NOTES COURS L2 2016 Table des matières 1. Subharmonic functions 1 2. Plurisubharmonic functions 4 3. Basic analysis of first order differential operators 10 4. Fundamental identities of Kähler geometry 15 5. L 2 -estimates for the -equation 18 6. Multiplier ideal sheaves and Nadel vanishing 22 7. The Ohsawa-Takegoshi extension theorem 24 8. Algebraic characterization of multiplier ideals 28 9. Subadditivity and Demailly approximation 32 10. Positivity of Bergman kernels 32 Références 32 Bibliography 32 1. Subharmonic functions 1.1. The Green-Riesz representation formula. We use [DemAG, 4.B] as a reference. Let Ω R m be a smooth bounded domain. The Green kernel G Ω (x, y) of Ω is defined so that G Ω (, y) solves the Dirichlet problem (1.1) { x G Ω (x, y) = δ y G Ω (, y) Ω 0 It shown to be symmetric, and smooth away from the diagonal of Ω Ω. Note also that G Ω 0, by the maximum principle. The Green-Riesz representation formula reconstructs a function u C 2 (Ω) in terms of u and the boundary values u Ω. Theorem 1.1. For each u C 2 (Ω) we have (1.2) u(x) = G Ω (x, y) u(y)dλ(y) + Ω Ω P Ω (x, y)u(y)dσ(y) where P Ω (x, y) (the Poisson kernel) is smooth on Ω Ω, and harmonic in x. This result follows by applying the following integration-by-part formula to u and G Ω (x, ), which also yields P Ω (x, y) = n(y) G Ω(x, y). Date: 29 avril 2016. 1
2 Lemma 1.2. For any two u, v C 2 (Ω), we have (1.3) (u v v u)dλ = Ω Ω ( u v n v u n ) dσ, with n the outward pointing normal unit vector. The result is also valid when v is only smooth near of the boundary. Exemple 1.3. The Poisson kernel P r of the ball B(0, r) R m is given by P r (x, y) = 1 r 2 x 2 σ m 1 r y x m. In particular, P r (0, )dσ is the uniform probability measure on S(0, r). On the other hand, the Green kernel G r satisfies G r (x, 0) = 1 σ m 1 r x dt t m 1. Injecting the latter formulae in (1.2), we get the Gauss-Jensen formula : Corollary 1.4. For u C 2 ( B(a, r)), the mean ffl S(a,r) u on the sphere S(a, r) satisfies (1.4) u u(a) = 1 r S(a,r) σ m 1 t=0 dt t m 1 u dλ. B(a,t) 1.2. Subharmonic functions. In what follows Ω R m is an open subset. Definition 1.5. A function u : Ω [, + ) is subharmonic if : (i) u is usc, with u not identically on each connected component of Ω ; (ii) for each ball B(a, r) Ω, the mean value inequality u(a) ffl S(a,r) u holds. Note that u is locally bounded above by (i), so that S(a,r) u dσ makes sense in [, + ). Since B(a,r) udλ = r 0 dt S(a,t) udσ, (ii) also implies u(a) ffl B(a,r) u. Using this, it is easy to see that u is locally integrable. Proposition 1.6. Let u be subharmonic on Ω. (i) If µ is a probability measure supported on a compact set K R m, then u µ on any open set U such that U + K Ω. (ii) If χ(t 1,..., t r ) is convex on R r and non-decreasing in each variable, then χ(u 1,..., u r ) is subharmonic for any subharmonic functions u i. Now pick a non-negative function ρ C (R), compactly supported in [ 1, 1] and such that σ m 1 1 0 ρ(r)rm 1 dr = 1, and set as usual (1.5) ρ ε (x) = ε m ρ(ε 1 x ) We then have the following key monotone regularization result. Theorem 1.7. Let u be suharmonic function on Ω. Then u ρ ε is smooth and subharmonic on Ω ε = {x Ω d(x, Ω) > ε}, and decreases to u as ε 0.
Démonstration. Note that 1 (1.6) (u ρ ε )(a) = σ m 1 ρ(r)r m 1 dr NOTES COURS L2 2016 3 0 u dσ. S(a,εr) When u is smooth, Jensen s formula shows that ffl S(a,r) u is a non-decreasing function of r, and u ρ ε is thus non-decreasing with respect to ε. Now consider the general case. By Proposition 1.6, u ρ ε is ubharmonic on Ω ε. As a result, (u ρ ε ) ρ δ = (u ρ δ ) ρ ε is non-decreasing in ε, and hence so is u ρ ε in the limit as δ 0. Finally, (1.6) shows that u u ρ ε, while lim ε 0 u ρ ε u since u is usc. Corollary 1.8. If u is subharmonic, then u 0. Conversely, if v L 1 loc (Ω) satisfies v 0, then there exists a unique subharmonic function u on Ω with u = v a.e. Démonstration. When u is smooth, the result is a consequence of the Jensen formula (Corollary 1.4). The general case follows by regularization. By monotone regularization, we can also prove that the Green representation fornula (1.2) and the Gauss-Jensen formula (1.4) remain valid for any u subharmonic in a neighborhood of Ω. In particular, a subharmonic function u on a neighborhood of B(a, r) satisfies u(x) 1 σ m 1 r S(a,r) r 2 x a 2 x y m u(y)dσ(y), for all x B(a, r). This imples the following Harnack inequality. Corollary 1.9. For any ε > 0, there exists a constant C = C(ε, m) such that sup u C B(a,(1 ε)r) u. S(a,r) for any subharmonic u 0 on a neighborhood of B(a, r). 1.3. Integrability properties. Proposition 1.10. Let v be a subharmonic function on an open subset in R m. Then v L p loc for 1 p < m m 2 and v Lp loc for 1 p < m m 1. For m = 2, we also have e cv L 1 loc near a given a C for c > 0 small enough. More precisely, this holds as soon as v({a}) < 2π c. Démonstration. We first note that the Newton kernel N satisfies these properties. Let χ be cut-off function, and U an open set on which χ 1. Then v N (χ v) is harmonic, and hence smooth, on U. The result now follows from Lemma 1.11. Lemma 1.11. Let χ : [0, + ) [0, + ) be a convex, nondecreasing function, and f a measurable function on R m with χ( f ) L 1 (R m ). Then f µ satisfies the same condition for each compactly supported probability measure µ on R m.
