General extension theorem for cohomology classes on non reduced analytic subspaces

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General extension theorem for cohomology classes on non reduced analytic subspaces

Jean-Pierre Demailly

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

Trends in Modern Geometry 2017 Graduate School of Mathematical Sciences The University of Tokyo, July 10–13, 2017

J.-P. Demailly (Grenoble), TIMG, Tokyo Univ., July 12, 2017 Extension of cohomology classes on nonreduced subspaces 1/21

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References

This is a joint work with Junyan Cao & Shin-ichi Matsumura J.-P. Demailly, Extension of holomorphic functions defined on non reduced analytic subvarieties, arXiv:1510.05230v1, Advanced Lectures in Mathematics Volume 35.1, the legacy of Bernhard Riemann after one hundred and fifty years, 2015.

J.Y. Cao, J.-P. Demailly, S-i. Matsumura, A general extension theorem for cohomology classes on non reduced analytic subspaces, arXiv:1703.00292v2, Science China Mathematics, 60(2017)

949–962

T. Ohsawa, K. Takegoshi, On the extension of L²holomorphic functions, Math. Zeitschrift 195 (1987), 197–204.

T. Ohsawa series of papers I – VI, and : On a curvature condition that implies a cohomology injectivity theorem of Koll´ar-Skoda type, Publ. Res. Inst. Math. Sci. 41 (2005), no. 3, 565–577.

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The extension problem

Let (X, ω) be a possibly noncompact n-dimensional K¨ahler manifold, J ⊂ O_X a coherent ideal sheaf, Y =V(J) its zero variety and

O_Y =O_X/J.

Here Y may be non reduced, i.e. O_Y may have nilpotent elements.

Also, let (L,h_L) be a hermitian holomorphic line bundle on X, and Θ_L,h_L =i∂∂logh⁻¹_L

its curvature current (we allow singular metrics, h_L =e^−ϕ, ϕ∈L¹_loc, Θ_L,h_L being computed in the sense of currents). Question

Under which conditions on X, Y =V(J), (L,h_L) is

H^q(X,K_X⊗L)→H^q(Y,(K_X⊗L)_|Y) =H^q(X,O_X(K_X⊗L)⊗O_X/J) a surjective restriction morphism?

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The extension problem

Let (X, ω) be a possibly noncompact n-dimensional K¨ahler manifold, J ⊂ O_X a coherent ideal sheaf, Y =V(J) its zero variety and

O_Y =O_X/J.

Here Y may be non reduced, i.e. O_Y may have nilpotent elements.

Also, let (L,h_L) be a hermitian holomorphic line bundle on X, and Θ_L,h_L =i∂∂logh⁻¹_L

its curvature current (we allow singular metrics, h_L =e^−ϕ, ϕ∈L¹_loc, Θ_L,h_L being computed in the sense of currents).

Question

Under which conditions on X, Y =V(J), (L,h_L) is

H^q(X,K_X⊗L)→H^q(Y,(K_X⊗L)_|Y) =H^q(X,O_X(K_X⊗L)⊗O_X/J) a surjective restriction morphism?

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Motivation: abundance conjecture and MMP

One potential application would be to study the Minimal Model Program (MMP) for arbitrary projective – or even K¨ahler – varieties, whereas only the case of general type varieties is known.

For a line bundle L, one defines the Kodaira-Iitaka dimension κ(L) = lim sup_m→+∞log dimH⁰(X,L^⊗m)/logm and the numerical dimension nd(L) = maximum exponentp of non zero “positive intersections” hT^pi of a positive current T ∈c₁(L) when L is psef (pseudoeffective), and nd(L) =−∞ otherwise. They always satisfy

−∞ ≤κ(L)≤nd(L)≤n= dimX.

Definition (abundance)

A line bundle L is said to be abundant if κ(L) = nd(L).

The fundamental abundance conjecture can be stated: for each nonsingular klt pair (X,∆) the Q-line bundleKX + ∆ is abundant.

