Overview of the Proof

Purely for the sake of illustration, i.e. so we can concentrate on the methodology rather than digress about what are the correct, [M3], statements about the hyperbolicity of algebraic surfaces, let's suppose that the goal is to prove the Green-Griffiths conjecture, viz: there are no Zariski dense entire curves on surfaces of general type. This was proved for surfaces with big co-tangent bundle in [M1]. Post factum, op. cit. is the key point in the classification, [M2], of foliations, $S\rightarrow [S/\mathcal{F}]$ on algebraic surfaces, where, strictly speaking, around the singularities the champ classifiant $[S/\mathcal{F}]$ is purely notational, and, otherwise as per [SGA.I] exposé VI. The general case of the conjecture was reduced by Green & Griffiths, [GG], to entire curves satisfying a $n-1$ dimensional order O.D.E., $n\in\mathbb{N}$, on an algebraic surface, so, in particular a Zariski dense entire curve invariant by a foliation by curves on a variety of dimension $n$. As such the basic strategy is to make this key point in the classification theory of foliations work in all dimensions. It now fits into a general structure, and may be succinctly summarised,

(a)

One first resolves the singularities. This means, functorially with respect to the ideas, that the relative dualising sheaf, $\omega_{S/[S/\mathcal{F}]}$, abbreviated to $K_{\mathcal{F}}$, has canonical singularities. This step is absolutely essential, otherwise $K_{\mathcal{F}}$ has absolutely no relation with the curvature of the leaves, and one might as well study giraffes or cockatoos, e.g. consider the foliation $C\rightarrow\mathrm{pt}$ for $C$ Gorenstein with singularities cusps, then we can easily ensure that the trichotomy of the uniformisation theorem for smooth curves (which for reasons of dimension is the same as having canonical singularities) is devoid of meaning if we only suppose Gorenstein.

(b)

The next step is to make a minimal model, i.e. either $S\rightarrow [S/\mathcal{F}]$ is a conic bundle or $K_{\mathcal{F}}$ nef. This can be done by blowing down chains of invariant rational curves. As a result to maintain an ambient smooth space, $S$, one needs to pass to the 2-category of champs de Deligne Mumford, albeit extremely simple ones. There is a related notion of canonical model, [M2], which leads to more precise results, but in the first instance we don't need it.

(c)

Now we need to measure the curvature of $\mathcal{F}$ invariant discs $f:\Delta\rightarrow S$. Basically the statement is,

$$f^*\mathrm{c}_1(K_{\mathcal{F}})\leq P_{\Delta}$$

where $P_{\Delta}$ is the Poincaré metric on the disc. However, the inequality should be understood as one of metricised line bundles on pointed (i.e. the Green's function is well defined, and one can take the degree in the sense of Nevanlinna theory) discs, so there are (necessary) subtleties about the proximity of the origin to the singularities and/or invariant curves through the same which are being glossed over, cf. [M4] III.2.2.

(d)

Let's say the goal is to prove that there is no Zariski dense invariant entire curve, albeit compactness of invariant discs with bubbles is the real theorem, and only marginally more difficult. Either way, should the goal be false, then by (c) we find an invariant measure $d\mu$ which must satisfy,

$$K_{\mathcal{F}}.d\mu\leq 0$$

Without loss of generality $d\mu$ has non-negative intersection with every effective Cartier divisor, so by (b),

$$K_{\mathcal{F}}.d\mu\geq 0$$

The Hodge index theorem then applies to conclude that $K_{\mathcal{F}}$ and $d\mu$ are parallel in $\mathrm{H}^2(S)$. Philosophically, this is the key difference between the Mori theory of foliations and that of varieties. In the latter case, when the situation is neither fano nor general type one invariably seeks a special representative of the canonical class in the form of an effective divisor. For foliations this is a known impossibility, e.g. poly-disc quotients, but, even if auto-duality in the middle dimension is obscuring the point on surfaces, one can find an invariant measure which is in duality with $K_{\mathcal{F}}$.

(e)

Now invariant measures do not grow on trees. They have very particular properties, amongst which is that their intersection with invariant bundles is governed by a residue at the singularities. Away from the singularities, transversals yield an étale covering of the champs classifiant, so $K_{[S/\mathcal{F}]}$ is such a bundle, and, of course,

$$K_S=K_{\mathcal{F}}\, +\, K_{[S/\mathcal{F}]}$$

An appropriate version of the residue lemma, albeit much stronger versions are available on surfaces, is:

$$\mathrm{s}_{Z,d\mu}=0\Longrightarrow K_{[S/\mathcal{F}]}.d\mu=0$$

where $\mathrm{s}_{Z,d\mu}$ is the Lelong number around the singular locus $Z$, while the notation $\mathrm{s}_*$ is to provide for its higher dimensional generalisation the Segre class, which is functorially, the ``usual'' Cauchy winding number around the singularities.

