Existence of logarithmic and orbifold jet differentials

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Existence of logarithmic and orbifold jet differentials

Jean-Pierre Demailly

Institut Fourier, Universit´e Grenoble Alpes & Acad´emie des Sciences de Paris

Joint work with F. Campana, L. Darondeau and E. Rousseau Summer School in Mathematics 2019

Foliations and Algebraic Geometry Institut Fourier, Universit´e Grenoble Alpes

June 17 – July 5, 2019

J.-P. Demailly (Grenoble), Institut Fourier, July 1, 2019 Existence of logarithmic and orbifold jet differentials 1/24

Aim of the lecture

Our goal is to study (nonconstant) entire curves f : C → X drawn in a projective variety/C. The variety X is said to be Brody (⇔

Kobayashi) hyperbolic if there are no such curves.

More generally, if ∆ = P

∆_j is a reduced normal crossing divisor in X, we want to study entire curves f : C → X r∆ drawn in the complement of ∆.

If there are no such curves, we say that the log pair (X,∆) is Brody hyperbolic.

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Aim of the lecture (continued)

Even more generally, if ∆ = P

(1− _ρ¹

j)∆_j ⊂ X is a normal crossing divisor, we want to study entire curves f : C → X meeting each component ∆_j of ∆ with multiplicity ≥ ρ_j.

The pair (X,∆) is called an orbifold (in the sense of Campana).

Here ρ_j ∈ ]1,∞], where ρ_j = ∞ corresponds to the logarithmic case. Usually ρ_j ∈ {2,3, ...,∞}, but ρ_j ∈ R^>1 will be allowed.

The strategy is to show that under suitable conditions, orbifold entire curves must satisfy algebraic differential equations.

k -jets of curves and k -jet bundles

Let X be a nonsingular n-dimensional projective variety over C. Definition of k-jets

For k ∈ N^∗, a k-jet of curve f_[k_] : (C,0)_k → X is an equivalence class of germs of holomorphic curves f : (C,0) → X, written f = (f₁, . . . ,f_n) in local coordinates (z₁, . . . ,z_n) on an open subset U ⊂ X, where two germs are declared to be equivalent if they have the same Taylor expansion of order k at 0 :

f (t) = x + tξ₁ +t²ξ₂ +· · ·+ t^kξ_k + O(t^k+1), t ∈ D(0, ε) ⊂ C, and x = f (0) ∈ U, ξ_s ∈ Cⁿ, 1 ≤ s ≤ k.

Notation

Let J^kX be the bundle of k-jets of curves, and π_k : J^kX → X the natural projection, where the fiber (J^kX)_x = π_k⁻¹(x) consists of k-jets of curves f_[k] such that f(0) = x.

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Algebraic differential operators

Let t 7→ z = f (t) be a germ of curve, f_[k_] = (f⁰,f⁰⁰, . . . ,f ^(k)) its k-jet at any point t = 0. Look at the C^∗-action induced by dilations

λ·f(t) := f(λt), λ ∈ C^∗, for f_[k_] ∈ J^kX.

Taking a (local) connection∇onT_X and putting ξ_s =f^(s)(0) =∇^sf (0), we get a trivialization J^kX ' (T_X)^⊕k and the C^∗ action is given by (∗) λ·(ξ₁, ξ₂, . . . , ξ_k) = (λξ₁, λ²ξ₂, . . . , λ^kξ_k).

We consider the Green-Griffiths sheaf E_k,m(X) of homogeneous polynomials of weighted degree m on J^kX defined by

P(x ; ξ₁, . . . , ξ_k) = P

a_α₁_α₂_...α_k(x)ξ₁^α¹ . . . ξ_k^α^k, Pk

s=1 s|α_s| = m.

Here, we assume the coefficients a_α₁_α₂_...α_k(x) to be holomorphic in x, and view P as a differential operator P(f) = P(f ; f⁰,f ⁰⁰, . . . ,f ^(k)),

P(f )(t) = P

a_α₁_α₂_...α_k(f(t)) f⁰(t)^α¹f⁰⁰(t)^α² . . .f^(k⁾(t)^α^k.

