Pluricomplex Green's functions and Fano manifolds

We show that if a Fano manifold does not admit Kahler-Einstein metrics then the Kahler potentials along the continuity method subconverge to a function with analytic singularities along a subvariety which solves the homogeneous complex Monge-Ampere equation on its complement, confirming an expectation of Tian-Yau.


Introduction
Let X n be a Fano manifold, i.e. a compact complex manifold with c 1 (X) > 0. A Kähler-Einstein metric on X is a Kähler metric ω which satisfies Ric(ω) = ω.
This implies that [ω] = c 1 (X). We assume throughout this paper that X does not admit a Kähler-Einstein metric. This is known to be equivalent to K-unstability by [13] (see also [40]), but we will not use this fact. We fix a Kähler metric ω with [ω] = c 1 (X), with Ricci potential F defined by Ric(ω) = ω + √ −1∂∂F (normalized by X (e F − 1)ω n = 0). We consider Kähler metrics ω t with [ω t ] = c 1 (X) which satisfy We can write ω t = ω + √ −1∂∂ϕ t and the functions ϕ t solve the complex Monge-Ampère equation [46] ω n t = e F−tϕ t ω n . (1.1) A solution ϕ t exists on [0, R(X)) where R(X) 1 is the greatest lower bound for the Ricci curvature of Kähler metrics in c 1 (X) [36]. It is known [35,38] that since X does not admit Kähler-Einstein metrics, we must have that lim t→R(X) sup X ϕ t = +∞. We fix a sequence t i → R(X) and write ϕ i := ϕ t i and ω i := ω t i . Using this result, together with multiplier ideal sheaves, Nadel [29,Proposition 4.1] proved that (up to passing to a subsequence) the measures ω n i converge to zero (as measures) on compact sets of X\V for some proper analytic subvariety V ⊂ X, and in [44] the second-named author improved this to uniform convergence.
By weak compactness of closed positive currents in a fixed cohomology class, up to subsequences we can extract a limit ρ of ϕ i − sup X ϕ i (which may depend on the subsequence), which is an unbounded ω-psh function, and the convergence happens in the L 1 topology.
In this note we confirm Tian-Yau's expectation: Theorem 1.1. Let X be a Fano manifold without a Kähler-Einstein metric, and let ω t = ω + √ −1∂∂ϕ t be the solutions of the continuity method (1.1). Given any sequence t i ∈ [0, R(X)) with t i → R(X), choose a subsequence such that ϕ t i − sup X ϕ t i converge in L 1 (X) to an ω-psh function ρ. Then we can find m 1 and an ω-psh function ψ on X with analytic singularities

2)
for some λ j ∈ (0, 1] and some sections S j ∈ H 0 (X, K −m X ), with nonempty common zero locus V ⊂ X such that ρ − ψ is bounded on X, and on X\V we have where the Monge-Ampère product is in the sense of Bedford-Taylor [6].
In particular, Theorem 1.1 implies that the non-pluripolar Monge-Ampère operator of ρ (defined in [12]) vanishes identically on X. On the other hand, there is another meaningful Monge-Ampère operator that can be applied to ρ. Indeed, the fact that ρ −ψ ∈ L ∞ (X) implies that ρ itself has analytic singularities. In [3] Andersson-Błocki-Wulcan defined a Monge-Ampère operator for ω-psh functions with analytic singularities (generalizing earlier work of Andersson-Wulcan [4] in the local setting). In general, applying this Monge-Ampère operator to ρ will produce a Radon measure µ on X (which may be identically zero in some cases), which by Theorem 1.1 is supported on the analytic set V , thus providing geometrically interesting examples of unbounded quasi-psh functions on compact Kähler manifolds with Monge-Ampère operator concentrated on a subvariety (see also [1,2] for related results in the local setting). In particular, this answers [11,Question 1 (c)], an open problem raised at the AIM workshop "The complex Monge-Ampère equation" in August 2016 (cf. the related [23,Question 12]).
Note that in general a formula for the total mass of µ is proved in [3, Theorem 1.2], and it satisfies X µ X ω n , with strict inequality in general (but it is not hard to see that if dim X = 2 and V is a finite set then equality holds). Therefore, the measure µ is in general different from the measures that one obtains as weak limits of (ω + √ −1∂∂ϕ i ) n (up to subsequences), whose total mass is always equal to X ω n .

