Entropy for Monge–Amp`ere Measures in the Prescribed Singularities Setting

. In this note, we generalize the notion of entropy for potentials in a relative full Monge–Amp`ere mass E ( X, θ, ϕ ), for a model potential ϕ . We then investigate stability properties of this condition with respect to blow-ups and perturbation of the cohomology class. We also prove a Moser–Trudinger type inequality with general weight and we show that functions with finite entropy lie in a relative energy class E nn − 1 ( X, θ, ϕ ) (provided n > 1), while they have the same singularities of ϕ when n = 1.

Let (X, ω) be a compact Kähler manifold of complex dimension n ≥ 1 and assume ω is normalized such that Vol(ω) := X ω n = 1.It is well known that Kähler metrics with constant scalar curvature are critical points of the Mabuchi K-energy defined as where the first two terms are energy terms while the latter is the entropy of the Monge-Ampère measure ω n u := (ω + dd c u) n .Here, given two positive Radon measures µ, ν, the relative entropy Ent(µ, ν) is defined as Ent(µ, ν) := X log dµ dν dµ, if µ is absolutely continuous with respect to ν, and +∞ otherwise.
This paper is a contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon for his 75th birthday.The full collection is available at https://www.emis.de/journals/SIGMA/Bourguignon.html The breakthrough result of Cheng and Chen [10,11] ensures that the existence of a cscK metric is equivalent to the properness of the Mabuchi functional.It is then crucial to relate the two notions of energy and entropy and to investigate them.
The first step in this direction has been done in [3], where the authors proved that full mass ω-psh functions whose Monge-Ampère measure has finite entropy have finite energy as well.In other words, they prove the following inclusion: Ent(X, ω) ∩ E(X, ω) ⊂ E 1 (X, ω), where Ent(X, ω) ∩ E(X, ω) is the set of ω-psh functions u such that ω n u has finite entropy with respect to a fixed volume form, say ω n .However, as all computable examples suggest, Ent(X, ω) is actually contained in a higher energy class E p (X, ω) for some p > 1 depending on the dimension.In [19], the authors indeed proved that Ent(X, ω) ∩ E(X, ω) ⊂ E n n−1 (X, ω).
In a series of papers [12,13,14], Darvas, Di Nezza and Lu developed the pluripotential theory in the relative big setting (see also [28,30] for the Kähler case).This proved to be very fruitful, and it allows us to work with potentials with not necessarily full mass in the general setting of a big cohomology class.In particular, given a big class {θ} and a "special" θ-psh function ϕ (called model potential ), they defined and studied relative Monge-Ampére energy classes E(X, θ, ϕ) and E χ (X, θ, ϕ), where χ : R + → R + is a weight function.These classes comprehend the energy classes defined in [9,24] for a particular choice of ϕ.Let us emphasize that such classes are at the heart of the variational approach for the search of (singular) Kähler-Einstein metrics.In the relative setting, the classes E(X, θ, ϕ) and E χ (X, θ, ϕ) are in turn crucial for the construction of Kähler-Einstein metrics with prescribed singularities, where the word "prescribed singularities" mean that the singularities of the potential of the metric are modeled by ϕ, as showed in [13,27,29].
The study of such classes, and their interplay with the (generalized) entropy is then useful to pursue the study of singular cscK metrics.
In this note, we define and study the entropy for Monge-Ampère measures θ n φ := (θ + dd c φ) n not necessarily with full mass, i.e., The function φ belongs to a relative full mass class E(X, θ, ϕ), where ϕ = P θ [φ] is a model potential with mass m.We refer to Section 2 for the definitions of all these notions.
We then define the θ-entropy of φ as We establish the stability properties of entropy with respect to proper bimeromorphic maps and we generalize the result in [19] relating finite entropy to finite relative energy.More precisely, given ϕ a model potential with positive mass, i.e., X θ n ϕ > 0, we prove: Theorem A (Theorem 3.7).With assumptions as above, we have Along the way, we obtain a Moser-Trudinger type inequality with a general weight, i.e., a continuous strictly increasing function χ : [0, +∞) → [0, +∞) such that χ(0) = 0 and χ(+∞) = +∞.
We end this note with a very natural question about stability of finiteness of the entropy under deformation of the cohomology class.More precisely, we ask the following: We emphasize that this kind of result does not seem to be known even when {θ} is a Kähler class: the (subtle) problem is that it is unknown if the absolute continuity of θ n φ with respect to ω n implies that (θ + εω + dd c φ) n has a density as well.

Preliminaries
We recall results from (relative) pluripotential theory of big cohomology classes.We borrow notation and terminology from [16].