Graded algebra of algebraic differential operators

In this way, we get a graded algebra L

mE_k,m(X) of differential operators. As sheaf of rings, in each coordinate chart U ⊂ X, it is a pure polynomial algebra isomorphic to

O_X[f_j^(s)]_1≤j≤n,_1≤s≤k where degf_j^(s) = s.

If a change of coordinates z 7→ w = ψ(z) is performed on U, the curve t 7→ f(t) becomes t 7→ ψ ◦f (t) and we have inductively

(ψ ◦f)^(s⁾ = (ψ⁰ ◦f )·f^(s⁾ + Q_ψ,s(f ⁰, . . . ,f ^(s−1)) where Q_ψ,s is a polynomial of weighted degree s.

By filtering by the partial degree of P(x;ξ₁, ..., ξ_k) successively in ξ_k, ξ_k₋₁, ..., ξ₁, one gets a multi-filtration on E_k_,m(X) such that the graded pieces are

G^•E_k,m(X) = M

`1+2`2+···+k`_k=m

S^`¹T_X^∗ ⊗ · · · ⊗S^`^kT_X^∗.

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Logarithmic jet differentials

Take a logarithmic pair (X,∆), ∆ = P

∆_j normal crossing divisor.

Fix a point x ∈ X which belongs exactly to p components, say

∆₁, ...,∆_p, and take coordinates (z₁, ...,z_n) so that ∆_j = {z_j = 0}.

=⇒ log differential operators : polynomials in the derivatives (logf_j)^(s), 1 ≤ j ≤ p and f_j^(s), p + 1 ≤ j ≤ n.

Alternatively, one gets an algebra of logarithmic jet differentials, denoted L

mE_k_,m(X,∆), that can be expressed locally as O_X

(f₁)⁻¹f₁^(s), ...,(f_p)⁻¹f_p^(s),f_p+1^(s⁾ , ...,f_n^(s)

1≤s≤k. One gets a multi-filtration on E_k,m(X,∆) with graded pieces

G^•E_k_,m(X,∆) = M

`1+2`2+···+k`_k=m

S^`¹T_X^∗h∆i ⊗ · · · ⊗S^`^kT_X^∗h∆i where T_X^∗h∆i is the logarithmic tangent bundle, i.e., the locally free sheaf generated by ^dz_z¹

1 , ..., ^dz_z^p

p ,dz_p+1, ...,dz_n.

Orbifold jet differentials

Consider an orbifold (X,∆), ∆ = P

(1− _ρ¹

j)∆_j a SNC divisor.

Assuming ∆₁ = {z₁ = 0} and f having multiplicity q ≥ ρ₁ > 1

along ∆₁, then f₁^(s) still vanishes at order ≥ (q − s)₊, thus (f₁)^−βf₁^(s) is bounded as soon as βq ≤ (q −s)+, i.e. β ≤ (1− _q^s)+. Thus, it is

sufficient to ask that β ≤ (1 − _ρ^s

1)₊. At a point x ∈ |∆₁| ∩...∩ |∆_p|, the condition for a monomial of the form

(∗) f₁^−β¹...f_p^−β^p Yk

s=1(f ^(s))^α^s, (f^(s⁾)^α^s = (f₁^(s⁾)^α^s,1...(f_n^(s))^α^s,n, α_s ∈ Nⁿ, β₁, ..., β_p ∈ N, to be bounded, is to require that

(∗∗) β_j ≤ Xk

s=1α_s,j

1− s ρ_j

+, 1 ≤ j ≤ p.

Definition

E_k_,m(X,∆) is taken to be the algebra generated by monomials (∗) of degree P

s|α_s| = m, satisfying partial degree inequalities (∗∗).

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Orbifold jet differentials [continued]

It is important to notice that if we consider the log pair (X,d∆e) with d∆e = P

∆_j, then M

m

E_k,m(X,∆) is a graded subalgebra of M

m

E_k_,m(X,d∆e).

