This is an expository text, originally intended for the ANR 'Hodgefun' workshop, twice reported, organised at Florence, villa Finaly, by B. Klingler. We show that holomorphic foliations on complex projective manifolds have algebraic leaves under a certain positivity property: the 'non pseudoeffectivity' of their duals. This permits to construct certain rational fibrations with fibres either rationally connected, or with trivial canonical bundle, of central importance in birational geometry. A considerable extension of the range of applicability is due to the fact that this positivity is preserved by the tensor powers of the tangent bundle. The results presented here are extracted from [
Citation: Frédéric Campana. Algebraicity of foliations on complex projective manifolds, applications[J]. Electronic Research Archive, 2022, 30(4): 1187-1208. doi: 10.3934/era.2022063
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This is an expository text, originally intended for the ANR 'Hodgefun' workshop, twice reported, organised at Florence, villa Finaly, by B. Klingler. We show that holomorphic foliations on complex projective manifolds have algebraic leaves under a certain positivity property: the 'non pseudoeffectivity' of their duals. This permits to construct certain rational fibrations with fibres either rationally connected, or with trivial canonical bundle, of central importance in birational geometry. A considerable extension of the range of applicability is due to the fact that this positivity is preserved by the tensor powers of the tangent bundle. The results presented here are extracted from [
In all the text, X will denote an n-dimensional connected complex projective manifold, A will be an ample line bundle on X. A Zariski open subset of X is said to be 'big' if its complement has codimension at least 2 in X.
Let F⊂TX, the holomorphic tangent bundle of X, be a coherent saturated subsheaf of rank r≤n.
Recall that F is saturated if TX/F has no torsion. Since F is saturated, the Zariski closed subset Sing(F)⊂X over which F is not a subbundle has codimension at least two, and any local section of F defined on the complement of Sing(F) extends across Sing(F).
Definition 1.1. We say that F is a foliation if, for any two germs of sections V,W of F, their Lie Bracket [V,W] is also a germ of section of F. Equivalently, the Lie bracket defines a morphism of sheaves of OX-modules*L:∧2F→TX/F which vanishes identically on X.
*Indeed: [fV,gW]=fg.[V,W]+(fV(g).W−g.W(f).V)≡fg.[V,W] modulo F for any local sections V,W of F, and holomorphic functions f,g.
This is the 'Frobenius' integrability condition. It implies that, over UF:=X∖Sing(F), each point has a local analytic open neighbourhood U=F×B, with F,B open subsets of Cr and Cn−r respectively such that FU=Ker(dβ), where β:U→B is the projection onto the second factor B. In particular, for each such x=(f,b)∈U, there is a unique germ of manifold Fx=F×{b} of dimension r, called the germ of the 'leaf' of F at x, which has in each of its points x′:=(f′,b), the subspace Fx′ as its tangent space.
We thus define an equivalence relation on UF for which two points are equivalent if they can be connected by a chain of germs of leaves of F locally defined as above. The classes of this relation are called the leaves of F. They are the maximal connected, r-dimensional manifolds immersed (but not necessarily closed) in UF, and tangent to F. Near the points of Sing(F), the leaves of F may have a chaotic behaviour.
Example 1.2. The simplest example of an everywhere regular foliation with non-closed leaves is the following: X:=C2/Λ is an Abelian surface, and F=X×{C.v}⊂TX=X×C2 is the trivial rank one subbundle generated by a vector v∈C2. Let Γ:=C.v∩Λ: this is a closed additive subgroup of C.v of rank ρ either 0,1 or 2. So all leaves are isomorphic to C.v/Γ, which is isomorphic to either C,C∗, or an elliptic curve when ρ=2, i.e., when Γ is a cocompact sublattice of C.v. If the leaves of F are not compact, they are Zariski-dense, but their topological closures are real tori of real dimension d either 3 or 4. It may happen that d=3 even if the leaves of F are isomorphic to C. This happens for example if v=e1+√2.e2, J.v=e1+√3.e3:=w, the lattice defining X being generated over Z by e1,…,e4, and the complex structure J on R4 extending the one defined above (J.v=w). One obtains so a two-dimensional compact complex torus X. It is less obvious whether this can be done in such a way that X is an abelian variety.
Example 1.3. Let X:=P2, with affine coordinates (x,y) and F be defined as the Kernel of the (rational) 1-form w:=dy−λ.dx, where λ∈C∗. The leaves are then defined by the equation y=xλ, and are thus algebraic if and only if λ∈Q∗. The Zariski and topological closures of the leaves can be easily described in terms of λ. Examples 1.2 and 1.3 show that algebraicity is an arithmetic property in families of foliations.
Example 1.4. The globally simplest foliations, however, are the ones arising from dominant rational fibrations , where Z is a manifold of dimension p:=n−r, and the generic fibre of f is connected. The associated foliation is then F:=(Ker(df))sat, where ∙sat denotes the saturation inside TX, and Sing(F) is the union of the indeterminacy locus of f, and of the singularities of the (reduction of the) fibres.
Definition 1.5. Let F⊂TX be a foliation on X. We say that F is 'algebraic' if all leaves of F are algebraic submanifolds of X, that is: if their Zariski and topological closures coincide, or equivalently, have the same dimension r.
Using Chow-scheme theory, one shows that F is algebraic if and only if it arises from a fibration, as in Example 1.4.
Problem: Find 'numerical' criteria for the algebraicity of foliations F⊂TX. The criteria we shall give and use below are the positivity of intersection numbers of det(F) and 'movable' classes of curves on either X or the projectivisation of F. We shall see that these conditions also imply restrictions on the structure of the leaves of F.
The criterion we shall prove and use here is the following, inspired by, and partially extending, former results [2,3,4]:
Theorem 1.6. (see [1,5]) The foliation F⊂TX is algebraic if the dual F∗:=HomOX(F,OX) of F is not pseudo-effective*.
*This is proved, but not stated explicitely in [1]. It is stated explicitely in [5].
We also have the next particular case:
Theorem 1.7. Let F⊂TX be a foliation with μα,min(F)>0 for some α∈Mov(X). Then F∗ is not pseudo-effective, F is an algebraic foliation. Moreover, its leaves have rationally connected closures.
The notions of rational connectedness and μα,min will be defined in sections 4 and 6 below. We now define the pseudo-effectivity.
Definition 1.8. A coherent sheaf G on X, is pseudo-effective if, for any j>0,c>0, H0(X,Sym[m](G)⊗Aj)≠{0} for some m>j.c. Here Sym[m](G) denotes the reflexive hull (i.e., the double dual) of Symm(G).
Otherwise, if H0(X,Sym[m](G)⊗Aj)={0},∀m>c(A).j, any j>0, and some constant c(A)>0, G is said to be not pseudo-effective.
Equivalently, by the next remark, G is not pseudo-effective if G is locally free on U=X∖S, where S⊂X is Zariski closed of codimension at least 2, and if H0(U,Symm(G)⊗Aj)={0},∀m>c(A).j.
Remark 1.9. Recall that F is reflexive if the natural morphism F→F∗∗ is an equality. Such a sheaf is torsionfree and normal (i.e., any section defined on a big Zariski open subset U⊂X extends to X). If rk(F)=1, reflexive means locally free on a smooth X. If F⊂E is of rank one, with E locally free, then F is reflexive if and only if it is saturated in E (but not necessarily a subbundle). Its powers Fk⊂SymkE,k>0 are then all saturated.
We shall see in §3 and §4 how to check the non-pseudo-effectivity of a sheaf G using negativity or non-positivity of intersection numbers with 'movable classes of curves'.
The following consequences of Theorem 1.6 can be stated without reference to pseudo-effectivity.
