L 2 extension theorem for sections defined on non reduced analytic subvarieties

Transcription

1 L 2 extension theorem for sections defined on non reduced analytic subvarieties Jean-Pierre D ly Institut Fourier, Université de Grenoble Alpes & Académie des Sciences de Paris Conference on Geometry and Topology, Harvard University, April 28 May 2, 2017, celebrating 50 years of the Journal of Differential Geometry J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 1/18

2 References This is a joint work with Junyan Cao & Shin-ichi Matsumura J.-P. D ly, Extension of holomorphic functions defined on non reduced analytic subvarieties, arxiv: v1, Advanced Lectures in Mathematics Volume 35.1, the legacy of Bernhard Riemann after one hundred and fifty years, J.Y. Cao, J.-P. D ly, S-i. Matsumura, A general extension theorem for cohomology classes on non reduced analytic subspaces, arxiv: v2, Science China Mathematics, 60 (2017) T. Ohsawa, On a curvature condition that implies a cohomology injectivity theorem of Kollár-Skoda type, Publ. Res. Inst. Math. Sci. 41 (2005), no. 3, T. Ohsawa, K. Takegoshi, On the extension of L 2 holomorphic functions, Math. Zeitschrift 195 (1987), J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 2/18

3 The extension problem Let (X, ω) be a possibly noncompact n-dimensional Kähler 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. J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 3/18

4 The extension problem Let (X, ω) be a possibly noncompact n-dimensional Kähler 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,hL = i log h 1 L its curvature current (we allow singular metrics, h L = e ϕ, ϕ L 1 loc, Θ L,h L being computed in the sense of currents). Question Under which conditions on X, Y = V (J ), (L, h L ) 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? is J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 3/18

5 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. J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 4/18

6 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 Čech cochains) J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 4/18

7 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 Čech cochains) Proof. H q (X, F) H 0 (S, R q p F) is a Fréchet space. J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 4/18

8 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 Čech cochains) Proof. H q (X, F) H 0 (S, R q p F) is a Fréchet space. Corollary To solve an equation u = v on a holomorphically convex manifold X, it is enough to solve it approximately: u ε = v + w ε, w ε 0 as ε 0 J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 4/18

9 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 x 0 X we have ψ(z) = c log g j (z) 2 + v(z), g j O X (V ), c>0, v C (V ), J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 5/18

10 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 x 0 X we have ψ(z) = c log g j (z) 2 + v(z), g j O X (V ), c>0, v C (V ), and I(e ψ ) O X is the multiplier ideal sheaf I(e ψ ) x0 = { f O X,x0 ; U x 0, U f 2 e ψ dλ < + } J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 5/18

11 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 x 0 X we have ψ(z) = c log g j (z) 2 + v(z), g j O X (V ), c>0, v C (V ), and I(e ψ ) O X is the multiplier ideal sheaf I(e ψ ) x0 = { f O X,x0 ; U x 0, U f 2 e ψ dλ < + } Claim (Nadel) I(e ψ ) is a coherent ideal sheaf. J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 5/18

12 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 x 0 X we have ψ(z) = c log g j (z) 2 + v(z), g j O X (V ), c>0, v C (V ), and I(e ψ ) O X is the multiplier ideal sheaf I(e ψ ) x0 = { f O X,x0 ; U x 0, U f 2 e ψ dλ < + } Claim (Nadel) I(e ψ ) is a coherent ideal sheaf. Moreover I(e ψ ) is always an integrally closed ideal. J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 5/18

13 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 x 0 X we have ψ(z) = c log g j (z) 2 + v(z), g j O X (V ), c>0, v C (V ), and I(e ψ ) O X is the multiplier ideal sheaf I(e ψ ) x0 = { f O X,x0 ; U x 0, U f 2 e ψ dλ < + } Claim (Nadel) I(e ψ ) is a coherent ideal sheaf. Moreover I(e ψ ) is always an integrally closed ideal. Typical choice: ψ(z) = c log s(z) 2 h E, c > 0, s H 0 (X, E). J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 5/18

