Characterizations of projective space and applications to complex symplectic manifolds.

*(English)*Zbl 1063.14065
Mori, Shigefumi (ed.) et al., Higher dimensional birational geometry. Papers from the international conference on higher dimensional algebraic varieties held at Kyoto University, Kyoto, Japan, June 2–6 and 9–13, 1997. Tokyo: Mathematical Society of Japan (ISBN 4-931469-19-1/hbk). Adv. Stud. Pure Math. 35, 1-88 (2002).

From the introduction: Projective \(n\)-space \(\mathbb{P}^n\) is the simplest \(n\)-dimensional algebraic variety and can accordingly be characterized in various ways. The main objectives of the present paper are:

A. To establish new characterizations of projective \(n\)-space in such a way that all the known characterizations are thereby systematically explained;

B. To apply our characterizations to morphisms from complex symplectic manifolds;

C. To provide a self-contained exposition of basic theory of families of rational curves, which is important for understanding the detailed structure of rationally connected varieties.

Let \(X\) be a projective variety and Chow \((X)\) the Chow scheme (see Section 1 below). Let \(S\subset\) Chow \((X)\) be an irreducible subvariety and \(\text{pr}_S:F\to S\) the associated universal family. We say that \(F\) is a closed family of rational curves if \(S\) is proper and the fibre \(F_s=\text{pr}^{-1}(s)\subset\{s\}\times X\simeq X\) over a general point \(s\in S\) is an irreducible, reduced rational curve as an effective 1-cycle. Any special fibre of a family of rational curves is a 1-cycle supported by a union of rational curves. A closed family of rational curves \(F\to S\) is called maximal if \(F\) is a union of irreducible components of \(F'\) for any family of rational curves \(F'\supset F\). When every fibre \(F_s\) is irreducible and reduced (as 1-cycles), we say that \(F\) is unsplitting. A family of rational curves \(F\) is dominant if the natural projection \(\text{pr}_X:F\to X\) is surjective. \(F\) is doubly dominant if \(\text{pr}_{X\times X}^{(2)}:F\times_SF\to X\times X\) is surjective.

Main Theorem 1. Let \(X\) be a normal projective variety defined over the complex number field \(\mathbb{C}\) (or over an algebraically closed field of characteristic zero). If \(X\) carries a closed, maximal, unsplitting, doubly dominant family \(\text{pr}_S:F\to S\) of rational curves, then \(X\) is isomorphic to \(\mathbb{P}^n\), and \(F\) is the family of the lines on \(X\) parameterized by \(S= \text{Grass} (\mathbb{P}^n,1)\).

Roughly speaking, this theorem means that \(X\) is a projective space if and only if its two general points can be joined by a single rational curve of minimum degree (i.e., a line) with respect to a polarization of \(X\). If we impose a slightly weaker condition than in Theorem 1, we have the following result.

Theorem 2. Let \(X\) be a normal projective variety of dimension \(n\) over \(\mathbb{C}\) and \(x\) a prescribed general point on it. Let \(\text{pr}_S:F\to S\) be a closed, maximal, doubly-dominant family of rational curves on \(X\), and write \(F\langle x\rangle \to S\langle x\rangle\) for the closed subfamily consisting of curves passing through \(x\). If \(F\langle x\rangle\) is unsplitting, then \(X\) is a quotient of \(\mathbb{P}^n\) by a finite group action without fixed point locus of codimension one. In particular, \(X\) is \(\mathbb{P}^n\) if it is smooth.

A smooth projective variety \(X\) is said to be a Fano manifold if its anticanonical divisor \(-K_X\) is ample. Our Main Theorem yields a simple numerical criterion for a Fano manifold to be projective space in terms of the length \(l(\cdot)\) of rational curves:

Corollary 3 (Conjecture of Mori and Mukai). Let \(X\) be a smooth complex Fano \(n\)-fold. Put \[ l(X)=\min\{(C,-K_X);\;C\subset X \text{ is a rational curve}\}. \] Then \(X\) is isomorphic to \(\mathbb{P}^n\) if and only if \(l(X)\geq n+1\).