Remark 1.2.
[Remark added in proof] After this work was posted on the arXiv, and partly prompted by it, Błocki [10] modified the definition of the Monge-Ampère operator for ω-psh functions with analytic singularities of [4,3], and with his definition the total mass is always equal to X ω n . It is an interesting question to determine whether this Monge-Ampère operator equals the weak limit of (ω + √ −1∂∂ϕ i ) n . Remark 1.3. As in the second-named author's previous work [44], Theorem 1.1 has a direct counterpart for solutions of the normalized Kähler-Ricci flow, instead of the continuity method (1.1). The statement is identical to Theorem 1.1, except that now the sequence t i goes to +∞. The proof is also almost verbatim the same, and the partial C 0 estimate along the flow is proved in [14,15] (see also [5]). All other ingredients used also have well-known counterparts for the flow (see [44]). We leave the simple details to the interested reader.
Remark 1.4. The behavior of the solutions ω t of (1.1) as t → R(X) has been investigated in the past. If the manifold is K-stable, [20] show that ω t converge smoothly to a Kähler-Einstein metric. If on the other hand no such metric exists, the blowup behavior of ω t has been investigated in [29,44] in the setting of this paper, and also in [20,26,33] by allowing reparametrizations of the metrics by diffeomorphisms.
The proof of Theorem 1.1 relies on the partial C 0 estimate for solutions of (1.1) which was established by Székelyhidi [37]. We recall this in section 2, together with a well-known reformulation of this estimate (Proposition 2.1). In section 3 we observe that this gives us the singularity model function ψ in (1.2), and it also implies that ρ has the same singularity type as ψ. In section 4 we show the general fact that every ω-psh function on X with the same singularity type as ψ has vanishing Monge-Ampère operator outside V , thus proving Theorem 1.1. This relies on a geometric understanding of the rational map defined by the sections {S j } as in Theorem 1.1. Lastly, in section 5 we discuss the pluricomplex Green's function with the same singularity type as ψ.

The partial C 0 estimate
To start we fix some notation. We choose a Hermitian metric h on K −1 X with curvature R h = ω (such h is unique up to scaling), and let h m be the induced metric on K −m X , for all m 1. Let N m = dim H 0 (X, K −m X ), and for any m 1 define the density of states function where S 1 , . . . , S N m are a basis of H 0 (X, K −m X ) which is orthonormal with respect to the L 2 inner product X S 1 , S 2 h m ω n . Clearly ρ m (ω) is independent of the choice of basis, and is also unchanged if we scale h by a constant. The integral X ρ m (ω)ω n equals N m , and if m is sufficiently large so that K −m X is very ample, then ρ m (ω) is strictly positive on X. If we apply this same construction to the metrics ω t and Hermitian metrics h t = he −ϕ t we get a density of states function ρ m (ω t ). Following [39], we say that a "partial C 0 estimate" holds if there exist m 1 and a constant C > 0 such that holds for all t ∈ [0, R(X)). The reason for this name is explained by the following proposition, which is essentially well-known (see [ In the rest of the paper we will fix a value of ε > 0 once and for all, for example ε = R(X)/2. The precise choice is irrelevant, since we are only interested in the behavior as t → R(X).
To see this, first observe for every S ∈ H 0 (X, K −m X ) we have and that since Ric(ω t ) tω t εω t , Myers' Theorem gives a uniform upper bound for diam(X, ω t ) and then Croke [19] and Li [27] show that the Sobolev constant of (X, ω t ) has a uniform upper bound. We can then apply Moser iteration to (2.4) to get provided we assume that X |S| 2 h m t ω n t = 1. Taking now an orthonormal basis of sections and summing we obtain (2.3).
Thanks to (2.3) we know that for t ∈ [ε, R(X)) a partial C 0 estimate is equivalent to We now take a basis {S j (t)} 1 j N m of H 0 (X, K −m X ) orthonormal with respect to the L 2 inner product of ω t , h m t and notice that since h m t = e −mϕ t h m we clearly have which is equivalent to It follows from (2.6) and (2.7) that that for t ∈ [ε, R(X)) a partial C 0 estimate is equivalent to an estimate We now choose another basis for some positive real numbers µ j (t), with µ 1 (t) . . . µ N m (t) > 0. We then let λ j (t) = µ j (t)/µ 1 (t) and we see that a partial C 0 estimate is equivalent to We now claim that if a partial C 0 estimate holds, then for all t ∈ [ε, R(X)) we also have Once this is proved, combining (2.8) and (2.9) we get (2.2). To prove (2.9), first use (2.5) to get On the other hand the partial C 0 estimate (2.1) implies that and we clearly have that sup since the sections {S j (t)} are just varying in a compact unitary group (or one can also repeat the Moser iteration argument of (2.3) for the fixed metric ω). This together with (2.10), evaluated at the point where ϕ t achieves its maximum, gives the reverse inequality which completes the proof of (2.9).