4 Démonstration. Since χ is non-decreasing and convex, we have ( ) χ ( f µ (x)) χ f(x y) dµ(y) χ ( f(x y) ) dµ(y) by Jensen s inequality, and hence ( ) χ( f µ (x))dx χ( f(x y) )dx dµ(y) = χ( f )(z)dz < +. 2.1. First properties. 2. Plurisubharmonic functions Definition 2.1. A function u : Ω [, + ) on an open subset of C n is plurisubharmonic (psh for short) if u is usc, not identically on any connected component of Ω, and the restriction of u to each complex line is subharmonic, i.e. u(a) 1 2π u(a + ξe iθ )dθ 2π 0 for all a Ω and ξ C n such that {a + zξ z D} Ω. Proposition 2.2. Let u : Ω [, + ). (i) If u is psh, then it is also subharmonic as a function of 2m real variables. (ii) If u is smooth, then u is psh iff its complex Hessian ( 2 u/ z j z k )1 j,k n is pointwise semipositive, i.e. i u 0. (iii) If u is psh and ρ ε is an in (1.5), then u ρ ε is smooth, psh, and decreases to u as ε 0. (iv) A locally integrable function v satisfies i v 0 in the sense of currents iff v coincides a.e. with a (unique) psh function. (v) f χ(t 1,..., t r ) is convex, non-decreasing in each variable, then χ(u 1,..., u r ) is psh for all u i psh.. In comparison with subharmonic functions, the key new property is that composing with a holomorphic function preserves plurisubharmonicity, as follows from (ii) and (iii). As a result, psh functions make sense on a complex manifold. Exemple 2.3. If f is a holomorphic function on a complex manifold, then log f is psh. The Lelong Poincaré formula states that dd c log f = δ D, the integration current on the divisor D = div(f). Here we have normalized d c so that dd c = i π. 2.2. Convexity properties. If u is independent of Im z, then u is psh iff it is convex as a function of Re z. In particular, a radial function function u(z) = u(r) is psh iff it is a convex function of log r. As a result, for each a Ω, ffl S(a,r) u and ffl B(a,r) u and sup B(a,r) u are convex functions of log r (generalized three-circle theorem)
NOTES COURS L2 2016 5 Definition 2.4. The Lelong number of u at a Ω is defined as sup B(a,r) u ν(u, a) := lim. r 0 log r Note that ν(u, a) 0 since u is locally bounded above. By convexity, we alternatively have (2.1) ν(u, a) = max{γ R + u(z) γ log z a + O(1) near a}. In other words, ν(u, a) is the vanishing order at a of e u. Proposition 2.5. The function f(r) := 1 r 2n 2 B(a,r) dd c u ( 1 2 ddc z 2) n 1 is non-decreasing, and converges to ν(u, a) as r 0. Recall that dd c = i π Démonstration. Using the expression λ = ( i 2 z 2) n /n! for the Lebesgue measure on C n, a computation shows that (2.2) 1 σ 2n 1 u λ = dd c u ( 1 2 ddc z 2) n 1. The Jensen formula of Corollary 1.4 may thus be written as (2.3) u u(a) = S(a,r) r 0 f(t) dt t. Since ffl S(a,r) u is a convex function of log r, it follows that f is non-decreasing, with lim f(r) = lim r 0 r 0 ffl S(a,r) u log r. Finally, the Harnack inequality of Corollary 1.9 shows that ffl S(a,r) (2.4) 0 u sup B(a,r) u C sup B(a,r/2) u sup B(a,r) u log r log r for 0 < r 1, which proves as desired that the left-hand side tends to 0. For later use, we prove Lemma 2.6. [BK, Lemma 5] Let u be a locally bounded psh function, and let ρ ε be a smoothing kernel as in (1.5). For any t > 0 we have ( ) locally uniformly wrt a. lim ε 0 (u ρ ε )(a) sup u B(a,tε) = 0
6 Démonstration. FI ME! By (1.6), it is enough to show that ffl S(a,r) u sup B(a,r) u tends to 0 locally uniformly wrt a. Using the Harnack inequality as in (2.4), this will follow from sup B(a,r) u sup B(a,r/2) u 0. Since log r sup B(a,r) u is convex and non-decreasing and u is locally bounded, we have for r r 0 sup B(a,r) u sup B(a,r/2) u which concludes the proof. log 2 log(2r 0 /r) ( sup u B(a,r 0 ) sup B(a,r/2) u ) C log(1/r), 2.3. Metrics and quasi-plurisubharmonic functions. Let L be a holomorphic line bundle on a complex manifold, and let be a (Hermitian) metric on L. Locally, L admits a trivializing (i.e. nowhere zero) section τ, and h is locally determined by the smooth function φ τ := log τ. Every other local trivializing section is of the form τ = uτ with u O, and we have the obvious compatibility (2.5) φ τ = φ τ log u. Conversely, any collection φ = (φ τ ) of local smooth functions satisfying (2.5) defines a metric φ on L. Indeed, every local section s of L is of the form s = fτ, and s φ := f e φτ is independent of the choice of τ. We shall simply say that φ itself is a smooth metric. This amounts to a convenient additive notation for metrics on line bundles : if φ, φ are smooth metrics on L, L, then φ+φ is smooth metric on L+L := L L, and φ = ( φ τ ) is a smooth metric on L := L. In particular, if φ, φ are smooth metrics on the same L, then φ φ = (φ τ φ τ ) is a smooth metric on the trivial line bundle O, and hence can be identified with a globally defined function in C (). In other words, the space of smooth metrics on L is an affine space modeled with underlying vector space C (). The curvature form of a smooth metric φ on L, denoted by dd c φ, is the global closed, real (1, 1)-form locally defined by dd c φ := dd c φ τ. This is indeed globally defined since log u in (2.5) is pluriharmonic. If φ is a smooth metric on L, then dd c (φ + φ ) = dd c φ + dd c φ. This convenient but possibly confusing notation should be used with care : dd c φ is of course not exact in general. Indeed, our normalization of d c is chosen so that θ represents the first Chern class in de Rham cohomology, i.e. the image of c 1 (L) H 2 (, Z) in H 2 (, C). We can similarly define a continuous (or more singular) metric φ on L as a collection of functions (φ τ ) satisfying (2.5). In particular, a psh metric φ on L is a singular metric such that φ τ is psh for each τ. The curvature current dd c φ is the globally defined closed positive (1, 1)-current locally given by dd c φ = dd c φ τ. It is convenient to interpret psh metrics on L as functions, which is accomplished as follows. Definition 2.7. A function ϕ : [, + ) on a complex manifold is quasipsh is ϕ is locally the sum of a psh function and a smooth function. If θ is a closed real (1, 1)-form, we say that ϕ is θ-psh is θ + dd c ϕ 0 in the sense of currents. We denote by PSH(, θ) the set of θ-psh functions.