The subalgebra E_k_,m(X,∆) still has a multi-filtration induced by the one on E_k_,m(X,d∆e), and, at least for ρ_j ∈ Q, we formally have

G^•E_k,m(X,∆) ⊂ M

`1+2`2+···+k`_k=m

S^`¹T_X^∗h∆⁽¹⁾i ⊗ · · · ⊗S^`^kT_X^∗h∆^(k)i,

where T_X^∗h∆^(s)i is the “s-th orbifold cotangent sheaf” generated by z_j^−(1−s^/ρ^j⁾⁺d^(s)z_j, 1 ≤ j ≤ p, d^(s⁾z_j, p + 1 ≤ j ≤ n

(which makes sense only after taking some Galois cover of X ramifying at sufficiently large order along ∆_j).

Projectivized jets and direct image formula

Green Griffiths bundles

Consider X_k := J^kX/C^∗ = ProjL

mE_k,m(X). This defines a bundle π_k : X_k → X of weighted projective spaces whose fibers are the

quotients of (Cⁿ)^k r{0} by the C^∗ action

λ·(ξ₁, . . . , ξ_k) = (λξ₁, λ²ξ₂, . . . , λ^kξ_k).

Correspondingly, there is a tautological rank 1 sheaf O_X_k(m) [only invertible when lcm(1, ...,k) | m], and a direct image formula

E_k_,m(X) = (π_k)_∗O_X_k(m) In the logarithmic case, we define similarly

X_kh∆i := ProjL

mE_k,m(X,∆)

and let O_X_k_h∆i(1) be the corresponding tautological sheaf, so that E_k,m(X,∆) = (π_k)_∗O_X_k_h∆i(m)

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Generalized Green-Griffiths-Lang conjecture

Generalized GGL conjecture (very optimistic ?)

If (X,∆) is an orbifold of general type, in the sense that K_X + ∆ is a big R-divisor, then there is a proper algebraic subvariety Y ( X

containing all orbifold entire curves f : C → (X,∆) (not contained in ∆ and having multiplicity ≥ ρ_j along ∆_j).

One possible strategy is to show that such orbifold entire curves f must satisfy a lot of algebraic differential equations of the form

P(f ;f ⁰, ...,f ^(k)) = 0 for k 1. This is based on:

Fundamental vanishing theorem

[Green-Griffiths 1979], [Demailly 1995], [Siu-Yeung 1996], ...

Let A be an ample divisor on X. Then, for all global jet differential operators on (X,∆) with coefficients vanishing on A, i.e.

P ∈ H⁰(X,E_k,m(X,∆)⊗ O(−A)), and for all orbifold entire curves f : C → (X,∆), one has P(f_[k_]) ≡ 0.

Proof of the fundamental vanishing theorem

Simple case. First consider the compact case (∆ = 0), and assume that f is a Brody curve, i.e. kf ⁰k_ω bounded for some hermitian metric ω on X. By raising P to a power, we can assume A very ample, and view P as a C valued differential operator whose coefficients vanish on a very ample divisor A.

The Cauchy inequalities imply that all derivatives f^(s⁾ are bounded in any relatively compact coordinate chart. Hence u_A(t) = P(f_[k])(t) is bounded, and must thus be

constant by Liouville’s theorem.

Since A is very ample, we can move A ∈ |A| such that A hits f(C) ⊂ X. But then u_A vanishes somewhere, and so u_A ≡ 0.

Logarithmic and orbifold cases. In the orbifold case, one must use instead an “orbifold metric” ω. Removing the hypothesis f ⁰ bounded is more tricky. One possible way is to use the Ahlfors lemma and some representation theory.

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Holomorphic Morse inequalities

Theorem (D, 1985, L. Bonavero 1996)

Let L → X be a holomorphic line bundle on a compact complex

manifold. Assume L equipped with a singular hermitian metric h = e^−ϕ with analytic singularities in Σ ⊂ X, and θ = _2πⁱ Θ_L,h. Let

X(θ,q) :=

x ∈ X r Σ ; θ(x) has signature (n −q,q) be the q-index set of the (1,1)-form θ, and

X(θ,≤ q) = S

j≤q X(θ,j).

