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Complex line fields on almost-complex manifolds

Nikola Sadovek, Baylee Schutte

TL;DR

This work develops a rigorous obstruction-theoretic framework to decide when a complex rank-$m$ bundle over a $2m$-dimensional CW complex contains a subbundle that is a direct sum of $r$ complex line bundles (with $r\le 3$). Using Moore–Postnikov towers, the authors translate the geometric problem into a sequence of obstructions, whose primary obstructions are given by top Chern classes of the stabilized complements, and whose higher obstructions are controlled by cohomology operations and $k$-invariants. They prove precise, often equivalent, étale criteria: for r=1 the criterion is $c_m(\xi-\ell)=0$; for r=2, $c_{m+1-i}(\xi-\ell_1\oplus\ell_2)=0$ for $i=1,2$ under suitable hypotheses; and for r=3 a parallel set of conditions with parity-sensitive refinements and additional cohomological hypotheses. The results specialize to classical cases on almost-complex manifolds (recovering Thom/Gilmore/Thomas-type theorems) and yield explicit descriptions in the complex projective space setting, including new insights into the complex span and projective span. Overall, the paper provides a concrete bridge between geometric splitting problems and computable cohomological obstructions, with potential applications to complex foliations and K"ahler geometry.

Abstract

We study linearly independent complex line fields on almost-complex manifolds, which is a topic of long-standing interest in differential topology and complex geometry. A necessary condition for the existence of such fields is the vanishing of appropriate virtual Chern classes. We prove that this condition is also sufficient for the existence of one, two, or three linearly independent complex line fields over certain manifolds. More generally, our results hold for a wider class of complex bundles over CW complexes.

Complex line fields on almost-complex manifolds

TL;DR

This work develops a rigorous obstruction-theoretic framework to decide when a complex rank- bundle over a -dimensional CW complex contains a subbundle that is a direct sum of complex line bundles (with ). Using Moore–Postnikov towers, the authors translate the geometric problem into a sequence of obstructions, whose primary obstructions are given by top Chern classes of the stabilized complements, and whose higher obstructions are controlled by cohomology operations and -invariants. They prove precise, often equivalent, étale criteria: for r=1 the criterion is ; for r=2, for under suitable hypotheses; and for r=3 a parallel set of conditions with parity-sensitive refinements and additional cohomological hypotheses. The results specialize to classical cases on almost-complex manifolds (recovering Thom/Gilmore/Thomas-type theorems) and yield explicit descriptions in the complex projective space setting, including new insights into the complex span and projective span. Overall, the paper provides a concrete bridge between geometric splitting problems and computable cohomological obstructions, with potential applications to complex foliations and K"ahler geometry.

Abstract

We study linearly independent complex line fields on almost-complex manifolds, which is a topic of long-standing interest in differential topology and complex geometry. A necessary condition for the existence of such fields is the vanishing of appropriate virtual Chern classes. We prove that this condition is also sufficient for the existence of one, two, or three linearly independent complex line fields over certain manifolds. More generally, our results hold for a wider class of complex bundles over CW complexes.

Paper Structure

This paper contains 39 sections, 57 theorems, 200 equations, 6 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Application
  3. Main Theorems
  4. Splitting off a single complex line bundle
  5. Splitting off two complex line bundles
  6. Splitting off three complex line bundles
  7. Preliminaries
  8. Some essentials of Moore--Postnikov theory
  9. A lemma about homotopy fibers
  10. Cohomology of certain Eilenberg--Maclane Spaces
  11. Some homotopy groups of complex Stiefel manifolds
  12. Relevant cohomology rings
  13. The lifting problem and the primary obstruction
  14. Two complex line bundles: $m$ odd
  15. The Primary Obstruction
  16. ...and 24 more sections

Key Result

Theorem 1.2

Let $M$ be a smooth closed connected $2m$-manifold and $\xi$ a complex $m$-plane bundle over $M$.

Figures (6)

  • Figure 1: An illustration of the morphism of Leray--Serre spectral sequences between $q$ and $q_{m,r}$ induced by \ref{['eq:morphism-q-q_m,r']}. Dashed differentials appear on later pages.
  • Figure 2: A portion of the Leray--Serre spectral sequence with $\mathbb{Z}$ coefficients of $s_1$ from \ref{['eq:s_1']} for (A) $m>3$, and (B) $m=3$.
  • Figure 3: A portion of the mod 2 Leray--Serre spectral sequence of (A) $p_1$ from \ref{['eq:p_1_even2']}, and (B) $p_2$ from \ref{['eq:p_2_even2']}, where dashed arrows represent zero differentials.
  • Figure 4: A portion of the mod 2 Leray--Serre spectral sequence of (A) $p_2$ from \ref{['eq:p2']}, and (B) $s_1$ from \ref{['eq:F[1]']}, where dashed arrows represent zero differentials.
  • Figure 5: A portion of the Leray--Serre spectral sequence of homotopy fibration $p_1$ from \ref{['eq:p1']} with (A) $\mathbb{Z}/2$, and (B) $\mathbb{Z}/3$ coefficients.
  • ...and 1 more figures

Theorems & Definitions (79)

  • Theorem 1.2: Main theorem
  • Theorem 2.1
  • proof
  • Proposition 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Corollary 3.4
  • Theorem 3.5
  • Corollary 3.6
  • proof
  • ...and 69 more