Let (X, ω) be a compact Kähler manifold of dimension n and let θ be a smooth closed (1, 1)form on X.A function φ : X → R ∪ {−∞} is quasi-plurisubharmonic (qpsh) if it can be locally written as the sum of a plurisubharmonic function and a smooth function, and φ is called θplurisubharmonic (θ-psh) if it is qpsh and θ + dd c φ ≥ 0 in the sense of currents.Here, d and d c are real differential operators defined as d := ∂ + ∂, d c := i 2π ∂ − ∂ .We let PSH(X, θ) denote the set of θ-psh functions that are not identically −∞.We also assume that {θ} is big, i.e., there exists ψ ∈ PSH(X, θ) such that θ + dd c ψ ≥ εω for some small constant ε > 0. The current T = θ + dd c ψ is called Kähler current.
We say that a θ-psh function φ has analytic singularities if there exists a constant c > 0 such that locally on X, where g is bounded and f 1 , . . ., f N are local holomorphic functions.We say that φ has analytic singularities with smooth remainder if moreover g is smooth.
The ample locus Amp(θ) of {θ} is the complement of the non-Kähler locus where ν(T, x) is the Lelong number of T at the point x.The ample locus Amp(θ) is a Zariski open subset, and it is nonempty [8].Also we note that Amp(θ) only depends only the cohomology class {θ}.
A θ-psh function φ has minimal singularity type if it is less singular than any other θ-psh function.Such θ-psh functions with minimal singularity type always exist, one can consider for example V θ := sup{φ θ-psh, φ ≤ 0 on X}.Trivially, a θ-psh function with minimal singularity type is locally bounded in Amp(θ).It follows from [21, Theorem 1.1] that V θ is C 1, 1 in the ample locus Amp(θ).Given θ 1 + dd c φ 1 , . . ., θ p + dd c φ p positive (1, 1)-currents, where θ j are closed smooth real (1, 1)-forms, following the construction of Bedford-Taylor [1] in the local setting, it has been shown in [9] that the sequence of currents is non-decreasing in k and converges weakly to the so called non-pluripolar product In the following, with a slight abuse of notation, we will denote the non-pluripolar product simply by θ 1 When p = n, the resulting positive (n, n)-current is a Borel measure that does not charge pluripolar sets.Pluripolar sets are Borel measurable sets that are locally contained in the −∞ locus of psh functions.As a consequence of [4,Corollary 2.11] for any pluripolar set A, there exists ψ ∈ PSH(X, θ) such that A ⊂ {ψ = −∞}.
For a θ-psh function φ, the non-pluripolar product θ n φ is said to be the non-pluripolar Monge-Ampère measure of φ.
The volume of a big class {θ} is defined by For notational convenience, we simply write Vol(θ), but keeping in mind that the volume is a cohomological constant.Note that Vol(θ) > 0 as {θ} is big (cf.[9]).A θ-psh function φ is said to have full Monge-Ampère mass if and we then write φ ∈ E(X, θ).By [9, Theorem 1.16], the set E(X, θ) strictly contains the set of θ-psh functions with minimal singularity type.An important property of the non-pluripolar product is that it is local with respect to the plurifine topology (see [1,Corollary 4.3], [9,Section 1.2]).This topology is the coarsest such that all qpsh functions on X are continuous.For convenience, we record the following version of this result for later use.
Lemma 2.1.Fix closed smooth real big (1, 1)-forms θ 1 , . . ., θ n .Assume that φ j , ψ j , j = 1, . . ., n are θ j -psh functions such that φ j = ψ j on U , an open set in the plurifine topology.Then Lemma 2.1 will be referred to as the plurifine locality property.We will often work with sets of the form {u < v}, where u, v are quasi-psh functions.These are always open in the plurifine topology.
It is easy to see that P θ [ψ](ϕ) only depends on the singularity type of ψ.When ϕ = V θ , we will simply write P θ [ψ] := P θ [ψ](V θ ), and we refer to this potential as the envelope of the singularity type ] and typically equality does not happen.When [ψ] = [P θ [ψ]], we say that ψ has model singularity type.In the (more particular) case ψ = P θ [ψ], we say that ψ is a model potential.
It is worth to mention that given any θ-psh function ψ with positive mass, the associated envelope P θ [ψ] is in fact a model potential [16,Theorem 3.14].Also, we recall that by [16,Remark 3.4], we know that X θ n ψ = X θ n P θ [ψ] .From now on (unless otherwise stated), ϕ will denote a model potential with positive mass, i.e., X θ n ϕ > 0. We say that a θ-psh function φ has ϕ-relative minimal singularities if φ ≃ ϕ.
