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Complex Monge-Ampere and Symplectic Manifolds
... In the Kähler setting, the GIT framework is naturally completed by a formal Riemannian metric, known as the Mabuchi-Semmes-Donaldson metric [20,56,60]. This metric formally gives K˛the structure of a symmetric space of nonpositive curvature. ...
... Such curves do not readily reduce to a Monge-Ampère equation, as exploited in the construction of (weak) geodesics in [15]. However, there is a generalization of the Semmes construction [60] expressing the geodesic equation as a prescribed volume form problem for a natural family of symplectic forms on an augmented spacetime track (cf. Remark 5.4). ...
... We now introduce a formal Riemannian metric on the space GK ;˛, generalizing the Mabuchi-Semmes-Donaldson Riemannian structure [20,56,60] on K˛. Recall that at any F 2 GK ;˛, we showed that the tangent space T F .GK ;˛/ is identified with C 1 .M; R/=R, see Remark 2.18. ...
On a compact complex manifold (M, J) endowed with a holomorphic Poisson tensor \pi_{J} and a de Rham class \alpha\in H^{2}(M, \mathbb{R}) , we study the space of generalized Kähler (GK) structures defined by a symplectic form F\in \alpha and whose holomorphic Poisson tensor is \pi_{J} . We define a notion of generalized Kähler class of such structures, and use the moment map framework of Boulanger (2019) and Goto (2020) to extend the Calabi program to GK geometry. We obtain generalizations of the Futaki–Mabuchi extremal vector field (1995) and the Calabi–Lichnerowicz–Matsushima result (1982, 1958, 1957) for the Lie algebra of the group of automorphisms of (M, J, \pi_{J}) . We define a closed 1 -form on a GK class, which yields a generalization of the Mabuchi energy and thus a variational characterization of GK structures of constant scalar curvature. Next we introduce a formal Riemannian metric on a given GK class, generalizing the fundamental construction of Mabuchi–Semmes–Donaldson (1987, 1992, 1997) We show that this metric has nonpositive sectional curvature, and that the Mabuchi energy is convex along geodesics, leading to a conditional uniqueness result for constant scalar curvature GK structures. We finally examine the toric case, proving the uniqueness of extremal generalized Kähler structures and showing that their existence is obstructed by the uniform relative K-stability of the corresponding Delzant polytope. Using the resolution of the Yau–Tian–Donaldson conjecture in the toric case by Chen–Cheng (2021) and He (2019), we show in some settings that this condition suffices for existence and thus construct new examples.
... In fact, Theorem 1.4 can be seen as a 2-dimensional toy model for this problem. As in this case, uniqueness is a delicate issue, related to the geodesic equation in the space of Kähler potentials [20,39,48]; see Section 5.3 for details and a discussion of the relevance of the homogeneous complex Monge-Ampère equation in this problem. Theorem 1.4 clarifies Yang's guess [61] that the location of the zeros of φ should "play an important role to global existence" and his observation that the condition corresponding to strict polystability is a "borderline situation" (with solutions preserved by an S 1 -action). ...
... This approach rests on the geometry of the infinite-dimensional space B, and the closely related space K of Kähler forms on Σ with fixed volume Vol(Σ). The space B is a symmetric space, as briefly reviewed in Section 5.1, and the space K is a Riemannian symmetric space, as shown by Semmes [48] and rediscovered by Mabuchi [39] and Donaldson [20]. Since the geodesic equation on K is the first equation in (5.1), i.e. the map ...
... As shown by Donaldson [20] and Semmes [48], for a suitable choice of Riemann surface D, the geodesic equation on K reduces to a homogeneous complex Monge-Ampère equation on the complex surface Σ × D. This method has been fruitfully applied in the context of the problem for constant scalar curvature Kähler metrics [9,12,16,47]. We expect that these results, and in particular the recent proof of the uniqueness of constant scalar curvature Kähler metrics by Berman and Berndtsson [7], can be adapted to the context of the geodesic equation (5.1). ...
In this work we consider the gravitating vortex equations. These equations couple a metric over a compact Riemann surface with a hermitian metric over a holomorphic line bundle equipped with a fixed global section --- the Higgs field ---, and have a symplectic interpretation as moment-map equations. As a particular case of the gravitating vortex equations on , we find the Einstein--Bogomol'nyi equations, previously studied in the theory of cosmic strings in physics. We prove two main results in this paper. Our first main result gives a converse to an existence theorem of Y. Yang for the Einstein--Bogomol'nyi equations, establishing in this way a correspondence with Geometric Invariant Theory for these equations. In particular, we prove a conjecture by Y. Yang about the non-existence of cosmic strings on superimposed at a single point. Our second main result is an existence and uniqueness result for the gravitating vortex equations in genus greater than one.