Corollary 1.10. (see [1], [3]) Let s be a nonzero section of ⊗mTX⊗L, for some m>0 and L a line bundle with c1(L)=0. Assume that s vanishes somewhere on X. Then X is uniruled (i.e., covered by rational curves).
Recall that a line bundle on X is said to be 'big' if h0(X,m.L)∼C.mn, for some C=C(X,L)>0, and m→+∞. A line bundle L is big if and only if mL=A+E for some m>0,A ample and E effective.
Corollary 1.11. (see [1]) Let L be big line bundle on X, and assume there exists a sheaf injection L⊂⊗mΩ1X, for some m>0. Then KX is big.
Remark 1.12. The statement of Corollary 1.11 is a form of stability specific to vector bundles E:=⊗mΩ1X, and fails for general E: on any positive-dimensional X, if E:=A⊕A⊗−2, with L=A ample of rank one, det(E)=−A, although A injects in E.
Products X=P1×Zn−1, with Zn−1 of general type, f:X→Z the projection, and L=f∗(KZ)⊂Ωn−1X, with κ(X,L)=(n−1) submaximal, show that KX may not be pseudo-effective. The positivity of subsheaves L⊂⊗mΩ1X is preserved by KX only when L is big.
We shall formulate a more general version of 'birational stability' for tensor powers of cotangent bundles in §.7.
We shall apply in §8 the Corollary 1.11 to moduli of canonically polarized manifolds.
Let S:=SingF⊂X be the Zariski closed set of codimension at least 2 over which F is not a subbundle of TX, and UF:=X∖S. For any x∈UF, let Ux=Fx×Bx be an open neighbourhood Ux of x in X such that the leaves of F|Ux are the fibres of the projection βx:Ux→Bx. If Δ⊂X×X is the diagonal of X, and if ΔF is the Zariski open set of Δ mapped to UF by (either) projections pi:X×X→X,i=1,2, then there exists a germ W of submanifold of dimension n+r of UF×UF such that for each x∈UF, W∩Ux×Ux=Ux×BxUx. In other words, the fibre Wx over x∈UF of the first projection p1|W:W→UF is a germ of the leaf of F through x.
∙ The leaves of F will thus be algebraic if (and only if) the manifold W is a germ of an algebraic subvariety of X×X, or equivalently, if the Zariski closure V of W has the same dimension n+r as W. Let d:=dim(V); if B is any ample line bundle on V, there exists a constant C=C(B,V)>0 such that h0(V,k.B)≅C.kd, when k→+∞. We thus only need to prove that h0(V,k.B)≤C.kn+r for some C>0, and some ample B on V in order to prove Theorem 1.6.
∙ Notice next that it is sufficient to show that h0(W,k.B)≤C.kn+r for some C>0, since the restriction map res:H0(V,k.B)→H0(W,k.B) is injective by the Zariski density of W in V. We now prove this last inequality.
∙ Let thus B be ample on X×X. Any section s of k.B on W admits a unique development in power series along the fibres of p1|W:W→UF of the form s=∑m≥0sm,k, where sm,k∈H0(UF,Symm(F∗)⊗k.B),∀m,k.
Since F∗ is not pseudo-effective, H0(UF,Symm(F∗)⊗k.B)={0} if m>C′.k, for some C′=C′(B).
Thus h0(W,k.B)≤∑m=C′.km=0h0(UF,Symm(F∗)⊗kB).
If UF=X (i.e., F is a subbundle of TX), h0(UF,Symm(F∗)⊗k.B)≤C".(m+k)n+r−1,∀k>0,∀m≥0, and some C">0. Indeed, h(m,k):=h0(X,Symm(F∗)⊗k.B)=h0(P(F),m.L+k.B′)), where L=OP(F)(1), B′ is the pullback of B, and so h(m,k)≤C".(m+k)n+r−1, since dim(P(F))=n+r−1, by dominating L,B′ by an ample line bundle A′ on P(F) such that A′−B′ and A′−L are effective. Thus:
∑m=C′.km=0h0(UF,Symm(F∗)⊗kB)≤∑m=C′.km=0C".(m+k)n+r−1≤(C′k+1).(C".(C′k+k)n+r−1)≤(C′+1)n+rC".kn+r, which concludes the proof of Theorem 1.6 if U=X.
The general case UF≠X is obtained by applying [10], III.5.10(3), which shows that the same inequalities h0(UF,Symm(F∗)⊗k.B)≤C".(m+k)n+r−1,∀k>0,∀m≥0, and some C">0 still hold true, by constructing a suitable modification of P(F), and of OP(F)(1).
More precisely: if p:P:=P(F):=Proj(Sym∙F)→X, with L:=OP(1) the tautological line bundle on P, p∗(Lm)=Symm(F),∀m>0. Moreover, if ρ:P′→P is a resolution of the singularities of P which coincides with P over UF, there exists an effective, ρ-exceptional divisor E⊂P′ such that (p∘r)∗(m.(L+E))=Sym[m](F),∀k>0. See Proposition 3.9 for a simplified proof.
Definition 3.1. Let (Ct)t∈T be an algebraic family of curves parametrised by an irreducible projective variety T. Assume that Ct is irreducible for t∈T generic, and that X is covered by the union of the C′ts. Then α:=[Ct]∈H2n−2(X,Z) is independent of t∈T. We call such a class a geometrically movable class on X.
The closed convex cone of H2n−2(X,R) generated by the geometrically movable classes is called the movable cone of X, denoted Mov(X), and its elements are the movable classes of X. From [7], we even have that α∈Hn−1,n−1(X,R) is in Mov(X) if and only if α.D≥0 for any irreducible effective divisor D on X.
Example 3.2. 1. If A is ample on X, then An−1∈Mov(X). There are many more examples, such as the following:
2. If C⊂X is an irreducible curve, locally complete intersection, with ideal sheaf IC and ample normal bundle (IC/I2C)∗, then α:=[C]∈Mov(X). Indeed: for any irreducible effective divisor D, we have D.C≥0, since D|C is a quotient of (IC/I2C)∗. See [8,9] for more on this situation.
Remark 3.3. If D is an effective Q-divisor on X, D.α≥0,∀α∈Mov(X). More generally, if m.D+A if effective for infinitely many m>0, and some ample A, D.α≥0,∀α∈Mov(X). Said otherwise: if D is pseudo-effective, then D.α≥0,∀α∈Mov(X).
An important result is the following converse.
Theorem 3.4. (see [7,10]) Let L be a line bundle on X. Then L is pseudo-effective if and only if L.α≥0,∀α∈Mov(X).
Equivalently, L is not pseudo-effective if and only if L.α<0 for some α∈Mov(X).
Pseudo-effectivity is fundamental in birational geometry when applied to L=KX because of the next:
Theorem 3.5. (see [11]) If KX is not pseudo-effective, then X is uniruled (i.e., covered by rational curves).
The only known proof rests on positive characteristic techniques. The original statement of Miyaoka-Mori is: if (Ct)t∈T is an algebraic family of curves (of possibly large genus) such that KX.Ct<0, it is possible to find such a family where the C′ts are rational curves!
Remark 3.6. There exists (Nagata, Mumford, see [12]) divisors D such that D.α≥0,∀α∈Mov(X), but which are not Q-effective. The main conjecture of birational geometry is that KX is Q-effective if KX.α≥0,∀α∈Mov(X).
The following birational invariance is crucial, here.
Corollary 3.7. Let π:X′→X be a birational morphism between smooth projective manifolds.
1. Let α∈Mov(X), then π∗(α)∈Mov(X′).
2. Let α′∈Mov(X′), then π∗(α′)∈Mov(X).
3. α∈Mov(X) if and only if α′:=π∗(α)∈Mov(X′).
