Nef line bundle

In algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line bundles are described by a convex cone, and the possible contractions of the variety correspond to certain faces of the nef cone. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of a nef divisor.

Definition

More generally, a line bundle L on a proper scheme X over a field k is said to be nef if it has nonnegative degree on every (closed irreducible) curve in X.[1] (The degree of a line bundle L on a proper curve C over k is the degree of the divisor (s) of any nonzero rational section s of L.) A line bundle may also be called an invertible sheaf.

The term "nef" was introduced by Miles Reid as a replacement for the older terms "arithmetically effective" (Zariski 1962, definition 7.6) and "numerically effective", as well as for the phrase "numerically eventually free".[2] The older terms were misleading, in view of the examples below.

Every line bundle L on a proper curve C over k which has a global section that is not identically zero has nonnegative degree. As a result, a basepoint-free line bundle on a proper scheme X over k has nonnegative degree on every curve in X; that is, it is nef.[3] More generally, a line bundle L is called semi-ample if some positive tensor power L ⊗ a {\displaystyle L^{\otimes a}} is basepoint-free. It follows that a semi-ample line bundle is nef. Semi-ample line bundles can be considered the main geometric source of nef line bundles, although the two concepts are not equivalent; see the examples below.

A Cartier divisor D on a proper scheme X over a field is said to be nef if the associated line bundle O(D) is nef on X. Equivalently, D is nef if the intersection number D ⋅ C {\displaystyle D\cdot C} is nonnegative for every curve C in X.

To go back from line bundles to divisors, the first Chern class is the isomorphism from the Picard group of line bundles on a variety X to the group of Cartier divisors modulo linear equivalence. Explicitly, the first Chern class c 1 ( L ) {\displaystyle c_{1}(L)} is the divisor (s) of any nonzero rational section s of L.[4]

The nef cone

To work with inequalities, it is convenient to consider R-divisors, meaning finite linear combinations of Cartier divisors with real coefficients. The R-divisors modulo numerical equivalence form a real vector space N 1 ( X ) {\displaystyle N^{1}(X)} of finite dimension, the Néron–Severi group tensored with the real numbers.[5] (Explicitly: two R-divisors are said to be numerically equivalent if they have the same intersection number with all curves in X.) An R-divisor is called nef if it has nonnegative degree on every curve. The nef R-divisors form a closed convex cone in N 1 ( X ) {\displaystyle N^{1}(X)} , the nef cone Nef(X).

The cone of curves is defined to be the convex cone of linear combinations of curves with nonnegative real coefficients in the real vector space N 1 ( X ) {\displaystyle N_{1}(X)} of 1-cycles modulo numerical equivalence. The vector spaces N 1 ( X ) {\displaystyle N^{1}(X)} and N 1 ( X ) {\displaystyle N_{1}(X)} are dual to each other by the intersection pairing, and the nef cone is (by definition) the dual cone of the cone of curves.[6]

A significant problem in algebraic geometry is to analyze which line bundles are ample, since that amounts to describing the different ways a variety can be embedded into projective space. One answer is Kleiman's criterion (1966): for a projective scheme X over a field, a line bundle (or R-divisor) is ample if and only if its class in N 1 ( X ) {\displaystyle N^{1}(X)} lies in the interior of the nef cone.[7] (An R-divisor is called ample if it can be written as a positive linear combination of ample Cartier divisors.) It follows from Kleiman's criterion that, for X projective, every nef R-divisor on X is a limit of ample R-divisors in N 1 ( X ) {\displaystyle N^{1}(X)} . Indeed, for D nef and A ample, D + cA is ample for all real numbers c > 0.