... The point of view taken in this paper is to exploit and develop the relation between test configurations and geodesic rays, which is a key idea going back to [19,4,20] and others. It is well known that, by solving a certain homogeneous complex Monge-Ampère equation [55,19,40], one may associate to a given test configuration a (unique, up to certain choices) geodesic ray in the space of Kähler potentials. The corresponding result in the Kähler case was proven in [56,38]. ...
... Fix a compact Kähler manifold (X, ω) and let (ϕ t ) t≥0 ⊂ PSH(X, ω) be a ray of ω-psh functions on X. Following Donaldson [41] and Semmes [55] we consider the standard correspondence between the family (ϕ t ) t≥0 and an associated S 1 -invariant function Φ on X ×∆ * , given by Φ(x, e −t+is ) = ϕ t (x), where the sign is chosen so that t → +∞ corresponds to τ := e −t+is → 0. Here∆ * ⊂ C denotes the punctured unit disc. The function Φ restricted to a fiber X × {τ } thus corresponds precisely to ϕ t on X. ...
... This terminology is motivated by the extensive study of (weak) geodesics in the space H, see e.g. [11,19,28,41,55,26]. ...
In this paper we study K-polystability of arbitrary (possibly non-projective) compact K\"ahler manifolds admitting holomorphic vector fields. As a main result, we show that existence of a constant scalar curvature K\"ahler (cscK) metric implies 'geodesic K-polystability', in a sense that is expected to be equivalent to K-polystability in general. In particular, in the spirit of an expectation of Chen-Tang we show that geodesic K-polystability implies algebraic K-polystability for polarized manifolds. Hence our main result recovers a possibly stronger version of results of Berman-Darvas-Lu in this case. As a key part of the proof we also study subgeodesic rays with singularity type prescribed by singular test configurations, and prove a result on asymptotics of the K-energy functional along such rays. In an appendix by R. Dervan it is moreover deduced that geodesic K-polystability implies equivariant K-polystability. This improves upon previous work and proves that existence of a cscK (or extremal) K\"ahler metric implies equivariant K-polystability (resp. relative K-stability).
... [61], [62], [63]), Tian made important contributions to the Calabi conjecture when c 1 (M) > 0; see also [2] and [64] for more references. More recently, Donaldson [23] made several conjectures concerning geodesics in the space of Kähler metrics which reduce to questions on special Dirichlet problems for the homogeneous complex Monge-Ampère (HCMA) equation; see also Mabuchi [51] and Semmes [57]. There has been interesting work in this direction, e.g. by Chen [16], Chen and Tian [17], Phong and Sturm [54], [55], [56], Blocki [10], and Berman and Demailly [7]. ...
... The tangent space T φ H of H at φ ∈ H is naturally identified to C 2 (M). Following [51], [57] and [23] one can define a Riemannian metric on H using the L 2 inner product on T φ H with respect to the volume form of ω φ : ...
... Accordingly, the length of a regular curve ϕ : [0, 1] → H is Here {g(ϕ) jk } is the inverse matrix of {g(ϕ) jk } = {g jk + ϕ jk }. It was observed by Donaldson [23], Mabuchi [51] and Semmes [57] that equation ( then ϕ is a geodesic in H. ...
We study the Dirichlet problem for complex Monge-Ampere equations in Hermitian manifolds with general (non-pseudoconvex) boundary. Our main result extends the classical theorem of Caffarelli, Kohn, Nirenberg and Spruck in the flat case. We also consider the equation on compact manifolds without boundary, attempting to generalize Yau's theorems in the Kaehler case. As applications of the main result we study some connections between the homogeneous complex Monge-Ampere ({\em HCMA}) equation and totally real submanifolds, and a special Dirichlet problem for the HCMA equation related to Donaldson's conjecture on geodesics in the space of Kaehler metrics.
... It is similar to the Kähler potential space H, H(ω, J) admits a natural Riemann metric of non-positive sectional curvature in same sense [24]. Recall that, for Kähler setting, it was discovered by Mabuchi [43] and rediscovered by Semmes [47] and Donaldson [24]. The real numbers act on H, by addition of constants, and we define H 0 = H/R, which can be viewed as the space of Kähler metrics on M 2n , in the given cohomology class. ...
... For almost Kähler setting, geodesics in H(ω, J) are related to a generalized Monge-Ampère equation (cf. [47]) as follows. Let S = {s ∈ C | 0 < Ims < 1}, and letω the pullback of ω by the projectionS × M 2n → M 2n . ...