NOTES COURS L2 2016 7 Note that PSH(, θ) identifies with a closed convex subset of L 1 loc (). Since θ is closed, locally we have θ = dd c ψ with ψ C, and ϕ is θ-psh iff u := ϕ + ψ is psh. Lemma 2.8. If L is a (holomorphic) line bundle and φ 0 is a smooth metric on L with curvature form θ := dd c φ 0, then φ φ φ 0 is a bijection between psh metrics on L and θ-psh functions on. The notion of Lelong number makes sense for quasi-psh functions. Definition 2.9. If ϕ is quasi-psh, we define its Lelong number ν(ϕ, a) at a as that of u = ϕ + ψ in some local coordinates centered at a. Since any two local coordinates centered at a are Lipschitz equivalent, (2.1) shows that the definition is independent of the choice of local coordinates. 2.4. Regularization of quasi-plurisubharmonic functions. Locally, quasipsh functions can easily be easily regularized by convolution with a smoothing kernel. Theorem 2.10. Let ϕ be quasi-psh function on a complex manifold, and pick a reference positive (1, 1)-form ω. Let ρ j := ρ 1/j be a smoothing kernel on C n as in (1.5), and denote by ϕ z ρ j the (locally) defined smooth function obtained by convolution in a given local coordinate system z centered at a point a. Then the following holds : (i) ϕ z ρ j + c j decreases pointwise to ϕ for some positive constants c j 0 ; (ii) given a continuous (1, 1)-form γ such that dd c ϕ γ, we have dd c (ϕ z ρ j ) γ ε j ω with ε j 0. (iii) if ϕ is locally bounded and z is another local coordinate system centered at a, then ϕ z ρ j ϕ z ρ j 0 locally uniformly. Démonstration. Locally there exists ψ C such that u := ϕ + ψ is psh. By Proposition 2.2 u z ρ j decreases pointwise to u, and (i) follows since ψ z ρ j ψ locally uniformly by continuity of ψ. The coefficientwise convolution γ z ρ j satisfies dd c (ϕ z ρ j ) γ z ρ j, and (ii) follows since γ z ρ j γ locally uniformly. To see (iii), write as before u = ϕ+ψ. Let B(a, ε) and B (a, ε) be the Euclidian balls in the respective coordinates. Since z and z are locally Lipschitz equivalent, we have B(a, C 1 ε) B (a, ε) B(a, Cε) for some locally uniform C > 0. By Lemma 2.6, it follows that u z ρ ε u z ρ ε converges locally uniformly to 0, and the same holds true for ϕ by continuity of ψ. In general, global regularization of quasi-psh functions can however not be achieved with an arbitrary small loss of positivity as in (ii). Exemple 2.11. Let E be the exceptional divisor of the blow-up of a point on a compact Kähler surface (, ω). Recall that its cohomology class [E] H 2 () satisfies [E] 2 = 1. Decompose the integration current on E as [E] = θ + dd c ϕ with θ as smooth closed (1, 1)-form, and suppose that ϕ can be written as the
8 distributional limit of smooth functions ϕ j C () such that θ+dd c ϕ j ε j ω. Then 0 (θ j + ε j ω) 2 = ([E] + ε j [ω]) 2 [E] 2 = 1, a contradiction. The following global regularization result is originally due to Demailly [Dem92] ; we present here is the elementary approach of [BK]. Theorem 2.12. Let ϕ be a locally bounded quasi-psh function on a complex manifold, and pick a reference positive (1, 1)-form ω. Then each compact subset of admits a neighborhood Ω on which ϕ can be written as the decreasing limit of a sequence ϕ j C (Ω) with the following property : given a continuous (1, 1)- form γ on such that dd c ϕ γ, we have dd c ϕ γ ε j ω on Ω for some sequence ε j 0. Corollary 2.13. Let θ be a semipositive (1, 1)-form and ϕ be an arbitrary θ- psh function. Then each compact subset of admits an open neighborhood Ω on which ϕ may be written as the pointwise limit of a decreasing sequence of smooth functions ϕ j C (Ω) such that θ + dd c ϕ j ε j ω. Corollary 2.14. Assume that is Stein. Let θ be an arbitrary (1, 1)-form and ϕ be a θ-psh function. Then each compact set of admits an open neighborhood Ω on which ϕ can be written as the decreasing limit of a sequence of smooth θ-psh functions. We begin with some simple facts on regularized max functions. Lemma 2.15. Suppose that χ : R p R is convex, non-decreasing in each variable and such that χ(t 1 + c,..., t p + c) = χ(t 1,..., t p ) + c for each c R. If ϕ 1,..., ϕ p are quasi-psh functions with dd c ϕ j γ for some continuous (1, 1)-form γ, then dd c χ(ϕ 1,..., ϕ p ) γ. Démonstration. Locally we can find ψ such that dd c ψ + εγ dd c ψ. Since each ϕ j ψ is psh, (v) in Proposition 2.2 implies that is psh, and the result follows. χ(ϕ 1 ψ,..., ϕ p ψ) = χ(ϕ 1,..., ϕ p ) ψ Definition 2.16. Given a non-negative function θ C (R) with supp θ [0, 1] and θ(t)dt = 1, we define a regularized max function max ε {x 1,... x p } as the convolution product max ε {x 1,..., x p } = max{x 1 + εt 1,..., x p + εt p }θ(t i )dt i. t [0,1] p The following properties are trivially satisfied : (i) max ε is smooth and convex on R p, non-decreasing in each variable, and for each c R. max ε {x 1 + c,..., x p + c} = max ε {x 1,..., x p } + c
(ii) If x i x j ε for some i j, then NOTES COURS L2 2016 9 max ε {x 1,..., x p } = max ε {x 1,..., x i,..., x p }. (iii) max ε decreases to max as ε 0, and max{x 1,..., x p } max ε {x 1,..., x p } max{x 1,..., x p } + ε. Proof of Theorem 2.12. Choose a finite open cover (V α ) of K by coordinate charts of containing open subsets U α U α V α with (U α) still covering K. For each α, Theorem 2.10 yields a decreasing sequence of smooth functions ϕ α,j on a neighborhood of U α, decreasing pointwise to ϕ, and with the following properties : (i) for each continuous (1, 1)-form γ such that dd c ϕ γ, we have dd c ϕ α,j γ δ j ω with δ j 0 ; (ii) sup Uα U β ϕ α,j ϕ β,j ε j 0. Following the Richberg patching procedure, choose a cut-off function χ α C c (U α ) with 0 χ α 1 and χ α 1 on a neighborhood of U α, and set for each z ϕ j (z) := max εj {ϕ α,j (z) + 2ε j (z)χ α (z) z U α }. Replacing ε j with sup k j ε k, we may assume that ε j is decreasing, and the monotonicity properties of the regularized max then imply that ϕ(z) ϕ j+1 (z) ϕ j (z) max{ϕ α,j (z) + 2ε j χ α (z) z U α } + ε j, which proves that ϕ j (z) decreases to ϕ(z). We next show that each z 0 K admits a neighborhood V on which (2.6) ϕ j = max εj {ϕ α,j + 2ε j χ α α A 0 } for a fixed finite set of indices A 0. Choosing C > 0 such that dd c χ α Cω near K, (2.6) will show that ϕ j is smooth and satisfies dd c ϕ γ = dd c ϕ j γ (δ j + 2Cε j )ω, thanks to Lemma 2.15. Let A 0 be the set of α with z 0 U α, and pick α A 0 such that z 0 U α. Let V U α be a neighborhood of z 0 such that V U β = for each β with z 0 / U β and V supp χ β = for each β with z 0 U β. For z V and β with z U β, we then have either z 0 U β, i.e. β A 0, or z 0 U β. In the latter case, χ β (z) = 0 while χ α (z) = 1, and the definition of δ j ε j yields ϕ β,j (z) + 2ε j χ α (z) = ϕ β,j (z) ϕ α,j (z) + ε j = ϕ α,j (z) + 2ε j χ α (z) ε j. By property (ii) of the regularized max, β does not contribute to ϕ j (z), and (2.6) therefore holds on V. Proof of Corollary 2.13. Assume that ϕ is an arbitrary θ-psh function with θ 0. Observe first that a decreasing sequence ϕ j of quasi-psh functions with dd c ϕ j Cω converges pointwise to ϕ iff ϕ j ϕ in L 1. Indeed, the latter condition implies that the two quasi-psh functions inf j ϕ j and ϕ coincide a.e., and hence are equal. The bounded functions ψ j := max{ϕ, j} are also θ-psh, and decrease pointwise to ϕ as j +. We are going to construct by induction on j a decreasing sequence of smooth functions ϕ j on a neighborhood of Ω such that
10 (a) ϕ j > ψ j ; (b) ϕ j ψ j L 1 j 1 ; (c) θ + dd c ϕ j j 1 ω. For j = 1, we get ϕ 1 by applying Step 1 to ψ 1. Suppose that ϕ j has been constructed. By Theorem 2.12, ψ j+1 is the decreasing limit of a sequence (ψ j+1,k ) k such that θ + dd c ψ j+1,k ε j+1,k ω with lim k ε j+1,k = 0 on some neighborhood of K. Since ϕ j > ψ j, each point x K satisfies ψ j+1,k (x) < ϕ j (x) for some k. By continuity of ψ j+1,k ϕ j, the compact set K therefore admits a neighborhood on which ψ j+1,k < ϕ j for some (and hence all) k 1. Setting ϕ j+1 := ψ j+1,k + k 1 with k 1 completes the induction. Since ψ j ϕ in L 1, (b) yields ϕ j ϕ in L 1, and hence pointwise on thanks to the observation above. Proof of Corollary 2.14. Let ϕ an arbitrary θ-psh on Stein. Let τ : R be a smooth strictly psh exhaustion function. Replacing with a neighborhood of K, we may assume that θ + Cdd c τ ω on for some C 1. By the case just treated, the (θ + Cdd c τ)-psh function ϕ Cτ may be written as the decreasing limit of a sequence of smooth functions ψ j on a neighborhood Ω of K with θ + Cdd c τ + dd c ψ j (1 ε j )ω ; setting ϕ j := ψ j + Cψ yields the desired approximation of ϕ. Remark 2.17. In the setting of the last proof, assume further given a (non necessarily positive) (1, 1)-form γ such that θ + dd c ϕ γ. We may then choose C 1 such that θ + Cdd c τ ω on a neighborhood of K. The approximants ϕ j will then also satisfy θ + dd c ϕ j γ. 3. Basic analysis of first order differential operators 3.1. Differential operators. We use [Dem1, 2] as a reference for what follows. Let M be a smooth manifold and E, F be smooth, complex vector bundles on M. A linear differential operator of order (at most) δ is a C-linear map A : C (M, E) C (M, F ) such that for any coordinate chart in which E, F are trivialized, we have (3.1) A = a α D α α N m, α δ with a α Hom(E, F ). More invariantly, the space of differential operator of order δ can be written as D δ (M; E, F ) = C (M, Hom(J δ E, F )) with J δ E = JM δ E the vector bundle of δ-jets of sections of E. The jet exact sequence 0 S δ TM JM d J δ 1 M 0 gives rise to an isomorphism D δ (M; E, F )/D δ 1 (M; E, F ) C (M, S δ T M Hom(E, F )).
