[Paper Review] Caractérisations des espaces projectifs et des quadriques

Matthieu Paris|arXiv (Cornell University)|Sep 7, 2010

Algebraic Geometry and Number Theory8 references3 citations

TL;DR

This paper generalizes a characterization of complex projective manifolds as projective spaces or smooth quadrics by weakening the positivity condition on the tangent bundle. Using rational curve families and fibrations over curves, it proves that if the space of global sections of the p-th tensor power of the tangent bundle twisted by the inverse of an ample line bundle is non-zero, then the manifold is isomorphic to $\mathbb{P}^n$, $\mathbb{P}^1 \times \mathbb{P}^1$, or a smooth quadric hypersurface in $\mathbb{P}^{n+1}$, under mild assumptions on the Picard number.

ABSTRACT

In this paper we prove the following result : if the p-th tensor power of the tangent bundle of a smooth projective variety contains the p-th power of an ample line bundle, then the variety is isomorphic either to the projective space or to a smooth quadric hypersurface.

Motivation & Objective

• To generalize a known characterization of projective spaces and quadrics using positivity of the tangent bundle.
• To weaken the hypotheses of the original theorem by Araujo, Druel, and Kovács, which required either wedge powers or tensor powers with specific line bundle twists.
• To establish that the non-vanishing of $H^0(X, T_X^{igotimes p} \otimes L^{-p})$ alone suffices to classify the variety as a projective space or quadric.
• To extend the classification to cases where the twist is stronger, i.e., $L^{-k}$ with $k > p$, yielding a sharper characterization.

Proposed method

• Use Miyaoka's theorem to deduce that $X$ is uniruled, hence admits a covering family of rational curves.
• Construct a minimal covering family of rational curves and prove it is complete.
• Define the $H$-rationally connected quotient $\pi_0: X_0 \to Y_0$ and analyze its structure.
• Reduce the problem to two cases: fibrations over curves that are either projective bundles or quadric fibrations.
• Apply two key vanishing theorems (Theorems 3.1 and 3.2) on these fibrations to rule out all but the expected cases.
• Use induction and base change techniques to extend morphisms and preserve fiber types, applying results from Fujita and others on polarized varieties.

Experimental results

Research questions

• RQ1 Under what conditions on the global sections of $T_X^{igotimes p} \otimes L^{-p}$ can one classify the underlying complex projective manifold $X$?
• RQ2 Can the characterization of $X$ as $\mathbb{P}^n$, $Q_n$, or $\mathbb{P}^1 \times \mathbb{P}^1$ be recovered from a weaker positivity condition than previously known?
• RQ3 What happens when the twist by the ample line bundle is stronger, i.e., $L^{-k}$ with $k > p$, and how does this affect the classification?
• RQ4 How do the structure of the $H$-rationally connected quotient and the geometry of the fibers constrain the possible varieties?

Key findings

• If $H^0(X, T_X^{igotimes p} \otimes L^{-p}) \neq 0$ for $p \geq 1$, then $X$ is isomorphic to $\mathbb{P}^n$, $\mathbb{P}^1 \times \mathbb{P}^1$, or a smooth quadric $Q_n \subset \mathbb{P}^{n+1}$, with $L$ corresponding to the hyperplane bundle in each case.
• The case $X \cong \mathbb{P}^1 \times \mathbb{P}^1$ arises only when $L \cong \mathcal{O}(2,2)$, but this is excluded under the assumption $\rho(X) \geq 2$, so it does not appear in the final classification.
• For $k > p$, the stronger condition $H^0(X, T_X^{igotimes p} \otimes L^{-k}) \neq 0$ forces $X \cong \mathbb{P}^n$ and $L \cong \mathcal{O}_{\mathbb{P}^n}(1)$, showing a sharper characterization.
• The proof relies on reducing to fibrations over curves: either projective bundles or quadric fibrations, and applying vanishing theorems to rule out all but the expected cases.
• The $H$-rationally connected quotient $\pi_0: X_0 \to Y_0$ is shown to be a fibration over a curve, and the fiber type determines the global structure.
• The argument uses the existence of a complete covering family of rational curves and extends morphisms to preserve fiber types, relying on results from Fujita and others on polarized varieties.