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Motivation: abundance conjecture and MMP

For a line bundle L, one defines the Kodaira-Iitaka dimension κ(L) = lim sup_m→+∞log dimH⁰(X,L^⊗m)/logm and the numerical dimension nd(L) = maximum exponent p of non zero “positive intersections” hT^pi of a positive current T ∈c₁(L) when L is psef (pseudoeffective), and nd(L) =−∞ otherwise. They always satisfy

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Motivation: abundance conjecture and MMP

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Motivation: abundance conjecture and MMP

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Generalized base point free theorem ?

One can try to investigate the abundance of L=K_X + ∆ by induction on the dimension n = dimX, by extending sections of K_X +L_m, L_m = (m−1)K_X +m∆ from subvarieties (noticing that K_X + ∆ psef implies L_m psef, and even L_m−∆ psef).

Cf. BCHM and recent work of Demailly-Hacon-P˘aun, Fujino, Gongyo, Takayama. Standard base point free theorem

Let (X,∆) be a projective klt pair, andL be a nef line bundle such that L−(K_X + ∆) is nef and big. Then L is semiample, i.e. |mL| is BPF for some m >0.

Question (weak positivity variant of the BPF property ?)

Assume that X is not uniruled, i.e. that K_X is pseudoeffective, and let L be a line bundle such thatL−εK_X is pseudoeffectivefor some 0< ε1. Does there existG ∈Pic⁰(X) such thatL+G is abundant ?

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Generalized base point free theorem ?

One can try to investigate the abundance of L=K_X + ∆ by induction on the dimension n = dimX, by extending sections of K_X +L_m, L_m = (m−1)K_X +m∆ from subvarieties (noticing that K_X + ∆ psef implies L_m psef, and even L_m−∆ psef). Cf. BCHM and recent work of Demailly-Hacon-P˘aun, Fujino, Gongyo, Takayama.

Standard base point free theorem

Let (X,∆) be a projective klt pair, and L be a nef line bundle such that L−(K_X + ∆) is nef and big. Then L is semiample, i.e. |mL|

is BPF for some m>0.

Question (weak positivity variant of the BPF property ?)

Assume that X is not uniruled, i.e. that K_X is pseudoeffective, and let L be a line bundle such thatL−εK_X is pseudoeffectivefor some 0< ε1. Does there existG ∈Pic⁰(X) such thatL+G is abundant ?

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First naive (and too restrictive) technique

Consider the exact sequence

0→ J → O_X → O_X/J → 0 twisted by O_X(K_X ⊗L),

and the corresponding long exact sequence of cohomology groups

· · · →H^q(X,K_X ⊗L)→H^q(X,O_X(K_X ⊗L)⊗ O_X/J)

→H^q+1(X,O_X(K_X⊗L)⊗ J) · · · It is therefore enough to have

H^q+1(X,OX(KX⊗L)⊗ J) = 0.

In order to kill H^q+1 it is enough to make a strict positivity (ampleness) assumption, e.g. by the Kodaira-Nakano / Nadel vanishing theorems.

But we only want to make a weak semipositivity assumption! In that case, one cannot expect to kill the cohomology group H^q+1.

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First naive (and too restrictive) technique

0→ J → O_X → O_X/J → 0

twisted by O_X(K_X ⊗L), and the corresponding long exact sequence of cohomology groups

→H^q+1(X,O_X(K_X⊗L)⊗ J) · · ·

It is therefore enough to have

H^q+1(X,OX(KX⊗L)⊗ J) = 0.

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First naive (and too restrictive) technique

0→ J → O_X → O_X/J → 0

H^q+1(X,OX(KX⊗L)⊗ J) = 0.

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First naive (and too restrictive) technique

0→ J → O_X → O_X/J → 0

H^q+1(X,OX(KX⊗L)⊗ J) = 0.

But we only want to make a weak semipositivity assumption!