NOTES COURS L2 2016 11 The image σ A C (M, S δ T M Hom(E, F )) of A D δ (M; E, F ) is called the principal symbol of A, and is more concretely described as the Hom(E, F )-valued homogeneous of degree δ on TM σ A (ξ) = a α ξ α when A is written in the form (3.1). α =δ Proposition 3.1. Let A : C (M, E) C (M, F ) be a differential operator of order δ. (i) If A : C (M, F ) C (M, G) is a differential operator of order δ, then A A is a differential operator of order δ + δ, and for each ξ TM we have σ A A(ξ) = σ A (ξ)σ A (ξ). (ii) For δ = 1, we may view σ A is an Hom(E, F )-valued vector field, and we have the commutator relation (3.2) [A, f]u := A(fu) fau = σ A (df)u = (σ A f)u for each f C (M) and u C (M, E). In particular, the space of order 1 differential operators E F with a given (degree 1) symbol is an affine space modeled on the space of bundle homomorphisms. Exemple 3.2. A connection on E induces a differential operator of order 1 with principal symbol ξ. D : C (M, Λ T M E) C (M, Λ T M E), 3.2. Adjoint. Now assume given a smooth volume form dv on M and Hermitian scalar products on E and F. If A : C (M, E) C (M, F ) is a differential operator of order δ, its formal adjoint A : C (M, F ) C (M, E) is defined by requiring (3.3) Au, v L 2 = u, A v L 2 for all u C (M, E), v C (M, F ) such that u (or v) has compact support. Lemma 3.3. The adjoint A is a linear differential operator of order δ and principal symbol σ A = ( 1) δ σ A. Further, given a section v C (M, F ), the value of A v at a given point only depends on the δ-jet of dv and the metrics of E and F. Démonstration. By linearity and a partition of unity, we may assume that E, F are trivial and A = ad α with a Hom(E, F ). Denote by h E, h F the matrices representing the scalar product of E, F and write dv = fdx with f > 0. Then Au, v L 2 = ad α u, v F dv = D α u, a v E dv = D α ( t u)h E a vfdx = ( 1) d t u D α ( h E a vf ) dx,
12 and hence A v = ( 1) δ t h 1 E Dα (t h E a vf ) f 1 = ( 1) δ a D α v + l.o.t. 3.3. Friedrich s lemma for complete metrics. A differential operator A D δ (M; E, F ) acts in a natural way on distributional sections, and (3.3) remains valid when u is a distribution and v Cc. In particular, A gives a densely defined, closed operator (i.e. with closed graph) L 2 (M, E) L 2 (M, F ), with u Dom A iff Au L 2. However, in general (3.3) does not extend to all u Dom A, v Dom A (i.e. Dom A does not coincide with the domain of the Hilbert space adjoint of A). Exemple 3.4. Let I R be a bounded open interval endowed with the Euclidian metric, and A = d/dx. Then A = A, Dom A = Dom A is the Sobolev space W 1,2 (I) C 0 (Ī), and Au, v = u, A v + [uv] I for u, v W 1,2. Theorem 3.5. If the metric g is complete, then any u L 2 (M, E) can be written as the L 2 -limit of a sequence of test sections u j Cc (M, E) with the following property : for each first order linear differential operator A D 1 (M; E, F ) with bounded symbol σ A, Au L 2 (M, F ) implies Au j Au in L 2. The regularizing sequence u j will obtained by truncation and local convolution. Lemma 3.6. Let (M, g) be a Riemannian manifold, possibly with boundary, fix p [1, ], and consider the following conditions : (i) M admits a smooth exhaustion function ψ : M R with dψ L p ; (ii) there exists an exhaustion by compact sets K j K j+1 M and a sequence of cut-off functions χ j Cc ( K j+1 ) with 0 χ j 1, χ j 1 on a neighborhood of K j, such that dχ j L p 0 ; (iii) there exists K j, χ j as in (ii) with dχ j bounded in L p. Then (i)= (ii)= (iii) ; when p > 1, we conversely have (iii)= (i). For p =, these conditions are also equivalent to the completeness of g. The case of M = (0, 1] shows that (iii)= (i) fails in general for p = 1. A Riemannian manifold (M, g) satisfying the above conditions with p = 2 is classically called parabolic, cf. [Gla83] and the references therein for more information. Démonstration. Suppose given ψ as in (i), and pick a smooth function cut-off function θ Cc (R) with 0 θ 1, θ 1 near 0. Then χ j (x) = θ(j 1 ψ(x)) satisfies (ii). (ii)= (iii) is trivial. Let χ j as in (iii) and p > 1. Then ψ := j 1 j 1 (1 χ j ) is a smooth exhaustion function, since we have ψ S j outside K j+1 with S j = j k=1 k 1 +. Since the supoorts of dχ j are disjoint, we have dψ p dv j p dχ j p < +, M j 1 M which proves (i).