(f)

Finally we put everything together. To begin with suppose at least one of $K_{\mathcal{F}}^2$, $K_S.K_{\mathcal{F}}$ is non-zero. In the former case there cannot be an invariant entire curve by the curvature estimate (c). In the latter case without the former $K_{\mathcal{F}}.K_{[S/\mathcal{F}]}\neq 0$, so by (e) $\mathrm{s}_{Z,d\mu}\neq 0$. On a surface it follows from [S] that if the class of $d\mu$ is not big, then it is effective. If, however, it were big then $K_{\mathcal{F}}=0\in\mathrm{H}^2(S)$ and up to an étale cover the foliation is given by a global vector field. If it were effective, then on the canonical model it is parallel to an invariant rational or elliptic curve, $C$ with $C^2=0$, and the foliation is what is termed Ricatti or turbulent, i.e. transverse to a conic, respectively elliptic, bundle. Whence,

$$K_{\mathcal{F}}^2=K_S.K_{\mathcal{F}}=0$$

and Riemann-Roch shows that the only other foliations that can admit entire invariant curves not factoring through a fixed sub-variety are elliptic fibrations. Irrespectively, for $K_S$ big, the original context of [M1], it's pretty trivial that $K_S.K_{\mathcal{F}}\neq 0$ and that there is no rational or elliptic curve with $C^2=0$.

This discussion is close to optimal, even from the more limited perspective of the hyperbolicity of algebraic surfaces as is evidenced by nos. 10-13 of the F.A.Q.. It applies whenever the co-tangent bundle of the surface is big, e.g. $\mathrm{c}_1^2>\mathrm{c}_2$. As has been said, Green & Griffiths themselves, [GG], established the reduction of their conjecture for surfaces to the study of foliations by curves on higher dimensional varieties. The dimension in question being limited as a function of the ratio of $\mathrm{c}_1^2$ to $\mathrm{c}_2$. Close to the Castelnuovo line, the dimension is potentially so large as to have the same practical effect as being arbitrary. As such, we need to understand the above schema for a foliation $X\rightarrow [X/{\mathcal{F}}]$ by curves in arbitrary dimension, viz:

($\alpha$)

Lemma 1: Resolution of Singularities. This means exactly as for surfaces, i.e. relative canonicity of the arrow $X\rightarrow [X/{\mathcal{F}}]$. In practice this means as follows: whenever the foliation is Gorenstein it is defined by a vector field $\partial$. Should $\mathfrak{m}$ be the maximal ideal of a singularity in its local ring, then the Leibniz rule implies that,

$$\partial: \frac{\mathfrak{m}}{\mathfrak{m}^2} \rightarrow \frac{\mathfrak{m}}{\mathfrak{m}^2}$$

is $k(\mathfrak{m})$ linear, and it may be shown [M5] I.6.12, that this endomorphism is linear iff the singularity is log-canonical. For foliations by curves, the difference between canonical and log-canonical is slight, [MP1] III.ii, and it's easy to go from a log-canonical resolution to a canonical one. More importantly to maintain a smooth ambient, resolution can only be achieved, op. cit. III.iii, in the 2-category of champs de Deligne-Mumford. A similar phenomenon was already encountered in the minimal model theory of surfaces, and poses no real difficulty to the programme, albeit by $X$ one should understand from now on a smooth champ with projective moduli rather than a variety. In any case op. cit. proves resolution in dimension 3, and arbitrary dimension is progressing well.

($\beta$)

Lemma 2: Minimal Models. Just as for surfaces, albeit after contractions and flips, there is a dichotomy, either, $X\rightarrow [X/{\mathcal{F}}]$ is a conic bundle, or $K_{\mathcal{F}}$ is nef. Supposing that one starts from some, $X\rightarrow [X/{\mathcal{F}}]$ with canonical singularities this is done in all dimensions, [M5], and the final result of the algorithm is again a smooth champs de Deligne-Mumford. This may surprise experts in usual Mori theory, but, nowadays this should really be called Kawamata theory, whereas the original emphasis of Mori on rational curves is what works well for foliations by curves.

($\gamma$)

Lemma 3: Tautological Inequality. As per (c) for surfaces, this is something that can only make sense if $X\rightarrow [X/{\mathcal{F}}]$ has canonical singularities. Up to an additional small error, $\varepsilon\mathrm{c}_1(H)$, for $H$ ample, and excluding the invariant curve from factoring through a sub-variety $Z_{\varepsilon}$, $\varepsilon >0$, the statement is exactly as per (c), and is a theorem in all dimensions, [M5] V.5, or [M4] §II.