In that case, one cannot expect to kill the cohomology group H^q+1.

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Assumptions (1)

We assume X to be holomorphically convex. By the Cartan-Remmert theorem, this is the case iff X admits a

proper holomorphic map p:X →S only a Stein complex space S.

Observation : cohomology is then always Hausdorff

Let X be a holomorphically convex complex space and F a coherent analytic sheaf over X. Then all cohomology groups H^q(X,F) are Hausdorff with respect to their natural topology (local uniform convergence of holomorphic ˇCech cochains) Proof. H^q(X,F)'H⁰(S,R^qp∗F) is a Fr´echet space. Corollary

To solve an equation ∂u=v on a holomorphically convex manifold X, it is enough to solve it approximately:

∂uε=v +wε, wε →0 asε→0

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Assumptions (1)

proper holomorphic map p:X →S only a Stein complex space S. Observation : cohomology is then always Hausdorff

Let X be a holomorphically convex complex space and F a coherent analytic sheaf over X. Then all cohomology groups H^q(X,F) are Hausdorff with respect to their natural topology (local uniform convergence of holomorphic ˇCech cochains)

Proof. H^q(X,F)'H⁰(S,R^qp∗F) is a Fr´echet space. Corollary

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Assumptions (1)

Let X be a holomorphically convex complex space and F a coherent analytic sheaf over X. Then all cohomology groups H^q(X,F) are Hausdorff with respect to their natural topology (local uniform convergence of holomorphic ˇCech cochains) Proof. H^q(X,F)'H⁰(S,R^qp∗F) is a Fr´echet space.

Corollary

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Assumptions (1)

Let X be a holomorphically convex complex space and F a coherent analytic sheaf over X. Then all cohomology groups H^q(X,F) are Hausdorff with respect to their natural topology (local uniform convergence of holomorphic ˇCech cochains) Proof. H^q(X,F)'H⁰(S,R^qp∗F) is a Fr´echet space.

Corollary

∂uε=v +wε, wε →0 as ε→0

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Assumptions (2)

We assume that the subvariety Y ⊂X is defined by Y =V(I(e^−ψ)), O_Y :=O_X/I(e^−ψ)

where ψ is a quasi-psh function with analytic singularities, i.e.

locally on a neighborhood V of an arbitrary point x₀ ∈X we have ψ(z) = clogP|g_j(z)|²+v(z), g_j∈O_X(V), c>0, v∈C^∞(V),

and I(e^−ψ)⊂ O_X is themultiplier ideal sheaf I(e^−ψ)x0 =

f ∈ OX,x0; ∃U 3x0, R

U|f|²e^−ψdλ <+∞ Claim (Nadel)

I(e^−ψ) is a coherent ideal sheaf.

Moreover I(e^−ψ) is always an integrally closed ideal. Typical choice: ψ(z) = clog|s(z)|²_h

E, c >0, s ∈H⁰(X,E).

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Assumptions (2)

locally on a neighborhood V of an arbitrary point x₀ ∈X we have ψ(z) = clogP|g_j(z)|²+v(z), g_j∈O_X(V), c>0, v∈C^∞(V), and I(e^−ψ)⊂ O_X is themultiplier ideal sheaf

I(e^−ψ)x0 =

f ∈ OX,x0; ∃U 3x0, R

U|f|²e^−ψdλ <+∞

Claim (Nadel)

E, c >0, s ∈H⁰(X,E).

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Assumptions (2)

I(e^−ψ)x0 =

f ∈ OX,x0; ∃U 3x0, R

Claim (Nadel)

E, c >0, s ∈H⁰(X,E).

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Assumptions (2)

I(e^−ψ)x0 =

f ∈ OX,x0; ∃U 3x0, R

Claim (Nadel)

Moreover I(e^−ψ) is always an integrally closed ideal.

Typical choice: ψ(z) = clog|s(z)|²_h

E, c >0, s ∈H⁰(X,E).