Proof. The first (resp. second) claim follows from the equality: π∗(α).D′=α.π∗(D′) for any divisor D′ on X′ (resp. π∗(α).π∗(D)=π∗(α).(D′+E′)=π∗(α).D′, where D′ is the strict transform of D, and E′ is π-exceptional, since π∗(α).E′=0,∀E′, π-exceptional divisor on X′. The third claim follows from the others since π∗(π∗(α))=α,∀α.
The following is an immediate consequence of Theorem 3.4 since p∗(A)+ε.OP(E)(1) is ample on P(E) for any ε>0 sufficiently small, and Symm(E)=p∗(OP(E)(m)),∀m>0:
Corollary 3.8. Let E be a rank r vector bundle on X. Then E (identified with its sheaf of sections) is pseudo-effective if and only if L:=OP(E)(1) is pseudo-effective on p:P(E)→X, that is, if L.α≥0,∀α∈Mov(P(E)).
The above Corollary 3.8 extends to the case when G=E is not locally free by considering a suitable modification of P(G) as constructed in [10,III,5.10]. We give a simplified proof, following the strategy of [10], in Proposition 3.9 below (to be applied to a smooth model P of the main component of P(F), as in [10,V,3.23]).
Proposition 3.9. Let p:P→X be a fibration with X normal, P smooth, both projective, let D be a Cartier divisor on P such that p∗(OP(D))≠0.
1. If E is an effective divisor on P supported on the exceptional divisor Exc(p) of p, and such that, for every divisorial component Γ of Exc(p), its restriction EΓ to Γ is not pseudo-effective, then p∗(OP(k.(D+ℓ.E)))=p∗(OP(k.D))∗∗,∀k>0, where ℓ>0 is chosen to be sufficiently large, so that (D+ℓ.E)Γ is not pseudo-effective for any Γ.
2. There exists a divisor E satisfying the previous properties.
Proof. 1. Let Σ⊂X be the codimension 2 or more locus of points over which p is not equidimensional, so that Exc(p)⊂p−1(Σ). There is some ℓ>0 such that (D+ℓ.E)Γ is not pseudo-effective for every Γ, since this is the case for E. We replace E by ℓ.E, so that ℓ=1, to simplify notations. Then every section of OP(k.(D+E)) vanishes on Exc(p), for every k>0, and has thus no pole there. Every element s of H0(P,p∗(p∗(OP(k.D))∗∗)) lies in H0(P,k.(D+E)+E′), for some divisor E′ supported on Exc(p), which may be supposed to be effective. But then E′≤t.E for some t>0 since the support of E coincides with the divisorial part of Exc(p) by our non pseudo-effectivity assumption. Since t.E is not pseudo-effective on each Γ, H0(P,k.(D+E)+t.E)=H0(P,k.(D+E))=H0(P,p∗(p∗(OP(k.D))∗∗)), which implies the claim.
2. Let v:X′→X be a modification, q:P′→X′,u:P′→P be the normalisation of the main component of P×XX′, so that p∘u=v∘q. We choose v to be projective birational, with X′ smooth, and moreover such that q is equidimensional, and such that there exists an effective divisor Δ⊂Exc(v) such that −Δ is ample on the divisorial part of Exc(v).
Let E:=u∗(q∗(Δ)), and let H be a sufficiently ample divisor on P, with HΓ its restriction to Γ, for each divisorial component Γ of Exc(p). Let αΓ:=Hd−2Γ, where d:=dim(P), then αΓ∈Mov(Γ). If Γ′⊂P′ is the strict transform of Γ by u, then q∗(Δ).u∗(αΓ)=E.αΓ since the generic member of the family αΓ does not meet the indeterminacy locus of u|Γ′:Γ′→Γ. We thus just have to show that q∗(Δ).u∗(αΓ)<0.
Moreover, q∗(Δ).u∗(αΓ)=Δ.q∗(u∗(αΓ))<0. Indeed: q∗(u∗(αΓ))∈Mov(q(Γ′)), −Δ is ample on the divisorial part of Exc(v), and q(Γ′) is a divisorial component of Exc(v), by the equidimensionality of q. Thus E.αΓ<0, and EΓ is not pseudo-effective.
The description of Mov(P(E)) is, in general, quite delicate. An important particular situation where the pseudo-effectivity can be tested on X rather than on P(E) is exposed in the next section.
The data here are X,G,α∈Mov(X),r:=rk(G)>0 as above. We assume always G to be nonzero, torsionfree. See [13], Chap. V and [14] for a detailed treatment.
The (always locally free, since X is smooth), rank-one, sheaf det(G) is defined as det(G):=∧r(G)∗∗.
Definition 4.1. The slope of G with respect to α is: μα(G):=det(G).αr. We say that G is α-stable (resp. α-semi-stable) if: μα(H)<μα(G) (resp.μα(H)≤μα(G)), ∀H⊊G coherent subsheaf.
Notice that if F⊊G is a saturated subsheaf of rank r with torsionfree quotient Q of rank s, we have: det(G)=det(F)+det(Q) and so:
(∗) μα(G)=rr+s.μα(F)+sr+s.μα(Q)∈[μα(F),μα(Q)].
If F is torsionfree, then μα(F)=μα(F∗∗)=−μα(F∗), since these equalities need to be checked only on a big Zariski-open subset of X.
These notions are classical when α=[A]n−1, where A is an ample line bundle. Just as in this classical case, we still have (with essentially the same proof, by induction on the rank, and the equality (∗)):
Lemma 4.2. For G,α as above, there is a unique maximum H⊂G with μα(H) maximum (i.e., for any H′⊂G, we have: μα(H′)≤μα(H):=μα,max(G), in case of equality: H′⊂H).
H is the α-maximal destabilising subsheaf of G, denoted Gα,max. It is, by construction, α-semi-stable, and saturated inside G, so its quotient Gα,min:=G/Gα,max is torsionfree (or zero iff G is semi-stable).
Proof. Obvious if rank(G)=1, and so for direct sum of copies of a line bundle. Then embedd G in the direct sum of a certain number of copies of A, sufficiently ample. Then μα(G′)≤μα(⊕NA)=μα(A) for any G′⊂G. This proves the boundedness of μα(G′), for G′⊂G. If the class α is rational, one easily gets the existence of a maximum for these slopes, because of the finiteness of the possible denominators, and one can choose the rank to be maximal. The conclusion then follows. If the class α is not rational, one needs a further (but still elementary) argument: choose Gi,i=1,2 both different from their intersection, of the same maximal rank r with α-slopes μi,i=1,2, approaching from below up to ε>0 the upper bound μ:=μα,max(G) of α-slopes of all G′⊂G. The sum G1+G2 has rank larger than r and slope at least each μ−2r.ε (by a simple computation using the exact sequence 0→G1∩G2→G1⊕G2→G1+G2→0). Contradiction to the maximality of r for the ranks of G′⊂G with slope at least μ−2r.ε if ε>0 is chosen sufficiently small.
Corollary 4.3. There are a unique integer s≥0, and an increasing filtration {0}=H0⊊H1⊊H2⊊⋯⊊Hs=G by saturated subsheaves such that: Hj+1/Hj is α-semistable for j=0,…,s−1, and μα(Hj+1/Hj)>μα(Hj+2/Hj+1) for j=0,…,s−2.
This filtration is called the α-Harder-Narasimhan filtration of G. We write μα,min(G):=μα(Gα,min):=μα(G/Hs−1), and μα,max(G):=μα(Gα,max).
Proof. Induction on s applied to G/Gα,max and H1=Gα,max.
Lemma 4.4. 1. Let H≠{0} be a torsionfree quotient sheaf of G on X. Then: μα(H)≥μα,min(G). If equality holds, H is a quotient of Gmin.
