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

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Aim of the lecture

Our goal is to study (nonconstant) entire curves f :C→X drawn in a projective variety/C. The varietyX 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→Xr∆ drawn in the complement of ∆.

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

J.-P. Demailly (Grenoble), Institut Fourier, July 1, 2019 Existence of logarithmic and orbifold jet differentials 2/24

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Aim of the lecture

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Aim of the lecture

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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 curvesf :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.

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Aim of the lecture (continued)

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.

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Aim of the lecture (continued)

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.

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k -jets of curves and k-jet bundles

Let X be a nonsingularn-dimensional projective variety overC.

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, ξs ∈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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k -jets of curves and k-jet bundles

Let X be a nonsingularn-dimensional projective variety overC. 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, ξs ∈Cⁿ, 1≤s ≤k.

Notation

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k -jets of curves and k-jet bundles

Let X be a nonsingularn-dimensional projective variety overC. 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, ξs ∈Cⁿ, 1≤s ≤k.

Notation

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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 pointt = 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_Xand putting ξ_s=f^(s)(0) =∇^sf(0), we get a trivialization J^kX '(TX)^⊕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=1s|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,f⁰⁰, . . . ,f^(k)),

P(f)(t) =P

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

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Algebraic differential operators

P(x; ξ₁, . . . , ξ_k) =P

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

s=1s|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,f⁰⁰, . . . ,f^(k)),

P(f)(t) =P

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Algebraic differential operators

P(x; ξ₁, . . . , ξ_k) =P

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

s=1s|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,f⁰⁰, . . . ,f^(k)),

P(f)(t) =P

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Algebraic differential operators

P(x; ξ₁, . . . , ξ_k) =P

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

s=1s|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,f⁰⁰, . . . ,f^(k)),

P(f)(t) =P

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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 onU, 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(f⁰, . . . ,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, ξk−1, ..., ξ₁, one gets a multi-filtration onE_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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Graded algebra of algebraic differential operators

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

By filtering by the partial degree of P(x;ξ₁, ..., ξ_k) successively in ξ_k, ξk−1, ..., ξ₁, one gets a multi-filtration onE_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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Graded algebra of algebraic differential operators

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

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

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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 top 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 Ek,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.

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Logarithmic jet differentials

=⇒ 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

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

G^•E_k_,m(X,∆) = M

`1+2`2+···+k`k=m

1 , ...,^dz_z^p

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

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Logarithmic jet differentials

=⇒ 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

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

G^•E_k_,m(X,∆) = M

`1+2`2+···+k`k=m

1 , ...,^dz_z^p

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

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Logarithmic jet differentials

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

1≤s≤k.

One gets a multi-filtration on Ek,m(X,∆) with graded pieces G^•E_k_,m(X,∆) = M

`1+2`2+···+k`k=m

1 , ...,^dz_z^p

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

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Logarithmic jet differentials

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

G^•E_k_,m(X,∆) = M

`1+2`2+···+k`k=m

1 , ...,^dz_z^p

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

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Orbifold jet differentials

Consider an orbifold (X,∆), ∆ = P (1− _ρ¹

j)∆_j a SNC divisor.

Assuming ∆₁ ={z₁ = 0}and f having multiplicityq ≥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− ^s_q)+. 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^pY^k

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 ≤X^k

s=1α_s,j 1− s

ρj

+

, 1≤j ≤p.

Definition

E_k,m(X,∆) is taken to be the algebra generated by monomials (∗) of degree Ps|α_s|=m, satisfying partial degree inequalities(∗∗).

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Orbifold jet differentials

Assuming ∆₁ ={z₁ = 0} and f having multiplicityq ≥ρ₁ >1 along ∆₁, then f₁^(s) still vanishes at order≥(q−s)₊, thus

(f1)^−βf₁^(s) is bounded as soon asβq ≤(q−s)+, i.e. β ≤(1− ^s_q)+. 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^pY^k

s=1α_s,j 1− s

ρj

+

, 1≤j ≤p.

Definition

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Orbifold jet differentials

s=1α_s,j 1− s

ρj

+

, 1≤j ≤p.