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Assumptions (2)

I(e^−ψ)x0 =

f ∈ OX,x0; ∃U 3x0, R

Claim (Nadel)

Moreover I(e^−ψ) is always an integrally closed ideal.

Typical choice: ψ(z) = clog|s(z)|²_h

E, c >0, s ∈H⁰(X,E).

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Log resolution / reduction to the divisorial case

The simplest case is when Y =P

m_jY_j is an effective simple normal crossing divisor and O_Y =O_X/O_X(−Y). We can then take

ψ(z) = X

c_jlog|σ_Y_j|²_h_j, c_j >0, bc_jc=m_j,

for some smooth hermitian metric h_j on O_X(Y_j). Then I(e^−ψ) =O_X −P

m_jY_j

, i∂∂ψ =P

c_j(2π[Y_j]−ΘO(Y_j),hj)

The case of a higher codimensional multiplier ideal scheme I(e^−ψ) can easily be reduced to the divisorial case by using a suitable log resolution (a composition of blow ups, thanks to Hironaka’s desingularization theorem).

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Log resolution / reduction to the divisorial case

The simplest case is when Y =P

m_jY_j is an effective simple normal crossing divisor and O_Y =O_X/O_X(−Y). We can then take

ψ(z) = X

c_jlog|σ_Y_j|²_h_j, c_j >0, bc_jc=m_j,

for some smooth hermitian metric h_j on O_X(Y_j). Then I(e^−ψ) =O_X −P

m_jY_j

, i∂∂ψ =P

c_j(2π[Y_j]−ΘO(Y_j),hj) The case of a higher codimensional multiplier ideal scheme I(e^−ψ) can easily be reduced to the divisorial case by using a suitable log resolution (a composition of blow ups, thanks to Hironaka’s desingularization theorem).

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Main results

Theorem (JY. Cao, D– , S-i. Matsumura, January 2017) Take (X, ω) to be K¨ahler and holomorphically convex,

and let (L,hL) be a hermitian line bundle such that

(∗∗) Θ_L,h_L+ (1 +αδ)i∂∂ψ≥0 in the sense of currents for some δ(x)>0 continuous and α= 0,1. Then:

the morphism induced by the natural inclusion I(h_Le^−ψ)→ I(h_L) H^q(X,K_X ⊗L⊗ I(h_Le^−ψ))→H^q(X,K_X ⊗L⊗ I(h_L)) is injective for every q≥0, in other words, the sheaf morphism I(h)→ I(h_L)/I(h_Le^−ψ) yields a surjection

H^q(X,K_X ⊗L⊗ I(h_L))→H^q(X,K_X ⊗L⊗ I(h_L)/I(h_Le^−ψ)).

Corollary (take h_L smooth⇒ I(h_L) = O_X, and Y =V(I(e^−ψ)) If h_L is smooth,O_Y =O_X/I(e^−ψ) and h_L, ψ satisfy (∗∗), then H^q(X,K_X ⊗L)→H^q(Y,(K_X ⊗L)|Y) is surjective.

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Main results

the morphism induced by the natural inclusion I(h_Le^−ψ)→ I(h_L) H^q(X,K_X ⊗L⊗ I(h_Le^−ψ))→H^q(X,K_X ⊗L⊗ I(h_L)) is injective for every q≥0,

in other words, the sheaf morphism I(h)→ I(h_L)/I(h_Le^−ψ) yields a surjection

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Main results

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Main results

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Comments / algebraic consequences

The exact sequence 0→ I(h_Le^−ψ)→ I(h_L)→ I(h_L)/I(h_Le^−ψ)→0 implies that both injectivity and surjectivity hold when

H^q(X,K_X ⊗L⊗ I(h_Le^−ψ) = 0,

and for this it is enough to have a strict curvature assumption (∗∗∗) ΘL,h_L +i∂∂ψ≥δω >0 in the sense of currents.