14 Log resolution / reduction to the divisorial case The simplest case is when Y = m j Y j is an effective simple normal crossing divisor and O Y = O X /O X ( Y ). We can then take ψ(z) = c j log σ Yj 2 h j, c j > 0, c j = m j, for some smooth hermitian metric h j on O X (Y j ). Then I(e ψ ) = O X ( mj Y j ), i ψ = cj (2π[Y j ] Θ O(Yj ),h j ) J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 6/18

15 Log resolution / reduction to the divisorial case The simplest case is when Y = m j Y j is an effective simple normal crossing divisor and O Y = O X /O X ( Y ). We can then take ψ(z) = c j log σ Yj 2 h j, c j > 0, c j = m j, for some smooth hermitian metric h j on O X (Y j ). Then I(e ψ ) = O X ( mj Y j ), i ψ = cj (2π[Y j ] Θ O(Yj ),h j ) 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). J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 6/18

16 Main results Theorem (JY. Cao, D, S-i. Matsumura, January 2017) Take (X, ω) to be Kähler and holomorphically convex, and let (L, h L ) be a hermitian line bundle such that ( ) Θ L,hL + (1 + αδ)i ψ 0 in the sense of currents for some δ(x) > 0 continuous and α = 0, 1. Then: J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 7/18

17 Main results Theorem (JY. Cao, D, S-i. Matsumura, January 2017) Take (X, ω) to be Kähler and holomorphically convex, and let (L, h L ) be a hermitian line bundle such that ( ) Θ L,hL + (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 L e ψ ) I(h L ) H q (X, K X L I(h L e ψ )) H q (X, K X L I(h L )) is injective for every q 0, J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 7/18

18 Main results Theorem (JY. Cao, D, S-i. Matsumura, January 2017) Take (X, ω) to be Kähler and holomorphically convex, and let (L, h L ) be a hermitian line bundle such that ( ) Θ L,hL + (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 L e ψ ) I(h L ) H q (X, K X L I(h L e 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 L e ψ ) yields a surjection H q (X, K X L I(h L )) H q (X, K X L I(h L )/I(h L e ψ )). J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 7/18

19 Main results Theorem (JY. Cao, D, S-i. Matsumura, January 2017) Take (X, ω) to be Kähler and holomorphically convex, and let (L, h L ) be a hermitian line bundle such that ( ) Θ L,hL + (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 L e ψ ) I(h L ) H q (X, K X L I(h L e 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 L e ψ ) yields a surjection H q (X, K X L I(h L )) H q (X, K X L I(h L )/I(h L e 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. J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 7/18

20 Comments / algebraic consequences The exact sequence 0 I(h L e ψ ) I(h L ) I(h L )/I(h L e ψ ) 0 implies that both injectivity and surjectivity hold when H q (X, K X L I(h L e ψ ) = 0, and for this it is enough to have a strict curvature assumption ( ) Θ L,hL + i ψ δω > 0 in the sense of currents. J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 8/18

21 Comments / algebraic consequences The exact sequence 0 I(h L e ψ ) I(h L ) I(h L )/I(h L e ψ ) 0 implies that both injectivity and surjectivity hold when H q (X, K X L I(h L e ψ ) = 0, and for this it is enough to have a strict curvature assumption ( ) Θ L,hL + 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 base S). Let Y = m j Y j be an effective divisor and O Y = O X /O X ( Y ). If (as Q-divisors) ( ) L (1 + δ) c j Y j = G δ + U δ, c j = m j with δ = 0 or δ 0 Q +, G δ semiample and U δ Pic 0 (X ), then is surjective. H q (X, K X L) H q (Y, (K X L) Y ) J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 8/18

22 Possible use for abundance by induction on dim X? For a line bundle L, one defines the Kodaira-Iitaka dimension κ(l) = lim sup m + log dim H 0 (X, L m )/ log m and the numerical dimension nd(l) = maximum power of non zero positive intersection of a positive current T c 1 (L), if L is pseudo-effective, and nd(l) = otherwise. They always satisfy κ(l) nd(l) n = dim X. J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 9/18