Definition 2.2.Given a model potential ϕ, the relative full mass class E(X, θ, ϕ) is the set of all θ-psh functions u such that u is more singular than ϕ and X θ n u = X θ n ϕ .
Proposition 2.3.Assume {θ} is a big cohomology class, and Then there exists a model potential u ∈ PSH(X, θ) such that X θ n u = m.Moreover, V θ is the only model potential with Monge-Ampère mass V , whereas there are infinitely many model potentials with Monge-Ampère mass m < V .
Proof .For the first statement, we need to treat the 1-dimensional case separately.When n = 1, the big cohomology class {θ} is actually Kähler, hence we can work with a Kähler form ω ∈ {θ} with X ω = V as a reference form.Fix a ∈ X, then the measures ω and V δ a define cohomologus closed positive (1, 1)-currents.Hence, there exists ψ a ∈ SH(X, ω) such that ω + dd c ψ a = V δ a and sup X ψ a = 0. Note that on a fixed local holomorphic coordinate chart centered in a we have that ψ a writes as the sum of V log(|z|) and a smooth function.Thus, ψ a (a) = −∞.Observe moreover that the non-pluripolar product (ω + dd c ψ a ) is given by the product of the current ω + dd c ψ a with the characteristic function 1 {ψa>−∞} .This implies that ω + dd c ψ a has mass zero.Since for t ∈ [0, 1], we have ω defined on [0, 1] is continuous and m(0) = V , m(1) = 0, we arrive at the conclusion.
Assume now n ≥ 2. Let us consider π : Y → X the blow up of X at a point p ∈ X with exceptional divisor E. Let [E] be the current of integration along E.
Therefore, for large t, {π * (θ )-current and, since the non-pluripolar product does not put mass of analytic subsets, we also get For the last statement, we observe that, since P [u t ] ≤ V θ for all t ∈ [0, 1], by [16, Theorem 3.14], V θ is the only model potential with Monge-Ampére mass V .Now, by [7, Corollary 1.18], the Lelong number of the function u 1 at p is strictly positive.Therefore, for 0 < t ≤ 1 also the Lelong number of u t is strictly positive at p.By [16, Lemma 5.1], so is the Lelong number of P [u t ].Therefore, fixing 0 ≤ m < V and varying the point p, we obtain infinitely many distinct model potentials all of the same Monge-Ampère mass m. ■ As pointed out in [24], and then in the relative setting in [26], it is natural to consider weighted subspaces of E(X, θ, ϕ).
We fix ϕ a model potential and we let E χ (X, θ, ϕ) denote the set of all u ∈ E(X, θ, ϕ) such that Compared to [24], we have changed the sign of the weight, but the weighted classes are the same.
Also, in the special case χ(t) = t p , p > 0, we simply denote the relative energy class with E p (X, θ, ϕ) and the corresponding relative energy E p (u, ϕ).
We now consider h : R + → R + such that h > 0 satisfying: Such a function h exists as we show below.We choose a sequence of distinct points {t k } k∈N ∈ R + such that ψ(t k ) ≥ k.Such a sequence exists since ψ is increasing and ψ → +∞ as t goes to +∞.Also, we can take t 1 = 1 since ψ(1) ≥ ψ(0) = 1.We then set I k := [t k , t k+1 [ and we define h to be the piece-wise continuous function constant on each We then define the function χ : R µ({φ<ϕ−s}) ds.We infer that the fundamental theorem of calculus holds almost everywhere.Indeed, for t / ∈ J ∪ {t k } k , given ε > 0 (small enough), the mean value theorem ensures that there exists c ε ∈ (t, t + ε) such that Sending ε → 0, and using the continuity of h • ψ on R + \ (J ∪ {t k } k ), we get that (χ ′ (t)) + = h(t)ψ(t).
Proof .For j ∈ N, by Lemma 2.1, we have The conclusion follows letting j → +∞.■

Entropy
We recall that given two positive probability measures µ, ν, the relative entropy Ent(µ, ν) is defined as if µ is absolutely continuous with respect to ν, and +∞ otherwise.
