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The topology of spaces of holomorphic maps to projective space

Alexis Aumonier

Abstract

We show that the spaces of holomorphic and continuous maps from a smooth complex projective variety to a projective space have the same homology in a range depending on the degree of the maps.

The topology of spaces of holomorphic maps to projective space

Abstract

We show that the spaces of holomorphic and continuous maps from a smooth complex projective variety to a projective space have the same homology in a range depending on the degree of the maps.
Paper Structure (11 sections, 12 theorems, 35 equations)

This paper contains 11 sections, 12 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Relations to previous work
  3. Open questions
  4. Proof strategy
  5. Outline
  6. Mapping spaces and point-set topology
  7. Interpolating line bundles
  8. Proof of Theorem \ref{['thm:main-intro']}
  9. An intermediate model for continuous maps
  10. Maps and fibrewise sections
  11. The proof of Theorem \ref{['thm:homology-isomorphism']}

Key Result

Theorem A

Let $\alpha \in H^2(X;\mathbb Z)$ be such that $\alpha - c_1(K_X)$ is ampleWe write $K_X$ for the canonical line bundle, and recall that ampleness is a numerical property by the Nakai--Moishezon criterion.. Then the inclusion of the subspace of holomorphic maps of degree $\alpha$ inside the space of continuous maps of the same degree, induces an integral homology isomorphism in the range of degre

Theorems & Definitions (28)

  • Theorem A
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of \ref{['prop:scheme-isomorphism']}
  • Definition 3.1
  • Example 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof
  • ...and 18 more