2. Let F,G be torsionfree coherent sheaves on X. Assume that μα,min(F)>μα,max(G). Then Hom(F,G)={0}.
3. If det(G).α>0, then μα,max(G)=μα,min(Gα,max)>0.
Proof. 1. Induction on r: if H is a nonzero quotient of G, it fits in a short exact sequence 0→K→H→Q→0, in which K (resp. Q) is a quotient of Gα,max (resp. G/Gα,max), hence the conclusion since both K and Q have slope at least μα,min(G). The second claim holds by induction on s, because μα,min(G/Gα,max)>μα,min(G) if G is not semi-stable.
2. If 0≠h∈Hom(F,G), its image H a subsheaf of G, and a quotient of F. We thus get (with μ=μα): μmax(G)≥μ(H)≥μmin(F), a contradiction.
3. is obvious.
Lemma 4.5. We have: (G∗)α,max=(Gα,min)∗, in particular: μα,min(G)=−μα,max(G∗), and HNα(G∗)=(HNα(G))∗ (i.e., the terms for G∗ are the dual of those for G in reverse order).
Proof. Dualising the projection G→Gmin, we get: (Gmin)∗⊂G∗, and so −μmin(G)≤μmax(G∗). Dualising (G∗)α,max⊂G∗, we get a generically surjective morphism G→G∗∗→((G∗)α,max)∗ from which the inequality −μmax(G∗)≥μmin(G) follows. We thus get the equality −μmax(G∗)=μmin(G). Hence the second claim. The first claim follows from the second part of Lemma 4.4.(1). The last claim is seen by induction on the number of terms of the HN filtration.
Let Fˆ⊗G:=(F⊗G)∗∗ be the reflexive tensor product of F and G.
Theorem 4.6. For X,G,H,α as before, we have:
1. μα,min(G)=−μα,max(G∗), (Gα,min)∗=(G∗)α,max.
2. (Gˆ⊗H)α,max=Gα,maxˆ⊗Hα,max/.
3. μα,max(Gˆ⊗H)=μα,max(G)+μα,max(H).
4.μα,max(ˆ⊗m(G))=μα,max(^Symm(G))=m.μα,max(G).
5. μα,max(ˆ∧p(G))=p.μα,max(G),∀p>0.
The first claim follows from the fact that dualisation exchanges subobjects and quotients. By contrast, the proof of claim 3 is quite deep, relying on the Kobayashi-Hitchin correspondance between α-stable bundles and Hermite-Einstein vector bundles (with respect to Gauduchon metrics). For this correspondance, originally due to Li-Yau in [15], see details in the book of Lübke-Teleman [16].
Lemma 4.7. Let F be a torsionfree coherent sheaf on X, and α∈Mov(X) such that μα,min(F)>0. Then F∗ is not pseudo-effective.
Proof. Let A be ample on X. Applying Theorem 4.6, we see that −μ:=μα,max(F∗)<0, and so μα,max(Sym[m](F)⊗Aj)=m.μ+j.A.α<0 for m>j.c, where c:=A.αμ, which implies that h0(X,Sym[m](F)⊗Aj)=0, as claimed.
The following criterion for foliations among distributions is crucial here:
Corollary 4.8. (see [4]) Let F⊂TX be a coherent subsheaf. If, for some α∈Mov(X), 2.μα,min(F)>μα,max(TX/F), then F is a foliation.
Proof. Let Λ:∧2(F)→TX/F be the sheaf morphism induced by the Lie bracket. The slopes assumption implies by claim 5 of Theorem 4.6 and claim 2 of Lemma 4.4 that it vanishes. The conclusion follows by Frobenius theorem.
From Theorem 1.6, and Lemmas 4.7 and 4.8, we get:
Theorem 4.9. Let F⊂TX be a distribution with μα,min(F)>0 for some α∈Mov(X). Assume that 2.μα,min(F)>μα,max(TX/F) (this is satisfied if F is a piece of the Harder-Narasimhan filtration of TX relative to α). Then F is an algebraic foliation.
Remark 4.10. In the situation of Theorem 4.9, we shall prove in the next two sections that the leaves of F have rationally connected closures.
Corollary 4.11. Assume that KX is not pseudo-effective. There then exists on X a nonzero foliation F⊂TX such that μα,min(F)>0, the dual F∗ is not pseudo-effective, and F is algebraic.
Proof. Just apply Theorem 4.9, choosing F:=TXα,max. The last claim requires new notions and techniques presented in the next two sections.
Remark 4.12. We thus obtain the existence of (all of the) fibrations with rationally connected fibres on any given X with KX not pseudo-effective without using the Minimal Model Program.
Another easy, but central, property is the birational invariance 'up' of the condition μα,min(F)>0 for subsheaves of the tangent bundles:
Lemma 4.13. Let F⊂TX be a subsheaf, let π:X′→X be a birational morphism between X,X′ projective smooth. Let F′⊂TX′ be the pullback distribution, defined as: F′:=π∗(F)∩TX′. Let α∈Mov(X),α′:=π∗(α). Then μα′(F′)=μα(F), and μα′,min(F′)=μα,min(F).
Proof. The statements hold for π∗(F), hence for F′ since det(F′) and det(π∗(F)) coincide outside the exceptional divisor of π, and α′.E=0 for each component E of this divisor.
Remark 4.14. The statement does not (always) hold 'down' (i.e., if α=π∗(α′), but α′≠π∗(α): let X=P2, let F be tangent to the conics C going through 4 points in general position, and let α′ be the class of the strict transforms C′ of these conics on the blow-up X′ of P2 in the 4 given points. Then μα′(F′)=2>0, since F′|C′=−KC′, but μα(F)=2−4=−2>0.
Let f:X→Z be a fibration between complex projective manifolds. Let Xz be its generic (smooth) fibre. Let f∗(Ω1Z)sat⊂Ω1X be the saturation of f∗(Ω1Z), and denote by f∗(KZ)+:=det(f∗(KZ)sat). Let finally K−X/Z:=KX−f∗(KZ)+.
If F:=Ker((df)sat)⊂TX, we thus have: det(F∗)=K−X/Z, by the consideration of the determinants in the exact sequence: 0→(f∗(Ω1Z))sat→Ω1X→F∗→0, dual to: 0→F→TX→TX/F→0.
Definition 5.1. We say that f is 'neat' if:
1. The discriminant locus D(f)⊂Z of singular fibres of f is a divisor of strict normal crossings.
2. f−1(D(f))⊂X is also a divisor of simple normal crossings.
3. There exists an equidimensional fibration f0:X0→Z and a birational morphism u:X→X0 such that f0∘u=f.
By suitable blow-ups of Z and X, any fibration can be made 'neat', as a consequence of Hironaka's resolution of singularities, and Hironaka's or Raynaud's flattening theorems.
Theorem 5.2. Let f:X→Z be a fibration between complex projective manifolds. Let Xz be its generic (smooth) fibre. If f is 'neat', and if KXz is pseudo-effective, K−X/Z is pseudo-effective.
This result strengthens a weak version of Viehweg's theorem on the weak positivity of direct images of pluricanonical sheaves. A particular case was also obtained by Miyaoka using deformations of rational curves, Mori's bend and break, and positive characteristic methods.
Corollary 5.3. Assume that KX is not pseudo-effective, let α∈Mov(X) be such that KX.α<0. Let F:=TXα,max⊂TX, thus μα,min(F)>0, and so F is a foliation, by corollary 4.11, and algebraic, by Theorem 1.6. Moreover, after suitable blow-ups of X, we have: if f:X→Z is a fibration such that F=Ker(df)sat⊂TX, then:
1. KXz is not pseudo-effective. More generally:
2. If f=h∘g for rational fibrations g:X→Y and h:Y→Z, then KYz is not pseudo-effective, Yz being the generic fibre of h (on suitable birational neat models of g,h).