Then

q

X

j=0

(−1)^q−jh^j(X,L^⊗m ⊗ I(mϕ)) ≤ mⁿ n!

Z

X(θ,≤q)

(−1)^qθⁿ +o(mⁿ),

where I(mϕ) ⊂ O_X denotes the multiplier ideal sheaf I(mϕ)_x =

f ∈ O_X_,x ; ∃U 3 x s.t. R

U |f|²e^−mϕdV < +∞ .

Holomorphic Morse inequalities [continued]

Consequence of the holomorphic Morse inequalities

For q= 1, with the same notation as above, we get a lower bound h⁰(X,L^⊗m) ≥ h⁰(x,L^⊗m ⊗ I(mϕ))

≥ h⁰(x,L^⊗m ⊗ I(mϕ))− h¹(x,L^⊗m ⊗ I(mϕ))

≥ mⁿ n!

Z

X(θ,≤1)

θⁿ − o(mⁿ).

here θ is a real (1,1) form of arbitrary signature on x.

when θ = α −β for some explicit (1,1)-forms α, β ≥ 0 (not necessarily closed), an easy lemma yields

1_X_{(α−β,≤1)} (α −β)ⁿ ≥ αⁿ −nαⁿ⁻¹ ∧β hence

h⁰(X,L^⊗m) ≥ mⁿ n!

Z

X

(αⁿ −nαⁿ⁻¹ ∧β) −o(mⁿ).

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Finsler metric on the k -jet bundles

Assume that T_X is equipped with a C^∞ connection ∇ and a hermitian metric h. One then defines a ”weighted Finsler metric” on J^kX by taking b = lcm(1,2, ...,k) and, at each point x =f (0),

Ψ_h_k(f_[k]) :=

X

1≤s≤k

ε_sk∇^sf(0)k^2b/s_h(x₎1/b

, 1 = ε₁ ε₂ · · · ε_k. Letting ξ_s =∇^sf(0), this can be viewed as a metric h_k onL_k :=O_X_k(1), and the curvature form of L_k is obtained by computing

i

2π∂∂log Ψ_h_k(f_[k_]) as a function of (x, ξ₁, . . . , ξ_k).

Modulo negligible error terms of the form O(ε_s₊₁/ε_s), this gives Θ_L_k_,h_k = ω_FS,k(ξ) + i

2π

X

1≤s≤k

1 s

|ξ_s|^2b/s P

t |ξ_t|^2b/t

X

i,j,α,β

c_ij_αβξ_s_αξ_sβ

|ξ_s|² dz_i ∧d z_j where (c_ij_αβ) are the coefficients of the curvature tensor Θ_T^∗

X,h^∗ andω_FS,k is the weighted Fubini-Study metric on the fibers of X_k → X.

Evaluation of Morse integrals

The above expression is simplified by using polar coordinates x_s = |ξ_s|^2b/s_h , u_s = ξ_s/|ξ_s|_h = ∇^sf (0)/|∇^sf (0)|.

In such polar coordinates, one gets the formula Θ_L_k_,h_k = ω_FS,k(ξ) + i

2π

X

1≤s≤k

1

sx_s X

i,j,α,β

c_ij_αβ(z)u_s_αu_sβ dz_i ∧d z_j

where ω_FS,k(ξ) is positive definite in ξ.

By holomorphic Morse inequalities, we need to evaluate an integral Z

X_k(Θ_Lh_,_hk,≤1)

Θ^N_L^k

k,h_k, N_k = dimX_k = n + (kn − 1),

and we have to integrate over the parameters z ∈ X, x_s ∈ R+ and u_s in the unit sphere bundle S(T_X,1) ⊂ T_X.