The singularity model function
The next goal is to use the partial C 0 estimate in Proposition 2.1 to construct a singular ω-psh function ψ which will have the same singularity type of any weak limit of the normalized solutions ϕ i − sup X ϕ i of the continuity method. Let the notation be as in Proposition 2.1, and in particular we fix once and for all a value of m 1 given there. We can find a sequence t i → R(X) and an ω-psh function ρ with sup X ρ = 0 such that ϕ i − sup X ϕ i → ρ in L 1 (X), and pointwise a.e. Passing to a subsequence, we can find a basis For ease of notation, write These functions are Kähler potentials for ω since Up to passing to a subsequence of t i , we may assume that λ j (t i ) → λ j as i → ∞ for all j, and we have for some 1 p < N m . The case p = N m is impossible because by (2.2) it would imply a uniform L ∞ bound for ϕ t and so X would admit a Kähler-Einstein metric. For the same reason, the set V := {S 1 = · · · = S p = 0} must be a nonempty proper analytic subvariety of X.
Note that thanks to (2.2) we can write ω n t = e F−t(ϕ t −sup X ϕ t ) e −t sup X ϕ t ω n Ce tψ t e −t sup X ϕ t ω n , and since the term e tψ t is uniformly bounded on compact sets of X\V , we see immediately that uniformly on compact sets of X\V (this result was proved in [44] without using the partial C 0 estimate, which was not available at the time, with weaker results established earlier in [29]). Let then which is a smooth function on X\V which approaches −∞ uniformly on V . Since e mψ t → e mψ smoothly on X, and since ψ t are smooth and ω-psh, it follows that ψ is ω-psh. This will be our singularity model function in the rest of the argument, as we now explain: of ω-psh functions with the same singularity type as ψ. Then we have that ρ ∈ C.
Proof. Recall that we have ϕ i − sup X ϕ i → ρ a.e. on X. Thanks to (2.2), the function ρ satisfies