Remark 3.1.Let µ, ν positive probability measure with µ := f ν absolutely continuous with respect to ν.Then Ent(µ, ν) < +∞ if and only if Once and for all, we normalize the Kähler form ω such that X ω n = 1.We consider u ∈ PSH(X, θ) such that θ n u = f ω n , 0 ≤ f and m u := X θ n u > 0. Then u ∈ E(X, θ, ϕ) for the model potential ϕ = P θ [u] [13, Theorem 1.3], and m −1 u θ n u is a probability measure.We then define the θ-entropy of u as By Jensen's inequality, we have Ent θ (u) ≥ 0. Also, observe that the definition of the θ-entropy does depend on the chosen volume form ω n but its finiteness does not.Also, the expression in (3.1) coincides with the definition of entropy in [19] when P [u] = V θ , i.e., when u ∈ E(X, θ).The definition in (3.1) is indeed a generalisation that allows to consider any θ-psh function not necessarily of full mass.
More generally, given two θ-psh functions u, v with m u , m v > 0 we define Also, if no confusion can arise, we simply write Ent(u) and Ent(u, v).
We start with the following observations ensuring that the set Ent(X, θ) is not empty.
Proof .By [16, Theorem 3.6], for some positive constant C. In particular, θ n ϕ = gω n for some g ∈ L ∞ (X), 0 ≤ g ≤ C.This proves (i).For (ii), we observe that since p − 1 > 0, we have that Given φ ∈ PSH(X, θ) with analytic singularities with smooth remainder, it follows from [13, Proposition 4.36] that θ n φ = f ω n with f ∈ L p (X).The previous step then gives (iii).We now prove the last statement.We first assume k = 1 and f 1 quasi-psh.If we let θ = θ + dd c f 1 , then Observe that since f 1 is C 1, 1 and quasi-psh, dd c f 1 is bounded.This means that θ is a (1, 1)-form with bounded coefficients.By [22, Corollary 3.4 (i)], we have θn Pθ(0) = g 1 θn for some non-negative bounded function g 1 on X.Since θ has bounded coefficients, we can ensure that there exists a non-negative bounded function g 2 such that θn Pθ(0) = g 2 ω n .It then follows that P θ (f 1 ) has finite entropy with respect to ω.
We now treat the general case of k functions f 1 , . . ., f k which are assumed to be only C 1, 1.We choose C > 0 such that θ ≤ Cω and we claim that for the other inequality, we have that and that P θ (f 1 , . . ., f k ) is θ-psh hence it is also Cω plurisubharomnic.This implies If we now apply P θ to both sides of the above inequality, we find By [17,Theorem 2.5], P Cω (f 1 , . . ., f k ) is C 1, 1 on X and quasi-psh.We can then apply the previous step to conclude.■ Since, by [16,Theorem 5.19], we know that [ϕ] = [φ], the above results could make a reader ask whether the property of having finite entropy is stable in the singularity class, i.e., if given φ 1 , φ 2 ∈ PSH(X, θ) with [φ 1 ] = [φ 2 ], then Ent θ (φ 1 ) < +∞ iff Ent θ (φ 2 ) < +∞.The answer is negative as the following example shows: Example 3.4.Let U ⊂ X be a local chart and write ω = dd c ρ in U and define where χ is a cut-off function such that χ ≡ 1 on B(r 1 ) and χ ≡ 0 on U \B(r 2 ), for r 1 , r 2 > 0 small enough so that B(r 2 ) ⋐ B(r 2 ) ⋐ U .Without loss of generality, we may assume r 1 = 1, r 2 = 2. Choosing C big enough, u induces a ω-psh function which we note by ũ.Then ũ is bounded, hence and the measure (dd c (max(log ∥z∥, 0)) n is the normalized Lebesgue measure on the torus S 1 n ⊂ C n (that is, a real analytic subspace of real dimension n).It then follows that (ω + dd c ũ) n is not even absolutely continuous with respect to ω n .

Entropy and energy
It was proved in [19,20] that As will be shown below, one can extend these results to the case of prescribed singularities.We start with an integrability result of Moser-Trudinger type for general weights: Let ϕ be a model potential with m ϕ > 0. Let χ 1 : R + → R + be a weight.Let χ 2 (t) := t 0 χ 1 (s) 1 n ds.Then there exist c > 0, C > 0 depending on X, θ, n, m ϕ and ω such that, for all φ ∈ E χ 1 (X, θ, ϕ) with sup X φ = −1, we have Proof .We first note that if we replace χ 1 by αχ 1 with α positive constant the left-hand side of (3.2) does not change, so we may assume χ 1 (1) = 1.We claim that it suffices to prove the above inequality for φ with relative minimal singularities, i.e., [φ] = [ϕ].Indeed, given φ ∈ PSH(X, θ), φ j := max(φ, ϕ − j) has the same singularity type as ϕ.This would mean that Moreover, it follows from Lemma 2.5 that E χ 1 (φ j , ϕ) ↗ E χ 1 (φ, ϕ) as j → +∞.Fatou's lemma will then give the desired estimate.Thus assume that φ has relative minimal singularities.Let ψ = −aχ 2 (ϕ − φ) + ϕ where a > 0 is a small constant to be suitably chosen.Then define where γ 2 : R + → R + denotes the inverse function of aχ 2 , which is concave and increasing.We observe that u is a θ-psh function with the same singularity type as ϕ and that with equality on the contact set C = {u = ψ}.