In this paper, we define operator that is a generalized operator on higher dimensional almost K\"{a}hler manifolds. In terms of operator, we study -problem in almost K\"{a}hler geometry and the generalized Monge-Amp\`{e}re equation on almost K\"{a}hler manifolds. Similarly to the K\"{a}hler case, we obtain a priori estimates for the solution of the generalized Monge-Amp\`{e}re equation on the almost K\"{a}hler manifold depended only on and J. Then as done in K\"{a}hler geometry, we study Calabi conjecture for almost K\"{a}hler manifolds. Finally, we will pose some questions in almost K\"{a}hler geometry.
... Let (ϕ t ) t≥0 ⊂ PSH(X, ω) be a ray of ω-psh functions. Following a useful point of view of Donaldson [Don02] and Semmes [Sem92], there is a basic correspondence between the family (ϕ t ) t≥0 and an associated S 1 -invariant function Φ on X ×∆ * , where∆ * ⊂ C denotes the pointed unit disc. We denote by τ the coordinate on ∆. ...
... We will use the following standard terminology, motivated by the extensive study of (weak) geodesics in the space H, see e.g. [Blo13], [Che00b], [Dar14], [Don02], [Sem92]. ...
We prove that constant scalar curvature K\"ahler (cscK) manifolds with transcendental cohomology class are K-semistable, naturally generalising the situation for polarised manifolds. Relying on a very recent result by R. Berman, T. Darvas and C. Lu regarding properness of the K-energy, it moreover follows that cscK manifolds with finite automorphism group are uniformly K-stable. As a main step of the proof we establish, in the general K\"ahler setting, a formula relating the (generalised) Donaldson-Futaki invariant to the asymptotic slope of the K-energy along weak geodesic rays.
... A striking feature of the present situation is that F − is not concave. However E is geodesically affine for the L 2 -metric on the space of strictly psh weights considered in [Mab87,Sem92,Don99], and it follows from Berndtsson's results on psh variation of Bergman kernels [Bern09a] that L − is geodesically convex with respect to the L 2 -metric. We thus see that F − is geodesically concave, which morally explains Ding-Tian's result (compare Donaldson's analogous result for the Mabuchi functional [Don05a]). ...
... where m := dim Y and dd c (x,y) acts on both variables (x, y). If Y is a radially symmetric domain in C and Φ is smooth on X ×Y such that ω +dd c x Φ(·, y) > 0 for each y ∈ Y then by definition Φ is flat iff Φ(e t ) is a geodesic for the Riemannian metric on {ϕ ∈ C ∞ (X), ω + dd c ϕ > 0} defined in [Mab87,Sem92,Don99]. ...
We show that degenerate complex Monge-Ampere equations in a big cohomology class of a compact Kaehler manifold can be solved using a variational method independent of Yau's theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kaehler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kaehler metrics and Berndtsson's positivity of direct images we extend Ding-Tian's variational characterization and Bando-Mabuchi's uniqueness result to singular Kaehler-Einstein metrics. Finally using our variational characterization we prove the existence, uniqueness and convergence of k-balanced metrics in the sense of Donaldson both in the (anti)canonical case and with respect to a measure of finite pluricomplex energy in our sense.
... In this section we will study the geometry of the space of plurisubharmonic functions in strongly pseudoconvex domain, based upon works of Mabuchi [Mab87], Semmes [Sem92] and Donaldson [Don99], as it was clarified through lecture notes of Guedj [G14] and Kolev [Kol12]. ...
... We will show in this section that the geodesic equation in H is equivalent to Monge-Ampère equation on Ω × A as in Semmes [Sem92]. ...
Let be a strongly pseudoconvex domain. We introduce the Mabuchi space of strongly plurisubharmonic functions in . We study metric properties of this space using Mabuchi geodesics and establish regularity properties of the latter, especially in the ball. As an application we study the existence of local K\"ahler-Einstein metrics.
... The definition in (1.2) is inspired by the Mabuchi-Semmes-Donaldson [27,32,14] metric of Kähler geometry, wherein a formal Riemann metric is put on a Kähler class by imposing on the tangent space to a given Kähler potential the L 2 metric with respect to the associated Kähler metric. As observed in [27], this metric enjoys many nice formal properties, for instance nonpositive sectional curvature. ...
... As remarked on above, in the setting of Mabuchi geodesics, as observed by Semmes [32] if one complexifies the time direction the equation admits an interpretation as a certain modification of the tensor A will show up naturally in the linearized operator. Let ...
We define a new formal Riemannian metric on a conformal classes of four-manifolds in the context of the -Yamabe problem. Exploiting this new variational structure we show that solutions are unique unless the manifold is conformally equivalent to the round sphere.