23 Possible use for abundance by induction on dim X? For a line bundle L, one defines the Kodaira-Iitaka dimension κ(l) = lim sup m + log dim H 0 (X, L m )/ log m and the numerical dimension nd(l) = maximum power of non zero positive intersection of a positive current T c 1 (L), if L is pseudo-effective, and nd(l) = otherwise. They always satisfy κ(l) nd(l) n = dim X. One says that L is abundant if κ(l) = nd(l). The abundance conjecture states that K X is always abundant if X is nonsingular and projective (or even compact Kähler). More generally, it is expected that K X + is abundant for every effective klt Q-divisor. J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 9/18

24 Possible use for abundance by induction on dim X? For a line bundle L, one defines the Kodaira-Iitaka dimension κ(l) = lim sup m + log dim H 0 (X, L m )/ log m and the numerical dimension nd(l) = maximum power of non zero positive intersection of a positive current T c 1 (L), if L is pseudo-effective, and nd(l) = otherwise. They always satisfy κ(l) nd(l) n = dim X. One says that L is abundant if κ(l) = nd(l). The abundance conjecture states that K X is always abundant if X is nonsingular and projective (or even compact Kähler). More generally, it is expected that K X + is abundant for every effective klt Q-divisor. When X is not uniruled, i.e. K X is pseudoeffective, one can ask whether the following generalized abundance property holds true: let L be a line bundle such that L εk X is pseudoeffective, 0 < ε 1; does there exist G Pic 0 (X ) such that L + G is abundant? J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 9/18

25 Twisted Bochner-Kodaira-Nakano inequality (Ohsawa-Takegoshi) Let (X, ω) be a Kähler manifold and let η, λ > 0 be smooth functions on X. For every compacted supported section u C c (X, Λ p,q T X L) with values in a hermitian line bundle (L, h L ), one has (η + λ) 1 2 u 2 + η 1 2 u 2 + λ 1 2 u λ 1 2 η u 2 B p,q L,h L,ω,η,λ u, u dv X,ω X where dv X,ω = 1 n! ωn is the Kähler volume element and B p,q the Hermitian operator on Λ p,q TX L such that B p,q L,h L,ω,η,λ = [η iθ L i η iλ 1 η η, Λ ω ]. L,h L,ω,η,λ is J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 10/18

26 Approximate solutions to -equations Main L 2 estimate Let (X, ω) be a Kähler manifold possessing a complete Kähler metric let (E, h E ) be a Hermitian vector bundle over X. Assume that B = B n,q E,h,ω,η,λ satisfies B + ε Id > 0 for some ε > 0 (so that B can be just semi-positive or even slightly negative). J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 11/18

27 Approximate solutions to -equations Main L 2 estimate Let (X, ω) be a Kähler manifold possessing a complete Kähler metric let (E, h E ) be a Hermitian vector bundle over X. Assume that B = B n,q E,h,ω,η,λ satisfies B + ε Id > 0 for some ε > 0 (so that B can be just semi-positive or even slightly negative). Take a section v L 2 (X, Λ n,q TX E) such that v = 0 and M(ε) := (B + ε Id) 1 v, v dv X,ω < +. X J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 11/18

28 Approximate solutions to -equations Main L 2 estimate Let (X, ω) be a Kähler manifold possessing a complete Kähler metric let (E, h E ) be a Hermitian vector bundle over X. Assume that B = B n,q E,h,ω,η,λ satisfies B + ε Id > 0 for some ε > 0 (so that B can be just semi-positive or even slightly negative). Take a section v L 2 (X, Λ n,q TX E) such that v = 0 and M(ε) := (B + ε Id) 1 v, v dv X,ω < +. X Then there exists an approximate solution u ε L 2 (X, Λ n,q 1 TX E) and a correction term w ε L 2 (X, Λ n,q TX E) such that u ε = v + w ε and X (η + λ) 1 u ε 2 dv X,ω + 1 ε X w ε 2 dv X,ω M(ε). J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 11/18

29 Proof: setting up the relevant equation (1) Every cohomology class in H q (X, O X (K X L) I(h L )/I(h L e ψ )) is represented by a holomorphic Čech q-cocycle with respect to a Stein covering U = (U i ), say (c i0...i q ), c i0...i q H 0( U i0... U iq, O X (K X L) I(h L )/I(h L e ψ ) ). J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 12/18