($\delta$)

Parenthesis: Duality. If one is only interested in the hyperbolicity of surfaces this step can be eschewed, but it would obscure the structure. Exactly as in (d) for surfaces, should there be a Zariski dense invariant entire curve, or indeed un-bounded discs not close to a sub-variety, one finds an invariant measure $d\mu$ meeting every effective Cartier divisor non-negatively, and $K_{\mathcal{F}}.d\mu=0$. By a theorem of Bogomolov, [B2], $K_{\mathcal{F}}$ is either big, or has numerical co-dimension 1. The former would imply the absurdity that $d\mu=0$, so we may suppose the latter. Quite generally, however, if $L$ is a nef. bundle on a variety of dimension $n$, of numerical dimension $n-1$, then the only class in $\mathrm{H}_2$ which meets every Cartier divisor non-negatively and $L$ in 0 is $L^{n-1}$, [M6] II.2. As such we may suppose that $K_{\mathcal{F}}\in\mathrm{H}^2(X)$ and $d\mu\in\mathrm{H}_2(X)$ are in duality.

($\epsilon$)

Lemma 4: The Residue Lemma. Exactly as per (e) the calculation of the intersection of the invariant bundle $K_{[X/\mathcal{F}]}$ with the invariant measure $d\mu$ localises at the singularities of the foliation, $Z$. However, this only means that there is a 1-form $\omega$ smooth off $Z$ such that,

$$K_{[X/\mathcal{F}]}.d\mu=\lim_{\varepsilon\rightarrow 0} \int_{\mathrm{dist}_Z=\varepsilon} \omega d\mu$$

It has a priori absolutely nothing to do with Cauchy residues, mixed Hodge theory, or, whatever. It is, rather, one of several invariants which control how far the term ``invariant measure'' may be un-ambiguously applied to $d\mu$ at the singularities, and the Residue lemma, the most important case of which is that encountered in (e), asserts that this is controlled by the simplest possible such invariant, $\mathrm{s}_{Z,d\mu}$, the Segre class/Cauchy winding number of $d\mu$ around $Z$. The matter is explained in detail in the introduction to [M7], which is the proof of the lemma in dimension 3. Philosophically, op. cit. is a huge advance on previous efforts to prove the residue lemma since it introduces a new idea, almost holonomy, which allows one to exploit the measure regularity (without which the residue lemma is wholly false) without knowing anything about the relation between the formal and analytic structure of the singularities, which is just as well, since already in dimension 3 this is far too complicated, [MP2], to have any relevance to this problem. Nevertheless, the application of this idea in [M7] makes extensive use of dimension 3, so, here, much more so than for resolution of singularities, there is still work to be done.

($\phi$)

Final Assembly: By Induction on dimension For $X$ of dimension $n$ the hypothesis of (f) become that at least one of $K_{\mathcal{F}}^n$, $K_{\mathcal{F}}^{n-1}.K_X$ is non-zero. These hypothesis are, of course, satisfied when the foliation comes from an order $n-1$ O.D.E. on an algebraic surface, and, as we've noted, the former trivially implies by ($\gamma$) that there cannot be a Zariski dense invariant entire curve, or families of unbounded invariant discs that are not close to an algebraic sub-variety. Thus we're in the latter case, and we reason exactly as per (f), but now by way of the higher dimensional ($\gamma$), and ($\epsilon)$) rather than (d) and (e), to deduce that the Segre class $s_{Z,d\mu}$ around the singular locus $Z$ is non-zero. Now, this is not an easy thing to happen for a closed positive current, $T$, let alone an invariant measure. Indeed quite generally, by way of counting sections of a small perturbation by an ample: if $L$ is nef., of numerical dimension $d$, $L.T=0$, and $\mathrm{s}_{Y,T}\neq 0$ for $Y$ of dimension less than $d$ then $T$ is supported in a countable union of proper algebraic sub-varieties. Given the minimal model theorem ($\beta$) this is as near to a reduction of dimension as makes no difference. For foliations coming from O.D.E.'s on an algebraic surface $S$, the details of the induction are in [M6], and we conclude that $d\mu$ pushes forward to a countable sum of rational and elliptic curves on $S$. So, certainly, the surface is hyperbolic iff it contains no rational or elliptic curves. To do better appears to require the construction of foliated flops, which, in one of life's ironies, is more difficult than the construction of flips in ($\beta$). A weak form of the termination of flops in dimension 3 is proved in [M6], and this allows one to conclude that the sum is finite. The impossibility of a finite sum, however, has been established in [M4] III.4, and whence for 2nd order O.D.E.'s on algebraic surfaces one has a serious of results which are best possible as explained in no. 1-5 of the F.A.Q..

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