... Mabuchi studied the corresponding geometry of H α Y , showing in particular that it can formally be seen as a locally symmetric space of non positive curvature. Semmes [Sem92] re-interpreted the geodesic equation as a complex homogeneous equation, while Donaldson [Don99] strongly motivated the search for smooth geodesics through its connection with the uniqueness of constant scalar curvature Kähler metrics. ...
... i.e. we associate to each path (ϕ t ) a function ϕ on the complex manifold M = X × S, which only depends on the real part of the strip coordinate: we consider S as a Riemann surface with boundary and use the complex coordinate z = t + is to parametrize the strip S. Set ω(x, z) := ω(x). Semmes observed in [Sem92] that the path ϕ t is a geodesic in H ω if and only if the associated function ϕ on X × S is a ω-psh solution of the homogeneous complex Monge-Ampère equation ...
Let Y be a compact K\"ahler normal space and a K\"ahler class. We study metric properties of the space of K\"ahler metrics in using Mabuchi geodesics. We extend several results by Calabi, Chen, Darvas previously established when the underlying space is smooth. As an application we analytically characterize the existence of K\"ahler-Einstein metrics on -Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties.
... This point of view on interpolation theory has had enormous influence not just on that theory itself but also on complex geometry. Indeed, a decade later Semmes [98,99] realized that these 1-parameter curves (real or complex) can be interpreted as geodesics in an infinite-dimensional space associated to a (weakly) Riemannian metric. The simplest case of Riesz-Thorin simply corresponds to the metric ...
... It turns out the answer is yes, and moreover, that the classical Legendre transform itself can be obtained and discovered that way. The first clue to that comes from a problem raised in the celebrated paper of Semmes [99]. Semmes observed that (very roughly) there is a certain duality operation on certain real-analytic Lagrangian submanifolds in the complexification of the twisted cotangent bundle of a Kähler manifold, whose unique self-dual point is a given such [99, p. 543]. ...
Convex geometry and complex geometry have long had fascinating interactions. This survey offers a tour of a few.
... The path length metric structure associated with (8.1) on the space of Kähler potentials was introduced by Mabuchi [48] for p = 2, and by Darvas [20] for any p ∈ [1, +∞[. To describe the geodesics in this space, we identify paths u t , t ∈ [0, 1], of Kähler potentials with rotationally-invariantû : X × D(e −1 , 1) → R, as followŝ u(x, τ ) = u t (x), where x ∈ X and t = − log |τ |. (8.2) According to [53], [29] smooth geodesic segments in Mabuchi space can be described as the only path of Kähler potentials u t , t ∈ [0, 1], connecting u 0 to u 1 , so thatû is the solution of the Dirichlet problem associated with the homogeneous Monge-Ampère equation ...
We study small eigenvalues of Toeplitz operators on polarized complex projective manifolds. For Toeplitz operators whose symbols are supported on proper subsets, we prove the existence of eigenvalues that decay exponentially with respect to the semiclassical parameter. We moreover, establish a connection between the logarithmic distribution of these eigenvalues and the Mabuchi geodesic between the fixed polarization and the Lebesgue envelope associated with the polarization and the non-zero set of the symbol. As an application of our approach, we also obtain analogous results for Toeplitz matrices.
... This concavity plays a substantial role in a priori estimates. It is worth mentioning that the geometry of Gursky-Streets' metric on the space of conformal metrics has a parallel theory with the geometry of the space of Kähler metrics, where the geodesic equation can be written as a homogeneous complex Monge-Ampère equation (see [1,20,19,21,4,2,5]). ...
We solve the modified Gursky-Streets equation, which is a fully nonlinear equation arising in conformal geometry, for all with uniform estimates.
... Donaldson's program and related problems have great impact to the Kähler geometry. A key ingredient is the geodesic equation, which can be written as a homogeneous complex Monge-Ampere equation by the work of Semmes [28] and Donaldson [18]. A foundational result is to solve the geodesic equation (the Dirichlet problem) by X. Chen [9], proving the existence of C 1,1 solution for any two given boundary datum. ...
