Almost-lines and quasi-lines on projective manifolds.

*(English)*Zbl 1078.14010
Peternell, Thomas (ed.) et al., Complex analysis and algebraic geometry. A volume in memory of Michael Schneider. Berlin: Walter de Gruyter (ISBN 3-11-016204-0/hbk). 1-27 (2000).

Let \(X\) be a complex projective manifold of dimension \(n\geq 2\), and let \(Y\) be a smooth closed (connected) subvariety of \(X\). For every integer \(m\geq 0\), the \(m\)th infinitesimal neighborhood of \(Y\) in \(X\) is denoted by \(X(m)\). According to the scheme structure of \(X(m)\) there is a natural restriction homomorphism \(\alpha_m: \text{Pic}(X)\to \text{Pic}(X(m))\) of Picard groups, whose properties should be of just as natural great interest.

As \(X(0)= Y\), the first non-classical case would be the case of \(m= 1\), and the question of under which conditions the restriction map \(\alpha_1\) is surjective has indeed been studied, to quite some extent, by several authors in the past. The particular case of a curve \(Y\) with positive self-intersection in a smooth projective surface \(X\) has been completely settled by J. d’Almeida [Enseign. Math., II. Sér. 41, No. 1–2, 135–139 (1995; Zbl 0862.14023)], who showed that the map \(\alpha_1\) is surjective essentially when \(Y\) is a line in \(\mathbb{P}^2\). In the paper under review, the authors generalize d’Almeida’s theorem to the case of a smooth connected curve \(Y\) in a projective manifold \(X\) of arbitrary dimension \(n\geq 2\). Their main result states that the map \(\alpha_1\) is surjective if and only if the following three conditions are satisfied:

(1) \(Y\cong\mathbb{P}^1\), (2) \(N_{Y|X}\cong(n- 1){\mathcal O}_{\mathbb{P}^2}(1)\), and (3) the Lefschetz map \(\alpha_0: \text{Pic}(X)\to \text{Pic}(Y)\) is surjective.

In view of this theorem, the authors call a curve \(Y\subset X\) a quasi-line if it satisfies conditions (1) and (2). If all three conditions (1), (2) and (3) hold for \(Y\) then the curve \(Y\) is called an almost-line. It is shown that in dimension \(n\geq 3\) a quasi-line or almost-line need not be a line in \(\mathbb{P}^n= X\), and therefore a thorough study of the geometry of quasi-lines and almost-lines is carried out in the second part of the article. The authors prove that projective manifolds containing quasi-lines and/or almost-lines are very special examples of rationally connected manifolds in the sense of Campana-Kollár-Miyaoka-Mori, and that these manifolds are stable under small deformations. Finally, in this context, a very special class of rational manifolds, which the authors call strongly rational, is introduced and analyzed in terms of properties of quasi-lines.

For the entire collection see [Zbl 0933.00031].

As \(X(0)= Y\), the first non-classical case would be the case of \(m= 1\), and the question of under which conditions the restriction map \(\alpha_1\) is surjective has indeed been studied, to quite some extent, by several authors in the past. The particular case of a curve \(Y\) with positive self-intersection in a smooth projective surface \(X\) has been completely settled by J. d’Almeida [Enseign. Math., II. Sér. 41, No. 1–2, 135–139 (1995; Zbl 0862.14023)], who showed that the map \(\alpha_1\) is surjective essentially when \(Y\) is a line in \(\mathbb{P}^2\). In the paper under review, the authors generalize d’Almeida’s theorem to the case of a smooth connected curve \(Y\) in a projective manifold \(X\) of arbitrary dimension \(n\geq 2\). Their main result states that the map \(\alpha_1\) is surjective if and only if the following three conditions are satisfied:

(1) \(Y\cong\mathbb{P}^1\), (2) \(N_{Y|X}\cong(n- 1){\mathcal O}_{\mathbb{P}^2}(1)\), and (3) the Lefschetz map \(\alpha_0: \text{Pic}(X)\to \text{Pic}(Y)\) is surjective.

In view of this theorem, the authors call a curve \(Y\subset X\) a quasi-line if it satisfies conditions (1) and (2). If all three conditions (1), (2) and (3) hold for \(Y\) then the curve \(Y\) is called an almost-line. It is shown that in dimension \(n\geq 3\) a quasi-line or almost-line need not be a line in \(\mathbb{P}^n= X\), and therefore a thorough study of the geometry of quasi-lines and almost-lines is carried out in the second part of the article. The authors prove that projective manifolds containing quasi-lines and/or almost-lines are very special examples of rationally connected manifolds in the sense of Campana-Kollár-Miyaoka-Mori, and that these manifolds are stable under small deformations. Finally, in this context, a very special class of rational manifolds, which the authors call strongly rational, is introduced and analyzed in terms of properties of quasi-lines.

For the entire collection see [Zbl 0933.00031].

Reviewer: Werner Kleinert (Berlin)