Definition

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Orbifold jet differentials

s=1α_s,j 1− s

ρj

+

, 1≤j ≤p.

Definition

E_k,m(X,∆) is taken to be the algebra generated by monomials(∗) of degree Ps|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)iis 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).

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Orbifold jet differentials [continued]

∆_j, then M

m

E_k,m(X,d∆e).

The subalgebra E_k,m(X,∆) still has a multi-filtration induced by the one on Ek,m(X,d∆e), and, at least for ρj ∈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)iis 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).

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Orbifold jet differentials [continued]

∆_j, then M

m

E_k,m(X,d∆e).

The subalgebra E_k,m(X,∆) still has a multi-filtration induced by the one on Ek,m(X,d∆e), and, at least for ρj ∈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)iis 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

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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)^kr{0} by theC^∗ 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_kh∆i(1) be the corresponding tautological sheaf, so that E_k,m(X,∆) = (π_k)∗O_X_kh∆i(m)

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Projectivized jets and direct image formula

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

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,∆)

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Projectivized jets and direct image formula

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

X_kh∆i:=ProjL

mE_k,m(X,∆)

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Projectivized jets and direct image formula

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

X_kh∆i:=ProjL

mE_k,m(X,∆)

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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 thatK_X + ∆ is a big R-divisor, then there is a proper algebraic subvarietyY (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;f⁰, ...,f^(k)) = 0 fork 1. This is based on:

Fundamental vanishing theorem

[Green-Griffiths 1979], [Demailly 1995], [Siu-Yeung 1996], ... Let Abe an ample divisor on X. Then, for all global jet differential operators on (X,∆) with coefficients vanishing on A, i.e.

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

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Generalized Green-Griffiths-Lang conjecture

[Green-Griffiths 1979], [Demailly 1995], [Siu-Yeung 1996], ... Let Abe an ample divisor on X. Then, for all global jet differential operators on (X,∆) with coefficients vanishing on A, i.e.

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

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Generalized Green-Griffiths-Lang conjecture

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

Let Abe 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.

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Proof of the fundamental vanishing theorem

Simple case. First consider the compact case (∆ = 0), and assume that f is a Brody curve, i.e.kf⁰k_ω bounded for some hermitian metric ω on X. By raising P to a power, we can assume A very ample, and viewP 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 Ais 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 f⁰ bounded is more tricky. One possible way is to use the Ahlfors lemma and some representation theory.

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Proof of the fundamental vanishing theorem

u_A(t) =P(f_[k])(t) is bounded, and must thus be constant by Liouville’s theorem.

Since Ais 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 f⁰ bounded is more tricky. One possible way is to use the Ahlfors lemma and some representation theory.

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Proof of the fundamental vanishing theorem

Since Ais 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 f⁰ bounded is more tricky. One possible way is to use the Ahlfors lemma and some representation theory.

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Proof of the fundamental vanishing theorem

Since Ais 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 f⁰ 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 asingular hermitian metric h =e^−ϕ with analytic singularities in Σ⊂X, and θ= _2πⁱ ΘL,h.

Let X(θ,q) :=

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

X(θ,≤q) = S

j≤qX(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 3x s.t. R

U|f|²e^−mϕdV <+∞ .

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Holomorphic Morse inequalities

Let L→X be a holomorphic line bundle on a compact complex manifold. Assume L equipped with asingular hermitian metric h =e^−ϕ with analytic singularities in Σ⊂X, and θ= _2πⁱ ΘL,h. Let

X(θ,q) :=

X(θ,≤q) = S

j≤qX(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 3x s.t. R

U|f|²e^−mϕdV <+∞ .

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Holomorphic Morse inequalities

Let L→X be a holomorphic line bundle on a compact complex manifold. Assume L equipped with asingular hermitian metric h =e^−ϕ with analytic singularities in Σ⊂X, and θ= _2πⁱ ΘL,h. Let

X(θ,q) :=

X(θ,≤q) = S

j≤qX(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),

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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).

(45)

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 onx.

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).