Gursky-Streets introduced a formal Riemannian metric on the space of conformal metrics in a fixed conformal class of a compact Riemannian four-manifold in the context of the -Yamabe problem. The geodesic equation of Gursky-Streets' metric is a fully nonlinear degenerate elliptic equation and Gursky-Streets have proved uniform regularity for a perturbed equation. Gursky-Streets apply the results and parabolic smoothing of Guan-Wang flow to show that the solution of -Yamabe problem is unique. A key ingredient is the convexity of Chang-Yang's \cF-functional along the (smooth) geodesic, in view of Gursky-Streets metric and a weighted Poincare inequality of B. Andrews on manifolds with positive Ricci curvature. In this paper we establish uniform regularity of the Gursky-Streets' equation. As an application, we can establish strictly the geometric structure in terms of Gursky-Streets' metric, in particular the convexity of \cF-functional along geodesic. This in particular gives a straightforward proof of the uniqueness of solutions of -Yamabe problem.
... For p = 2 one recovers the Riemannian structure of Mabuchi which turns H into a Riemannian symmetric space of constant negative curvature [Mab,Do1,Se], but as will be explained below, the Finsler case p = 1 will also play a key role in the present paper. ...
Let be a compact connected K\"ahler manifold and denote by the metric completion of the space of K\"ahler potentials with respect to the -type path length metric . First, we show that the natural analytic extension of the (twisted) Mabuchi K-energy to is a -lsc functional that is convex along finite energy geodesics. Second, following the program of J. Streets, we use this to study the asymptotics of the weak (twisted) Calabi flow inside the CAT(0) metric space . This flow exists for all times and coincides with the usual smooth (twisted) Calabi flow whenever the latter exists. We show that the weak (twisted) Calabi flow either diverges with respect to the -metric or it -converges to some minimizer of the K-energy inside . This gives the first concrete result about the long time convergence of this flow on general K\"ahler manifolds, partially confirming a conjecture of Donaldson. Finally, we investigate the possibility of constructing destabilizing geodesic rays asymptotic to diverging weak (twisted) Calabi trajectories, and give a result in the case when the twisting form is K\"ahler. If the twisting form is only smooth, we reduce this problem to a conjecture on the regularity of minimizers of the K-energy on , known to hold in case of Fano manifolds.
... Following the work of Mabuchi [Mab87], Semmes [Sem92] and Donaldson [Don99], we endow the space H with the structure of an infinite dimensional Riemannian manifold. Define an inner product on T ϕ H ∼ = C ∞ (X; R) by ...
We introduce the inverse Monge-Ampere flow as the gradient flow of the Ding energy functional on the space of Kahler metrics in for . We prove the long-time existence of the flow. In the canonically polarized case, we show that the flow converges smoothly to the unique Kahler-Einstein metric with negative Ricci curvature. In the Fano case, assuming X admits a Kahler-Einstein metric, we prove the weak convergence of the flow to a Kahler-Einstein metric. In general, we expect that the limit of the flow is related with the optimally destabilizing test configuration for the -normalized non-Archimedean Ding functional. We confirm this expectation in the case of toric Fano manifolds.
... There is a natural notion of distance on the space of Hermitian metrics GL(n + 1, C)/U(n + 1), and indeed as k increases this distance function is expected [PS06] to approximate that on the infinite dimensional space of Kähler metrics [Mab87], [Sem92], [Don99], [Che00]. A natural question one might ask is: Are the T, T ν , or T K iterations distance reducing on the space of metrics? ...
In a recent paper Donaldson defines three operators on a space of Hermitian metrics on a complex projective manifold: Iterations of these operators converge to balanced metrics, and these themselves approximate constant scalar curvature metrics. In this paper we investigate the convergence properties of these iterations by examining the case of the Riemann sphere as well as higher dimensional .
... In §5 we observe a number of formal properties for solutions of this flow in the set C + (A) . These are in line with the properties established for Calabi flow in relation to the Mabuchi metric in Kähler geometry [5,9,16,19]: Theorem 1.4. Let (M 2m , g) be a closed, even-dimensional Riemannian manifold, and suppose u = u(t) is a solution to inverse v m -flow with g u = e −2u g and g u ∈ C + (A) . ...
We define a new formal Riemannian metric on a conformal class in the context of the -Yamabe problem. Our construction leads to a new variational characterization and a new parabolic flow approach to this problem. Moreover, this variational framework suggests that solutions to this problem are unique in a given conformal class.
... Let H denote the space of Kähler potentials in a given Kähler class (M, [ω]). T. Mabuchi [51], S. Semmes [52] and S. K. Donaldson [37] set up an L 2 metric in the space of Kähler potentials: ...
In this paper, we generalize our apriori estimates on cscK(constant scalar curvature K\"ahler) metric equation to more general scalar curvature type equations (e.g., twisted cscK metric equation). As applications, under the assumption that the automorphism group is discrete, we prove the celebrated Donaldson's conjecture that the non-existence of cscK metric is equivalent to the existence of a destabilized geodesic ray where the K-energy is non-increasing. Moreover, we prove that the properness of K-energy in terms of geodesic distance in the space of K\"ahler potentials implies the existence of cscK metric. Finally, we prove that weak minimizers of the K-energy in are smooth.