Proof. We may blow-up X,Z0 for any rational fibration with generic fibre the closure of a generic leaf of F in such a way that the birational model f:X→Z (we keep the notation X for the blown-up manifold, so as to save notations) so obtained is regular and 'neat'. By Lemma 4.13, the property μα,min(F) is preserved on the blown-up manifold by lifting both the foliation and the movable class in the natural manner, so that the Theorem 5.2 can be applied to the new f:X→Z.
Claim 1. By contradiction, assume not. Then K−X/Z=det(F∗) is pseudo-effective, by Theorem 5.2. Thus 0≤K−X/Z.α=det(F∗).α=−μα(F)<0. Contradiction.
Claim 2. By contradiction again, assume that KYz is pseudo-effective. By applying Theorem 5.2 to h, det(H∗), and so also g∗(det(H)∗) were pseudo-effective, with H:=Ker(dh)sat⊂TY. Thus μα(g∗(H∗))=−μα(g∗(H))≥0. On the other hand, the fibration g|Xz:=gz:Xz→Yz implies that g∗(H) is a quotient of F, so μα(g∗(H))≥μα,min(F)>0. Contradiction.
Remark 5.4. With (much) more work, it is possible to show more: for general z∈Z, there exists αz∈Mov(Xz) such that μαz,min(TXz)>0. We shall prove this less directly, but more simply, by using the theory of rational curves in the next section.
Definition 6.1. X is said to be uniruled (resp. rationally connected) if through any generic point (resp. any two generic points) of X pass an irreducible rational curve (i.e., a non-constant image of P1).
Example 6.2. 1. Pn is rationally connected, so also any X birational to Pn, or rationally dominated by Pn.
2. X is rationally connected if −KX is ample (i.e., if X is Fano).
3. Smooth hypersurfaces of Pn+1 of degree at most n+1 are Fano (by adjunction), hence rationally connected.
4. If X→Z is a fibration with Z and Xz rationally connected, then X is rationally connected (a deep result of [17]).
The present birational theory of rational curves on projective manifolds is based on [17] and the following two results:
Theorem 6.3. (see [11]) X is uniruled if (and only if) KX is not pseudo-effective.
Theorem 6.4. There is a unique fibration r:X→R (called the Maximally Rationally Connected (or MRC-) fibration of X) such that:
1. its fibres are rationally connected.
2. its base is not uniruled (i.e., KR is pseudo-effective).
Although in general, r:X→R is rational (almost holomorphic, in the sense that its generic fibre does not meet the indeterminacy locus), since all the properties considered here are of birational nature, we may and shall assume that r is regular.
The two extreme cases are: X=R, or R is a point, meaning respectively that X is not uniruled, and that X is rationally connected.
Using Chow-scheme theory, one gets a relative version as well, on suitable birational models:
Corollary 6.5. If f:X→Z is a fibration, there exists g:X→Y and h:Y→Z such that for general z∈Z, gz:=g|Xz:Xz→Yz is the MRC of Xz, and so KYz is pseudo-effective.
Theorem 6.6. Let F⊂TX be a foliation such that μα,min(F)>0, for some α∈Mov(X). Then F is algebraic, and its leaves have rationally connected closures.
Proof. The algebraicity of F follows from Theorem 4.9. Let f=h∘g be the relative MRC of f, a suitable fibration such that F=Ker(df)sat. By Corollary 5.3.1, and Theorem 6.3, Xz is uniruled and so dim(Y)<dim(X). By Corollary 5.3.2, KYz is not pseudo-effective if dim(Y)>dim(Z), contradicting Corollary 6.5. Thus Y=Z, and the claim.
Corollary 6.7. The following are equivalent:
1. X is uniruled.
2. KX is not pseudo-effective.
3. h0(X,mKX+A)=0 for any ample A and large m.
4. μα,min(Ω1X)<0, for some α∈Mov(X).
We have: 1 ⟹ 2, 3 ⟹ 4 ⟹ 1 (the last two implications by Corollaries 4.11 and 6.6 respectively).
Corollary 6.8. The following are equivalent:
1. X is rationally connected
2. ΩpX is not pseudo-effective for any p>0.
3. h0(X,Symm(Ω1X)⊗A)=0 for any p>0, A ample and m large.
4. μα,max(Ω1X)<0 for some α∈Mov(X).
Proof. The implications 1 ⟹ 2, 3 are easy by taking a rational curve with ample normal bundle in X, 3 ⟹ 4 is seen by contradiction, from Theorem 6.4, taking L:=r∗(KR)⊂ΩpX if p:=dim(R)>0 is a pseudo-effective line bundle, thus such that 0<h0(X,m.L+A)≤h0(X,Symm(ΩpX)⊗r∗(A)) for infinitely many m′s. That 4 ⟹ 1 follows from Theorem 6.6 applied to F=TX.
Remark 6.9. 1. Theorem 6.3 (resp. Corollary 6.8) claims that X is uniruled (resp. rationally connected) if and only if there is a covering family of curves (possibly of large genus) on which det(TX) (resp. TX) is ample.
2. The 'uniruledness conjecture' claims that X is uniruled if KX is not Q-effective. It implies that in the above two corollaries, the ample A could be removed in the assertions 3, and that 'pseudo-effective' could be replaced by 'Q-effective' in assertions 2.
Corollary 6.10. (see [1,3]) Let s be a nonzero section of ⊗mTX⊗L, for some m>0 and L a line bundle with c1(L)=0. Assume that s vanishes somewhere on X. Then X is uniruled (i.e., covered by rational curves).
Proof. It is sufficient to show that μα,max(TX)>0, or equivalently, that μα,max(⊗mTX⊗L)>0 for some α∈Mov(X). Let L be the subsheaf of rank one of ⊗mTX⊗L generated by s, and let α be the geometrically movable class defined by a family of ample curves going through some point where s vanishes. Thus 0<μα(L)≤μα,max(⊗mTX⊗L) as claimed.
Corollary 6.11. Let C⊂X be an irreducible projective curve such that TX|C is ample. ThenX is rationally connected.
This was observed in [18] by a different approach.
Proof. Apply Theorem 6.6, observing that [C]∈Mov(X), by considering the graph C" of the embedding j:C′⊂X in C′×X,C′ the normalisation of C, which has normal bundle TX|C′, so that Corollary 6.8 applies, and [C"]∈Mov(C′×X), so that [C]=p∗([C"])∈Mov(X).
Theorem 7.1. (see [1]) Assume that KX is pseudo-effective. Let Q be a torsionfree quotient of ⊗mΩ1X, for some m>0. Then det(Q) is pseudo-effective.
Proof. KX being pseudo-effective, μα,min(Ω1X)≥0,∀α∈Mov(X), by Corollary 6.7. By Theorem 4.6, μα,min(⊗mΩ1X)=m.μα,min(Ω1X)≥0, ∀α∈Mov(X), which means that any torsionfree quotient of ⊗m(Ω1X) is pseudo-effective.
Theorem 7.2. (see [1]) Let L a pseudo-effective line bundle on X. If there exists a nonzero sheaf morphism λ:L→⊗mΩ1X⊗K⊗pX for some m≥0,p>0, KX is pseudo-effective.
Proof. We proceed by induction on n=dim(X), the case n=1 being obvious. Assume KX is not pseudo-effective, there then exists α∈Mov(X) such that KX.α<0, and F⊂TX a foliation such that μα,min(F)>0. Thus F is algebraic, of the form Ker(df)sat for some fibration f:X→Z with rationally connected fibres, by Theorem 6.6.