30 Proof: setting up the relevant equation (1) Every cohomology class in H q (X, O X (K X L) I(h L )/I(h L e ψ )) is represented by a holomorphic Čech q-cocycle with respect to a Stein covering U = (U i ), say (c i0...i q ), c i0...i q H 0( U i0... U iq, O X (K X L) I(h L )/I(h L e ψ ) ). By the standard sheaf theoretic isomorphism with Dolbeault cohomology, this class is represented by a smooth (n, q)-form f = i 0,...,i q c i0...i q ρ i0 ρ i1... ρ iq 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. J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 12/18

31 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 f = i 0,...,i q c i0...i q ρ i0 ρ i1... ρ iq where c i0...i q = extension of c i0...i q from U i0... U iq Y to U i0... U iq J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 13/18

32 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 f = i 0,...,i q c i0...i q ρ i0 ρ i1... ρ iq where c i0...i q = extension of c i0...i q from U i0... U iq Y to U i0... U iq {x X / t<ψ(x)<t+1} 1 θ(s) Y 0 1 s J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 13/18

33 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 f = i 0,...,i q c i0...i q ρ i0 ρ i1... ρ iq where c i0...i q = extension of c i0...i q from U i0... U iq Y to U i0... U iq {x X / t<ψ(x)<t+1} 1 θ(s) Y 0 1 s Now, truncate f as θ(ψ t) f on the green hollow tubular neighborhood, and solve an approximate -equation ( ) u t,ε = (θ(ψ t) f ) + w t,ε J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 13/18

34 Proof: setting up the relevant equation (3) Here we have (θ(ψ t) f ) = θ (ψ t) ψ f + θ(ψ t) f where the first term vanishes near Y and the second one is L 2 with respect to h L e ψ (as the image of f in I(h L )/I(h L e ψ ) is f = 0). J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 14/18

35 Proof: setting up the relevant equation (3) Here we have (θ(ψ t) f ) = θ (ψ t) ψ f + θ(ψ t) f where the first term vanishes near Y and the second one is L 2 with respect to h L e ψ (as the image of f in I(h L )/I(h L e ψ ) is f = 0). With ad hoc twisting functions η = η t := 1 δχ t (ψ), λ := π(1 + δ 2 ψ 2 ) and a suitable adjustment ε = e (1+β)t, β 1, we can find approximate L 2 solutions of the -equation such that u t,ε = (θ(ψ t) f ) + w t,ε, u t,ε 2 ω,h L e ψ dv X,ω < + and lim w t,ε 2 ω,h t L e ψ dv X,ω = 0. X X J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 14/18

36 Proof: setting up the relevant equation (3) Here we have (θ(ψ t) f ) = θ (ψ t) ψ f + θ(ψ t) f where the first term vanishes near Y and the second one is L 2 with respect to h L e ψ (as the image of f in I(h L )/I(h L e ψ ) is f = 0). With ad hoc twisting functions η = η t := 1 δχ t (ψ), λ := π(1 + δ 2 ψ 2 ) and a suitable adjustment ε = e (1+β)t, β 1, we can find approximate L 2 solutions of the -equation such that u t,ε = (θ(ψ t) f ) + w t,ε, u t,ε 2 ω,h L e ψ dv X,ω < + and lim w t,ε 2 ω,h t L e ψ dv X,ω = 0. X The estimate on u t,ε with respect to the weight h L e ψ shows that θ(ψ t) f u t,ε is an approximate extension of f. J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 14/18 X

37 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. J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 15/18

38 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. 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 of Y as follows. If g C c (Y ) is a compactly supported continuous function on Y and g a compactly supported extension of g to X, one sets Y g dv Y,ω[ψ] = lim t {x X, t<ψ(x)<t+1} ge ψ dv X,ω J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 15/18

39 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. 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 of Y as follows. If g C c (Y ) is a compactly supported continuous function on Y and g a compactly supported extension of g to X, one sets Y g dv Y,ω[ψ] = lim t {x X, t<ψ(x)<t+1} ge ψ dv X,ω Theorem If ψ is lc and the curvature hypothesis is satisfied, for any f in H 0 (Y, K X L I(h L )/I(h L e ψ )) s.t. f 2 Y ω,h L dv Y,ω[ψ] < +, there exists f H 0 (X, K X L I(h L )) which extends f, such that (1 + δ 2 ψ 2 ) 1 e ψ f 2 ω,h L dv X,ω 34 f 2 ω,h δ L dv Y,ω[ψ]. Y X J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 15/18