... This turns H into a smooth Riemannian manifold (see [L2, section 5 and Example 2.1]), whose study was initiated by Mabuchi,Semmes,and Donaldson [Do1,M,Se]. The curvature of H is covariantly constant, a property that for finite dimensional manifolds would imply the existence of local symmetries, self-isometries of neighborhoods of an arbitrary point, that act on tangent vectors issued from the point by −id. ...
The space of K\"ahler potentials in a compact K\"ahler manifold, endowed with Mabuchi's metric, is an infinite dimensional Riemannian manifold. We characterize local isometries between spaces of K\"ahler potentials, and prove existence and uniqueness for such isometries.
... whereφ ωu = 1 V X φω n u . The case p = 2 gives the Riemannian structure of Mabuchi-Semmes-Donaldson [27,32,23], a space with non-positive sectional curvature, with close ties to canonical Kähler metrics. As shown in [20,21], the case p = 1 gives a geometry with good compactness properties, suitable for the variational study of canonical metrics by way of infinite-dimensional convex optimization. ...
Suppose is a compact K\"ahler manifold. We introduce and explore the metric geometry of the -Calabi Finsler structure on the space of K\"ahler metrics . After noticing that the -Calabi and -Mabuchi path length topologies on do not typically dominate each other, we focus on the finite entropy space , contained in the intersection of the -Calabi and -Mabuchi completions of and find that after a natural strengthening, the -Calabi and -Mabuchi topologies coincide on . As applications to our results, we give new convergence results for the K\"ahler--Ricci flow and the weak Calabi flow.
... Later on Donaldson [24] described a beautiful geometric picture of H ω with the Mabuchi metric, and set up a program which tights up the K-energy, the existence and uniqueness of CSCK and the notion of stability in Geometric Invariant Theory (GIT). The geodesic equation in H ω can be written as a homogeneous complex Monge-Ampere equation, observed by Semmes [38] and Donaldson [24]. Chen [9] confirmed a conjecture of Donaldson by proving that given any two points in H ω , there exists a unique solution to the homogenous complex-Ampere equation with C 1,1 regularity, which are potentials of a Kähler current in [ω] with L ∞ coefficients. ...
In this paper we study constant scalar curvature equation (CSCK), a nonlinear fourth order elliptic equation, and its weak solutions on K\"ahler manifolds. We first define a notion of weak solution of CSCK for an K\"ahler metric. The main result is to show that such a weak solution (with uniform bound) is smooth. As an application, this answers in part a conjecture of Chen regarding the regularity of K-energy minimizers. The new technical ingredient is a regularity result for the Laplacian equation on K\"ahler manifolds, where the metric has only coefficients. It is well-known that such a regularity ( regularity for any ) fails in general (except for dimension two) for uniform elliptic equations of the form for , without certain smallness assumptions on the local oscillation of . We observe that the K\"ahler condition plays an essential role to obtain a regularity for elliptic equations with only elliptic coefficients on compact manifolds.
... Consider nowX = X × [0, 1] × S 1 . Donaldson [Don99] and Semmes [Sem92] observed that, by extending Φ(p, t, s) := ϕ t (p) then the geodesic equation can be given as ...
We present classical and recent results on K\"ahler-Einstein metrics on compact complex manifolds, focusing on existence, obstructions and relations to algebraic geometric notions of stability (K-stability). These are the notes for the SMI course "K\"ahler-Einstein metrics" given by C.S. in Cortona (Italy), May 2017. The material is not intended to be original.
... Note that (1.4) implies that ∂∂ω k = 0 for all 1 ≤ k ≤ n − 1 (see, for example, [FT]). Also, Guan-Li apply their estimates to the problem of finding geodesics in the space of Hermitian metrics via the homogeneous complex Monge-Ampère equation (for some related works, see [CLN,BT,Ma,Se,Do1,GuB,Chn,GuP,PS], for example). ...
We generalize Yau's estimates for the complex Monge-Ampere equation on compact manifolds in the case when the background metric is no longer Kahler. We prove a priori estimates for a solution of the complex Monge-Ampere equation when the background metric is Hermitian (in complex dimension two) or balanced (in higher dimensions), giving an alternative proof of a theorem of Cherrier. We relate this to recent results of Guan-Li.
