Holomorphic sections of line bundles vanishing along subvarieties - PowerPoint PPT Presentation

Holomorphic sections of line bundles vanishing along subvarieties Dan Coman, George Marinescu and Viˆ et-Anh Nguyˆ en Department of Mathematics, Syracuse University Syracuse, NY 13244-1150, USA 2019 Taipei Conference on Complex Geometry Institute of Mathematics, Academia Sinica, Taipei, Taiwan December 15 - 19, 2019 Coman, Marinescu, Nguyˆ en Sections vanishing along subvarieties 12/17/2019 1 / 28

Plan of the talk: 1. Preliminaries and notation 2. Dimension of spaces of holomorphic sections vanishing along subvarieties 3. Envelopes of quasiplurisubharmonic functions with poles along a divisor 4. Convergence of Fubini-Study currents 5. Zeros of random sequences of holomorphic sections Coman, Marinescu, Nguyˆ en Sections vanishing along subvarieties 12/17/2019 2 / 28

1. Preliminaries and notation Pluripotential theory on compact complex manifolds X compact complex manifold, dim X = n , ω Hermitian form on X d = ∂ + ∂ , d c = 2 π i ( ∂ − ∂ ) , dd c = i 1 π ∂∂ ν ( T , x ) = Lelong number of a positive closed current T on X at x ∈ X ϕ : X → R ∪ {−∞} is called quasiplurisubharmonic (qpsh) if ϕ = u + χ , near each x ∈ X , where u is plurisubharmonic (psh) and χ is smooth. If α is a smooth real closed (1 , 1)-form on X , we let PSH ( X , α ) = { ϕ : X → R ∪ {−∞} : ϕ qpsh, α + dd c ϕ ≥ 0 } . Coman, Marinescu, Nguyˆ en Sections vanishing along subvarieties 12/17/2019 3 / 28

ν ( ϕ, x ) := ν ( α + dd c ϕ, x ) = ν ( u , x ) Lelong number of ϕ at x ∈ X : { α } ∂∂ := ∂∂ -cohomology class of a smooth real closed (1 , 1)-form α on X H 1 , 1 ∂∂ ( X , R ) := {{ α } ∂∂ : α smooth real closed (1 , 1)-form on X } Since X is compact, H 1 , 1 ∂∂ ( X , R ) is finite dimensional. If X is a compact K¨ ahler manifold then, by the ∂∂ -lemma, H 1 , 1 ∂∂ ( X , R ) = H 1 , 1 ( X , R ) . Definition 1 A positive closed current T of bidegree (1 , 1) is called a K¨ ahler current if T ≥ εω , ε > 0. A class { α } ∂∂ is big if it contains a K¨ ahler current. Coman, Marinescu, Nguyˆ en Sections vanishing along subvarieties 12/17/2019 4 / 28

If { α } ∂∂ is big then, by Demailly’s regularization theorem, ∃ T ∈ { α } ∂∂ K¨ ahler current with analytic singularities , i.e. � | g j | 2 � N � T = α + dd c ϕ ≥ ǫω , where ǫ > 0, ϕ = c log + χ locally on X , j =1 with c > 0, χ a smooth function and g j holomorphic functions. Non-ample locus of { α } ∂∂ : � � � � � NAmp { α } ∂∂ = E + ( T ) : T ∈ { α } ∂∂ K¨ ahler current � � � = E + ( T ) : T ∈ { α } ∂∂ K¨ ahler current with analytic singularities , where E + ( T ) = { x ∈ X : ν ( T , x ) > 0 } . � � Hence NAmp { α } ∂∂ is an analytic subset of X . Boucksom: ∃ T ∈ { α } ∂∂ K¨ ahler current with analytic singularities such � � that E + ( T ) = NAmp { α } ∂∂ . Coman, Marinescu, Nguyˆ en Sections vanishing along subvarieties 12/17/2019 5 / 28

Plurisubharmonic (psh) functions and currents on analytic spaces Let X be an irreducible complex space, dim X = n X reg := set of regular points of X , X sing := set of singular points of X Call τ : U ⊂ X → G ⊂ C N a local embedding if U is open in X , G is open in C N , and τ : U → V is a biholomorphism, where V ⊂ G is a (closed) analytic subvariety. ϕ : X → [ −∞ , ∞ ) is (strictly) psh if for every x ∈ X there exist: τ : U x ⊂ X → G ⊂ C N local embedding, and ϕ : G → [ −∞ , ∞ ) (strictly) psh with ϕ | U x = � ϕ ◦ τ . � If � ϕ can be chosen continuous (resp. smooth), then ϕ is called a continuous (resp. smooth) psh function. Coman, Marinescu, Nguyˆ en Sections vanishing along subvarieties 12/17/2019 6 / 28

If τ : U ⊂ X → G ⊂ C N is a local embedding, then p , q ( G )) , where τ ∗ : Ω ∞ Ω ∞ p , q ( U ) := τ ∗ (Ω ∞ p , q ( G ) → Ω ∞ p , q ( U ∩ X reg ) . p , q ( X ) is the space of C ∞ smooth ( p , q )-forms with compact D p , q ( X ) ⊂ Ω ∞ support. The dual D ′ p , q ( X ) of D p , q ( X ) is the space of currents of bidimension ( p , q ), or bidegree ( n − p , n − q ), on X . T ( X ) ⊂ D ′ n − 1 , n − 1 ( X ) is the space of positive closed currents with local psh potentials: T ∈ T ( X ) if for every x ∈ X there exist U x ⊂ X (depending on T ) and v psh on U x such that T = dd c v on U x ∩ X reg . A K¨ ahler form on X is a current ω ∈ T ( X ) whose local potentials are smooth strictly psh functions. Coman, Marinescu, Nguyˆ en Sections vanishing along subvarieties 12/17/2019 7 / 28

