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A configuration space model for algebraic function spaces

Oishee Banerjee

TL;DR

The paper develops a configuration-space–type model for the moduli space $\mathrm{Mor}_{\mathbf{d}}(X,Y)$ of algebraic morphisms by constructing a natural compactification and a proper hypercover augmented over a discriminant locus. It leverages $\Delta S$-sheaves and cohomological descent to produce a first-quadrant spectral sequence computing $H_c^*(\mathrm{Mor}_{\mathbf{d}}(X,Y);\mathbf{Q})$, with $E_1$-terms given by a twisted configuration-space cohomology factor, the Picard variety, and the cohomology of auxiliary fibers $Y(D_p)$. In the special case $Y=\mathbb{P}^N$, the spectral sequence collapses at $E_2$ in a stability range $p\le r(\mathbf{d})+1$, informed by purity and Lefschetz hyperplane arguments, yielding concrete bounds on the growth of $\mathrm{Mor}_{\mathbf{d}}(X,Y)$. The approach connects algebro-geometric function spaces to topological configuration-space phenomena via a Koszul-type complex arising from descent, while preserving Galois/mixed Hodge structures on both sides and emphasizing intersection-theoretic data of $X$.

Abstract

We prove that the space of algebraic maps between two smooth projective varieties, under certain conditions, admit a configuration space model, thereby obtaining an algebro-geometric analogue of Bendersky-Gitler's result on topological function spaces. Our result is a natural higher dimensional counterpart of \cite[Theorem 3]{Ban24}.

A configuration space model for algebraic function spaces

TL;DR

The paper develops a configuration-space–type model for the moduli space of algebraic morphisms by constructing a natural compactification and a proper hypercover augmented over a discriminant locus. It leverages -sheaves and cohomological descent to produce a first-quadrant spectral sequence computing , with -terms given by a twisted configuration-space cohomology factor, the Picard variety, and the cohomology of auxiliary fibers . In the special case , the spectral sequence collapses at in a stability range , informed by purity and Lefschetz hyperplane arguments, yielding concrete bounds on the growth of . The approach connects algebro-geometric function spaces to topological configuration-space phenomena via a Koszul-type complex arising from descent, while preserving Galois/mixed Hodge structures on both sides and emphasizing intersection-theoretic data of .

Abstract

We prove that the space of algebraic maps between two smooth projective varieties, under certain conditions, admit a configuration space model, thereby obtaining an algebro-geometric analogue of Bendersky-Gitler's result on topological function spaces. Our result is a natural higher dimensional counterpart of \cite[Theorem 3]{Ban24}.
Paper Structure (19 sections, 10 theorems, 65 equations)

This paper contains 19 sections, 10 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Motivation.
  3. Setup.
  4. Some remarks and context
  5. Acknowledgement.
  6. The space of morphisms from $X$ to $Y$
  7. Preliminaries on positivity of line bundles
  8. Geometry and topology of $\mathrm{Mor}_{\mathbf{d}}(X,\mathbb{P}^N)$
  9. Compactification of $\mathrm{Mor}_{\mathbf{d}}(X,\mathbb{P}^N)$
  10. Geometry and topology of $\mathrm{Mor}_{\mathbf{d}}(X,Y)$
  11. Compactification of $\mathrm{Mor}_{\mathbf{d}}(X,Y)$
  12. Hypercover over $\mathcal{Z}_{\mathbf{d}}(X,Y)$
  13. Symmetric (co)simplicial sheaves
  14. $\Delta S$-hypercovers
  15. Sheaves on schemes
  16. ...and 4 more sections

Key Result

Theorem 1.0.1

Let $X$ be a smooth projective variety of dimension $n$, $(Y,\Upsilon)$ a polarized smooth projective variety, and $N:=\dim |\Upsilon|$. Let $\mathbf{d}$ be acyclic, and let $r(\mathbf{d})$ be as in eq:rd. Then:

Theorems & Definitions (19)

  • Theorem 1.0.1
  • Definition 2.2
  • Theorem 2.2.1
  • Definition 2.3
  • Lemma 2.3.1
  • proof
  • Lemma 2.4.4
  • proof : Proof of Lemma \ref{['lemma:X0']}
  • Lemma 2.4.5
  • Lemma 2.4.6
  • ...and 9 more