aa r X i v : . [ m a t h . C V ] J a n Loss of mass of non-pluripolar products
Duc-Viet VuJanuary 15, 2021
Abstract
It is a well-known fact that the non-pluripolar self-products of a closed positive (1 , -current in a big nef cohomology class on a compact K¨ahler manifold are not offull mass in the presence of positive Lelong numbers of the current in consideration.In this paper, we give a quantitative version of the last property. Our proof involvesa generalization of Demailly’s comparison of Lelong numbers to the setting of the-ory of density currents, a reversed Alexandrov-Fenchel inequality for non-pluripolarproducts and the notion of relative non-pluripolar products. Keywords: relative non-pluripolar product, density current, tangent current, Lelongnumber, full mass intersection, Alexandrov-Fenchel inequality.
Mathematics Subject Classification 2010:
Let X be a compact K¨ahler manifold of dimension n . Let ω be a K¨ahler form on X . Forevery closed positive ( p, p ) -current S on X , we recall that the mass k S k of S is givenby R X S ∧ ω n − p . A cohomology ( p, p ) -class α is said to be pseudoeffective if it is the classof a closed positive ( p, p ) -current. For pseudoeffective ( p, p ) -classes α and α on X , wewrite α ≤ α if α − α is pseudoeffective. For a pseudoeffective ( p, p ) -class α , we usethe notation k α k to denote the norm of a closed positive current S representing α . Thisquantity is independent of the choice of S .Let ≤ m ≤ n be an integer. Let α , . . . , α m be big nef cohomology classes on X . Let T j be a closed positive current of bi-degree (1 , in α j for ≤ j ≤ m . The non-pluripolarproduct h T ∧ · · · ∧ T m i plays an important role in complex geometry; see [2, 4, 5, 14, 25]and references therein. It generalizes the classical product of (1 , -currents of boundedpotentials (see [1]). A key phenomenon about that notion is that the non-pluripolarproducts don’t preserve the mass, i.e, in general, we have (cid:13)(cid:13) h T ∧ · · · ∧ T m i (cid:13)(cid:13) ≤ (cid:13)(cid:13) α ∧ · · · ∧ α m (cid:13)(cid:13) . (1.1)We indeed have a much stronger property that the cohomology class of the current h T ∧· · · ∧ T m i is less than or equal to α ∧ · · · ∧ α m , see [25, Theorem 1.1] and also [4, 6, 26]for the case where m = n . 1hen the equality in (1.1) occurs, the currents T , . . . , T m are said to be of full massintersection . The last notion is at the heart of the theory of non-pluripolar products.Characterizing such currents is hence important. So far we have known that the positivityof Lelong numbers of T j ’s is an obstruction to being of full mass intersection; see [7, 14,24] and references therein. We are interested in understanding this property from aquantitative point of view. Although we think that this question is worth studying, therehas not been much research in this direction. To our best knowledge, the first availableresult is probably [24, Theorem 1.2] which treated the case when m = n ; see also [8,Corollary 7.6] for a related result. Unfortunately, the method in [24] is not good enoughto obtain appropriate quantitative estimates when m < n , see comments after Theorem1.1 below. The following is our main result giving such an quantitative estimate for every m in the case of self-intersection. Theorem 1.1.
Let N be a compact subset in the intersection of the big and nef cones of X . Let α be a cohomology (1 , -class in N . Let T be a closed positive current in α . Let ≤ m ≤ n be an integer. Let V be the set of maximal irreducible analytic subsets V ofdimension at least n − m of X such that the generic Lelong number ν ( T, V ) of T along V isstrictly positive. Then, there exists a constant C > independent of α such that (cid:13)(cid:13) α m − {h T m i} (cid:13)(cid:13) ≥ C (cid:18) X V ∈ V ν ( T, V ) n − dim V vol( V ) (cid:19) m . (1.2)Here we put vol( V ) := R V ω dim V . The maximality of V in the above result means thatin the set of irreducible analytic subsets V ′ of X such that ν ( T, V ′ ) > , we considerthe partial order given by the inclusion of sets, and V is maximal if it is so with respectto that order. Theorem 1.1 no longer holds if one enlarges V to include non-maximalanalytic sets. One can see it as follows: take for example T to be a current with analyticsingularities along analytic set V of dimension > n − m , for a suitable choice of V (e.g, V is biholomorphic to a complex projective space), we see that the volume of an irreducibleset V of dimension n − m in V can be as big as we want.To get motivated about Theorem 1.1, one can consider an ideal classical examplewhere T has analytic singularities along an irreducible analytic set V of dimension n − m ,in this case, the right-hand side of (4.1) can be replaced by ν ( T, V ) m vol( V ) . When N iscompact in the K¨ahler cone of X , we have a more precise estimate; see Theorem 4.1 fordetails.We underline that the arguments used in [24, Theorem 1.2] and [25, Theorem 1.2]are not sufficient to get (1.2) because the proofs there use desingularizations of V andthe blowup along V ; hence the lower bounds obtained there depend intrinsically on V .We don’t know what should be an optimal lower bound for the left-hand side of (1.2).As a direct consequence of Theorem 1.1, we infer that if {h T m i} = α m , then for every(not necessarily maximal) irreducible analytic subset V of dimension at least n − m , wehave ν ( T, V ) = 0 . This recovers a previous known result in [24] (see Theorem 1.3 there).As a byproduct of our method, we also generalize some results from [7], see Corollaries3.5 and 3.6 below.
Finally we note that Theorem 1.1 is new even when m = n and α isK¨ahler.