40 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 L e m p 1ψ ) I(h L e mpψ ) = I(h L e ψ ) and assume f H 0 (Y, K X L I(h L e m p 1ψ )/I(h L e mpψ )), i.e., f vanishes just a little bit less than prescribed by the sheaf I(h L e ψ )). Then there is still a corresponding residue measure: J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 16/18

41 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 L e m p 1ψ ) I(h L e mpψ ) = I(h L e ψ ) and assume f H 0 (Y, K X L I(h L e m p 1ψ )/I(h L e mpψ )), i.e., f vanishes just a little bit less than prescribed by the sheaf I(h L e ψ )). Then there is still a corresponding residue measure: Higher multiplicity residue measure If f is as above, and f is a local extension, one can associate a higher multiplicity residue measure f 2 dv Y,ω[ψ] (formal notation) as follows. If g C c (Y ) and g a compactly supported extension of g to X, one sets Y g f 2 dv Y,ω[ψ] = lim g f 2 e mpψ dv X,ω t {x X, t<ψ(x)<t+1} J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 16/18

42 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 L e m p 1ψ ) I(h L e mpψ ) = I(h L e ψ ) and assume f H 0 (Y, K X L I(h L e m p 1ψ )/I(h L e mpψ )), i.e., f vanishes just a little bit less than prescribed by the sheaf I(h L e ψ )). Then there is still a corresponding residue measure: Higher multiplicity residue measure If f is as above, and f is a local extension, one can associate a higher multiplicity residue measure f 2 dv Y,ω[ψ] (formal notation) as follows. If g C c (Y ) and g a compactly supported extension of g to X, one sets Y g f 2 dv Y,ω[ψ] = lim g f 2 e mpψ dv X,ω t {x X, t<ψ(x)<t+1} Then a global extension f H 0 (X, K X L I(h L e m p 1ψ )) exists, that satisfies the expected L 2 estimate. J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 16/18

43 Special case / limitations of the L 2 estimates In the special case when ψ is given by ψ(z) = r log s(z) 2 h E for a section s H 0 (X, E) generically transverse to the zero section of a rank r vector vector E on X, the subvariety Y = s 1 (0) has codimension r, and one can check easily that dv Y,ω[ψ] = dv Y,ω Λ r (ds) 2 ω,h E. J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 17/18

44 Special case / limitations of the L 2 estimates In the special case when ψ is given by ψ(z) = r log s(z) 2 h E for a section s H 0 (X, E) generically transverse to the zero section of a rank r vector vector E on X, the subvariety Y = s 1 (0) has codimension r, and one can check easily that dv Y,ω[ψ] = dv Y,ω Λ r (ds) 2 ω,h E. Thus one sees that the residue measure takes into account in a very precise manner the singularities of Y. It may happen that dv Y,ω[ψ] has infinite mass near the singularities of Y, as is the case when Y is a simple normal crossing divisor. J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 17/18

45 Special case / limitations of the L 2 estimates In the special case when ψ is given by ψ(z) = r log s(z) 2 h E for a section s H 0 (X, E) generically transverse to the zero section of a rank r vector vector E on X, the subvariety Y = s 1 (0) has codimension r, and one can check easily that dv Y,ω[ψ] = dv Y,ω Λ r (ds) 2 ω,h E. Thus one sees that the residue measure takes into account in a very precise manner the singularities of Y. It may happen that dv Y,ω[ψ] has infinite mass near the singularities of Y, as is the case when Y is a simple normal crossing divisor. Therefore, sections s H 0 (Y, (K X L) Y may not be L 2 with respect to dv Y,ω[ψ]), and the L 2 estimate of the approximate extension can blow up as ε 0. The surprising fact is this is however sufficient to prove the qualitative extension theorem, but without any effective L 2 estimate in the limit. J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 17/18

46 The end Thank you! Tentative image of a Calabi-Yau manifold (K X = O X ). It would be important to know about the generalized abundance property in that case. J.-P. D ly (Grenoble), JDG Conference 2017, Harvard L 2 extension theorem from nonreduced analytic subvarieties 18/18