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Weiss derivatives of holomorphic maps

Alexis Aumonier

TL;DR

This paper develops a Weiss/unitary calculus framework to analyze the stable homotopy type of spaces of holomorphic maps to projective spaces. By proving that the unitary functor Holα(X, P(C^{n+1} ⊕ V)) is N-polynomial, it constructs a finite, functorial Weiss tower whose layers are governed by derivatives Θ_k and their U(k) actions, and it relates these to a corresponding Goodwillie tower for continuous sections. The authors give explicit identifications of the k-th derivatives, including a top derivative Θ_N, and provide concrete computations in case studies (notably degree 1 and degree (1,1) maps) that illustrate the polynomial structure and the resulting stable splittings. An application yields a conceptual proof of the Cohen–Cohen–Mann–Milgram stable splitting for rational holomorphic maps on P^1, and the work offers a roadmap for understanding unstable homology via finitely many spectral pieces and extension problems. The framework also integrates Picard-variety variations and a canonical U(1) action, enabling a robust treatment of holomorphic map spaces across line-bundle parameters.

Abstract

We propose an orthogonal approach to the stable homotopy type of spaces of holomorphic maps to projective space. We study the Weiss towers of the unitary functors of holomorphic and continuous maps to $\mathbb{P}(V)$, and show that the former is polynomial and completely compute the latter. As an application we give a new proof of a stable splitting of Cohen--Cohen--Mann--Milgram.

Weiss derivatives of holomorphic maps

TL;DR

This paper develops a Weiss/unitary calculus framework to analyze the stable homotopy type of spaces of holomorphic maps to projective spaces. By proving that the unitary functor Holα(X, P(C^{n+1} ⊕ V)) is N-polynomial, it constructs a finite, functorial Weiss tower whose layers are governed by derivatives Θ_k and their U(k) actions, and it relates these to a corresponding Goodwillie tower for continuous sections. The authors give explicit identifications of the k-th derivatives, including a top derivative Θ_N, and provide concrete computations in case studies (notably degree 1 and degree (1,1) maps) that illustrate the polynomial structure and the resulting stable splittings. An application yields a conceptual proof of the Cohen–Cohen–Mann–Milgram stable splitting for rational holomorphic maps on P^1, and the work offers a roadmap for understanding unstable homology via finitely many spectral pieces and extension problems. The framework also integrates Picard-variety variations and a canonical U(1) action, enabling a robust treatment of holomorphic map spaces across line-bundle parameters.

Abstract

We propose an orthogonal approach to the stable homotopy type of spaces of holomorphic maps to projective space. We study the Weiss towers of the unitary functors of holomorphic and continuous maps to , and show that the former is polynomial and completely compute the latter. As an application we give a new proof of a stable splitting of Cohen--Cohen--Mann--Milgram.
Paper Structure (27 sections, 44 theorems, 209 equations)

This paper contains 27 sections, 44 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Applications
  3. Unitary calculus
  4. Aide-mémoire on calculus
  5. Shifts and colimits
  6. Parameterised unitary calculus
  7. Polynomiality of holomorphic maps
  8. The rank filtration on linear maps
  9. Polynomiality of holomorphic sections
  10. Polynomiality of holomorphic maps
  11. The canonical U(1)-action
  12. The Picard parameter
  13. Goodwillie calculus for continuous sections
  14. Reminders on parameterised spectra
  15. Recollections on Malkiewich's tower for section spaces
  16. ...and 12 more sections

Key Result

Theorem A

Let $X$ be a connected smooth projective complex variety. Let $\alpha \in H^2(X;\mathbb Z)$ and suppose that there exists a very ample line bundle $\mathcal{L}$ on $X$ with $c_1(\mathcal{L}) = \alpha$ and such that $c_1(\mathcal{L} \otimes K_X^\vee)$ is ample, where $K_X^\vee$ is the dual of the can is $N$-polynomialThe dimension $N$ is independent of the chosen $\mathcal{L}$ by the Kodaira vanish

Theorems & Definitions (94)

  • Theorem A
  • Theorem B
  • Definition 2.1: Weiss
  • Definition 2.2: Weiss
  • Proposition 2.3: Weiss and Taggart
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 84 more