Since the weighted projective space can be viewed as a circle quotient of the pseudosphere P

|ξ_s|^2b/s = 1, we can take here P

x_s = 1, i.e. (x_s) in the (k − 1)-dimensional simplex ∆^/ ^k−1.

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Probabilistic interpretation of the curvature

Now, the signature of Θ_L_k_,h_k depends only on the vertical terms, i.e.

X

1≤s≤k

1

sx_sq(u_s), q(u_s) := i 2π

X

i,j,α,β

c_ij_αβ(z)u_sαu_s_β dz_i ∧d z_j. After averaging over (x_s) ∈ ∆^/ ^k−1 and computing the rational number R ω_FS,k(ξ)^nk⁻¹ = _(k!)¹ n, what is left is to evaluate Morse integrals with respect to (u_s) of “horizontal” (1,1)-forms given by sums P ₁

sq(u_s), where u_s are “random points” on the unit sphere.

As k→+∞, this sum yields asymptotically a “Monte-Carlo” integral

1 + 1

2 +· · ·+ 1 k

Z

u∈S(T_X,1)

q(u)du.

Since q is quadratic in u, we have Z

u∈S(T_X,1)

q(u)du = 1

n Tr(q) and Tr(q) = Tr(Θ_T^∗

X,h^∗) = Θ_det_T^∗

X,deth^∗ = Θ_K_X_,det_h^∗.

Probabilistic cohomology estimate

Theorem 1 (D-, Pure and Applied Math. Quarterly 2011)

Fix A ample line bundle on X, (T_X,h), (A,h_A) hermitian structures on T_X, A, and ω_A = Θ_A,h_A > 0. Let η_ε = Θ_K_X_,det_h^∗ − εω_A and

L_k = O_X_k(1) ⊗π_k^∗O_X

− 1 kn

1 + 1

2 + · · ·+ 1 k

εA

, ε ∈ Q⁺. Then for m sufficiently divisible, we have a lower bound

h⁰(X_k,L^⊗m_k ) = h⁰

X,E_k,m(X) ⊗ O_X

− mε kn

1 +1

2 + . . . + 1 k

A

≥ m^n+kn−1 (n + kr − 1)!

(logk)ⁿ n! (k!)ⁿ

Z

X(η,≤1)

η_εⁿ − C logk

.

Corollary

If K_X is big and ε > 0 is small, then ηε can be taken > 0, so

h⁰(X_k,L^⊗m_k ) ≥ C_n,k_,η,εm^n+kn−1 with C_n,k,η,ε > 0, for m k 1.

There are in fact similar upper/lower bounds for all h^q(X_k,L^⊗m_k ).

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Non probabilistic cohomology estimate

The Monte-Carlo estimate can be replaced by a non probabilistic one, if one assumes an explicit lower bound for the curvature tensor

Θ_T^∗

X,h^∗ ≥ −γ ⊗ Id, where γ ≥ 0 is a smooth (1,1)-form on X.

In case X ⊂ P^N and A = O(1), one can always take γ = 2ω_A where ω_A = Θ_A,h_A > 0.

By Morse inequalities for differences 1_X_{(α−β,≤1)} (α− β)ⁿ, one gets Theorem 2 (D-, Acta Math. Vietnamica 2012)

Assume k ≥ n and m 1. With the same notation as in Theorem 1, the dimensions h⁰(X_k,L^⊗m_k ) are bounded below by

m^n+kn−1 n!k!ⁿ(n +kn −1)!

Z

X

Θ_K_X +nγn

−c_n,k Θ_K_X +nγn−1

∧(εω_A+nγ), with c_n,k ∈ Q^>0 explicit, c_n,k ≤ 4ⁿ⁻¹n! 1 + ¹₂ +· · ·+ _k¹n

.