NOTES COURS L2 2016 13 Fix a basepoint x 0 M. By the Hopf-Rinow Lemma, g is complete iff d(, x 0 ) is an exhaustion function. If ψ is a smooth exhaustion function with bounded gradient, then ψ(x) ψ(x 0 )+Cd(x, x 0 ), and d(, x 0 ) is thus exhaustive. Conversely, if the Lipschitz continuous function d(, x 0 ) is exhaustive, then a regularization argument shows the existence of a smooth exhaustion function with bounded gradient. Proof of Theorem 3.5. By Lemma 3.6, we may find an exhausting sequence of cut-off functions χ j Cc (M) with dχ j g 1. For each u L 2 (M, E), we have χ j u u in L 2. Further, (3.2) yields A(χ j u) = χ j Au + σ A (dχ j )u with σ A (dχ j )u C u since dχ j g 1 and σ A is bounded. By dominated convergence, we thus have σ A (dχ j )u 0 in L 2, and hence A(χ j u) Au in L 2 whenever the latter is in L 2. We are thus reduced to the case where u has compact support. Using a partition of unity, we can further assume that u is compactly supported in a coordinate chart. We then conclude using Lemma 3.7 below, usually known as Friedrich s lemma. Lemma 3.7. Pick ρ Cc (R m ) with ρ(x)dx = 1, and set ρ ε (x) = ε m ρ(ε 1 x). Let also A : C (R m ) C (R m ) be a first order linear differential operator with smooth, Lipschitz continuous coefficients. For each f L 2, we then have A(f ρ ε ) (Af) ρ ε 0 in L 2 as ε 0. Démonstration. The result is clearly true if f is a test function, since f ρ ε f in C. By density of test funtions in L 2, it is thus enough to prove the existence of C > 0 such that A(f ρ ε ) (Af) ρ ε L 2 C f L 2 for all f L 2. To do this, we may assume by linearity that A = a j where a C has bounded gradient. Write (a j f) ρ ε = (af) j ρ ε (( j a)f) ρ ε. Since L 2 L 1 L 2, the second term satisfies (( j a)f) ρ ε L 2 ( j a)f L 2 ρ ε L 1 C f L 2. Next, ((af) j ρ ε ) (x) a(x)( j f ρ ε )(x) = (a(x y) a(x))f(x y)( j ρ ε )(y)dy yields (af) j ρ ε a( j f ρ ε ) C f (x j ρ ε ), whose L 2 norm is bounded above by a uniform multiple of f L 2 since x j ρ ε L 1 = x j ρ L 1 is bounded.
14 3.4. Removable singularities for first order PDEs. Proposition 3.8. Let A D 1 (M; E, F ) be a first order differential operator, u L 2 loc (M, E), v L1 loc (M, F ), and assume that Au = v holds outside a closed submanifold N M of codimension at least 2. Then Au = v on M. Lemma 3.9. If M is a compact Riemannian manifold with boundary and N M is a closed submanifold of codimension at least 2, then M \ N is parabolic, i.e. it satisfies the equivalent condition of Lemma 3.6 with p = 2. Démonstration. Using a partition of unity, we are reduced to a local statement. We may then choose local coordinates x = (x, x ) R n R m such that N = {x = 0}. Pick any smooth function χ : R R + with χ = 0 near 0, χ 1 on [1, + ), and set χ j (x) := χ ( j x ). Then dχ j 2 dv Cj 2 Vol( x j 1 ) Cj 2 n 0 since n 2 by assumption. Proof of Proposition 3.8. Let χ j be a sequence of cut-off functions as in (ii) of Lemma 3.6, and write A(χ j u) = χ j v + σ A (dχ j )u. We have χ j u u and χ j v v in L 1 loc, which implies A(χ ju) Au in the sense of distributions. Further, σ A (dχ j ) L 2 0 since σ A is bounded and dχ j 0 in L 2. Since u L 2 loc, it follows that σ A (dχ j )u 0 in L 1 loc, and we are done. More generally, we prove : m Proposition 3.10. Let p (1, m] with conjugate exponent q [ m 1, ). Let A D 1 (M; E, F ) be a first order differential operator, u L q loc (M, E), v L 1 loc (M, F ), and assume that Au = v holds outside a closed subset F M of finite (m p)-hausdorff measure. Then Au = v on M. The Heaviside function shows that result fails in general for p = 1. As before, Proposition 3.10 will be a direct consequence of the following result. Lemma 3.11. Let M be a compact Riemannian manifold with boundary, and let F M be a closed subset of (m p)-dimensional Hausdorff measure for some p [1, m]. Then M \ F admits an exhausting sequence of cut-off functions with gradient bounded in L p, as in (iii) of Lemma 3.6. Démonstration. Following [EG92, p.154, Theorem 3], we are going to show that any compact set K M \ F, we can find a cut-off function χ Cc (M) with 0 χ 1, χ 1 on a neighborhood of K, and χ L p C for a uniform constant C > 0. Since F is compact and has finite H m p -measure, there exists a constant C > 0 such that F can be covered by finitely many open balls B(x i, r i ) of radius r i 1 such that i rm p i C. Let θ : R R be the piecewise affine function defined by θ(t) = 0 for t 1, θ(t) = t 1 for 1 t 2, and χ(t) = 1 for t 2, and consider the Lipschitz continuous function χ i (x) = θ ( ri 1 d(x, x i ) ).
NOTES COURS L2 2016 15 Then χ := max i χ i is also Lipschitz continuous, it has compact support in M \ F, and χ 1 on a neighborhood of K. Further, we have dχ max i a.e. on M, hence dχ p dv C M i dχ i max r 1 i i 1 {x ri d(x,x i ) 2r i } r p i Vol B(x i, 2r i ) C i r m p i C with C > 0 independent of K. This proves the claim, after regularizing χ near the boundary of {χ = 1}. 4. Fundamental identities of Kähler geometry 4.1. Some Hermitian algebra. Let V C n be a complex vector space, and denote by Λ = p,q N Λp,q the corresponding bigraded exterior algebra. The graded commutator of graded endomorphisms A, B of Λ is defined by It satisfies the graded Jacobi identity [A, B] = AB ( 1) A B BA (4.1) [[A, B], C] = [A, [B, C]] ( 1) A B [B, [A, C]], which expresses that A [A, ] is a graded Lie algebra homomorphism. For each (constant) differential form η on V, we also denote by η the associated endomorphism η of Λ. Note that [η, η ] = 0 for any two η, η Λ. Now suppose that V is equipped with a Hermitian scalar product, and denote by ω Λ 1,1 the corresponding positive (1, 1)-form. Given an orthonormal basis (ξ i ) of Λ 1,0 V, (ξ I ξ J ) I =p, J =q is an orthonormal basis of Λ p,q, and (4.2) ω = i j ξ j ξ j. For η Λ p,q, denote by η : Λ Λ the adjoint of η, which has bidegree ( p, q). As we shall see, the following result contains the linear algebra part of the Kähler commutation identities. Lemma 4.1. For each ξ Λ 1,0 V, the following identities hold : (i) [ξ, ω] = i ξ ; (ii) [ξ, ω ] = i ξ ; (iii) [ω, ξ ] = iξ ; (iv) [ω, ξ] = iξ. Démonstration. A direct check shows that ξ is given by contraction against the dual vector ξ # V. As a result, ξ is an anti-derivation of Λ, i.e. (4.3) ξ (α β) = ξ (α) β + ( 1) α α ξ (β), or, equivalently, [ξ, α] = ξ (α). (ii) follows from ξ (ω) = i ξ, which is easily seen using (4.2). Applying adjunction and conjugation yields the remaining identities.
