[Paper Review] Convex rationally connected varieties

Rahul Pandharipande|ArXiv.org|Jan 7, 2004

Geometric and Algebraic Topology4 references4 citations

TL;DR

This paper investigates the conjecture that convex, rationally connected projective varieties over ℂ are homogeneous spaces. Using quantum cohomology, Leray spectral sequences, and Poincaré polynomial analysis of complete intersections, it proves that nonsingular complete intersections in ℙⁿ satisfying convexity and rational connectedness must be of type (1,…,1) or (1,…,1,2), which are homogeneous varieties—thus confirming the conjecture in this case.

ABSTRACT

Nonsingular projective varieties which are both convex and rationally connected are considered. We ask whether such varieties must be algebraic homogeneous spaces G/P. In case X is a complete intersection, an affirmative answer is obtained by an elementary argument.

Motivation & Objective

• To investigate the conjecture that convex and rationally connected projective varieties over ℂ are homogeneous spaces.
• To determine whether convexity and rational connectedness imply homogeneity in the case of complete intersections in projective space.
• To analyze the geometry of lines in complete intersections using normal bundle semipositivity and fibrations.
• To use the Leray spectral sequence and Poincaré polynomial vanishing to constrain possible types of complete intersections.
• To classify which complete intersection types satisfy the necessary topological and cohomological conditions for convexity and rational connectedness.

Proposed method

• Analyzes the parameter space M of lines in a complete intersection X ⊂ ℙⁿ, showing it is nonsingular and non-empty for generic X.
• Applies the condition of convexity by requiring H¹(ℙ¹, μ*TX) = 0, which implies the normal bundle NL of any line L ⊂ X is semi-positive.
• Uses the Leray spectral sequence for the universal family ν: U → X, with fiber F over x being the space of lines through x, to relate cohomologies of U, F, and X.
• Derives the relation p_U(t) = p_F(t) · p_X(t) using the degeneration of the Leray spectral sequence and simple connectivity of X.
• Applies a lemma on Poincaré polynomials: p_Y(i) = 0 only if the complete intersection Y has type (1,…,1) with odd dimension, or (1,…,1,2) with specific dimension conditions.
• Uses cohomological bounds on Betti numbers via Lefschetz theorems and hypersurface cohomology formulas to show that only types (1,…,1) and (1,…,1,2) satisfy p_F(i) = 0 or p_X(i) = 0.

Experimental results

Research questions

• RQ1 Are all convex and rationally connected complete intersections in ℙⁿ homogeneous varieties?
• RQ2 What constraints do convexity and rational connectedness impose on the type of a complete intersection in projective space?
• RQ3 Under what conditions does the Poincaré polynomial of a fiber or variety vanish at t = i, and how does this affect the geometry?
• RQ4 Can the Leray spectral sequence and cohomological vanishing be used to classify complete intersections with semi-positive normal bundles?
• RQ5 Which complete intersection types (d₁,…,dₗ) allow both convexity and rational connectedness?

Key findings

• The only convex, rationally connected, nonsingular complete intersections in ℙⁿ are those of type (1,…,1) or (1,…,1,2), which are homogeneous spaces.
• If a complete intersection X ⊂ ℙⁿ is convex and rationally connected, then its normal bundle on every line must be semi-positive, implying the degree of the normal bundle is n - d - 1 ≤ 0.
• The fiber F of the universal line family over a point x ∈ X is a complete intersection of type (1,2,…,d₁,1,2,…,d₂,…,1,2,…,dₗ) in ℙⁿ⁻¹.
• The condition that 1 + t² divides p_U(t) implies that either p_F(i) = 0 or p_X(i) = 0, which restricts the possible types to (1,…,1) or (1,…,1,2).
• For complete intersections of type (1,…,1,2), both p_F(i) = 0 and p_X(i) = 0 can hold if dimensions are odd, but only one vanishes if dimensions are even, consistent with the classification.
• The lemma on Poincaré polynomials shows that p_Y(i) = 0 only if the complete intersection has type (1,…,1) with odd dimension, or (1,…,1,2) with dimension ≡ 2 mod 4 or odd dimension.