A simple computation gives where in the above we used that γ 2 is concave.We consider the set G := {γ ′ 2 (ϕ − u) < 1}.We are going to show by contradiction that for a suitable choice of a we have G ̸ = X.So, we assume G = X.It then follow that v is θ-psh and, by construction, we infer that v has the same singularity type as ϕ.
Recall that sup X φ = sup X (φ−ϕ) as ϕ is a model potential [16, Lemma 3.5], hence ϕ−φ ≥ 1.This implies that χ 1 is increasing and χ 1 (1) = 1 giving that By [16, Theorem 2.7], the non-pluripolar Monge-Ampère measure (θ + dd c u) n is supported on C, hence The last identity follows from the fact that, since (aχ [16,Lemma 5.19] that the above inequality between positive currents implies an inequality between the non-pluripolar measures (observe that this is not trivial since (aχ ′ 2 (ϕ− φ)) −1 (θ + dd c u) is not closed).Thus, we can infer that where the last inequality follows from [15,Lemma 4.5].The above is equivalent to where in the last inequality we used that (χ ′ 2 ) n = χ 1 .We now choose a so that Observe that a ∈ (0, 1) since we observed that E χ 1 (φ, ϕ) ≥ m ϕ .Integrating both sides of (3.3) over X, we obtain Since X θ n u = m ϕ , we arrive at a contradiction.So we can infer that G ̸ = X, or equivalently that G c ̸ = ∅.This means that there exists x 0 ∈ X such that (ϕ−u)(x 0 ) ≤ τ 2 (1) where τ 2 is the inverse function of γ ′ 2 which we note to be decreasing (since so in γ ′ 2 ).In particular, sup Applying the uniform version of Skoda's integrability theorem [31] to PSH(X, Cω) for C > 0 such that θ ≤ Cω, we know that there exist uniform constants c 0 , C 0 > 0 such that, for all h ∈ PSH(X, θ) with sup X h = 0 we have X e −c 0 h ω n ≤ C 0 .For h := u + τ 2 (1), we have sup X h ≥ 0, hence where in the first inequality we used ϕ ≤ 0 and u ≤ ψ.
It follows that Let ϕ be a model potential with X θ n ϕ > 0. Let χ 1 : R + → R + be a weight.Let χ 2 (t) := t 0 χ 1 (s) and let m(t) be the mass of Ω(t) with respect to ω n .Then there exists S > 1 depending only on X, θ, n, ω such that Proof .Observe that again all inequalities are unchanged if we replace χ 1 with αχ 1 where α is a positive constant, so we assume χ 1 Since ω has volume 1, by Jensen's inequality and by Theorem 3.5, we find By the Fubini theorem, we obtain the inequality if we recall the value of the constants C and c given in the proof of 3.5 and we let S = 2 log(C 0 (e C 0 +1)) Also, observe that, if we choose φ = ϕ − 1 and χ 1 (t) = t p with p > 0, we find S m ϕ ≥ n n (p+n) n m ϕ .If we let p go to zero, we conclude that S m ϕ ≥ 1 m ϕ , i.e., S ≥ 1.If we apply the inequality choosing φ as in the statement of the corollary with φ ̸ = ϕ − 1, we see that in fact S > 1. ■ Remark 3.6.Observe that χ 1 (t) = t p and χ 2 (t) = q −1 t q with q = 1 + p/n satisfy the assumptions of Theorem 3.5.With this particular choice of weights, we have that Thus, in this particular case τ 2 (1) = a − n p q −1 , that is, (up to a power of q) the constant appearing in [20,Theorem 2.11].

Here Ent
We also emphasize the constants c, C > 0 in the statement do not depend on ϕ.
Using (3.5) again, we see that > 0 is a uniform constant.
Moreover, by construction we have that ω + dd c ψ k a is the Lebesgue measure σ k on the sphere ∥z| = e −k (see [25,Example 3.13]) normalized such that {∥z∥=e −k } dσ k = 1.

Stability of the entropy
We collect some results on how the property of having finite entropy changes when the reference probability measure changes, when we perturb the reference big cohomology class and when we pull-back through bimeromorphic holomorphic maps.