... When dim M = 1, the equation c(φ) = 0 is equivalent to the famous homogenous complex Monge-Ampère equation ( √ −1∂∂φ) n+1 = 0 (cf. [9], [23]), which plays a crucial role in a lot of related important problems. Note that in this case, the equation c(φ) = 0 can be also written as tr ω c(φ) = 0 for any metric ω on M . ...
In this paper, we introduce notions of nonlinear stabilities for a relative ample line bundle over a holomorphic fibration and define the notion of a geodesic-Einstein metric on this line bundle, which generalize the classical stabilities and Hermitian-Einstein metrics of holomorphic vector bundles. We introduce a Donaldson type functional and show that this functional attains its absolute minimum at geodesic-Einstein metrics, and we also discuss the relations between the existence of geodesic-Einstein metrics and the nonlinear stabilities of the line bundle. As an application, we will prove that a holomorphic vector bundle admits a Finsler-Einstein metric if and only if it admits a Hermitian-Einstein metric, which answers a problem posed by S. Kobayashi.
... Questions concerning the regularity of the solution to the Dirichlet problem for the kind of complex HMAE we consider here go back at least as far as the work of Semmes [24] and Donaldson [10], and has been the focus of much interest due to it being the geodesic equation in the space of Kähler metrics. By the work of Chen [8] with complements by Błocki [4] we know such a solution always has bounded Laplacian (so in particular is C 1,α for any α < 1). ...
We give examples of regular boundary data for the Dirichlet problem for the Complex Homogeneous Monge-Amp\`ere Equation over the unit disc, whose solution is completely degenerate on a non-empty open set and thus fails to have maximal rank.
... We first recall the metric structure on H, the space of Kahler potentials. Mabuchi [11] introduced a Riemannian metric on H and he also proves that this is formally a symmetric space with nonpositive curvature (See [13,8] also). Chen [3] proved that H is actually a metric space and it is convex by C 1,1 geodesics (potential with bounded mixed derivative), by confirming partially conjectures of Donaldson [8], in which he set up an ambitious program which tights up the existence of constant scalar curvature with the geometry of H. ...
Suppose there is a constant scalar curvature metric on a compact Kahler manifold without holomorphic vector field. We prove that the Calabi flow, if it is assumed to exist for all time with bounded Ricci curvature, will converge to the constant scalar curvature metric.
... This holds for any ball, so the theorem follows. [60], Donaldson [21], Chen [10] and Blocki [7]. It follows from the comparison principle (an extension of Lemma 3.10 for M in the place of B), that there exists at most one continuous solution to the equation. ...
We solve the classical Dirichlet problem for a general complex Hessian equation on a small ball in \bC^n. Then, we show that there is a continuous solution, in pluripotential theory sense, to the Dirichlet problem on compact Hermitian manifolds with boundary that equipped locally conformal K\"ahler metrics, provided a subsolution.
... When trying to find weak solutions for such equations, one is naturally led to the space E 1 (X, ω), introduced by Guedj and Zeriahi [GZ07] building on previous constructions of Cegrell in the local case [Ce98]. Later, in the work of the first named author [Da14,Da15] it was discovered that E 1 (X, ω) has a natural metric geometry, arising as the completion of a certain L 1 Finsler metric on the space of smooth Kähler potentials, an open subset of C ∞ (X), reminiscent of the L 2 Riemannian metric of Se92,Do99]). The exploration of the space E 1 (X, ω) and its metric structure led to numerous applications concerning existence of Kähler-Einstein and constant scalar curvature metrics (see [BBEGZ11,BBGZ13,BBJ15,BDL16,CC18,DR17,Da16,DNG16] as well as references in the recent survey [Da17]). ...
Suppose is a compact K\"ahler manifold of dimension n, and is closed (1,1)-form representing a big cohomology class. We introduce a metric on the finite energy space , making it a complete geodesic metric space. This construction is potentially more rigid compared to its analog from the K\"ahler case, as it only relies on pluripotential theory, with no reference to infinite dimensional Finsler geometry. Lastly, by adapting the results of Ross and Witt Nystr\"om to the big case, we show that one can construct geodesic rays in this space in a flexible manner.
... Starting with pioneer work by Mabuchi [19], a notion of geodesics in the space of Kähler metrics on compact complex manifolds has been playing a prominent role in Kähler geometry and has found a lot of applications. We will not give here any detailed account on this subject; the interested reader can consult, for example, [23], [10], [14], [1], [5], [15], and the bibliography therein. In particular, geodesics in the space of metrics on a compact n-dimensional Kähler manifold (X, ω) have been characterized as solutions to a complex homogeneous equation, which implies linearity of the Mabuchi functional ...