2n order to prove Theorem 1.1, we will prove a reserved Alexandrov-Fenchel inequal-ity for non-pluripolar products, see Proposition 3.2 below. This will reduce the questionto the K¨ahler case. Next, we establish a generalization of Demailly’s lower bound forLelong numbers of intersection of currents to the setting of the theory of density currentsintroduced by Dinh-Sibony in [13], see Corollary 2.6. This is a key in our treatmentof the K¨ahler case. Moreover, we emphasize that the notion of relative non-pluripolarproducts in [25] will play a crucial role in our proof. It serves as a bridge from densitycurrents to non-pluripolar products.In the next section, we prove the above-mentioned lower bound for Lelong numbersof density currents. In Section 3, we establish a reversed Alexandrov-Fenchel type in-equality for non-pluripolar products. Theorems 1.1 is proved in Sections 4.
Acknowledgments.
The author would like to thank Tien-Cuong Dinh and Nessim Sibonyfor fruitful discussions. This research is supported by a postdoctoral fellowship of theAlexander von Humboldt Foundation.
We first recall some basic properties of density currents. The last notion was introducedin [13].Let X be a complex K¨ahler manifold of dimension n and V a smooth complex sub-manifold of X of dimension l. Let T be a closed positive ( p, p ) -current on X, where ≤ p ≤ n. Denote by π : E → V the normal bundle of V in X and E := P ( E ⊕ C ) the projective compactification of E. By abuse of notation, we also use π to denote thenatural projection from E to V .Let U be an open subset of X with U ∩ V = ∅ . Let τ be a smooth diffeomorphismfrom U to an open neighborhood of V ∩ U in E which is identity on V ∩ U such that therestriction of its differential dτ to E | V ∩ U is identity. Such a map is called an admissiblemap . Note that in [13], to define an admissible map, it is required furthermore that dτ is C -linear at every point of V . This difference doesn’t affect what follows. When U is asmall enough tubular neighborhood of V, there always exists an admissible map τ by [13,Lemma 4.2]. In general, τ is not holomorphic. When U is a small enough local chart, wecan choose a holomorphic admissible map by using suitable holomorphic coordinates on U . For λ ∈ C ∗ , let A λ : E → E be the multiplication by λ on fibers of E. We recall thefollowing crucial result.
Theorem 2.1. ([13, Theorem 4.6]) Let τ be an admissible map defined on a tubular neigh-borhood of V . Then, the family ( A λ ) ∗ τ ∗ T is of mass uniformly bounded in λ , and if S is alimit current of the last family as λ → ∞ , then S is a current on E which can be extendedtrivially through E \ E to be a closed positive current on E such that the cohomology class { S } of S in E is independent of the choice of S , and { S }| V = { T }| V , and k S k ≤ C k T k forsome constant C independent of S and T . We call S a tangent current to T along V , and its cohomology class is called the total angent class of T along V and is denoted by κ V ( T ) . By [13, Theorem 4.6] again, if S = lim k →∞ ( A λ k ) ∗ τ ∗ T for some sequence ( λ k ) k converging to ∞ , then for every open subset U of X and everyadmissible map τ ′ : U ′ → E , we also have S = lim k →∞ ( A λ k ) ∗ τ ′∗ T. This is equivalent to saying that tangent currents are independent of the choice of theadmissible map τ . Definition 2.2. ([13]) Let F be a complex manifold and π F : F → V a holomorphicsubmersion. Let S be a positive current of bi-degree ( p, p ) on F . The h-dimension of S withrespect to π F is the biggest integer q such that S ∧ π ∗ F θ q = 0 for some Hermitian metric θ on V . By a bi-degree reason, the h-dimension of S is in [max { l − p, } , min { dim F − p, l } ] .We have the following description of currents with minimal h-dimension. Lemma 2.3. ([13, Lemma 3.4]) Let π F : F → V be a submersion. Let S be a closed positivecurrent of bi-degree ( p, p ) on F of h-dimension ( l − p ) with respect to π F . Then S = π ∗ S ′ forsome closed positive current S ′ on V . By [13], the h-dimensions of tangent currents to T along V are the same and thisnumber is called the tangential h-dimension of T along V .Let m ≥ be an integer. Let T j be a closed positive current of bi-degree ( p j , p j ) for ≤ j ≤ m on X and let T ⊗ · · · ⊗ T m be the tensor current of T , . . . , T m which is acurrent on X m . A density current associated to T , . . . , T m is a tangent current to ⊗ mj =1 T j along the diagonal ∆ m of X m . Let π m : E m → ∆ be the normal bundle of ∆ m in X m .Denote by [ V ] the current of integration along V . When m = 2 and T = [ V ] , the densitycurrents of T and T are naturally identified with the tangent currents to T along V (see[21, Lemma 2.3]).The unique cohomology class of density currents associated to T , . . . , T m is called thetotal density class of T , . . . , T m . We denote the last class by κ ( T , . . . , T m ) . The tangentialh-dimension of T ⊗ · · · ⊗ T m along ∆ m is called the density h-dimension of T , . . . T m . Lemma 2.4. ([13, Section 5]) Let T j be a closed positive current of bi-degree ( p j , p j ) on X for ≤ j ≤ m such that P mj =1 p j ≤ n . Assume that the density h-dimension of T , . . . , T m is minimal, i.e , equal to n − P mj =1 p j . Then the total density class of T , . . . , T m is equal to π ∗ m ( ∧ mj =1 { T j } ) . We recall the following result.
Theorem 2.5. ([13, Proposition 4.13]) Let V ′ be a submanifold of V and let T be a closedpositive current on X . Let T ∞ be a tangent current to T along V . Denote by s the densityh-dimension of T ∞ and [ V ′ ] . Then, the density h-dimension of T and [ V ′ ] is at most s , andwe have κ s ( T, [ V ′ ]) ≤ κ s ( T ∞ , [ V ′ ]) . The inequality still holds if we replace s by the tangential h-dimension of T along V .
4s a consequence, we obtain the following result generalizing the well-known lowerbound of Lelong numbers of intersection of (1 , -currents due to Demailly [9, Page 169]in the compact setting. Corollary 2.6.
Let T j be a closed positive current on X for ≤ j ≤ m . Then, for every x ∈ X and for every density current S associated to T , . . . , T m , we have ν ( S, x ) ≥ ν ( T , x ) · · · ν ( T m , x ) . (2.1) Proof.