Singular Hermitian holomorphic line bundles X compact, irreducible, normal complex space, dim X = n π : L − → X holomorphic line bundle on X : X = � U α , U α open, g αβ ∈ O ∗ X ( U α ∩ U β ) are the transition functions . H 0 ( X , L ) = space of global holomorphic sections of L , dim H 0 ( X , L ) < ∞ { ϕ α ∈ L 1 loc ( U α , ω n ) } α such that Singular Hermitian metric h on L : ϕ α = ϕ β + log | g αβ | on U α ∩ U β , | e α | h = e − ϕ α ( e α local frame on U α ). The curvature current c 1 ( L , h ) ∈ D ′ n − 1 , n − 1 ( X ) of h : c 1 ( L , h ) = dd c ϕ α on U α ∩ X reg . If c 1 ( L , h ) ≥ 0 then ϕ α is psh on U α ∩ X reg , hence on U α ( X is normal). So c 1 ( L , h ) ∈ T ( X ) . Coman, Marinescu, Nguyˆ en Sections vanishing along subvarieties 12/17/2019 8 / 28

2. Dimension of spaces of holomorphic sections vanishing along subvarieties X compact complex manifold, dim X = n , L → X holomorphic line bundle L p := L ⊗ p , H 0 ( X , L p ) = space of global holomorphic sections of L p Siegel’s Lemma: ∃ C > 0 such that dim H 0 ( X , L p ) ≤ Cp n for all p ≥ 1 . 1 p n dim H 0 ( X , L p ) > 0. A line bundle L is called big if lim sup p →∞ If L is big one can show that ∃ c > 0, p 0 ≥ 1, such that dim H 0 ( X , L p ) ≥ cp n for all p ≥ p 0 . L is big if and only if ∃ h singular Hermitian metric on L Ji-Shiffman: such that c 1 ( L , h ) is a K¨ ahler current. Coman, Marinescu, Nguyˆ en Sections vanishing along subvarieties 12/17/2019 9 / 28

Setting: (A) X is a compact, irreducible, normal complex space, dim X = n , X reg is the set of regular points of X , X sing is the set of singular points of X . (B) L is a holomorphic line bundle on X . (C) Σ = (Σ 1 , . . . , Σ ℓ ), Σ j �⊂ X sing , are distinct irreducible proper analytic subsets of X . (D) τ = ( τ 1 , . . . , τ ℓ ), τ j ∈ (0 , + ∞ ), and τ j > τ k if Σ j ⊂ Σ k . Coman, Marinescu, Nguyˆ en Sections vanishing along subvarieties 12/17/2019 10 / 28

Let: � if τ j p ∈ N τ j p t j , p = if τ j p �∈ N , 1 ≤ j ≤ ℓ , p ≥ 1 . ⌊ τ j p ⌋ + 1 � � H 0 0 ( X , L p ) = H 0 S ∈ H 0 ( X , L p ) : ord( S , Σ j ) ≥ t j , p 0 ( X , L p , Σ , τ ) := So S ∈ H 0 0 ( X , L p ) vanishes to order at least τ j p along Σ j , 1 ≤ j ≤ ℓ . Definition 2 1 p n dim H 0 0 ( X , L p ) > 0. We say that the triplet ( L , Σ , τ ) is big if lim sup p →∞ Coman, Marinescu, Nguyˆ en Sections vanishing along subvarieties 12/17/2019 11 / 28

Characterization when X is a complex manifold and dim Σ j = n − 1: Theorem 3 Let: X compact complex manifold, dim X = n, L holomorphic line bundle on X, Σ = (Σ 1 , . . . , Σ ℓ ) , τ = ( τ 1 , . . . , τ ℓ ) , where Σ j are distinct irreducible complex hypersurfaces in X, τ j ∈ (0 , + ∞ ) . The following are equivalent: (i) ( L , Σ , τ ) is big; (ii) There exists a singular Hermitian metric h on L such that ℓ � c 1 ( L , h ) − τ j [Σ j ] j =1 is a K¨ ahler current on X, where [Σ j ] is the current of integration along Σ j ; 0 ( X , L p ) ≥ cp n for all p ≥ p 0 . (iii) ∃ c > 0 , p 0 ≥ 1 , such that dim H 0 Coman, Marinescu, Nguyˆ en Sections vanishing along subvarieties 12/17/2019 12 / 28

Proposition 4 Let X, Σ verify (A), (C). Then there exist a compact complex manifold � X, dim � X = n, and a surjective holomorphic map π : � X → X, given as the composition of finitely many blow-ups with smooth center, such that: (i) ∃ Y ⊂ X analytic subset such that dim Y ≤ n − 2 , X sing ⊂ Y , Σ j ⊂ Y if dim Σ j ≤ n − 2 , Y ⊂ X sing ∪ � ℓ j =1 Σ j , E = π − 1 ( Y ) is a divisor in � X that has only normal crossings, and π : � X \ E → X \ Y is a biholomorphism. (ii) There exist smooth complex hypersurfaces � Σ 1 , . . . , � Σ ℓ in � X such that π ( � Σ j ) = Σ j . If dim Σ j = n − 1 then � Σ j is the final strict transform of Σ j , and if dim Σ j ≤ n − 2 then � Σ j is an irreducible component of E. (iii) If F → X is a holomorphic line bundle and S ∈ H 0 ( X , F ) then ord( S , Σ j ) = ord( π ⋆ S , � Σ j ) , for all j = 1 , . . . , ℓ . Coman, Marinescu, Nguyˆ en Sections vanishing along subvarieties 12/17/2019 13 / 28