If dim(Z)=0, we get F=TX, and so: μα,min(TX)>0, which implies that 0≤L.α≤μα,max((⊗mΩ1X)⊗K⊗pX)=−m.μα,min(TX)+p.KX.α<0. Contradiction.
If dim(Z)>0, then dim(Xz)<n and we may apply induction. Let λz:=λ|Xz:Lz→(⊗mΩ1X⊗K⊗pX)|Xz be the restriction. The exact sequence on Xz: 0→T→Ω1X|Xz→Ω1Xz→0, with T a trivial bundle of rank dim(Z), induces a filtration on (⊗mΩ1X⊗K⊗pX)|Xz with graded pieces isomorphic to T⊗(m−j)⊗jΩ1Xz⊗K⊗pXz, and we thus get for some j∈{0,…,m} a nonzero sheaf morphism LXz→⊗jΩ1Xz⊗K⊗pXz. The induction then implies that KXz is pseudo-effective, since LXz is, like L on X, pseudo-effective on Xz, for z general in Z (i.e., ouside countably many strict Zariski-closed subsets). This contradicts the non-pseudoeffectivity of KX.
Corollary 7.3. (see [1]) Let L a big line bundle on X. If there exists a nonzero sheaf morphism L→⊗mΩ1X for some m>0, KX is big.
Proof. We may assume that L is saturated, let Q:=(⊗mΩ1X/L). If KX is pseudo-effective, det(⊗mΩ1X)=N.KX=L+det(Q). Since L is big and det(Q) is pseudo-effective by Theorem 7.1, N.KX, and so KX, is big.
We show that KX is pseudo-effective, which will imply the claim. Since L is big, k.L+KX is effective for some large k>0, and −KX thus admits a nonzero morphism in kL=(kL+KX)−KX, and so by composition with λ⊗k, −KX maps nontrivially into ⊗mkΩ1X. Equivalently, OX injects into ⊗mkΩ1X⊗KX. Theorem 7.2 implies that KX is pseudo-effective.
Remark 7.4. Let X:=Pk×Z, where k>0, and Z is of general type and dimension (n−k)≥0. Then KX is not pseudo-effective, although Ωn−kX⊂Ω1⊗(n−k)X contains L:=p∗(KZ), a line bundle of Kodaira dimension 0≤(n−k)≤(n−1). The assumption that L is big cannot be weakened to any smaller Kodaira dimension in order to imply the pseudoeffectivity of KX.
The argument used in Theorem 7.1 however extends to arbitrary numerical dimensions when KX is assumed to be pseudo-effective.
Definition 7.5. Let L,P,A be line bundles on X, A ample.
1. κ(X,L):=max{k∈Z|¯limm>0(h0(X,m.L)mk)>0}∈{−∞,0,…,n}
2. ν(X,L):=max{A,k∈Z|¯limm>0(h0(X,m.L+A)mk)>0}∈{−∞,0,…,n}
The elementary properties of these two invariants are:
Proposition 7.6. 0. ν(X,L)≥κ(X,L)
1. L is pseudo-effective (resp. Q-effective) iff ν(X,L)≥0 (resp. κ(X,L)≥0.
2. L is big (i.e., κ(X,L)=n) iff ν(X,L)=n.
3. If P is pseudo-effective, ν(X,L+P)≥ν(X,L).
4. If L is big and if P is pseudo-effective, L+P is big.
Theorem 7.7. Assume that KX is pseudo-effective. Let L be a line bundle on X, and L⊂⊗mΩ1X a nonzero sheaf morphism. Then ν(X,L)≤ν(X,KX).
Proof. Let Q:=(⊗mΩ1X)/Lsat. Since det(Q) is pseudo-effective by Theorem 7.1, N.KX=det(Lsat)+det(Q) is the sum of Lsat and P:=det(Q) which is pseudo-effective.
Thus ν(X,KX)=ν(X,N.KX)=ν(X,Lsat+P)≥ν(X,L).
Remark 7.8. 1. The analogue of Proposition 7.6.(3) is false in general for the Kodaira dimension. The proof of Theorem 7.7 thus does not apply to prove κ(X,L)≤κ(X,KX) in the same situation. However:
2. A central conjecture of birational geometry claims that ν(X,KX)=κ(X,KX) for any X. It thus implies, together with Theorem 7.7, that κ(X,KX)≥ν(X,L) for any m>0,L⊂⊗mΩ1X) if ν(X,KX)≥0.
3. The Remark 7.4 shows that the assumption that KX is pseudo-effective cannot be removed or weakened.
When KX is not pseudo-effective (i.e., when X is uniruled), we have the general version:
Corollary 7.9. Let f:X→Z be the MRC of X. Let L⊂⊗mΩ1X be a subsheaf of rank 1. Then ν(X,L)≤ν(Z,KZ).
Proof. We can and shall assume that L is saturated in ⊗mΩ1X:=E. Let Xz:=f−1(z),z∈Z is a generic smooth fibre of f, and let A be an ample line bundle on X, with LXz:=L|Xz.
Lemma 7.10. ν(X,L)=−∞, unless LXz⊂f∗(⊗mΩ1Z,z) over Xz.
Proof. The exact sequence 0→f∗(Ω1Z,z)→Ω1X|Xz→Ω1Xz→0 induces on ⊗mΩ1X|Xz a filtration with graded pieces isomorphic, for 0≤j≤m, to Gj:=⊗m−jf∗(Ω1Z,z)⊗(⊗jΩ1Xz). Since Xz is rationally connected, we may choose α∈Mov(X) such that f∗(α)=0, and α restricts to Xz such that μα,max(⊗jΩ1Xz)=j.μα,max(Ω1Xz)<0, for any j>0. If L is not contained in ⊗m(f∗(Ω1Z,z)), there exists j>0 and an injective sheaf map L→Gj. We thus get: μα(L)=L.α≤μα,max(Gj)=j.μα,max(Ω1Xz)<0, since μα,max(⊗m−j(f∗(Ω1Z,z)))=0, the bundle f∗(Ω1Z,z) being trivial. Thus L.α<0, and ν(X,L)=−∞.
From lemma 7.10, we deduce that L⊂f∗(⊗mΩ1Z)sat⊂⊗mΩ1X. We can now conclude from the following Lemma 7.11, analogous to Lemma 7.1, using the very same arguments used to deduce Theorem 7.7 from Lemma 7.1.
Lemma 7.11. Let f:X→Z be the MRC fibration of X, and let G⊊f∗(⊗mΩ1Z)sat be a saturated subsheaf. Then det(Q) is pseudo-effective on X, where Q is the quotient of (f∗(⊗mΩ1Z))sat by G.
Proof. We may and shall assume Z to be smooth. By contradiction, if the conclusion does not hold, μα,min(f∗(⊗mΩ1Z)sat))<0 for some α∈Mov(X), and −μα,max((f∗(TZ))sat)=μα,min((f∗(Ω1Z)sat))<0. There is then F⊂(f∗(TZ))sat, α-semi-stable with μα,min(F)>0 which is an algebraic foliation on X with closures of leaves rationally connected, and mapped non-trivially on Z by f, contradicting the non-uniruledness of Z.
Let f:Y→B be a projective holomorphic submersion with connected fibres between connected complex quasi-projective manifolds Y,B.
We assume that all fibres of f have an ample canonical bundle, and that the 'variation' of f, defined by: Var(f)=rk(ks:TB→R1f∗(TY/B)) is maximal, equal to dim(B).
The situation considered by I. Shafarevich in 1962, was when f was a non-isotrivial family of curves of genus at least 2 on a curve B. His conjecture (proved by Parshin and Arakelov) was (formulated differently) that KˉB+D was big on ˉB=B∪D, a compactification of B.