Understanding the class C
We now exploit the geometry of our setting to gain a better understanding of the class of functions C. The sections {λ j S j } 1 j p define a rational map Φ : X CP p−1 , with indeterminacy locus Z ⊂ V (this inclusion is in general strict, since codimZ 2 while V may contain divisorial components). Let Y be the image of Φ, i.e. the closure of Φ(X\Z) in CP p−1 , which is an irreducible projective variety. By resolving the indeterminacies of Φ we get a modification µ :X → X, obtained as a sequence of blowups with smooth centers, and a holomorphic map Ψ :X → Y such that Ψ = Φ • µ holds onX\µ −1 (Z). We may also assume without loss of generality that µ principalizes the ideal sheaf generated by {S j } 1 j p , so that we have where E is an effective R-divisor with µ(E) ⊂ V , and θ is a smooth closed semipositive (1, 1) form onX. We will denote by ω FS,p the Fubini-Study metric on CP p−1 . To identify θ, note that on X\V we have by  LetX ν →Ỹ q → Y be the Stein factorization of Ψ , whereỸ is an irreducible projective variety, the map ν has connected fibers, and q is a finite morphism.
We have that q * ω FS,p is a smooth semipositive (1, 1) form onỸ , in the sense of analytic spaces. Since ν has compact connected fibers, a standard argument shows that the set of -psh function to any fiber of ν is plurisubharmonic and hence constant on that fiber). We will use this standard argument several other times in the following.
Here and in the following, as in [21], a weakly quasi-psh function on a compact analytic space means a quasi-psh function on its regular part which is locally bounded above near the singular set. As shown in [21, §1], weakly quasi-psh functions are the same as usual quasi-psh functions if the analytic space is normal, and otherwise they can be identified with quasi-psh functions on its normalization.
Conversely, given any bounded weakly q * ω FS,p m -psh function u onỸ there is a unique function η ∈ C such that (4.2) holds.
The relation in (4.2) thus allows us to identify the class C with the class of bounded weakly q * ω FS,p m -psh functions onỸ .
Next, we observe that This is a consequence of our assumption that X does not admit a Kähler-Einstein metric. Lastly, every function η ∈ C belongs to L ∞ loc (X\V ), and so its Monge-Ampère operator (ω + √ −1∂∂η) n is well-defined on X\V thanks to Bedford-Taylor [6]. Combining the results in Propositions 4.1 and 4.2 we will obtain: Theorem 4.3. For every η ∈ C we have that In particular, this holds for the function ρ, thanks to Lemma 3.1, and Theorem 1.1 thus follows from these.
Conversely, given a bounded weakly q * ω FS,p m -psh function u onỸ , we have that ν * u is Ψ * ω FS,p m -psh and bounded onX and so and so µ * ψ + ν * u descends to an ω-psh function η u on X with η u − ψ ∈ L ∞ (X), i.e. η u ∈ C. These two constructions are inverses to each other, and so we obtain the desired bijective correspondence between functions in C and bounded weakly q * ω FS,p m -psh functions onỸ . Proof of Proposition 4.2. On X we have the estimate which is a direct consequence of the partial C 0 estimate (see e.g. [24,Lemma 4.2]). We can also give a direct proof by calculating if A is sufficiently large, and applying the maximum principle together with the partial C 0 estimate (2.2) (for this calculation we used that the bisectional curvature of the metrics have a uniform upper bound independent of t).
If we had dim Y = dim X then the rational map Φ would be generically finite, so there would be a nonempty open subset U X\V such that Φ| U is a biholomorphism with its image. Recall that Φ is the rational map defined by the sections {λ j S j } 1 j p , while ι : X → CP N m −1 is the embedding defined by the sections {S j } 1 j N m , and so Φ =τ • P • ι where P : CP N m −1 CP p−1 is the linear projection given by [z 1 : · · · : z N m ] → [z 1 : · · · : z p ] andτ : CP p−1 → CP p−1 is the automorphism given by [z 1 : · · · : z p ] → [λ 1 z 1 : · · · : λ p z p ].
In particular, on the embedded open n-fold ι(U ), we have that P | ι(U ) is also a biholomorphism with its image. The automorphisms τ(t i ) descend to automorphismsτ(t i ) on CP p−1 , and now as i → ∞ these converge smoothly to the automorphismτ.
Since Φ is an isomorphism on U , smooth convergence gives us that P • τ(t i ) • σ (t i ) • ι is a local isomorphism. Thus, after possibly shrinking U , is an isomorphism, and for i large the open sets (τ( for all i large, and still P −1 is well-defined on V (and P : On U we also have that for all i large, which implies that U ω n i C −1 , which is absurd thanks to (3.2). 5. The pluricomplex Green's function 10 5. The pluricomplex Green's function

Remark 4.4.
In particular we see that if dim Y = 0 (i.e. Y is a point) then we have C = {ψ + s} s∈R . On the other hand as long as dim Y > 0 the class C is always rather large.
Proof of Theorem 4.3. Thanks to Proposition 4.1, every η ∈ C satisfies µ * η = µ * ψ + ν * u for some bounded weakly q * ω FS,p m -psh function u onỸ . Then using (4.1) we have and so if K is any compact subset of X\V , since µ is an isomorphism on µ −1 (K), we get