Let x ∈ X . Let ∆ m be the diagonal of X m . Let π : E → ∆ m be the naturalprojection from the normal bundle of the diagonal ∆ m of X m in X m . We identify x witha point in ∆ m via the natural identification ∆ m with X . Put T := ⊗ mj =1 T j and V ′ := { x } .By [20, Lemma 2.4], we have ν ( T, x ) ≥ ν ( T , x ) · · · ν ( T m , x ) . By [13, Proposition 5.6], wehave κ ( S, [ V ′ ]) = ν ( S, x ) , κ ( T, [ V ′ ]) = ν ( T, x ) (notice here dim V ′ = 0 ). This combined by Theorem 2.5 applied to X m , T := ⊗ mj =1 T j , ∆ m the diagonal of X m and V ′ := { x } implies ν ( S, x ) ≥ ν ( T, x ) ≥ ν ( T , x ) · · · ν ( T m , x ) . Hence, the desired inequality follows. The proof is finished.In the next part, we will use density currents to study the loss of mass of non-pluripolar products. We need to recall basic properties of relative non-pluripolar prod-ucts. Let T , . . . , T m be closed positive (1 , -currents on X . By [25], the T -relativenon-pluripolar product h∧ mj =1 T j ˙ ∧ T i is defined in a way similar to that of the usual non-pluripolar product. For readers’ convenience, we explain briefly how to do it.Write T j = dd c u j + θ j , where θ j is a smooth form and u j is a θ j -psh function. Put R k := ∩ mj =1 { u j > − k } ∧ mj =1 ( dd c max { u j , − k } + θ j ) ∧ T for k ∈ N . By the strong quasi-continuity of bounded psh functions ([25, Theorems 2.4and 2.9]), we have R k = ∩ mj =1 { u j > − k } ∧ mj =1 ( dd c max { u j , − l } + θ j ) ∧ T for every l ≥ k ≥ . Although it is not an immediate fact, one can check that R k ispositive (see [25, Lemma 3.2]).As in [4], we have that R k is of mass bounded uniformly in k and ( R k ) k convergesto a closed positive current as k → ∞ . This limit is denoted by h∧ mj =1 T j ˙ ∧ T i . The lastproduct is, hence, a well-defined closed positive current of bi-degree ( m + p, m + p ) ; andit is symmetric with respect to T , . . . , T m and homogeneous. We refer to [25, Proposition3.5] for more properties of relative non-pluripolar products.For every closed positive (1 , -current P , we denote by I P the set of x ∈ U so thatlocal potentials of P are equal to −∞ at x . Note that I P is a locally complete pluripolarset. It is clear from the definition that h∧ mj =1 T j ˙ ∧ T i has no mass on ∪ mj =1 I T j . Furthermore,for every locally complete pluripolar set A ( i.e, A is locally equal to { ψ = −∞} for somepsh function ψ ), if T has no mass on A , then so does h∧ mj =1 T j ˙ ∧ T i . The following isdeduced from [25, Proposition 3.5]. 5 roposition 2.7. ( i ) For R := h∧ mj = l +1 T j ˙ ∧ T i , we have h∧ mj =1 T j ˙ ∧ T i = h∧ lj =1 T j ˙ ∧ R i . ( ii ) For every complete pluripolar set A , we have X \ A h T ∧ T ∧ · · · ∧ T m ˙ ∧ T i = (cid:10) T ∧ T ∧ · · · ∧ T m ˙ ∧ ( X \ A T ) (cid:11) . In particular, the equality h∧ mj =1 T j ˙ ∧ T i = h∧ mj =1 T j ˙ ∧ T ′ i holds, where T ′ := X \∪ mj =1 I Tj T . Here is a crucial property of relative non-pluripolar products.
Theorem 2.8. ([25, Theorem 1.1]) Let T ′ j be closed positive (1 , -current in the cohomologyclass of T j on X such that T ′ j is less singular than T j for ≤ j ≤ m . Then we have {h T ∧ · · · ∧ T m ˙ ∧ T i} ≤ {h T ′ ∧ · · · ∧ T ′ m ˙ ∧ T i} . Weaker versions of the above result were proved in [4, 6, 26]. Regarding the relationbetween relative non-pluripolar products and density currents, the following fact wasproved in [23, Theorem 3.5], see also [16, 17].
Theorem 2.9.
Let R ∞ be a density current associated to T , . . . , T m , T . Then we have π ∗ h∧ mj =1 T j ˙ ∧ T i ≤ R ∞ , (2.2) where π is the natural projection from the normal bundle of the diagonal ∆ of X m +1 to ∆ ,and as usual we identified ∆ with X . We will need the following to estimate the density h-dimension of currents.
Proposition 2.10. ([23, Proposition 3.6]) Let A be a Borel subset of X . Assume that forevery density current R J associated to ( T j ) j ∈ J , T with m J := | J | < m , we have that R J hasno mass on the set π ∗ m J +1 ( ∩ j J { x ∈ A : ν ( T j , x ) > } ) . Then for every density current S associated to T , . . . , T m , T , the h-dimension of the current A S is equal to n − p − m . Let P and T be closed positive current of bi-degree (1 , and ( p, p ) respectively on X ,where ≤ p ≤ n − . Lemma 2.11.