Corollary (purely algebraic)

Assume that X is projective (or that one has a projective morphism X →S over an affine algebraic baseS). Let Y =P

m_jY_j be an effective divisor and O_Y =O_X/O_X(−Y). If (as Q-divisors) (∗∗) L−(1 +δ)P

cjYj =Gδ+Uδ, bcjc=mj

with δ= 0 orδ₀ ∈Q^∗+, G_δ semiample and U_δ ∈Pic⁰(X), then H^q(X,K_X ⊗L)→H^q(Y,(K_X ⊗L)|Y)

is surjective.

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Comments / algebraic consequences

The exact sequence 0→ I(h_Le^−ψ)→ I(h_L)→ I(h_L)/I(h_Le^−ψ)→0 implies that both injectivity and surjectivity hold when

H^q(X,K_X ⊗L⊗ I(h_Le^−ψ) = 0,

and for this it is enough to have a strict curvature assumption (∗∗∗) ΘL,h_L +i∂∂ψ≥δω >0 in the sense of currents.

Corollary (purely algebraic)

Assume that X is projective (or that one has a projective morphism X →S over an affine algebraic baseS). Let Y =P

m_jY_j be an effective divisor and O_Y =O_X/O_X(−Y). If (as Q-divisors) (∗∗) L−(1 +δ)P

cjYj =Gδ+Uδ, bcjc=mj

with δ= 0 orδ₀ ∈Q^∗+, G_δ semiample and U_δ ∈Pic⁰(X), then H^q(X,K_X ⊗L)→H^q(Y,(K_X ⊗L)|Y)

is surjective.

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Twisted Bochner-Kodaira-Nakano inequality (Ohsawa-Takegoshi)

Let (X, ω) be a K¨ahler manifold and let η, λ >0 be smooth functions on X.

For every compacted supported section u ∈ C_c^∞(X,Λ^p,qT_X^∗ ⊗L) with values in a hermitian line bundle (L,h_L), one has

k(η+λ)¹²∂^∗uk²+kη¹²∂uk²+kλ¹²∂uk²+ 2kλ⁻¹²∂η∧uk²

≥ Z

X

hB_L,h^p,q

L,ω,η,λu,uidV_X_,ω

where dV_X_,ω = _n!¹ωⁿ is the K¨ahler volume element andB_L,h^p,q

L,ω,η,λ is the Hermitian operator on Λ^p,qT_X^∗ ⊗L such that

B_L,h^p,q

L,ω,η,λ = [ηiΘ_L−i∂∂η−iλ⁻¹∂η∧∂η , Λ_ω].

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Approximate solutions to ∂-equations

Main L² estimate

Let (X, ω) be a K¨ahler manifold possessing a complete K¨ahler metric let (E,hE) be a Hermitian vector bundle over X. Assume that B =B_E,h,ω,η,λ^n,q satisfies B+εId>0for some ε >0 (so that B can be just semi-positive or even slightly negative).

Take a section v ∈L²(X,Λ^n,qT_X^∗ ⊗E) such that ∂v = 0 and M(ε) :=

Z

X

h(B+εId)⁻¹v,vidV_X_,ω <+∞.

Then there exists an approximate solution u_ε∈L²(X,Λ^n,q−1T_X^∗ ⊗E) and a correction term w_ε ∈L²(X,Λ^n,qT_X^∗ ⊗E) such that

∂u_ε =v +w_ε and Z

X

(η+λ)⁻¹|u_ε|²dV_X_,ω +1 ε

Z

X

|w_ε|²dV_X_,ω ≤M(ε).

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Approximate solutions to ∂-equations

Main L² estimate

Z

X

h(B +εId)⁻¹v,vidV_X_,ω <+∞.

Then there exists an approximate solution u_ε∈L²(X,Λ^n,q−1T_X^∗ ⊗E) and a correction term w_ε ∈L²(X,Λ^n,qT_X^∗ ⊗E) such that

X

(η+λ)⁻¹|u_ε|²dV_X_,ω +1 ε

Z

X

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Approximate solutions to ∂-equations

Main L² estimate

Z

X

h(B +εId)⁻¹v,vidV_X_,ω <+∞.

