[Paper Review] Cohomological characterizations of the complex projective space

Olivier Debarre|arXiv (Cornell University)|Dec 14, 2015

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper investigates whether the complex projective space ℂPⁿ can be uniquely characterized by its integral cohomology ring among compact Kähler manifolds. Using Hirzebruch–Riemann–Roch computations and Chern class analysis, it proves that for dimensions n ≤ 6, any compact Kähler manifold with the same integral cohomology ring as ℂPⁿ is either isomorphic to ℂPⁿ or a quotient of the unit ball in ℂⁿ, with no known such quotients existing for even-dimensional balls with torsion-free cohomology.

ABSTRACT

In this survey, we discuss whether the complex projective space can be characterized by its integral cohomology ring among compact complex manifolds.

Motivation & Objective

• To determine whether the complex projective space ℂPⁿ is uniquely characterized by its integral cohomology ring among compact Kähler manifolds.
• To investigate whether compact Kähler manifolds with the same Betti numbers as ℂPⁿ must be biholomorphic to ℂPⁿ or are quotients of unit balls.
• To extend computational methods based on Hirzebruch–Riemann–Roch polynomials to analyze Chern classes and cohomology constraints in higher dimensions.
• To explore the possibility of generalizing the characterization beyond dimension 6, particularly for Fano and general type manifolds.

Proposed method

• Computing the tₙ(z; c₁,…,cₙ) polynomials derived from the Hirzebruch–Riemann–Roch theorem to relate Chern classes to cohomological invariants.
• Using the equality of integral cohomology rings to deduce that Pic(X) ≅ ℤL with Lⁿ = 1 and c₁(L) generating H²(X, ℤ).
• Applying the Hirzebruch–Riemann–Roch formula to express χₚ(X) and χ_y(X) in terms of Chern classes, enabling algebraic constraints on cᵢ(X).
• Analyzing the sign of c₁(X) to distinguish between Fano manifolds (c₁ > 0) and general type manifolds (c₁ < 0), using Yau’s inequality for the latter.
• Computing explicit expressions for tₙ(z; c₁,…,cₙ) up to n = 9 using Sage, extending results from Libgober–Wood (1982) to higher dimensions.
• Applying the Hirzebruch proportionality principle to rule out ball quotients unless all Chern numbers match those of ℂPⁿ, which only occurs for even n.

Experimental results

Research questions

• RQ1 Is any compact Kähler manifold with the same integral cohomology ring as ℂPⁿ necessarily isomorphic to ℂPⁿ?
• RQ2 For n ≤ 6, can compact Kähler manifolds with the same cohomology ring as ℂPⁿ be classified as either ℂPⁿ or quotients of the unit ball in ℂⁿ?
• RQ3 Are there any known compact Kähler manifolds that are quotients of the unit ball B²ᵐ with the same integral cohomology ring as ℙ²ᵐ and trivial H²-torsion?
• RQ4 Can the computational approach used for n ≤ 6 be extended to characterize ℂPⁿ in higher dimensions using cohomological data?
• RQ5 What constraints do the tₙ(z; c₁,…,cₙ) polynomials impose on the Chern classes of manifolds with cohomology isomorphic to that of ℂPⁿ?

Key findings

• For n ≤ 6, any compact Kähler manifold with the same integral cohomology ring as ℂPⁿ is either isomorphic to ℂPⁿ or a quotient of the unit ball in ℂⁿ.
• No known examples of quotients of even-dimensional unit balls B²ᵐ exist that have the same integral cohomology ring as ℙ²ᵐ and trivial H²-torsion.
• The tₙ(z; c₁,…,cₙ) polynomials are explicitly computed up to n = 9, extending prior work by Libgober–Wood.
• When c₁(X) < 0, the inequality (2(n+1)/n c₂(X) − c₁(X)²) · (−c₁(X))ⁿ⁻² ≥ 0 holds, with equality iff X is a ball quotient, which only occurs for even n.
• For Fano manifolds (c₁(X) > 0), the inequality c₁(X) ≤ n+1 holds, with equality if and only if X ≅ ℂPⁿ.
• The Chern number cₙ(X) = n+1 and c₁(X)cₙ₋₁(X) = ½n(n+1)² are necessary conditions for cohomological equivalence to ℂPⁿ.