Assume that T has no mass on I P . Then, the cohomology class γ := { P } ∧ { T } − {h P ˙ ∧ T i} is pseudoeffective and we have k γ k ≥ X V ν ( P, V ) ν ( T, V ) vol( V ) , (2.3) where the sum is taken over every irreducible subset V of dimension at least n − p − in X . Here ν ( P, V ) and ν ( T, V ) denote the generic Lelong numbers of P and T along V ,respectively. 6 roof. Let V be the set of irreducible analytic subsets V of dimension at least n − p − in X such that ν ( T, V ) > and ν ( P, V ) > . We note that in (2.3), it is enough to consider V ∈ V . We will see below that V has at most countable elements.Observe that if ν ( P, x ) > , then x ∈ I P . Hence, by hypothesis, the trace measure of T has no mass on the set { x ∈ X : ν ( P, x ) > } . This allows us to apply Proposition 2.10to P and T to obtain that the density h-dimension of P and T is minimal. Using this andLemma 2.4 gives κ ( P, T ) = π ∗ ( { P } ∧ { T } ) , (2.4)where π is the natural projection from the normal bundle of the diagonal ∆ of X to ∆ .Let S be a density current associated to P and T . Since the h-dimension of S isminimal, using Lemma 2.3, we get that there exists a current S ′ on X such that S = π ∗ S ′ (recall ∆ is identified with X ). Since the relative non-pluripolar product is dominated bydensity currents (Theorem 2.9), the current S ′ − h P ˙ ∧ T i is closed and positive. Moreover,by (2.4), the cohomology class of the last current is equal to γ . It follows that γ ispseudoeffective.It remains to prove (2.3). Let V ∈ V . By definition, the generic Lelong number of T along V is positive. Since T is of bi-degree ( p, p ) , the dimension of V must be at most n − p . Hence, we have two possibilities: either dim V = n − p − or dim V = n − p . Thelatter case cannot happen because T would have mass on V which is contained in I P (for ν ( P, V ) > ), this contradicts the hypothesis that T has no mass on I P . Hence, for every V ∈ V , we have dim V = n − p − . We also deduce that for V, V ′ ∈ V , then V V ′ .Applying Corollary 2.6 to P, T and generic x ∈ V gives ν ( S, V ) ≥ ν ( P, V ) ν ( T, V ) . This combined with the fact that dim V = n − p − implies S ≥ ν ( P, V ) ν ( T, V ) [ V ] . Wededuce that S ≥ X V ∈ V ν ( P, V ) ν ( T, V ) [ V ] . The desired assertion follows. The proof is finished.
We first recall an integration by parts formula for relative non-pluripolar products from[22] generalizing those given in [4, 19, 28].Let X be a compact K¨ahler manifold. Recall that a dsh function on X is the differenceof two quasi-plurisubharmonic (quasi-psh for short) functions on X (see [12]). Thesefunctions are well-defined outside pluripolar sets. Let v be a dsh function on X . The lastfunction is said to be bounded in X if there exists a constant C such that | v | ≤ C on X (outside certain pluripolar set).Let T be a closed positive current on X . We say that v is T -admissible if there existquasi-psh functions ϕ , ϕ on X such that v = ϕ − ϕ and T has no mass on { ϕ j = −∞} j = 1 , . In particular, if T has no mass on pluripolar sets, then every dsh function is T -admissible.Assume now that v is bounded T -admissible. Let ϕ , ϕ be quasi-psh functions suchthat v = ϕ − ϕ and T has no mass on { ϕ j = −∞} for j = 1 , . Let ϕ j,k := max { ϕ j , − k } for every j = 1 , and k ∈ N . Put v k := ϕ ,k − ϕ ,k and Q k := dv k ∧ d c v k ∧ T = dd c v k ∧ T − v k dd c v k ∧ T. By the plurifine locality with respect to T ([25, Theorem 2.9]) applied to the right-handside of the last equality, we have ∩ j =1 { ϕ j > − k } Q k = ∩ j =1 { ϕ j > − k } Q k ′ (3.1)for every k ′ ≥ k . By [22, Lemma 2.5], the mass of Q k on X is bounded uniformly in k .This combined with (3.1) implies that there exists a positive current Q on X such thatfor every bounded Borel form Φ with compact support on X such that h Q, Φ i = lim k →∞ h Q k , Φ i . We define h dv ∧ d c v ˙ ∧ T i to be the current Q . This agrees with the classical definition if v is the difference of two bounded quasi-psh functions. One can check that this definitionis independent of the choice of ϕ , ϕ .Let w be another bounded T -admissible dsh function. If T is of bi-degree ( n − , n − ,we can also define the current h dv ∧ d c w ˙ ∧ T i by a similar procedure as above. We put h dd c v ˙ ∧ T i := h dd c ϕ ˙ ∧ T i − h dd c ϕ ˙ ∧ T i which is independent of the choice of ϕ , ϕ . By T -admissibility, we have h dd c ( v + w ) ˙ ∧ T i = h dd c v ˙ ∧ T i + h dd c w ˙ ∧ T i . Here is an integration by parts formula for relative non-pluripolar products.
Theorem 3.1. ([22, Theorem 2.6]) Let T a closed positive current of bi-degree ( n − , n − on X . Let v and w be bounded T -admissible dsh functions on X . Then, we have Z X w h dd c v ˙ ∧ T i = Z X v h dd c w ˙ ∧ T i = − Z X h dw ∧ d c v ˙ ∧ T i . (3.2)Let θ be a closed smooth (1 , -form on X . Given a θ -psh function u , we use the usualnotation that θ u := dd c u + θ. Let T be a closed positive current of bi-degree ( n − , n − on X . Let u, v, ϕ be θ -pshfunctions such that u ≤ ϕ and v ≤ ϕ .The following can be regarded as an inequality of reversed Alexandrov-Fenchel type.Several related estimates for mixed Monge-Amp`ere operators were obtained in the localsetting; see [10, 18]. 8 roposition 3.2. Assume that Z X h θ ϕ ˙ ∧ T i = Z X h θ u ∧ θ ϕ ˙ ∧ T i , (3.3) and T has no mass on { u = −∞} ∪ { v = −∞} . Then, we have Z X (cid:0) h θ ϕ ∧ θ v ˙ ∧ T i − h θ u ∧ θ v ˙ ∧ T i (cid:1) ≤ (cid:18) Z X (cid:0) h θ ϕ ˙ ∧ T i − h θ u ˙ ∧ T i (cid:1)(cid:19) / (cid:18) Z X (cid:0) h θ ϕ ˙ ∧ T i − h θ v ˙ ∧ T i (cid:1)(cid:19) / (3.4)Notice that by the monotonicity of relative non-pluripolar products, the left-hand sideof (3.4) is non-negative. Proof.