Then there exists an approximate solution u_ε∈L²(X,Λ^n,q−1T_X^∗ ⊗E) and a correction termw_ε ∈L²(X,Λ^n,qT_X^∗ ⊗E) such that

X

(η+λ)⁻¹|u_ε|²dV_X_,ω +1 ε

Z

X

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Proof: setting up the relevant ∂ equation (1)

Every cohomology class in

H^q(X,O_X(K_X ⊗L)⊗ I(h_L)/I(h_Le^−ψ))

is represented by a holomorphic ˇCech q-cocycle with respect to a Stein covering U = (U_i), say (c_i₀_...i_q),

c_i₀_...i_q ∈H⁰ U_i₀ ∩. . .∩U_i_q,O_X(K_X ⊗L)⊗ I(h_L)/I(h_Le^−ψ) .

By the standard sheaf theoretic isomorphism with Dolbeault cohomology, this class is represented by a smooth (n,q)-form

f = X

i0,...,iq

c_i₀_...i_qρ_i₀∂ρ_i₁∧. . . ∂ρ_i_q

by means of a partition of unity (ρ_i) subordinate to (U_i). This form is to be interpreted as a form on the (non reduced) analytic subvariety Y associated with the colon ideal sheaf

J =I(he^−ψ) :I(h) and the structure sheaf O_Y =O_X/J.

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Proof: setting up the relevant ∂ equation (1)

Every cohomology class in

H^q(X,O_X(K_X ⊗L)⊗ I(h_L)/I(h_Le^−ψ))

is represented by a holomorphic ˇCech q-cocycle with respect to a Stein covering U = (U_i), say (c_i₀_...i_q),

c_i₀_...i_q ∈H⁰ U_i₀ ∩. . .∩U_i_q,O_X(K_X ⊗L)⊗ I(h_L)/I(h_Le^−ψ) . By the standard sheaf theoretic isomorphism with Dolbeault cohomology, this class is represented by a smooth (n,q)-form

f = X

i0,...,iq

ci0...iqρi0∂ρi1∧. . . ∂ρiq

by means of a partition of unity (ρ_i) subordinate to (U_i). This form is to be interpreted as a form on the (non reduced) analytic subvariety Y associated with the colon ideal sheaf

J =I(he^−ψ) :I(h) and the structure sheaf O_Y =O_X/J.

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Proof: setting up the relevant ∂ equation (2)

We get an extension of f as a smooth (no longer ∂-closed) (n,q)-form on X by taking

fe= X

i0,...,iq

ec_i₀_...i_qρ_i₀∂ρ_i₁∧. . . ∂ρ_i_q

where ec_i₀_...i_q = extension of c_i₀_...i_q from U_i₀∩. . .∩U_i_q∩Y to U_i₀∩. . .∩U_i_q

Y

{x∈X/t<ψ(x)<t+1}

1

0

θ(s)

1 s Now, truncate feas θ(ψ−t)·feon the green hollow tubular neighborhood, and solve an approximate ∂-equation

(∗) ∂ut,ε =∂(θ(ψ−t)·fe) +wt,ε

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Proof: setting up the relevant ∂ equation (2)

fe= X

i0,...,iq

ec_i₀_...i_qρ_i₀∂ρ_i₁∧. . . ∂ρ_i_q

Y

{x∈X/t<ψ(x)<t+1}

1

0

θ(s) 1 s

Now, truncate feas θ(ψ−t)·feon the green hollow tubular neighborhood, and solve an approximate ∂-equation

(∗) ∂ut,ε =∂(θ(ψ−t)·fe) +wt,ε

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Proof: setting up the relevant ∂ equation (2)

fe= X

i0,...,iq

ec_i₀_...i_qρ_i₀∂ρ_i₁∧. . . ∂ρ_i_q

Y

{x∈X/t<ψ(x)<t+1}

1

0

θ(s)