Our criterion (Theorem 1), stated in terms of the geometry of rational curves, is strong enough to yield a whole series of characterizations of projective \(n\)-space expressed in very different languages, including the Hirzebruch-Kodaira-Yau condition, the Kobayashi-Ochiai condition, the Frankel-Siu-Yau condition, and various other conditions. Although our result is far stronger than the results known before, we are not completely independent of the preceding works. Our basic strategy is in fact very similar to the argument used in [S. Mori, Ann. Math. (2) 110, 593–606 (1979; Zbl 0423.14006)]. Given a closed, unsplitting, doubly dominant family \(F\to S\) of rational curves, consider the subfamily \(F\langle x\rangle\to S\langle x\rangle\). We prove that the projection \(\text{pr}_X:F\langle x\rangle\to X\) is actually the blow-up \(\text{Bl}_x(X)\) of \(X\) at \(x\), the base variety \(S\langle x \rangle\) being isomorphic to the associated exceptional divisor \(E_x \simeq \mathbb{P}^{n-1}\).

If one knows that every point of \(S\langle x \rangle\) represents a curve smooth at \(x\), then the birationality of \(\text{pr}_X\) follows from an elementary argument, but it is by no means obvious that this smoothness condition is always satisfied. On the contrary, when \(S\langle x\rangle\) happens to contain a point which represents a curve singular at \(x\), then \(F\langle x\rangle\) is never birational to \(X\). Thus we need to rule out the existence of such bad points in \(S\), which is done with the aid of a theorem of Kebekus saying that no point of \(S\) can represent a curve which has a cuspidal singularity at the base point \(x\). Our characterization of projective \(n\)-space provides intriguing information on complex symplectic manifolds. Given a compact complex symplectic manifold \(Y\) of dimension \(2n\) and an arbitrary non-constant morphism \(f:\mathbb{P}^1\to Y\), one can show that \(\dim_{[f]}\text{Hom}(\mathbb{P}^1,Y)\geq 2n+1\). If one knows that \(f_t (\mathbb{P}^1)\) stays in a fixed \(n\)-dimensional subvariety \(X\subset Y\) for any (small) deformation \(f_t\) of \(f\), then theorem 1 implies that the normalization of \(X\) is necessarily \(\mathbb{P}^n\). This is indeed the case in some important situations imposing very restrictive constraints on the fibre space structure of, or birational contractions from, complex symplectic manifolds. Specifically, we completely understand the symplectic resolutions of a normal projective variety with only isolated singularities.

For the entire collection see [Zbl 1001.00023].

A. To establish new characterizations of projective \(n\)-space in such a way that all the known characterizations are thereby systematically explained;

B. To apply our characterizations to morphisms from complex symplectic manifolds;

C. To provide a self-contained exposition of basic theory of families of rational curves, which is important for understanding the detailed structure of rationally connected varieties.

Let \(X\) be a projective variety and Chow \((X)\) the Chow scheme (see Section 1 below). Let \(S\subset\) Chow \((X)\) be an irreducible subvariety and \(\text{pr}_S:F\to S\) the associated universal family. We say that \(F\) is a closed family of rational curves if \(S\) is proper and the fibre \(F_s=\text{pr}^{-1}(s)\subset\{s\}\times X\simeq X\) over a general point \(s\in S\) is an irreducible, reduced rational curve as an effective 1-cycle. Any special fibre of a family of rational curves is a 1-cycle supported by a union of rational curves. A closed family of rational curves \(F\to S\) is called maximal if \(F\) is a union of irreducible components of \(F'\) for any family of rational curves \(F'\supset F\). When every fibre \(F_s\) is irreducible and reduced (as 1-cycles), we say that \(F\) is unsplitting. A family of rational curves \(F\) is dominant if the natural projection \(\text{pr}_X:F\to X\) is surjective. \(F\) is doubly dominant if \(\text{pr}_{X\times X}^{(2)}:F\times_SF\to X\times X\) is surjective.

Main Theorem 1. Let \(X\) be a normal projective variety defined over the complex number field \(\mathbb{C}\) (or over an algebraically closed field of characteristic zero). If \(X\) carries a closed, maximal, unsplitting, doubly dominant family \(\text{pr}_S:F\to S\) of rational curves, then \(X\) is isomorphic to \(\mathbb{P}^n\), and \(F\) is the family of the lines on \(X\) parameterized by \(S= \text{Grass} (\mathbb{P}^n,1)\).