By hypothesis, one has { u = −∞}∪{ v = −∞} T = 0 . (3.5)Note here that { ϕ = −∞} ⊂ { u = −∞} ∩ { v = −∞} . Let u k := max { u, ϕ − k } − ( ϕ − k ) and ψ k := k − max { u + v − ϕ, − k } + 1 = k − max { u + v, ϕ − k } − k − ϕ + 1 Define v k similarly. We have ψ k = 0 on { u ≤ ϕ − k } ∪ { v ≤ ϕ − k } . Note also that ≤ u k , v k ≤ k for j = 1 , (hence u k , v k , ψ k are bounded T -admissible dsh functions by(3.5)) and dd c ψ k + k − η ≥ , where η := 2 θ ϕ . We can check that h θ u ∧ θ v ˙ ∧ T i = lim k →∞ ψ k (cid:10) ( dd c u k + θ ϕ ) ∧ ( dd c v k + θ ϕ ) ˙ ∧ T (cid:11) . Put B k := Z X ψ k (cid:10) θ ϕ ∧ ( dd c v k + θ ϕ ) ˙ ∧ T (cid:11) − Z X ψ k (cid:10) ( dd c u k + θ ϕ ) ∧ ( dd c v k + θ ϕ ) ˙ ∧ T (cid:11) and A := Z X (cid:0) h θ ϕ ∧ θ v ˙ ∧ T i − h θ u ∧ θ v ˙ ∧ T i (cid:1) By (3.5) and Proposition 2.7 ( ii ) , we have A = lim k →∞ B k . B k = − Z X ψ k h dd c u k ∧ ( dd c v k + θ ϕ ) ˙ ∧ T (cid:11) = − Z X ψ k h dd c u k ∧ dd c v k ˙ ∧ T (cid:11) − Z X ψ k h dd c u k ∧ θ ϕ ˙ ∧ T (cid:11) . Denote by I , I the first and second term in the right-hand side of the last equality. Using(3.5) gives lim k →∞ I = lim k →∞ (cid:18) − Z X ψ k h ( dd c u k + θ ϕ ) ∧ θ ϕ ˙ ∧ T (cid:11) + Z X ψ k h θ ϕ ˙ ∧ T (cid:11)(cid:19) = − Z X h θ u ∧ θ ϕ ˙ ∧ T (cid:11) + Z X h θ ϕ ˙ ∧ T (cid:11) = 0 by (3.3). Thus we get B k = I + o k →∞ (1) . (3.6)Theorem 3.1 applied to the formula defining I gives − I = Z X u k (cid:10) dd c ψ k ∧ dd c v k ˙ ∧ T (cid:11) (3.7) = − Z X (cid:10) du k ∧ d c v k ∧ dd c ψ k ˙ ∧ T (cid:11) = − Z X (cid:10) du k ∧ d c v k ∧ ( dd c ψ k + k − η ) ˙ ∧ T (cid:11) + k − Z X (cid:10) du k ∧ d c v k ∧ η ˙ ∧ T (cid:11) . Denote by J , J the first and second term in the right-hand side of the last equality. Wetreat J . By integration by parts (Theorem 3.1), we obtain J = − k − Z X u k h dd c v k ∧ η ˙ ∧ T i = − k − Z X u k h ( dd c v k + θ ϕ ) ∧ η ˙ ∧ T i + o k →∞ (1) (3.8)because of (3.5) and the fact that u k /k converges to on { u > −∞} and to − otherwise.Using this and noticing that u k ≤ , we infer lim inf k →∞ J ≥ . (3.9)On the other hand, using the Cauchy-Schwarz inequality, we obtain J ≤ Z X (cid:10) du k ∧ d c u k ∧ ( dd c ψ k + k − η ) ˙ ∧ T (cid:11) Z X (cid:10) dv k ∧ d c v k ∧ ( dd c ψ k + k − η ) ˙ ∧ T (cid:11) . (3.10)Denote by J , J the first and second term in the right-hand side of the last inequality.Put J ′ := k − Z X (cid:10) du k ∧ d c u k ∧ η ˙ ∧ T (cid:11) . J ′ = − k − Z X u k (cid:10) dd c u k ∧ η ˙ ∧ T (cid:11) (3.11) = − k − Z X u k (cid:10) ( dd c u k + θ ϕ ) ∧ η ˙ ∧ T (cid:11) + o k →∞ (1)= Z X (cid:10) ( dd c u k + θ ϕ ) ∧ η ˙ ∧ T (cid:11) − Z X ( u k /k + 1) (cid:10) ( dd c u k + θ ϕ ) ∧ η ˙ ∧ T (cid:11) + o k →∞ (1)= Z X (cid:10) θ ϕ ∧ η ˙ ∧ T (cid:11) − Z X ( u k /k + 1) (cid:10) θ u ∧ η ˙ ∧ T (cid:11) + o k →∞ (1) because u k /k + 1 = 0 on { u ≤ ϕ − k } . Letting k → ∞ in (3.11) gives lim k →∞ J ′ = Z X (cid:10) θ ϕ ∧ η ˙ ∧ T (cid:11) − Z X (cid:10) θ u ∧ η ˙ ∧ T (cid:11) = 2 Z X (cid:10) θ ϕ ˙ ∧ T (cid:11) − Z X (cid:10) θ u ∧ θ ϕ ˙ ∧ T (cid:11) . (3.12)Using Theorem 3.1 again and arguing as in (3.11), we have J = Z X (cid:10) du k ∧ d c u k ∧ dd c ψ k ˙ ∧ T (cid:11) + J ′ = − Z X ψ k (cid:10) dd c u k ∧ dd c u k ˙ ∧ T (cid:11) + J ′ = − Z X ψ k (cid:10) ( dd c u k + θ ϕ ) ˙ ∧ T (cid:11) + 2 Z X ψ k (cid:10) dd c u k ∧ θ ϕ ˙ ∧ T (cid:11) + Z X ψ k (cid:10) θ ϕ ˙ ∧ T (cid:11) + J ′ = Z X (cid:10) θ ϕ ˙ ∧ T (cid:11) − Z X (cid:10) θ u ˙ ∧ T (cid:11) + 2 Z X (cid:10) θ u ∧ θ ϕ ˙ ∧ T (cid:11) − Z X (cid:10) θ ϕ ˙ ∧ T (cid:11) + o k →∞ (1) + J ′ This combined with (3.12) yields that lim sup k →∞ J ≤ Z X (cid:10) θ ϕ ˙ ∧ T (cid:11) − Z X (cid:10) θ u ˙ ∧ T (cid:11) . By similar computations, we also get lim sup k →∞ J ≤ Z X (cid:10) θ ϕ ˙ ∧ T (cid:11) − Z X (cid:10) θ v ˙ ∧ T (cid:11) . This together with (3.7) and (3.6) gives A = lim sup k →∞ B k = lim sup k →∞ I = − lim inf k →∞ − I ≤ C, where C is the right-hand side of (3.4). This finishes the proof. Corollary 3.3.