1 s Now, truncate feas θ(ψ−t)·feon the green hollow tubular neighborhood, and solve an approximate ∂-equation

(∗) ∂ut,ε =∂(θ(ψ−t)·fe) +wt,ε

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Proof: setting up the relevant ∂ equation (3)

Here we have

∂(θ(ψ−t)·fe) =θ⁰(ψ−t)∂ψ∧fe+θ(ψ−t)·∂fe

where the first term vanishes near Y and the second one is L² with respect to h_Le^−ψ (as the image of ∂fein I(h_L)/I(h_Le^−ψ) is ∂f = 0).

With ad hoc “twisting functions” η=η_t := 1−δχ_t(ψ),

λ :=π(1 +δ²ψ²) and a suitable adjustmentε=e^(1+β)t, β 1, we can find approximate L² solutions of the∂-equation such that

∂u_t,ε =∂(θ(ψ−t)·fe) +w_t,ε , Z

X

|u_t,ε|²_ω,h_Le^−ψdV_X_,ω <+∞ and

t→−∞lim Z

X

|w_t,ε|²_ω,h

Le^−ψdV_X_,ω = 0.

The estimate on u_t,ε with respect to the weight h_Le^−ψ shows that θ(ψ−t)·fe−u_t,ε is an approximate extension of f.

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Proof: setting up the relevant ∂ equation (3)

Here we have

∂u_t,ε =∂(θ(ψ−t)·fe) +w_t,ε , Z

X

|u_t,ε|²_ω,h

Le^−ψdV_X_,ω <+∞

and

t→−∞lim Z

X

|wt,ε|²_ω,h_Le^−ψdVX,ω = 0.

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Proof: setting up the relevant ∂ equation (3)

Here we have

∂u_t,ε =∂(θ(ψ−t)·fe) +w_t,ε , Z

X

|u_t,ε|²_ω,h

Le^−ψdV_X_,ω <+∞

and

t→−∞lim Z

X

|wt,ε|²_ω,h_Le^−ψdVX,ω = 0.

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Can one get estimates for the extension ?

The answer is yes if ψ is log canonical, namely I(e^{−(1−ε)ψ}) =O_X for all ε >0. Then Y =V(I(e^−ψ)) is easily seen to be reduced.

Ohsawa’s residue measure

If ψ is log canonical, one can also associate in a natural way a measure dV_Y^◦_,ω[ψ] on the set Y^◦ of regular points ofY as follows. If g ∈ C_c(Y^◦) is a compactly supported continuous function on Y^◦ and eg a compactly supported extension of g to X, one sets

Z

Y^◦

g dV_Y^◦_,ω[ψ] = lim

t→−∞

Z

{x∈X,t<ψ(x)<t+1}g ee ^−ψdV_X_,ω Theorem

If ψ is lc and the curvature hypothesis is satisfied, for any f in H⁰(Y,K_X ⊗L⊗ I(h_L)/I(h_Le^−ψ)) s.t. R

Y^◦|f|²_ω,h

LdV_Y^◦_,ω[ψ]<+∞, there exists fe∈H⁰(X,KX ⊗L⊗ I(hL)) which extends f, such that

Z

X

(1 +δ²ψ²)⁻¹e^−ψ|ef|²_ω,h

LdV_X_,ω ≤ 34 δ

Z

Y^◦

|f|²_ω,h

LdV_Y^◦_,ω[ψ].

(45)

Can one get estimates for the extension ?

If ψ is log canonical, one can also associate in a natural way a measure dV_Y^◦_,ω[ψ] on the set Y^◦ of regular points ofY as follows.