The conjecture of Shafarevich was extended by Viehweg in the following form: in the above situation, KˉB+D is big if ˉB is a smooth projective compactification of B obtained by adding to B a divisor D of simple normal crossings.
In this situation, Viehweg-Zuo [19] proved the existence of a big line bundle (the 'Viehweg-Zuo sheaf') L⊂⊗mΩ1ˉB(LogD) which, combined with the following Theorem, implies Viehweg's conjecture (proved first in [20]):
Theorem 8.1. Let X be a connected complex projective and D a divisor of simple normal crossings on X. Assume that a big line bundle L on X has an injective sheaf morphism L⊂⊗mΩ1X(LogD). Then KX+D is big.
This is the logarithmic analogue of Theorem 7.3. It is proved by extending to the logarithmic setting the theorems 1.6 and 7.3, which is done very straightforwardly, by replacing in the proofs the (co)tangent bundles by their Log-analogs.
The Viehweg's conjecture has been extended, and proofs given, in various directions (fibres of general type, or with semi-ample canonical bundle, or B 'special', and then f is isotrivial). In another direction, inspired by Lang's conjectures, the Brody, Kobayashi, or Picard hyperbolicity of the corresponding moduli spaces have been proved in various situations. See [21] for references on this topic.
Let F⊂TX be a numerically trivial and pseudo-effective foliation, that is: such that det(F).An−1=0, for some ample line bundle A. Since c1(F)=0, the pseudo-effectivity of F and F∗ are equivalent.
When X is an abelian variety and F is linear (that is: F=Ker(w), w∈H0(X,Ωn−r)), F is numerically flat and pseudo-effective, but the leaves of F are in general not algebraic. In such cases however, F is flat (i.e., given by a linear representation of the fundamental group), and so semi-stable but not stable.
The following question/conjecture was raised by Pereira-Touzet (who gave several affirmative answers, mostly in the corank-one case): if F is A-stable and numerically trivial, is it algebraic?
Theorem 1.6, reduces this question to the non-pseudoeffectivity of F∗ when c1(F)=0, and F is A-stable.
A partial solution is given in the following:
Theorem 9.1. (see [22,23]) Let E be a vector bundle on X, assume that c1(E)=0≠c2(E), and that Symm(E) is A-stable for some ample A and some m>r=rk(E). Then E is not pseudo-effective*.
*The proof works for X compact Kähler, and E a reflexive sheaf on X.
The proof of [23] relies on analytic and metric methods. The previous proof given in [22] by Höring-Peternell is mostly algebro-geometric. It runs along the following lines, starting with E pseudo-effective, with c1(E)=0, and all Symm(E) A-stable, in order to show that c2(E)=0. The first step shows, by induction on the codimension of W⊂P:=P(E), a component of the restricted base locus B−(L),L:=OP(1), that p(W) has codimension at least 2 in X, p:P→X being the natural projection. The key ingredient in this step is following lemma, due to Mumford when r=rk(E)=2:
Lemma 9.2. (see [22]) Let E be a vector bundle on a smooth projective curve C. Let L:=OP(E)(1). Assume that c1(E)=0, and Symm(E) is stable for all m>0. Then Ld.Z>0 for all d>0, and any irreducible Z⊂P(E) of codimension d.
The second step then deduces from the first one that the restriction ES of E to a general complete intersection surface S⊂X of large degree, is nef. So that E and det(E∗)=det(E)=0 are nef, and E∗ is thus nef too. Hence E is numerically flat, so in particular all Chern classes of E vanish.
From Theorem 9.1 and Theorem 1.6, we get:
Corollary 9.3. Let F⊂TX be a foliation with c1(F)=0≠c2(F). If Sym[m](F) is A-stable for some m>rk(F), F is algebraic.
The Corollary 9.3 (or the slightly weaker version of [22]) now permits to give an alternative proof of the Beauville-Bogomolov-Yau* (BBY) decomposition theorem in the projective and klt singular case.
*Usually called the Bogomolov-Beauville decomposition. Since Yau's Ricci-flat metric is essential here, BBY seems justified.
Let X be an n-dimensional compact Kähler manifold with c1(X)=0. By Yau's theorem, any such X carries a Ricci-flat Kähler metric. There are 3 basic examples of such manifolds, each class characterised by the restricted holonomy group Hol0 of any of their Kähler, Ricci-flat, metrics: Hol0 is trivial for 1, is Sp(4r) for 2, and is SU(n) for 3.
1. The compact complex tori are the quotients Cn/Γ, where Γ is a cocompact lattice of Cn.
2. The irreducible hyperkähler (HK) manifolds are defined as the ones which are simply-connected, even-dimensional: n=2r, and admit a holomorphic symplectic 2-form σ generating H0(X,Ω2X), and such that s∧r is a nowhere vanishing section of KX. There are few known examples: 2 deformation classes in each even dimension due to Beauville (and Fujiki in dimension 4), and one additional in dimensions 6 and 10, found by O'Grady.
3. The Calabi-Yau (CY) manifolds are those which are simply connected, with hp,0=0,∀p=1,…,(n−1), and with KX trivial. Many deformation families known in each dimension. Among these complete intersections of suitable multidegrees in the projective spaces.
The BBY decomposition is the following:
Theorem 10.1. (see [24]) Let X be a connected compact Kähler manifold with c1(X)=0. A suitable finite étale cover of X is a product of a complex compact torus T, and of simply-connected CY or HK manifolds.
Let us sketch the classical proof given in [24]: We equip X with some Ricci-flat Kähler metric [38]. Let Hol0 be its restricted holonomy representation, and TX=F⊕(⊕iTi) be the splitting of the tangent bundle of X into factors which are irreducible for the action of Hol0. This splitting is well-defined locally on X, and globally only up to possible reordering. These local factors correspond also to a local splitting of X into a direct product of Kähler submanifolds. By Berger general classification of the holonomy representations, only a flat factor F appears, together with other factors Ti having restricted holonomy either Sp or SU (the ones susceptible to preserve a section of KX by parallel transport).
We lift this decomposition to the universal cover ˜X of X on which the metric is complete. By De Rham decomposition theorem, ˜X splits as a Kähler product, corresponding to these Hol0-factors [25,Theorem 10.43]. The Cheeger-Gromoll theorem then says that ˜X=Ck×P, where Ck corresponds the flat factor, the other factor P being compact, and corresponding to the product of the factors with Hol0 either Sp or SU [25,Theorem 6.65]. Bieberbach's theorem [26] then concludes the proof.
Notice that this shows in particular that the (everywhere regular) foliations defined on X by the factors Ti have compact leaves (with finite fundamental groups), and are thus 'algebraic' (i.e., with equal Zariski and topological closures of their leaves).
We shall now give, but only in the projective case, an alternative proof which does not require the consideration of the universal cover. This is done by proving directly, using Theorem 9.1, that the foliations defined by the T′is are all algebraic. This follows from the following:
Lemma 10.2. For each i, and each m>0, c1(Ti)=0, and Symm(Ti) is A-stable for any ample line bundle A on X.
Proof. Since c1(TX)=c1(F)=0, the first claim follows from the second one. The second claim is a consequence of representation theory of the groups SU and Sp [27].
If we take thus the (regular and submersive) fibration fi:X→B with fibres the (projective) leaves of Ti, it is isotrivial of fibre Fi (by the existence of the transversal foliation defined by F⊕j≠iTj), and Fi has a vanishing irregularity and a discrete automorphism group (because the restricted holonomy is either SU or Sp), which implies that after a finite étale base change B′→B, X splits as a product Fi×B′.
