[Paper Review] Transcendental submanifolds of RP^n

Selman Akbulut, Henry King|ArXiv.org|Apr 26, 2004

Geometry and complex manifolds6 references4 citations

TL;DR

This paper constructs closed smooth submanifolds of real projective space ℝℙⁿ that are isotopic to nonsingular projective algebraic subvarieties but cannot be isotoped to the real parts of nonsingular complex algebraic subvarieties in ℂℙⁿ. Using Stiefel-Whitney classes and cohomological obstructions, it proves that certain manifolds—such as ℝℙᵐ×S¹ for specific m and s—admit real algebraic models in ℝℙⁿ but fail to lift to smooth complex algebraic models due to topological incompatibilities in cohomology.

ABSTRACT

In this paper we give examples of closed smooth submanifolds of RP^n which are isotopic to nonsingular projective subvarieties of RP^n but they can not be isotopic to the real parts of nonsingular complex projective subvarieties of CP^n.

Motivation & Objective

• To construct smooth submanifolds of ℝℙⁿ that are isotopic to nonsingular real projective algebraic subvarieties but not to the real parts of nonsingular complex algebraic subvarieties in ℂℙⁿ.
• To establish topological obstructions—beyond projective closure issues—to realizing smooth submanifolds as real parts of complex algebraic sets.
• To extend earlier affine examples of transcendental submanifolds to the projective setting using cohomological invariants.
• To demonstrate that certain manifolds, such as ℝℙᵐ×S¹, cannot be isotoped to smooth complex algebraic subvarieties due to nonvanishing algebraic cohomology classes incompatible with complexification.

Proposed method

• Utilizes the embedding of ℝℙᵐ into ℝ²ᵐ⁻ˢ for s≥3 (when m even) or s≥5 (when m=4k+1), leveraging results from [MM] to ensure such embeddings exist.
• Applies Lemma 2 to isotopically approximate the submanifold M=ℝℙᵐ×S¹ in ℝℙⁿ to a nonsingular projectively closed algebraic subvariety V⊂ℝℙⁿ.
• Employs the Lefschetz hyperplane theorem (via Larsen [Hr]) to deduce that Hⁱ(Vℂ;ℤ) ≅ Hⁱ(ℂℙⁿ;ℤ) for i≤s−1, restricting the image of complex algebraic cohomology classes.
• Analyzes the mod 2 cohomology of V and Vℂ using the commutative diagram involving restriction maps j*: Hⁱ(ℂℙⁿ;ℤ₂) → Hⁱ(ℝℙⁿ;ℤ₂) and j*: Hⁱ(Vℂ;ℤ₂) → Hⁱ(V;ℤ₂), showing that Ĥℂ−algⁱ(V;ℤ₂)=0 for i≤s−1.
• Applies Theorem 1 from [AK5] to equate Ĥℂ−alg²(V;ℤ₂) with H²ₐ(V;ℤ₂)², the square of the algebraic cohomology classes.
• Uses the fact that Stiefel-Whitney classes w₁(V) and w₂(V) are algebraic for real algebraic sets (from [AK1]), and shows that w₁²(V)≠0 contradicts the vanishing of Ĥℂ−alg²(V;ℤ₂) when s≥3 or s≥5.

Experimental results

Research questions

• RQ1 Can every smooth submanifold of ℝℙⁿ that is isotopic to a nonsingular real projective algebraic subvariety be isotoped to the real part of a nonsingular complex algebraic subvariety in ℂℙⁿ?
• RQ2 What topological obstructions prevent a smooth submanifold of ℝℙⁿ from being isotopic to the real part of a nonsingular complex algebraic subvariety, beyond failure of projective closure?
• RQ3 How do Stiefel-Whitney classes and their squares in mod 2 cohomology relate to the existence of complex algebraic models for real algebraic submanifolds?
• RQ4 Under what conditions on dimension and embedding codimension does the Lefschetz hyperplane theorem restrict the algebraic cohomology of a real algebraic submanifold in ℂℙⁿ?
• RQ5 Can the construction of transcendental submanifolds in ℝⁿ be extended to ℝℙⁿ while preserving the obstruction to complexification?

Key findings

• There exist closed smooth submanifolds M⊂ℝℙⁿ that are isotopic to nonsingular projective algebraic subvarieties of ℝℙⁿ but cannot be isotoped to the real parts of nonsingular complex algebraic subvarieties of ℂℙⁿ.
• For M=ℝℙᵐ×S¹ with m even and s≥3, or m=4k+1 and s≥5, the submanifold M is isotopic to a nonsingular projectively closed algebraic subvariety V⊂ℝℙⁿ.
• The mod 2 cohomology class w₁²(V)≠0 contradicts the vanishing of Ĥℂ−alg²(V;ℤ₂) when s≥3 (m even) or s≥5 (m=4k+1), violating the condition H²ₐ(V;ℤ₂)²=Ĥℂ−alg²(V;ℤ₂).
• The obstruction arises from the fact that w₁²(V) is algebraic in H²ₐ(V;ℤ₂)² by [AK1], but Ĥℂ−alg²(V;ℤ₂)=0 due to the Lefschetz theorem and the embedding codimension s−1.
• The result generalizes earlier affine examples of transcendental submanifolds to the projective setting, showing that topological obstructions to complexification are intrinsic and not merely artifacts of projective closure.
• The construction is valid for all m even with s≥3, and m=4k+1 with s≥5, as ensured by the existence of embeddings ℝℙᵐ↪ℝ²ᵐ⁻ˢ from [MM].