Metric definition of nef line bundles

Let X be a compact complex manifold with a fixed Hermitian metric, viewed as a positive (1,1)-form ω {\displaystyle \omega } . Following Jean-Pierre Demailly, Thomas Peternell and Michael Schneider, a holomorphic line bundle L on X is said to be nef if for every ϵ > 0 {\displaystyle \epsilon >0} there is a smooth Hermitian metric h ϵ {\displaystyle h_{\epsilon }} on L whose curvature satisfies Θ h ϵ ( L ) ≥ − ϵ ω {\displaystyle \Theta _{h_{\epsilon }}(L)\geq -\epsilon \omega } . When X is projective over C, this is equivalent to the previous definition (that L has nonnegative degree on all curves in X).[8]

Even for X projective over C, a nef line bundle L need not have a Hermitian metric h with curvature Θ h ( L ) ≥ 0 {\displaystyle \Theta _{h}(L)\geq 0} , which explains the more complicated definition just given.[9]

Examples

  • If X is a smooth projective surface and C is an (irreducible) curve in X with self-intersection number C 2 ≥ 0 {\displaystyle C^{2}\geq 0} , then C is nef on X, because any two distinct curves on a surface have nonnegative intersection number. If C 2 < 0 {\displaystyle C^{2}<0} , then C is effective but not nef on X. For example, if X is the blow-up of a smooth projective surface Y at a point, then the exceptional curve E of the blow-up π : X → Y {\displaystyle \pi \colon X\to Y} has E 2 = − 1 {\displaystyle E^{2}=-1} .
  • Every effective divisor on a flag manifold or abelian variety is nef, using that these varieties have a transitive action of a connected algebraic group.[10]
  • Every line bundle L of degree 0 on a smooth complex projective curve X is nef, but L is semi-ample if and only if L is torsion in the Picard group of X. For X of genus g at least 1, most line bundles of degree 0 are not torsion, using that the Jacobian of X is an abelian variety of dimension g.
  • Every semi-ample line bundle is nef, but not every nef line bundle is even numerically equivalent to a semi-ample line bundle. For example, David Mumford constructed a line bundle L on a suitable ruled surface X such that L has positive degree on all curves, but the intersection number c 1 ( L ) 2 {\displaystyle c_{1}(L)^{2}} is zero.[11] It follows that L is nef, but no positive multiple of c 1 ( L ) {\displaystyle c_{1}(L)} is numerically equivalent to an effective divisor. In particular, the space of global sections H 0 ( X , L ⊗ a ) {\displaystyle H^{0}(X,L^{\otimes a})} is zero for all positive integers a.

Contractions and the nef cone

A contraction of a normal projective variety X over a field k is a surjective morphism f : X → Y {\displaystyle f\colon X\to Y} with Y a normal projective variety over k such that f ∗ O X = O Y {\displaystyle f_{*}O_{X}=O_{Y}} . (The latter condition implies that f has connected fibers, and it is equivalent to f having connected fibers if k has characteristic zero.[12]) A contraction is called a fibration if dim(Y) < dim(X). A contraction with dim(Y) = dim(X) is automatically a birational morphism.[13] (For example, X could be the blow-up of a smooth projective surface Y at a point.)

A face F of a convex cone N means a convex subcone such that any two points of N whose sum is in F must themselves be in F. A contraction of X determines a face F of the nef cone of X, namely the intersection of Nef(X) with the pullback f ∗ ( N 1 ( Y ) ) ⊂ N 1 ( X ) {\displaystyle f^{*}(N^{1}(Y))\subset N^{1}(X)} . Conversely, given the variety X, the face F of the nef cone determines the contraction f : X → Y {\displaystyle f\colon X\to Y} up to isomorphism. Indeed, there is a semi-ample line bundle L on X whose class in N 1 ( X ) {\displaystyle N^{1}(X)} is in the interior of F (for example, take L to be the pullback to X of any ample line bundle on Y). Any such line bundle determines Y by the Proj construction:[14]

Y = Proj  ⨁ a ≥ 0 H 0 ( X , L ⊗ a ) . {\displaystyle Y={\text{Proj }}\bigoplus _{a\geq 0}H^{0}(X,L^{\otimes a}).}

To describe Y in geometric terms: a curve C in X maps to a point in Y if and only if L has degree zero on C.