Let T be closed positive current of bi-dimension ( m, m ) on X and let θ be aclosed smooth (1 , -form on X . Let u, v, ϕ be θ -psh functions such that u, v ≤ ϕ , and T hasno mass on { u = −∞} and { v = −∞} . Let M be a positive constant greater than R X θ mϕ ˙ ∧ T .Then, there exists a constant C M depending only on M, m, n such that (cid:12)(cid:12)(cid:12)(cid:12) Z X h θ m − lϕ ∧ θ lv ˙ ∧ T i − Z X h θ m − lu ∧ θ lv ˙ ∧ T i (cid:12)(cid:12)(cid:12)(cid:12) ≤ C M (cid:18) Z X h θ mϕ ˙ ∧ T i − Z X h θ mu ˙ ∧ T i (cid:19) − l . (3.13)11 n particular, if Z X h θ mϕ ˙ ∧ T i = Z X h θ mu ˙ ∧ T i , then, for every integer ≤ l ≤ m , we have Z X h θ m − lϕ ∧ θ lv ˙ ∧ T i = Z X h θ m − lu ∧ θ lv ˙ ∧ T i . Proof.
In what follows, we denote by C M a positive constant depending only on M and n , and the value of C M might vary from line to line. Put I := Z X h θ mϕ ˙ ∧ T i − Z X h θ mu ˙ ∧ T i . By the monotonicity of relative non-pluripolar products, we get ≤ Z X h θ mϕ ˙ ∧ T i − Z X h θ m − lϕ ∧ θ lu ˙ ∧ T i ≤ I (3.14)for every ≤ l ≤ m . Observe that the desired assertion in the case where m = 2 is a direct consequence of Proposition 3.2. We prove by induction on l ′ that for every ≤ l , l ≤ m with l + l ≤ l ′ we have Z X h θ m − l ϕ ∧ θ l v ˙ ∧ T i − Z X h θ m − l − l ϕ ∧ θ l u ∧ θ l v ˙ ∧ T i ≤ C M I − l , (3.15)for some constant C M depending only on M, m and n .The desired inequality is a special case of (3.15) when l = m − l . When l ′ = 0 , theinequality (3.15) is clear. Suppose that (3.15) holds for every l , l with l + l ≤ l ′ − .We need to prove it for l ′ in place of l ′ − . To this end, we now use another inductionon ≤ l ≤ l ′ to prove the statement ( ∗ ) that (3.15) holds for every l with l + l ≤ l ′ .When l = 0 , the statement ( ∗ ) is a direct consequence of (3.14). Assume now that ( ∗ ) holds for l − . We now prove it for l . Let T ′ := h θ m − l − l ϕ ∧ θ l − u ∧ θ l − v ˙ ∧ T i . We have h θ m − l − l ϕ ∧ θ l u ∧ θ l v ˙ ∧ T i = h θ u ∧ θ v ˙ ∧ T ′ i , and Z X h θ u ˙ ∧ T ′ i = Z X h θ m − l − l ϕ ∧ θ l +1 u ∧ θ l − v ˙ ∧ T i and Z X h θ ϕ ˙ ∧ T ′ i = Z X h θ m − l − l +2 ϕ ∧ θ l − u ∧ θ l − v ˙ ∧ T i .
12y this and the induction hypothesis on l that ( ∗ ) holds for l − and the monotonicityof relative non-pluripolar products, we obtain that Z X h θ ϕ ˙ ∧ T ′ i − Z X h θ u ˙ ∧ T ′ i ≤ Z X h θ m − l +1 ϕ ∧ θ l − v ˙ ∧ T i − Z X h θ m − l − l ϕ ∧ θ l +1 u ∧ θ l − v ˙ ∧ T i≤ C M I − l . Note that T ′ has no mass on { u = −∞} and { v = −∞} because T does so; see [25,Lemma 2.1]. Applying Proposition 3.3 to u, v, ϕ, T ′ gives Z X h θ ϕ ∧ θ v ˙ ∧ T ′ i − Z X h θ u ∧ θ v ˙ ∧ T ′ i ≤ C M (cid:18) Z X h θ ϕ ˙ ∧ T ′ i − Z X h θ u ˙ ∧ T ′ i (cid:19) / ≤ C M I − l . We deduce that Z X h θ m − l − l +1 ϕ ∧ θ l − u ∧ θ l v ˙ ∧ T i − Z X h θ m − l − l ϕ ∧ θ l u ∧ θ l v ˙ ∧ T i ≤ C M I − l . On the other hand, using the induction hypothesis on l ′ − gives Z X h θ m − l ϕ ∧ θ l v ˙ ∧ T i − Z X h θ m − l − l +1 ϕ ∧ θ l − u ∧ θ l v ˙ ∧ T i ≤ C M I − l . Summing up the last two inequalities gives the desired assertion ( ∗ ) for l . In otherwords, (3.15) holds for every l , l with l + l ≤ l ′ . This is what we want to prove. Theproof is finished.We say that a closed positive (1 , -current P is a current with analytic singularitiesassociated to a coherent analytic ideal sheaf S on X if locally on X , every local potential u of P satisfies u = c log M X j =1 | f j | + O (1) , where c > is a constant, and { f , . . . , f M } are local generators of S ; see [3, Definition9.1]. By Demailly’s analytic approximation of psh functions, every big cohomology classcontains such a current P which is K¨ahler.We fix a smooth K¨ahler form ω on X with R X ω n = 1 . For every positive ( p, p ) -current S , the mass of S is defined by k S k := Z X S ∧ ω n − p . Similarly, we can define the mass norm of a pseudoeffective cohomology classes.