Induction on dim(X) now reduces the proof to the case where TX=F, the flat factor. Bieberbach theorem [26] then permits to conclude. For details, see [28].
This BBY decomposition is still valid for projective varieties with klt singularities, but the proof, obtained first in [22], and using a combination of the works [5,29,30,31], is much more involved, both technically and conceptually.
A simplified proof, which closely follows the above alternative proof in the smooth case, and avoids the delicate arguments in positive characteristic used in [5], can be found in [28]. The result has then been extended to the Kähler case in [6], by deformation to the projective case.
We denote by X a smooth and connected complex projective manifold*, of dimension n. Let E be a torsionfree coherent sheaf on X.
*The questions raised below could be extended to the compact Kähler case, and to smooth or klt orbifold pairs.
Definition 11.1. We say that E is pseudo-effective if so is det(Q), for each nonzero quotient of E∗∗. Equivalently, μα,min(E∗∗)≥0,∀α∈Mov(X).
Example 11.2. 1. If E is pseudo-effective, so is any of its quotients, as well as ⊗mE,SymmE,∧mE,∀m>0, and more generally the tensors deduced from E by linear representations, by Theorem 4.6.
2. If KX=det(Ω1X) is pseudo-effective, so is Ω1X, as seen in Theorem 7.1. In [1], the same property is shown, more generally, for any quotient of ⊗m(Ω1X) with pseudo-effective determinant.
We ask the same question also for subsheaves of ⊗m(Ω1X):
Question 11.3. Let E⊂⊗mΩ1X be such that det(E) is pseudo-effective. Is E pseudo-effective? This is a birational version of stability.
Notice that if E⊂ΩpX is saturated and defines an algebraic foliation of rank r=(n−p), the answer is yes by [1], since E is then the cotangent sheaf of the space of leaves.
This question might be answered by showing the relative algebraicity of two transcendental foliations deduced from suitable quotients of Ω1X, using a suitable extension of the next algebraicity criterion in [1].
Recall this algebraicity criterion (theorem 1.7): if F⊂TX is a foliation such that F∗ (which is a quotient of Ω1X) is not pseudo-effective, the leaves of F are algebraic. Moreover, if μα,max(F∗)<0 for some α∈Mov(X), the leaves of F are rationally connected.
Question 11.4. If F∗ is not pseudo-effective, are there restrictions on the leaves of F, and which ones?
Already when F=TX, the answer is nontrivial, and consists in the classification of manifolds X with Ω1X not pseudo-effective, or equivalently with ν1(X)=−∞, using the notations of the next subsection. Although such manifolds may have an ample, or a trivial canonical bundle, as shown by hypersurfaces in Pn+1, they share some common properties with the rationally connected manifolds. For example, rationally connected manifolds are simply-connected, and it is shown in [32] that the linear representations of X have a finite image if Ω1X is not pseudo-effective. For manifolds of general type, even for surfaces, the invariant κ1 (see below), does not seem to be described by, or even related to, other algebro-geometric invariants. For special manifolds, in the sense of definition 11.11 below, the situation might be much simpler (see conjecture 11.13).
An additional problem is that if the foliation F has F∗ not pseudo-effective, and so if F is algebraic, it is not clear whether or not the general fibres of an associated fibration have Ω1 not pseudo-effective. The proof when μα,min(F)>0 relies on the MRC, and it is unknown whether a similar fibration may be expected to exist on any X, with fibres having Ω1 not pseudo-effective, and base with Ω1 pseudo-effective.
Let E be torsionfree coherent on X, of rank r, and let P(E),OE(1) be the corresponding (Grothendieck) projectivisation, together with its tautological line bundle.
Definition 11.5. We write κ(E):=κ(P(E),OE(1))−(r−1), and ν(E):=ν(P(E),OE(1))−(r−1). We also write νp(X):=ν(ΩpX), and κp(X):=κ(ΩpX), so that: ν(X)=νn(X), and κ(X):=κn(X).
The definition fits with the original one for line bundles. Moreover, the invariants κp and νp are preserved under finite étale covers. The invariant κ1 has been introduced and studied by F. Sakai in [33]*.
*F. Sakai uses the 'invariant λ', which takes the value −n when κ1=−∞, but otherwise coincides with κ1. His convention simplifies the formulations in some situations, such as products.
The classical 'Abundance conjecture' νn(X)=κn(X) extends as:
Conjecture 11.6. For any X and p>0, one has: νp(X)=κp(X).
Remark 11.7. 1. In [34], this is stated when κp(X)=−∞, and proved for elliptic surfaces (except for some isotrivial ones), and some other additional cases.
2. If κp(X)=−∞,∀p>0, and if we assume the classical 'Abundance conjecture', we get by the MRC fibration that X is rationally connected, and so that −∞=κp(X)=νp(X),∀p>0. Conjecture 11.6 is thus a consequence of its 'classical' version in this situation.
Example 11.8. 1. If c1(X)=0, from the Beauville-Bogomolov-Yau decomposition theorem, and [22], one can easily see* that ν1(X)=κ1(X)=˜q(X)−n, where ˜q(X) is the maximum of q(X′) when X′ runs through the finite étale covers of X. The same properties can be still shown with more work†when X is a smooth model of a klt variety with c1=0. Thus κ1(X)=ν1(X)=˜q(X)−n whenever κ(X)=0, if one assumes the existence of good minimal models.
*In the next formula, if ˜q(X)=0, replace Sakai's −n by −∞.
†Because one needs to carefully distinguish étale from quasi-étale covers, as shown by the Kummer K3′s.
Remark 11.9. The Abundance conjecture ν=κ fails in general for rank-one subsheaves L of ΩpX, by the examples of M. Brunella and M. Mc Quillan of the (duals of the) tautological foliations on irreducible free quotients of the bidisc, which have κ=−∞ and ν=1.
Question 11.10. Are there rank-one subsheaves L of ΩpX for suitable X,p>0, with p=ν(X,L)>κ(X,L)≥0? As seen just above, there are examples with p=ν(X,L)>κ(X,L)=−∞.
Recall that from Bogomolov's theorem that κ(X,L)≤p,∀X,∀p>0, if L is any rank-one subsheaf of ΩpX.
Definition 11.11. We say that X is special‡ if κ(X,L)<p, for any such L, and any p>0;
‡We may assume X to be compact Kähler, here.
Bogomolov's theorem has been strengthened by C. Mourougane and S. Boucksom to ν(X,L)≤p,∀X,∀p>0.
Conjecture 11.12. If X is special, ν(X,L)<p,∀L⊂ΩpX,∀p>0, L of rank one.
This has been proved in [35] for p=1. Notice that if L⊂Ω1X is of rank one with ν(X,L)=1, there may be no L′⊂Ω1X with κ(X,L′)=1, as shown by the example of Brunella and Mc Quillan in remark 11.9, since no finite étale cover of the ambiant surface maps onto a curve of positive genus.
Conjecture 11.13. Assume X (smooth compact Kähler) is special. Do we have§ : ν1(X)=κ1(X)=˜q(X)−n?
§If ˜q(X)=0, replace the RHS −n by −∞.
If yes, then ν1(X)=κ1(X)=−∞, that is Ω1X not pseudo-effective, if and only if ˜q(X)=0, if and only if π1(X) is finite.
Recall that the Abelianity conjecture claims that π1(X) is virtually abelian if X is special, so that ˜q(X)=0 should then mean that π1(X) is finite.
Notice that if X is a compact Kähler complex manifold with all of its finite étale covers having a surjective Albanese map (as is the case when X is special), then κ1(X)≥˜q(X)−n.
We thank the reviewers for their careful readings, corrections and suggestions, which lead to improvements of the text.
The author declares there is no conflicts of interest.
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