As a result, there is a one-to-one correspondence between the contractions of X and some of the faces of the nef cone of X.[15] (This correspondence can also be formulated dually, in terms of faces of the cone of curves.) Knowing which nef line bundles are semi-ample would determine which faces correspond to contractions. The cone theorem describes a significant class of faces that do correspond to contractions, and the abundance conjecture would give more.

Example: Let X be the blow-up of the complex projective plane P 2 {\displaystyle \mathbb {P} ^{2}} at a point p. Let H be the pullback to X of a line on P 2 {\displaystyle \mathbb {P} ^{2}} , and let E be the exceptional curve of the blow-up π : X → P 2 {\displaystyle \pi \colon X\to \mathbb {P} ^{2}} . Then X has Picard number 2, meaning that the real vector space N 1 ( X ) {\displaystyle N^{1}(X)} has dimension 2. By the geometry of convex cones of dimension 2, the nef cone must be spanned by two rays; explicitly, these are the rays spanned by H and H − E.[16] In this example, both rays correspond to contractions of X: H gives the birational morphism X → P 2 {\displaystyle X\to \mathbb {P} ^{2}} , and H − E gives a fibration X → P 1 {\displaystyle X\to \mathbb {P} ^{1}} with fibers isomorphic to P 1 {\displaystyle \mathbb {P} ^{1}} (corresponding to the lines in P 2 {\displaystyle \mathbb {P} ^{2}} through the point p). Since the nef cone of X has no other nontrivial faces, these are the only nontrivial contractions of X; that would be harder to see without the relation to convex cones.

Notes

  1. ^ Lazarsfeld (2004), Definition 1.4.1.
  2. ^ Reid (1983), section 0.12f.
  3. ^ Lazarsfeld (2004), Example 1.4.5.
  4. ^ Lazarsfeld (2004), Example 1.1.5.
  5. ^ Lazarsfeld (2004), Example 1.3.10.
  6. ^ Lazarsfeld (2004), Definition 1.4.25.
  7. ^ Lazarsfeld (2004), Theorem 1.4.23.
  8. ^ Demailly et al. (1994), section 1.
  9. ^ Demailly et al. (1994), Example 1.7.
  10. ^ Lazarsfeld (2004), Example 1.4.7.
  11. ^ Lazarsfeld (2004), Example 1.5.2.
  12. ^ Lazarsfeld (2004), Definition 2.1.11.
  13. ^ Lazarsfeld (2004), Example 2.1.12.
  14. ^ Lazarsfeld (2004), Theorem 2.1.27.
  15. ^ Kollár & Mori (1998), Remark 1.26.
  16. ^ Kollár & Mori (1998), Lemma 1.22 and Example 1.23(1).

References

  • Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael (1994), "Compact complex manifolds with numerically effective tangent bundles" (PDF), Journal of Algebraic Geometry, 3: 295–345, MR 1257325
  • Kollár, János; Mori, Shigefumi (1998), Birational geometry of algebraic varieties, Cambridge University Press, doi:10.1017/CBO9780511662560, ISBN 978-0-521-63277-5, MR 1658959
  • Lazarsfeld, Robert (2004), Positivity in algebraic geometry, vol. 1, Berlin: Springer-Verlag, doi:10.1007/978-3-642-18808-4, ISBN 3-540-22533-1, MR 2095471
  • Reid, Miles (1983), "Minimal models of canonical 3-folds", Algebraic varieties and analytic varieties (Tokyo, 1981), Advanced Studies in Pure Mathematics, vol. 1, North-Holland, pp. 131–180, doi:10.2969/aspm/00110131, ISBN 0-444-86612-4, MR 0715649
  • Zariski, Oscar (1962), "The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface", Annals of Mathematics, 2, 76 (3): 560–615, doi:10.2307/1970376, JSTOR 1970376, MR 0141668