If g ∈ C_c(Y^◦) is a compactly supported continuous function on Y^◦ and eg a compactly supported extension of g to X, one sets

Z

Y^◦

t→−∞

Z

{x∈X,t<ψ(x)<t+1}g ee ^−ψdV_X_,ω

Theorem

Y^◦|f|²_ω,h

LdV_Y^◦_,ω[ψ]<+∞, there exists fe∈H⁰(X,KX ⊗L⊗ I(hL)) which extends f, such that

Z

X

(1 +δ²ψ²)⁻¹e^−ψ|ef|²_ω,h

LdV_X_,ω ≤ 34 δ

Z

Y^◦

|f|²_ω,h

LdV_Y^◦_,ω[ψ].

(46)

Can one get estimates for the extension ?

If ψ is log canonical, one can also associate in a natural way a measure dV_Y^◦_,ω[ψ] on the set Y^◦ of regular points ofY as follows.

If g ∈ C_c(Y^◦) is a compactly supported continuous function on Y^◦ and eg a compactly supported extension of g to X, one sets

Z

Y^◦

t→−∞

Z

{x∈X,t<ψ(x)<t+1}g ee ^−ψdV_X_,ω Theorem

Y^◦|f|²_ω,h

LdV_Y^◦_,ω[ψ]<+∞, there exists fe∈H⁰(X,K_X ⊗L⊗ I(h_L)) which extends f, such that

Z

X

(1 +δ²ψ²)⁻¹e^−ψ|ef|²_ω,h

LdV_X_,ω ≤ 34 δ

Z

Y^◦

|f|²_ω,h

LdV_Y^◦_,ω[ψ].

(47)

Can one get estimates for the extension ? (sequel)

If ψ is not log canonical, consider the “last jumps”m_p−1 <m_p ≤1 such that I(h_Le^−m^p−1^ψ))I(h_Le^−m^p^ψ) =I(h_Le^−ψ) and assume

f ∈H⁰(Y,K_X ⊗L⊗ I(h_Le^−m^p−1^ψ)/I(h_Le^−m^p^ψ)), i.e., f vanishes just a little bit less than prescribed by the sheaf I(h_Le^−ψ)). Then there is still a corresponding residue measure:

Higher multiplicity residue measure

If f is as above, and feis a local extension, one can associate a higher multiplicity residue measure |f|²dV_Y^◦_,ω[ψ] (formal notation) as follows. If g ∈ C_c(Y^◦) and ge a compactly supported extension of g to X, one sets

Z

Y^◦

g|f|²dV_Y^◦_,ω[ψ] = lim

t→−∞

Z

{x∈X,t<ψ(x)<t+1}eg|fe|²e^−m^p^ψdV_X_,ω Then a global extension fe∈H⁰(X,K_X ⊗L⊗ I(h_Le^−m^p−1^ψ)) exists, that satisfies the expected L² estimate.

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Can one get estimates for the extension ? (sequel)

If ψ is not log canonical, consider the “last jumps”m_p−1 <m_p ≤1 such that I(h_Le^−m^p−1^ψ))I(h_Le^−m^p^ψ) =I(h_Le^−ψ) and assume

f ∈H⁰(Y,K_X ⊗L⊗ I(h_Le^−m^p−1^ψ)/I(h_Le^−m^p^ψ)), i.e., f vanishes just a little bit less than prescribed by the sheaf I(h_Le^−ψ)). Then there is still a corresponding residue measure:

Higher multiplicity residue measure

If f is as above, and feis a local extension, one can associate a higher multiplicity residue measure |f|²dVY^◦,ω[ψ] (formal notation) as follows. If g ∈ C_c(Y^◦) and ge a compactly supported extension of g to X, one sets

Z

Y^◦

g|f|²dV_Y^◦_,ω[ψ] = lim

t→−∞

Z

{x∈X,t<ψ(x)<t+1}eg|fe|²e^−m^p^ψdV_X_,ω

Then a global extension fe∈H⁰(X,K_X ⊗L⊗ I(h_Le^−m^p−1^ψ)) exists, that satisfies the expected L² estimate.