16 Now fix a real (1, 1)-form θ Λ 1,1. The operator [θ, ω ] will play a crucial role in the L 2 estimates for, and we therefore study it in some detail. Proposition 4.2. [Dem82, Lemme 3.2] Assume that θ Λ 1,1 is positive, and fix an integer q 1. (i) The endomorphism [θ, ω ] = θω of Λ n,q is positive definite. More precisely, θ εω implies [θ, ω ] qε id on Λ n,q V ; (ii) for each u Λ n,q, the [θ, ω ] 1 u, u dv ω is a non-increasing function of ω and θ ; (iii) for u Λ n,1, we have [θ, ω ] 1 u, u dv ω = i n2 {u, ū} θ, where {u, ū} θ denotes the the image of iu ū under the contraction morphism Λ n,1 V Λ 1,n V Λ 1,1 Λ n,n V tr θ Λ n,n V. Proof of Proposition 4.2. Setting ξ := ξ 1... ξ n, a direct check shows that (4.4) θω (ξ ξ J ) = ( j J λ j )ξ ξ J, which implies (i). Pick u Λ n,1 V, and write u = ξ v with v = j v j ξ j in Λ 0,1. By (4.4), we have [θ, ω ] 1 u, u = j λ 1 j v j 2 = tr θ (iv v), and (ii) follows since dv ω = j iξ j ξ j = i n2 ξ ξ. Monotonicity with respect to θ in (iii) is clear, in view of (4.4). Observe that the quadratic form q ω (u) := [θ, ω ] 1 u, u ω satisfies u, v ω 2 (4.5) q ω (u) = sup 0 v Λ n,q V θω. v, v ω Now let ω ω be another positive (1, 1)-form with eigenvalues µ i 1 with respect to ω, and let (ζ j ) be an ω-orthonormal basis of Λ 1,0 V such that ω = j µ jζ j ζ j (which is however not necessarily diagonalizing θ). Then dv ω = µ 1... µ n dv ω, and our goal is thus to show that (4.6) q ω (u)µ 1... µ n q ω (u). Setting µ = diag(µ j ), we have ω = µ 1/2 ω µ 1/2, and (4.5) yields q ω (u) = u, v ω 2 sup 0 v Λ n,q V θω v, v ω Using µ j 1, one easily sees that = sup 0 v Λ n,q V u, v ω 2 θµ 1/2 ω µ 1/2 v, v ω. θµ 1/2 ω µ 1/2 v, v ω µ 1... µ n θω v, v ω,
NOTES COURS L2 2016 17 which yields as desired (4.6). 4.2. Kähler identities. In what follows, (, ω) is a Kähler manifold, and E is a Hermitian holomorphic vector bundle on, with Chern connection E + E. We start with the existence of local normal forms for ω and E. Lemma 4.3. Near each point of, we may find : (i) holomorphic coordinates (z 0,..., z n ) such that, ω = δ jk + O( z 2 ). z j z k (ii) a holomorphic basis (e j ) of E such that e j, e k E = δ jk + O( z 2 ). Conversely, note that a Hermitian metric ω admitting local holomorphic coordinates as in (i) is closed, and hence Kähler. Démonstration. Locally, we have ω = dd c ϕ for a smooth (strictly psh) function ϕ. Choose holomorphic coordinates w such that ω = j idw j d w j at 0. The Taylor expansion of ϕ at 0 then writes ϕ = w 2 + Re j P j (w) w j + Re P (w) + O( w 4 ) where each P j (w) is a holomorphic quadratic polynomial, and P (w) is a holomorphic polynomial of degree at most 3. Setting z j = w j 1 2 P j(w) yields (i). The proof of (ii) is similar and left to the reader. Theorem 4.4. Assume that (, ω) is Kähler, let E be a Hermitian holomorphic vector bundle on, with Chern connection E + E. Then [ E, ω] =i E [ E, ω] = i E [ω, E ] = i E [ω, E ] =i E. Démonstration. Since the adjoint of a first order operator only depends on the 1-jets of the metrics involves, the normal forms of Lemma 4.3 reduces us to the case of C n with its Euclidian metric and E the trivial line bundle. In that case, the first order operators involved have constant coefficients. It is thus enough to prove that their symbols coincide, which is the content of Lemma 4.1. The key consequence for us is the following Bochner-type formula, which goes back to the work of Kodaira and Nakano. Corollary 4.5. [Bochner-Kodaira-Nakano identity] With the previous notation, the Laplacians of E and E satisfy E = E + [θ E, ω ] with θ E C (, Λ 1,1 T Hom(E, E)) the curvature tensor.
18 Démonstration. The above Kähler identities combined with the Jacobi identity (4.1) yield E = [ E, E ] = i[[ E, ω ], E ] = i[ E, [ω, E ]] i[ω, [ E, E ]] = [ E, E] + i[[ E, E ], ω ] = E + i[θ E, ω ]. As a first application, we obtain the following classical cohomology vanishing result. Theorem 4.6 (Kodaira vanishing). If L is a Hermitian holomorphic line bundle with positive curvature form θ L > 0, then for each q 1. H n,q (, L) = H q (, K + L) = 0 Démonstration. By Hodge theory, it is enough to show that for every L-valued (n, q)-form u with L u = 0 vanishes. The Bochner-Kodaira-Nakano equality yields 0 = L u 2 + Lu 2 + [θ L, ω ]u, u L 2. By Proposition 4.2, the operator [θ, ω ] is positive definite on (n, q)-forms, hence the result. 5. L 2 -estimates for the -equation Hörmander, Adnreotti-Vesentini, Skoda, Demailly. 5.1. The smooth, complete case. Theorem 5.1. Let (, ω) be a Kähler manifold, and let be L a Hermitian holomorphic line bundle on with positive curvature form θ > 0. Assume also that admits a complete Kähler metric (not necessarily equal to ω or θ). If satisfies v = 0 and v L 2 loc (, Λn,q T L) [θ, ω ] 1 v, v dv ω < +, then v = u for some u L 2 (, Λ n,q 1 T L) such that u 2 L = u 2 dv 2 ω [θ, ω ] 1 v, v dv ω. If v is smooth, then u can be chosen smooth. Recall that the endomorphism [θ, ω ] = θω of Λ n,q T is positive definite, by Proposition 4.2.