Roughly speaking, this theorem means that \(X\) is a projective space if and only if its two general points can be joined by a single rational curve of minimum degree (i.e., a line) with respect to a polarization of \(X\). If we impose a slightly weaker condition than in Theorem 1, we have the following result.

Theorem 2. Let \(X\) be a normal projective variety of dimension \(n\) over \(\mathbb{C}\) and \(x\) a prescribed general point on it. Let \(\text{pr}_S:F\to S\) be a closed, maximal, doubly-dominant family of rational curves on \(X\), and write \(F\langle x\rangle \to S\langle x\rangle\) for the closed subfamily consisting of curves passing through \(x\). If \(F\langle x\rangle\) is unsplitting, then \(X\) is a quotient of \(\mathbb{P}^n\) by a finite group action without fixed point locus of codimension one. In particular, \(X\) is \(\mathbb{P}^n\) if it is smooth.

A smooth projective variety \(X\) is said to be a Fano manifold if its anticanonical divisor \(-K_X\) is ample. Our Main Theorem yields a simple numerical criterion for a Fano manifold to be projective space in terms of the length \(l(\cdot)\) of rational curves:

Corollary 3 (Conjecture of Mori and Mukai). Let \(X\) be a smooth complex Fano \(n\)-fold. Put \[ l(X)=\min\{(C,-K_X);\;C\subset X \text{ is a rational curve}\}. \] Then \(X\) is isomorphic to \(\mathbb{P}^n\) if and only if \(l(X)\geq n+1\).

Our criterion (Theorem 1), stated in terms of the geometry of rational curves, is strong enough to yield a whole series of characterizations of projective \(n\)-space expressed in very different languages, including the Hirzebruch-Kodaira-Yau condition, the Kobayashi-Ochiai condition, the Frankel-Siu-Yau condition, and various other conditions. Although our result is far stronger than the results known before, we are not completely independent of the preceding works. Our basic strategy is in fact very similar to the argument used in [S. Mori, Ann. Math. (2) 110, 593–606 (1979; Zbl 0423.14006)]. Given a closed, unsplitting, doubly dominant family \(F\to S\) of rational curves, consider the subfamily \(F\langle x\rangle\to S\langle x\rangle\). We prove that the projection \(\text{pr}_X:F\langle x\rangle\to X\) is actually the blow-up \(\text{Bl}_x(X)\) of \(X\) at \(x\), the base variety \(S\langle x \rangle\) being isomorphic to the associated exceptional divisor \(E_x \simeq \mathbb{P}^{n-1}\).

If one knows that every point of \(S\langle x \rangle\) represents a curve smooth at \(x\), then the birationality of \(\text{pr}_X\) follows from an elementary argument, but it is by no means obvious that this smoothness condition is always satisfied. On the contrary, when \(S\langle x\rangle\) happens to contain a point which represents a curve singular at \(x\), then \(F\langle x\rangle\) is never birational to \(X\). Thus we need to rule out the existence of such bad points in \(S\), which is done with the aid of a theorem of Kebekus saying that no point of \(S\) can represent a curve which has a cuspidal singularity at the base point \(x\). Our characterization of projective \(n\)-space provides intriguing information on complex symplectic manifolds. Given a compact complex symplectic manifold \(Y\) of dimension \(2n\) and an arbitrary non-constant morphism \(f:\mathbb{P}^1\to Y\), one can show that \(\dim_{[f]}\text{Hom}(\mathbb{P}^1,Y)\geq 2n+1\). If one knows that \(f_t (\mathbb{P}^1)\) stays in a fixed \(n\)-dimensional subvariety \(X\subset Y\) for any (small) deformation \(f_t\) of \(f\), then theorem 1 implies that the normalization of \(X\) is necessarily \(\mathbb{P}^n\). This is indeed the case in some important situations imposing very restrictive constraints on the fibre space structure of, or birational contractions from, complex symplectic manifolds. Specifically, we completely understand the symplectic resolutions of a normal projective variety with only isolated singularities.

For the entire collection see [Zbl 1001.00023].