Logarithmic situation

In the case of a log pair (X,∆), one reproduce essentially the same calculations, by replacing the cotangent bundle T_X^∗ with the logarithmic cotangent bundle T_X^∗h∆i. This gives

Theorem 3 (probabilistic estimate)

Put η_ε = Θ_K_X_+∆,det_h^∗ − εω_A. For m k 1, the dimensions h⁰ X,E_k_,m(X,∆) ⊗ O_X − ^mε_kn 1 + ¹₂ +· · ·+ _k¹

A are bounded below by

m^n+kn−1 (n + kr − 1)!

(logk)ⁿ n! (k!)ⁿ

Z

X(η,≤1)

ηⁿ_ε − C logk

, C > 0.

Theorem 4 (non probabilistic estimate) Assume Θ_T^∗

Xh∆i ≥ −γ ⊗ Id. For k ≥ n,m 1, there are bounds m^n+kn−1

n!k!ⁿ(n + kn− 1)!

Z

X

Θ_K_X_+∆+nγn

−c_n,k Θ_K_X_+∆+nγn−1

∧(εω_A+nγ).

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Orbifold situation

Consider now the orbifold case (X,∆), ∆ = P

(1− _ρ¹

j)∆_j.

In this case, the solution is to work on the logarithmic projectivized jet bundle X_khd∆ei, with Finsler metrics Ψ_h_k(f_[k]) of the form

X

1≤s≤k

ε_s

p

X

j=1

|f_j|⁻²⁽¹⁻

s ρj)+

|f_j^(s)(0)|² +

n

X

j=p+1

|f_j^(s⁾(0)|²

!b/s

hs(f(0))

!1/b

,

where h_s is a hermitian metric on the s-th orbifold bundle T_X^∗h∆^(s)i.

Theorem 5 (non probabilistic estimate [probabilistic doesn’t work]) Assume Θ_T^∗

Xh∆^(s)i ≥ −γ_sω ⊗ Id in the sense of Griffiths, with ω = Θ_A (A ample), γs ≥ 0, and let Θs = Θ_K

X+∆^(s) for s = 1, ...,k. Then, for k ≥ n and m 1, h⁰ X,E_k,m(X,∆)⊗ O_X(−mεA)

≥ m^n+kn−1

n!(k!)ⁿ(n + kn− 1)!

Z

X

^ⁿ

s=1 Θ_s + nγ_sω

− (2n −1)!

(n −1)!² × X^k

s=1

γ_s s

X^k

s=1

1

s(Θ_s + nγ_sω) n−1

∧ω − O(ε)

.

Application to projective space

Consider Pⁿ equipped with an orbifold divisor ∆ = PN

j=1(1− _ρ¹

j)∆_j. Lemma: lower bound on the curvature of the cotangent bundle

Put A = O_Pⁿ(1), d_j = deg ∆_j and γ₀ = max ^d_ρ^j

j,2

. Then ∀γ > γ₀, there exists a suitable hermitian metric on T^∗

Pⁿh∆i such that Θ_T^∗

Pnh∆i + γ ω_A ⊗ Id > 0 (in the sense of Griffiths).

Corollary: sufficient condition of existence of orbifold differentials A sufficient condition for the existence of negatively twisted orbifold order k = n jet differentials on Pⁿh∆i is

ρ_j ≥ ρ > n, X^N

j=1 d_j ≥ c_n max d_j

ρ_j,2 ⁿ

Y

s=1

1− s ρ

−1

. with c_n = O((2nlogn)ⁿ) an explicit constant.

Example: N = 1, ρ₁ ≥ 2c_n, d₁ ≥ 4c_n.

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Generalization: case of orbifold directed varieties

One can also consider a smooth directed variety (X,V) with a subbundle or subsheaf V ⊂ T_X (e.g. a foliation), equipped with an orbifold divisor ∆ transverse to V.

One then looks at entire curves f : C → X that are tangent to V and satisfy

the ramification conditions specified by ∆.

It is possible to define orbifold directed structures Vh∆^(s⁾i ⊂T_Xh∆^(s)i and corresponding jet differential bundles E_k,m(X,V,∆).

Theorem 6

An existence criterion for sections of E_k_,m(X,V,∆) holds as well.

Existence of logarithmic and orbifold jet differentials