Theorem 3.4.
Let ≤ m ≤ n be an integer. Let α be a big cohomology (1 , -class in X . Let T and T ′ be closed positive currents in α such that T ′ is more singular than T . Assume thatthere exists a closed positive current P with analytic singularities associated to a coherentideal sheaf such that P is more singular than T . Let θ be a closed continuous real (1 , -form. Let M > be constant greater than k α k , kh α m ik and k θ k C such that P ≥ M − ω .Then, there exists a constant C M > depending only on M, m, ω, X such that (cid:13)(cid:13) {h ( T ′ + θ ) m i} − {h ( T + θ ) m i} (cid:13)(cid:13) ≤ C M (cid:13)(cid:13) {h T ′ m i} − {h T m i} (cid:13)(cid:13) − m . (3.16)13 n particular, if T + θ and T ′ + θ are positive, then there holds {h T ′ m i} = {h T m i} if and only if {h ( T ′ + θ ) m i} = {h ( T + θ ) m i} . (3.17)Examples of T in the above result are currents with minimal singularities in α . Theassertion (3.17) was known when α is K¨ahler (see [11, 25]) and seems to be new for bigclasses even when m = n . Proof.
As above we denote by C M a positive constant depending only on M, m, ω, X whose value can vary from line to line. We can choose such a C M big enough such that ω ′ := θ + C M ω ≥ . By the multi-linearity of non-pluripolar products, it is sufficient toprove (3.16) for ω ′ in place of θ . In order to do so, we only need to check that (cid:13)(cid:13) {h T ′ l i} − {h T l i} (cid:13)(cid:13) ≤ C M (cid:13)(cid:13) {h T ′ m i} − {h T m i} (cid:13)(cid:13) − m (3.18)for every ≤ l ≤ m .Principalizing the ideal sheaf associated to P (see [15, 27]), we get a smooth modifi-cation σ : b X → X such that b P := σ ∗ P = P + P , where P is a linear combination withnonnegative coefficients of currents of integration along smooth hypersurfaces, and P isa closed positive smooth form. Since P ≥ M − ω , we get P ≥ M − σ ∗ ω (3.19)Let b T := σ ∗ T and b T ′ := b T ′ . Since non-pluripolar products have no mass on analytic sets,using the hypothesis, we have Z b X h b T m i ∧ σ ∗ ω n − m = Z X h T m i ∧ ω n − m , Z X h T ′ m i ∧ ω n − m = Z b X h b T ′ m i ∧ σ ∗ ω n − m . Using this and applying Corollary 3.3 to potentials of b T , b T ′ , b P give Z b X (cid:0) h b T l ∧ b P m − l i − h b T ′ l ∧ b P m − l i (cid:1) ∧ σ ∗ ω n − m ≤ C M kh T m i − h T ′ m ik − m + l for every ≤ l ≤ m . Let R be a closed positive current representing the cohomologyclass {h b T l i} − {h b T ′ l i} . By the above inequality and the smoothness of P , we obtain Z b X R ∧ P m − l ∧ σ ∗ ω n − m ≤ C M kh T m i − h T ′ m ik − m + l . This combined with (3.19) gives Z b X R ∧ σ ∗ ω n − l ≤ C M kh T m i − h T ′ m ik − m + l . It follows that Z X (cid:0) h T l i − h T ′ l i (cid:1) ∧ ω n − l = Z b X (cid:0) h b T l i − h b T ′ l i ) ∧ σ ∗ ω n − l = Z b X R ∧ σ ∗ ω n − l ≤ C M kh T m i − h T ′ m ik − m + l . kh T m ik ≤ kh α m ik .We now prove (3.17). The implication ⇒ is clear. The other one is obtained from thefirst one by applying to T + θ, T ′ + θ in place of T, T ′ respectively, and − θ in place of θ .The proof is complete. Corollary 3.5.
Let ≤ m ≤ n be an integer. Let α be a big cohomology (1 , -class in X .Let T be a closed positive current in α such that {h T m i} = h α m i . Then we have {h T l i} = h α l i for every ≤ l ≤ m . In particular, if α is big nef and θ is a closed continuous real (1 , -formrepresenting a cohomology class γ , then {h ( T + θ ) m i} = ( α + γ ) m . (3.20)The equality (3.20) was proved in [7, Theorem 1.1] in the case where m = n . Notethat in the last case, if additionally T + θ ≥ , then using [8, Corollary 6.4] and the factthat the Lelong numbers of T vanish, we get that α + γ is nef. Proof.
Let T min be a current with minimal singularity in α . By Theorem 3.4 for T and T min and the hypothesis, we get {h T + ω ) m i} = {h ( T min + ω ) m i} . Expanding the sums in both sides and using the fact that {h T l i} ≤ h T l min i for every ≤ l ≤ m , we get the first desired equality. The second desired equality follows immediatelyfrom the first one. This finishes the proof. Corollary 3.6.
