mathcal{E})$. Specializing to $X=\mathbb{P}^1_{\mathbb{C}}$ and rank-2 unstable linearisation, the moduli space becomes the topological cokernel of a morphism over the projective space of binary forms of degree $l=-2-n_2$, fibered with vector spaces, with fibre dimensions computable from $d=n_1-n_2$ and the cactus rank of the form. A striking duality ties these fibres to apolar ideals via a differential-operator correspondence, enabling explicit dimension formulas and revealing a stratification of $\mathbb{P}(\mathbb{C}[X_0,X_1]_{l})$ by cactus rank. The appendix formalizes the notion of topological cokernels for linear spaces and develops the Sylvester/apo­lar theory, providing explicit descriptions of cactus strata for small $l$ (notably $l=2,3,4$) and linking classical invariants to the geometry of the moduli spaces. Overall, the work uncovers a deep link between the classification of affine bundles on $\mathbb{P}^1_{\mathbb{C}}$ and apolar theory, offering concrete tools to compute fibre dimensions and to understand the geometry of the associated moduli spaces.

Abstract

We study the classification of affine holomorphic bundles over a compact complex manifold $X$ in general, and we apply the general theory to the case $X=\mathbb{P}^1_\mathbb{C}$. We study the moduli space of framed, non-degenerate rank 2 affine bundles over $\mathbb{P}^1_\mathbb{C}$ whose linearisation, viewed as locally free sheaf, is isomorphic to $ {\mathcal O}_{\mathbb{P}^1_\mathbb{C}}(n_1)\oplus {\mathcal O}_{\mathbb{P}^1_\mathbb{C}}(n_2)$ where $n_1>n_2$. We show that this moduli space can be identified with the "topological cokernel" of a morphism of linear spaces over the projective space $\mathbb{P}(\mathbb{C}[X_0,X_1]_{l})$ of binary forms of degree $l:= -2-n_2$, in particular it fibres over this projective space with vector spaces as fibres. We show that the stratification of $\mathbb{P}(\mathbb{C}[X_0,X_1]_{l})$ defined by the level sets of the fibre dimension map is determined explicitly by $d:= n_1-n_2$ and the cactus rank stratification of $\mathbb{P}(\mathbb{C}[X_0,X_1]_{l})$.

Affine holomorphic bundles over $\mathbb{P}^1_\mathbb{C}$ and apolar ideals

TL;DR

The paper develops a general classification framework for affine holomorphic bundles by translating the problem into extensions and the cohomology invariant , yielding a bijection between isomorphism classes with fixed linearisation and the Aut-orbit of . Specializing to and rank-2 unstable linearisation, the moduli space becomes the topological cokernel of a morphism over the projective space of binary forms of degree , fibered with vector spaces, with fibre dimensions computable from and the cactus rank of the form. A striking duality ties these fibres to apolar ideals via a differential-operator correspondence, enabling explicit dimension formulas and revealing a stratification of by cactus rank. The appendix formalizes the notion of topological cokernels for linear spaces and develops the Sylvester/apo­lar theory, providing explicit descriptions of cactus strata for small (notably ) and linking classical invariants to the geometry of the moduli spaces. Overall, the work uncovers a deep link between the classification of affine bundles on and apolar theory, offering concrete tools to compute fibre dimensions and to understand the geometry of the associated moduli spaces.

Abstract

We study the classification of affine holomorphic bundles over a compact complex manifold in general, and we apply the general theory to the case . We study the moduli space of framed, non-degenerate rank 2 affine bundles over whose linearisation, viewed as locally free sheaf, is isomorphic to where . We show that this moduli space can be identified with the "topological cokernel" of a morphism of linear spaces over the projective space of binary forms of degree , in particular it fibres over this projective space with vector spaces as fibres. We show that the stratification of defined by the level sets of the fibre dimension map is determined explicitly by and the cactus rank stratification of .

Paper Structure

This paper contains 18 sections, 20 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Previous work on holomorphic affine bundles. Our goals
  3. Preliminaries and main results
  4. The classification theorem for affine bundles
  5. Affine bundles of rank 2 over Riemann surfaces
  6. Affine bundles over 0
  7. The general case
  8. The case 1. General classification results
  9. The case 2. Framed non-degenerate affine bundles
  10. The duality theorem, the fibres of $\mathrm{Coker}(H)$ over ${\mathcal{P}}$ and apolar ideals
  11. Appendix
  12. The cokernel problem for linear spaces
  13. Apolar ideals and the Sylvester theorem. Cactus rank and the associated stratification
  14. General Theory
  15. The cactus rank stratification of C for small l
  16. ...and 3 more sections

Key Result

Proposition 1.8

The functors $A\mapsto (\tilde{L}(A),\varphi_A)$, ${\mathfrak a}\mapsto (A_{\mathfrak a},{\mathcal{T}}_{\mathfrak a})$ define an equivalence of groupoides between the groupoid of holomorphic affine rank $r$-bundles on $X$ and the groupoid of augmented holomorphic rank $(r+1)$-vector bundles on $X$.

Theorems & Definitions (62)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Remark 1.5
  • Definition 1.6
  • Remark 1.7
  • proof
  • Proposition 1.8
  • Theorem 1.9
  • ...and 52 more