NOTES COURS L2 2016 19 Remark 5.2. For q = 1, the result does in fact not involve ω, since and [θ, ω ] 1 v, v dv = i n2 {v, v} u 2 dv = i n2 u ū. As the next result shows, the assumption on is satisfied by a fairly general class of Kähler manifolds. Lemma 5.3. Assume that is weakly pseudoconvex, i.e. admits a smooth, psh exhaustion function. Then admits a complete Kähler metric. In particular, any Stein manifold admits a complete Kähler metric. Démonstration. Let ϕ : R be a smooth, psh exhaustion function. Since ϕ is bounded below, we may assume that ϕ 0. Then i (ϕ 2 ) = 2ϕ i ϕ + 2i ϕ ϕ shows the gradient of ϕ is bounded with respect to the Kähler metric ω+i (ϕ 2 ), which is thus complete. The key tool in the proof of Theorem 5.1 the following consequence of the Bochner-Kodaira-Nakano identity. Lemma 5.4. If ω is complete, then each u L 2 (, Λ n,q T L) with u, u L 2 satisfies u 2 L + u 2 2 L [θ, ω ]u, u dv. 2 Démonstration. The symbol of : C (, Λ n,q T L) C (, Λ n,q+1 T L) is given by σ (ξ) = ξ, and is thus bounded. The symbol of its adjoint σ (ξ) = σ (ξ) is thus also bounded, and Theorem 3.5 yields a sequence u j Cc (, Λ n,q T L) with u j u, u j u and u j u in L 2 norm. After passing to a subsequence, we may assume that u j u a.e. and hence [θ, ω ]u, u dv lim inf [θ, ω ]u j, u j dv, j by Fatou s lemma. For each j, the Bochner-Kodaira-Nakano identity implies u j 2 L + u 2 j 2 L [θ, ω ]u 2 j, u j dv, and we get the desired inequality in the limit. Proof of Theorem 5.1. Assume first that ω itself is complete. The existence of u clearly implies (5.1) w, v 2 C w 2 L 2
20 for all test forms w Cc (, Λ n,q T L). Assume conversely that this estimate holds. Then w w, v defines a continuous linear form of norm at most C 1/2 on Cc (, Λ n,q T L) L 2 (, Λ n,q 1 T L). By the Hahn-Banach theorem, this linear form admits a continuous extension to L 2 (, Λ n,q 1 T L) of norm at most C1/2. By the Riesz representation theorem, we may thus find u L 2 (, Λ n,q 1 T L) with u L 2 C1/2 such that w, v = w, u = w, u for all test forms w, and hence u = v. In order to establish (5.1), consider the orthogonal decomposition w = w + w in L 2 (, Λ n,q 1 T L) with w ker and w (ker ). For each test form η Cc (, Λ n,q 2 T L), η ker implies w, η = 0, i.e. w = 0 in the sense of distributions. Since v = 0, the Cauchy-Schwarz inequality Lemma 5.4 yield ( ) ( ) v, w 2 = v, w 2 [θ, ω ] 1 v, v dv [θ, ω ]w, w dv C w 2 = C w 2, which proves (5.1). Next, assume that ω is arbitrary, and let ω be a complete Kähler metric. For each ε > 0, ω ε := ω + εω is then complete as well. The monotonicity property of Proposition 4.2 implies [θ, ωε] 1 v, v ωε dv ωε C, and the first part of the proof yields the existence of u ε L 2 loc (, Λn,q 1 T L) with u ε = v and u ε 2 ω ε dv ωε C. In particular, u ε remains bounded in L 2 loc, and an easy weak compactness argument yields the desired solution u. To see the final point, note that we can always replace u by its orthogonal projection in (ker ). As above, this implies u = 0 and hence u = v in the sense of distributions. If v is smooth, then u is smooth by ellipticity of the Laplacian. 5.2. The singular case. Theorem 5.5. Let (, ω) be a Kähler manifold, and assume that contains a Stein Zariski open subset. Let L be a Hermitian holomorphic line bundle on with curvature form θ, and let also ϕ be a quasi-psh function such that θ+i ϕ η > 0 in the sense of currents for some positive (1, 1)-form η. If v L 2 loc (, Λn,q T L) satisfies v = 0 and [η, ω ] 1 v, v e 2ϕ dv ω < +,
NOTES COURS L2 2016 21 then v = u for some u L 2 (, Λ n,q 1 T L) such that u 2 e 2ϕ dv ω [η, ω ] 1 v, v e 2ϕ dv ω. Note that θ is not assumed to be positive here. The current θ + dd c ϕ should be interpreted as the curvature of the singular metric L e ϕ. As in Remark 5.2, the result does in fact not involve ω when q = 1. Lemma 5.6. If u, are L 2 loc L-valued forms on a complex manifold such that u = v outside a closed analytic subset A, then u = v on. Démonstration. Arguing by induction on dim A, it is enough to prove that u = v on \A sing, which follows from Proposition 3.8 since A reg is a closed submanifold of \ A sing, of real codimension at least 2. Proof of Theorem 5.5. By Lemma 5.6, we may assume that is Stein. By Corollary 2.14, we may then find an exhaustion of by weakly pseudoconvex open subsets Ω j such that ϕ Ωj is the decreasing limit of a sequence ϕ j,k C (Ω j ) with θ + dd c ϕ j,k η. By Lemma 5.3, Ω j admits a complete Kähler metric, and Theorem 5.1 yields the existence of u j,k L 2 (Ω j, Λ n,q 1 TΩ j L) such that u j,k = v on Ω j and u j,k 2 e 2ϕ j,k dv ω Ω j [η, ω ] 1 v, v e 2ϕ j,k dv ω Ω j C := [η, ω ] 1 v, v e 2ϕ dv ω, By monotonicity of (ϕ j,k ) k, we get for k l Ω j u j,k 2 e 2ϕ j,ldv ω C, which shows in particular that (u j,k ) k is bounded in L 2 (Ω j, e 2ϕ j,l). After passing to a subsequence, we thus assume that that u j,k converges weakly in L 2 (Ω j, e 2ϕ j,l) to u j, which may further be asumed to be the same for all l, by a diagonal argument. We then have u j = v, and Ω j u j 2 e 2ϕ j,ldv ω C for all l, hence Ω j u j 2 e 2ϕ dv ω C by monotone convergence of ϕ j,l ϕ. By a diagonal argument, we may finally arrange that u j u weakly in L 2 (K, e 2ϕ ) for each compact K, and we obtain the desired solution. Corollary 5.7. Let (, ω) be a Kähler manifold, and assume that contains a Stein Zariski open subset. Let L be a Hermitian holomorphic line bundle on with curvature form θ, and let also ϕ be a quasi-psh function such that θ+i ϕ εω (resp. θ + i ϕ + Ric(ω) εω > 0) in the sense of currents for some positive (1, 1)-form η. If v L 2 (, Λ n,1 T L) (resp. v L2 (, Λ 0,1 T L) satisfies v = 0, then v = u for some u L 2 (, Λ n,q 1 T L) (resp. L2 (, Λ 0,q 1 T L)) such that u 2 e 2ϕ dv ω 1 qε v 2 e 2ϕ dv ω. Démonstration. The case of (n, q)-forms is a direct consequence of Proposition 4.2 and Theorem 5.5. The case of (0, q)-forms follows by viewing a (0, q)-form v
22 with values in L as an (n, 1)-form with values in L K, which has curvature θ + Ric(ω). 6. Multiplier ideal sheaves and Nadel vanishing 6.1. Multiplier ideal sheaves. Definition 6.1. The multiplier ideal sheaf of a quasi-psh function ϕ on a complex manifold is the ideal sheaf J(ϕ) O of germs of holomorphic function f with f e ϕ L 2 loc. As a first key property, we have : Theorem 6.2. Multiplier ideal sheaves are coherent. Lemma 6.3. If ν(ϕ, x) n + k, with k N, then I(ϕ) x m k x. Démonstration. Arguing in local coordinates, we may assume that (, x) = (C n, 0). The assumption means that ϕ (k + n) log z + O(1) near x. For each f J(ϕ) x, we thus have Br f 2 z (2(n+k) dz 2 < + for 0 < r 1, and the result is now an easy consequence of the Parseval lemma. Lemma 6.4. Let A be an arbitrary set of holomorphic functions, and a O the ideal sheaf locally generated by A. Then a is coherent. Démonstration. Let I be the directed set of finite subsets of A. For each i I, the corresponding (globally) finitely generated sheaf a i is coherent. We then have a = i a i, and for each compact set K, the strong noetherian property of coherent sheaves shows that a = a i near K for some i, hence the result. Proof of Theorem 6.2. Arguing locally, we may argue in the ball B C n, and assume that ϕ is psh. Let H = H(B, ϕ) be the Bergman space of holomorphic functions f O(B) such that B f 2 e ϕ < +. By Lemma 6.4, the ideal sheaf a generated by H is coherent, and it will thus be enough to show that J(ϕ) = a. Fix x. By the Rees lemma, we need to show that J(ϕ) x a x + m k x for any given k N. Let thus U be an open neighborhood of x and f O(U) with U f 2 e 2ϕ < +. Pick a cut-off function χ Cc (U) with χ 1 near x. For C 1, the function ψ(z) := ϕ(z) + (n + k)χ(z) log z x + C z 2 is satisfies i ψ + Ric(ω) = i ψ εω. Note that v := (χf) = f χ is L 2 with respect to ψ. By Theorem 5.5, we may thus find a function u such that B u 2 e 2ψ dv < + and u = v. Note that g := u χf belongs to H. Since u is holomorphic near x and ν(ψ, x) n + k, Lemma 6.3 shows that u m k x, and we are done.