Let α and α be big nef cohomology classes on X . Let T j be a closed positivecurrent in α j for j = 1 , . Then, (cid:8)(cid:10) ( T + T ) m (cid:11)(cid:9) = h ( α + α ) m i (3.21) if and only if {h T mj i} = h α mj i (3.22) for j = 1 , . The las result was obtained in [6, Theorem 1.3 and Corollary 4.2] when m = n . Proof of Corollary 3.6.
At this point, the proof is similar to that of [6, Corollary 4.2].The implication from (3.21) to (3.22) is clear thanks to the monotonicity. Assume now(3.22).Let ω ′ be a big enough K¨ahler form such that { ω ′ } + α j is K¨ahler for j = 1 , . Usingthis, the fact that α j is big nef, and Theorem 3.4, we deduce that {h ( T j + ω ′ ) m i} = ( α j + { ω ′ } ) m for j = 1 , . Now the convexity of the class of currents of full mass intersection in K¨ahlercohomology classes (see, for example, [25, Theorem 1.3] or [11]) gives that (cid:8)(cid:10) ( T + T + 2 ω ′ ) m (cid:11)(cid:9) = ( α + α + 2 { ω ′ } ) m . By this and Theorem 3.4 again, we get the desired equality. The proof is finished.15
Proof of Theorem 1.1
In this section, we will give a proof of Theorem 1.1. The following result implies Theorem1.1 in the case where α is K¨ahler. Theorem 4.1.
Let K be a compact subset in the K¨ahler cone of (1 , -classes on X . Let α ∈ K and let T be a closed positive current in α . Let ≤ m ≤ n be an integer. Let V bethe set of maximal irreducible analytic subsets V of dimension at least n − m of X such thatthe generic Lelong number ν ( T, V ) of T along V is strictly positive. Then, we have (cid:13)(cid:13) α m − {h T m i} (cid:13)(cid:13) ≥ C X V ∈ V (cid:0) ν ( T, V ) (cid:1) n − dim V vol( V ) , (4.1) for some constant C > independent of α (but depending on K ). By the proof below, one can see that if the sum in the right-hand of (4.1) is takenonly on V ∈ V such that the dimension of V is equal to n − m , then the factor C can bereplaced by . Proof.
By Demailly’s analytic approximation of psh functions (see [9]), it is enough tocheck (4.1) when T has analytic singularities (note that the maximality of V is preservedhere). Hence, the polar locus I T of T is an analytic subset on X .Let ≤ l ≤ n − be an integer. Let V l be the subset of V consisting of V of dimension l . Let V ∈ V l . By the definition of V , we have l ≥ n − m . Put S := h T n − l − i . Notethat h T n − l i = h T ˙ ∧ S i by Proposition 2.7. Using this and monotonicity of non-pluripolarproducts, we get (cid:13)(cid:13) α m − {h T m i} (cid:13)(cid:13) ≥ (cid:13)(cid:13) α m − {h T n − l i ∧ α l + m − n } (cid:13)(cid:13) & (cid:13)(cid:13) α n − l − {h T ˙ ∧ S i} (cid:13)(cid:13) . (4.2)Let ≤ s ≤ n − l − be an integer. Since V is of dimension l and V is a maximalirreducible analytic subset of X such that ν ( T, V ) > , we see that the self-intersection T s is classically well-defined in an open neighborhood U of V \ Sing ( I T ) in X , whereSing ( I T ) is the singular part of the analytic set I T . Moreover, T s has no mass on V because s < n − l . By [9, Page 169] or Corollary 2.6, we have ν ( T s , V ∩ U ) ≥ ν ( T, V ) s . (4.3)Now using the fact that T has analytic singularities and the maximality of V , we get h T s i = U \ V T s = T s on U (we choose U such that U doesn’t intersect the singularity of I T ). Using theseobservations for s = n − l − gives ν ( S, V ) ≥ ν ( T, V ) n − l − . (4.4)Now recall that non-pluripolar products have no mass on pluripolar sets. In particular, S has no mass on I T . This allows us to apply Lemma 2.11 to T and S . As a result, weobtain (cid:13)(cid:13) α ∧ { S } − {h T ˙ ∧ S i} (cid:13)(cid:13) ≥ X V ∈ V l ν ( S, V ) ν ( T, V ) vol( V ) ≥ X V ∈ V l ν ( T, V ) n − l vol( V )
16y (4.4). On the other hand, by monotonicity of non-pluripolar products again, we get { S } ≤ α n − l − . This combined with the fact that α is K¨ahler implies α ∧ { S } ≤ α n − l .Hence, we deduce that (cid:13)(cid:13) α n − l − {h T ˙ ∧ S i} (cid:13)(cid:13) ≥ X V ∈ V l ν ( T, V ) n − l vol( V ) . This coupled with (4.2) gives (4.1). The proof is finished.We now prove Theorem 1.1. Let the notations be as in the statement of that result.
End of the proof of Theorem 1.1.
Let T min be a current with minimal singularities in α . Let θ be a smooth closed (1 , -form such that β := α + { θ } is K¨ahler. We can choose θ todepend only on k α k . Applying Theorem 3.4 to T, T min , θ gives (cid:13)(cid:13) {h ( T min + θ ) m i} − {h ( T + θ ) m i} (cid:13)(cid:13) ≤ C (cid:13)(cid:13) α m − {h T m i} (cid:13)(cid:13) − m , for some constant C depending only on N . Using this and the fact that α is big nef, weinfer (cid:13)(cid:13) β m − {h ( T + θ ) m i} (cid:13)(cid:13) ≤ C (cid:13)(cid:13) α m − {h T m i} (cid:13)(cid:13) − m . Now applying Theorem 4.1 to β, T + θ shows that the left-hand side of the last inequalityis greater than or equal to m X V ∈ V (cid:0) ν ( T, V ) (cid:1) n − dim V vol( V ) . Hence (1.2) follows. The proof is complete.
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E-mail address : [email protected]@math.uni-koeln.de