The pluricomplex Green's function
We can also consider the pluricomplex Green's function with singularity type determined by ψ, namely which is the compact manifold analog of the construction in [31], and has been studied in detail in [18,30,32] and references therein. In particular, since ψ has analytic singularities, it follows from [31,32] that G ∈ C. Thanks to Proposition 4.1 we can write for a bounded weakly q * ω FS,p m -psh function F onỸ . The function F is itself given by a suitable envelope. Proposition 5.1. The pluricomplex Green's function G satisfies (5.2) where F is the envelope onỸ given by and where we are writing ν * (f )(y) = sup for any function f onX, y ∈Ỹ .
In other words, F is given by a quasi-psh envelope with obstacle −ν * µ * ψ onỸ .
then we have thatG = µ * G (this is again because every µ * ω-psh function onX is in fact the pullback of an ω-psh function on X).
As in [28], we use a trick from [8,Section 4] (see also [31]), to show that For the reader's convenience, we supply the simple proof. Denote the right hand side byĜ. For one direction, if v is (µ * ω − R h )-psh and satisfies v − log |s| 2 h , then u := v + log |s| 2 h satisfies u 0 but also since v C onX, we see that u log |s| 2 h + C, and also and soĜ G . Conversely, if u is µ * ω-psh and satisfies u 0 and u log |s| 2 h + C for some C, then the Siu decomposition of µ * ω + √ −1∂∂u contains [E] and so and so v := u − log |s| 2 h is (µ * ω − R h )-psh and satisfies v − log |s| 2 h , and it follows thatG Ĝ , which proves our claim.
But finally note that for all x ∈X we have and taking the upper-semicontinuous regularization and using the claim above gives µ * G = µ * ψ + ν * F, which completes the proof.
Using Proposition 5.1 we can see that F is continuous on a Zariski open subset ofỸ , using the following argument. Let g : Y →Ỹ be a resolution of the singularities ofỸ . Then we have: Note that g * q * ω FS,p /m is semi-positive and big, and that −g * ν * µ * ψ is continuous off of g −1 (ν(µ −1 (ψ −1 (−∞)))), where it is unbounded. Using the trick in [28], we can replace the obstacle −g * ν * µ * ψ with a globally continuous obstacle h without changing g * F. Now, approximate h uniformly by smooth functions h j . It is easy to see that the envelopes: converge uniformly to g * F. But then by [8], the F j are continuous away from the non-Kähler locus of g * q * ω FS,p /m (a proper Zariski closed subset, see e.g. [12]), so we are done.

Remark 5.2.
One is naturally led to wonder about what the optimal regularity of G is. The sharp C 1,1 regularity (on a Zariski open subset) of envelopes of the form (5.3) has been recently obtained in [17,45] in Kähler classes and in [16] in nef and big classes (see also [7,8,9]) when the obstacle is smooth (or at least C 1,1 ), but in our case the regularity of −ν * µ * ψ does not seem to be very good, especially near the points where ν is not a submersion.
On the other hand, the first-named author [28] has very recently obtained C 1,1 regularity (on a Zariski open subset) of envelopes with prescribed analytic singularities, which include those of the form (5.1), generalizing results in [32] in the case of line bundles. In our situation, the results of [28,32] do not apply since in (5.1) the functions u and ψ are both ω-psh (while for these results one would need them to be quasi-psh with respect to two different (1, 1)-forms such that the cohomology class of their difference is big). Moreover, the main result of [28] also allows for u and ψ being both ω-psh, but then needs the condition that the total mass of the non-pluripolar Monge-Ampère operator of ψ be strictly positive. This is obviously not the case in our situation however, by Theorem 4.3. where here V Ω is the global (Siciak) extremal function for Ω. In particular, one sees that Ω is regular. There is then a well-developed theory about Hölder continuous regularity for such functions (the so called HCP property), see e.g. [34]. It may be possible to use this theory to study G, if one can first show that it is continuous in at least a neighborhood of V . Another possibility may be to study regularity of the boundary of Ω -see the very end of [28].

Remark 5.4.
On can also naturally ask whether the function ρ (and therefore also its singularity type ψ) in Theorem 1.1 is actually independent of the choice of subsequence t i , and also how regular ρ is on X\V . Our guess is that ρ is indeed uniquely determined, and is smooth on X\V . These properties would both follow if one could show that the map Φ : X Y is independent of the chosen subsequence, and that the corresponding function u onỸ given by Lemma 3.1 and Proposition 4.1 which satisfies µ * ρ = µ * ψ + ν * u, actually solves a suitable complex Monge-Ampère equation onỸ . In a related setting of Calabi-Yau manifolds fibered over lower-dimensional spaces, such a limiting equation after collapsing the fibers was obtained by the second-named author in [42, Theorem 4.1].
Remark 5.5. Lastly, we can also ask whether the limit ρ (if it is unique) is necessarily equal to the pluricomplex Green's function G up to addition of a constant. By remark 4.4 this is the case if the rational map Φ is constant, so that Y is a point. In general though this seems rather likely false.