NOTES COURS L2 2016 23 Theorem 6.5 (Nadel vanishing). Let (, ω) be Kähler manifold, and assume that is weakly pseudonvex and contains a Stein Zariski open subset (e.g. projective). Let L be a Hermitian holomorphic line bundle on with curvature form θ, and let also ϕ be a quasi-psh function such that θ + i ϕ εω for some ε > 0. Then H q (, O(K + L) J(ϕ)) = 0 for all q 1. Démonstration. For each q 0, denote by A q the sheaf of germs u of measurable sections of Λ n,q T L such that both ue ϕ and ue ϕ are L 2 loc. The operator induces a complex of sheaves 0 O(K + L) J(ϕ) A 0 A n. An application of Theorem 5.5 to a small ball in shows that this complex is exact. Since each A q is a C -module, the de Rham-Weil theorem implies that H q (, O(K +L) J(ϕ)) is isomorphic to the q-th cohomology group of the complex of global sections of A q. Let v be a -closed global section on of A q, q 1, denote by U the Stein Zariski open subset of, and pick a (strictly) psh exhaustion function ψ : R +. For each t 0, {ψ<t} v 2 e 2ϕ dv is finite, and we may thus choose χ : R + R + convex, increasing such that v 2 e 2(ϕ+χ ψ) dv < +. By Theorem 5.5, there exists u L 2 loc (, Λn,q 1 T L) such that u = v and u 2 e 2(ϕ+χ ψ) dv < +. In particular, u defines a global section of A q 1 on, and the result follows. Application : Kodaira embedding. CANNOT really do it because of assumption on Stein Zar open! 6.2. An algebraic formulation. We start with a few preliminary facts. A modification of is a proper bimeromorphic map µ : Y, with Y a complex manifold.the Jacobian of µ, locally defined in coordinate charts, yields a global section of the relative canonical bundle K Y/ := K Y µ K. The corresponding divisor is identified with K Y/, and its support coincides with the exceptional locus Exc(µ). We may then show that A := µ(exc(µ)) has codimension at least 2 in, and µ induces an isomorphism over \ A. The main example is the blow-up µ : Y of a submanifold Z. Then Exc(µ) = E = P(N Z/ ), and K / = (codim Z 1)E. For each prime divisor E in a modification Y of, we set and define the log discrepancy of E as ord E (a c ) := c min f a ord E(f µ), A (E) := 1 + ord E (K /). If Y is a modification of Y, the strict transform E of E on Y, then A (E) = A (E ). Definition 6.6. A quasi-psh function ϕ on a complex manifold is said to have analytic singularities if there exists c > 0 and a coherent ideal sheaf a O such
24 that we locally have ϕ = c log j f j + O(1) for some (and hence, any) choice of local generators (f j ) of a. We sometimes say more precisely that ϕ has analytic singularities of type a c. It is clear that J(ϕ) = J(a c ) only depends on a c. Definition 6.7. A log resolution of a coherent ideal sheaf a O is a proper bimeromorphic morphism µ : with a complex manifold, such that (i) a O = O ( D) with D an effective divisor on ; (ii) D + K / has simple normal crossing support. The existence of a log resolution is guaranteed by Hironaka, later simplfiied by Bierstone-Milman, Wlodarczyk, etc... Proposition 6.8. Assume that ϕ has analytic singularities of type a c, and let µ : Y be a log resolution of a, with SNC divisor i E i. Then J(ϕ) = J(a c ) = {f O ord Ei (f) > ord Ei (a c ) A (ord Ei ) for all i}. Démonstration. Change of variable. J(ϕ) = µ O Y (K Y/ D ). Define klt divisor. Kawamata-Viehweg? L=ample+klt implies H q (, K +L) = 0. Relative case, adjoint exact sequence? 7. The Ohsawa-Takegoshi extension theorem Extension theorems with L 2 estimates have a rather long history, going back to an original result of Ohsawa and Takegoshi. The following version is a slight refinement of [Ber, Theorem 2.5] (compare [Dem2, Theorem 13.6]). Recall that for a smooth hypersurface S in a complex manifold, we have a canonical isomorphism K S (K + S) S, given by the Poincaré residue map Res S. Theorem 7.1. Let be a complex manifold containing a Stein Zariski open subset, and consider the following data : (i) a smooth complex hypersurface S with canonical section s H 0 (, O(S)), and a smooth metric on O(S) with curvature form θ S, such that s e 1 ; (ii) a Hermitian holomorphic line bundle L on with curvature form θ L, and a quasi-psh function ϕ such that θ L + i ϕ 0 and θ L + i ϕ θ S ; Then for each σ H 0 (S, K S + L S ) such that S σ 2 e 2ϕ < +, there exists σ H 0 (, K + S + L) with Res S ( σ) = σ and (7.1) σ 2 s 2 log 2 s e 2ϕ C with C > 0 a purely numerical constant. S σ 2 e 2ϕ
1 The singular weight s 2 log 2 s growth (by analogy with the Poincaré metric NOTES COURS L2 2016 25 in the left-hand integral is said to have Poincaré dz 2 z 2 log 2 z of the punctured disc). In particular, it is integrable, but more singular that s 2c for any c < 1. The proof to follow is essentially an adaptation to the global setting of Blocki s exposition [B lo] of B.Y.Chen s proof [Chen]. 7.1. Reduction to a twisted L 2 -estimate. By the removable singularity property of L 2 holomorphic functions, it is enough to prove Theorem 7.1 when itself is Stein, so that σ admits a holomorphic extension σ H 0 (, K +F ) with F := S + L. Arguing by compactness as in the proof of Theorem 5.5, we may σ 2 s 2 log 2 s < + (but further assume that ϕ = 0, θ L > 0 and θ L > θ S, and without any uniform estimate!). A natural approach to prove Theorem 7.1 is to introduce a cut-off function χ ε := χ ( ε 1 s 2) with χ C (R) such that 0 χ 1, χ(t) = 1 for t 1/2 and χ(t) = 0 for t 1, and solve (7.2) u ε = (χ ε σ) = ε 1 χ ( ε 1 s 2) ( s 2) σ with an adequate L 2 estimate with respect to singular weight log s. The F -valued (n, 0)-form σ ε := χ ε σ u ε is then also a holomorphic extension of σ, since u ε is holomorphic near S, and hence vanishes on S (as a section of K + F ) thanks to the L 2 -estimate. Since χ ε σ 2 lim ε 0 s 2 log 2 s = 0 by integrability of the Poincaré weight, it would thus be enough to solve (7.2) with an estimate of the form u ε 2 s 2 (7.3) log 2 ( s 2 + ε) σ 2 s 2. supp u ε Indeed, since lim sup σ 2 s 2 C σ 2 ε 0 supp u ε S supp u ε {ε/2 s 2 ε}, and we would get the desired extension of σ as a weak L 2 loc limit of σ ε. In view of (7.2), a direct application of Theorem 5.5 with weight function ψ = log s yields instead u ε 2 s 2 C {ε/2 s 2 ε} ε 1 ( s 2) 2 θ σ 2 s 2,