We study geodesics for plurisubharmonic functions from the Cegrell class on a bounded hyperconvex domain of and show that, as in the case of metrics on K\"{a}hler compact menifolds, they linearize an energy functional. As a consequence, we get a uniqueness theorem for functions from in terms of total masses of certain mixed Monge-Amp\`ere currents. Geodesics of relative extremal functions are considered and a reverse Brunn-Minkowski inequality is proved for capacities of multiplicative combinations of multi-circled compact sets. We also show that functions with strong singularities generally cannot be connected by (sub)geodesic arks.
... The L 2 analog of this metric is the much studied Riemannian metric of Mabuchi-Semmes-Donaldson [Mab,Se,Do1]. Given this Finsler metric one can compute the length l(γ t ) of smooth curves t → γ t of H ω (X), ultimately yielding the path length metric: [Da2,Corollary 4.14] we showed that d ω 1 thus defined satisfies (5). ...
Let be a compact normal K\"ahler space, with Hodge metric . In this paper, the last in a sequence of works studying the relationship between energy properness and canonical K\"ahler metrics, we introduce a geodesic metric structure on , the space of K\"ahler potentials, whose completion is the finite energy space . Using this metric structure and the results of Berman-Boucksom-Eyssidieux-Guedj-Zeriahi as ingredients in the existence/properness principle of Rubinstein and the author, we show that existence of K\"ahler-Einstein metrics on log Fano pairs is equivalent to properness of the K-energy in a suitable sense. To our knowledge, this result represents the first characterization of general log Fano pairs admitting K\"ahler-Einstein metrics. We also discuss the analogous result for K\"ahler-Ricci solitons on Fano varieties.
... It turns out that the situation can be analyzed by estimating integrals of the form Xτ e 2Ψ| Xτ as τ → 0, where X → C is an snc test configuration for X, and Ψ is a smooth metric on the (logarithmic) relative canonical bundle of X near the central fiber, see Lemma 3.11. Donaldson [Don99] (see also [Mab87,Sem92]) has advocated the point of view that the space H of positive metrics on L is an infinite-dimensional symmetric space. One can view the space H NA of positive non-Archimedean metrics on L as (a subset of) the associated (conical) Tits building. ...
Consider a polarized complex manifold (X,L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X,L). For most of the common functionals in K\"ahler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as defined in our earlier paper) at the non-Archimedean metric on L defined by the test configuration. Using this asymptotic result, we show that coercivity of the Mabuchi functional implies uniform K-stability.
... According to [66], [36] smooth geodesic segments in Mabuchi space can be described as the only path u t ∈ H ω , t ∈ [0, 1], connecting u 0 to u 1 , so thatû is the solution of the Dirichlet problem associated with the homogeneous Monge-Ampère equation ...
We demonstrate that the weight operator associated with a submultiplicative filtration on the section ring of a polarized complex projective manifold is a Toeplitz operator. We further analyze the asymptotics of the associated weighted Bergman kernel, presenting the local refinement of earlier results on the convergence of jumping measures for submultiplicative filtrations towards the pushforward measure defined by the corresponding geodesic ray.
... Rooftop envelopes have recently emerged as a fundamental tool in constructing plurisubharmonic geodesics within geometry. These geodesics are defined as the upper envelopes of sub-geodesics, an idea that builds upon Mabuchi's seminal work [40] and the findings of Semmes [50] and Donaldson [27]. This approach has been further developed and adapted to local contexts by Berman and Berndtsson [12,13], among others. ...
We initiate the study of m-subharmonic functions with respect to a semipositive (1, 1)-form in Euclidean domains, providing a significant element in understanding geodesics within the context of complex Hessian equations. Based on the foundational Perron envelope construction, we prove a decomposition of m-subharmonic solutions, and a general comparison principle that effectively manages singular Hessian measures. Additionally, we establish a rooftop equality and an analogue of the Kiselman minimum principle, which are crucial ingredients in establishing a criterion for geodesic connectivity among m-subharmonic functions, expressed in terms of their asymptotic envelopes.
... S. Semmes [25] observed that the geodesic can be seen as a S 1 invariant function on X. We will use this perspective when we prove the convexity of the K-energy. ...
Let D be a smooth divisor on a closed K\"ahler manifold X. First, we prove that Poincar\'e type constant scalar curvature K\"ahler (cscK) metric with a singularity at D is unique up to a holomorphic transformation on X that preserves D, if there are no nontrivial holomorphic vector fields on D. For the general case, we propose a conjecture relating the uniqueness of Poincar\'e type cscK metric to its asymptotic behavior near D. We give an affirmative answer to this conjecture for those Poincar\'e type cscK metrics whose asymptotic behavior is invariant under any holomorphic transformation of X that preserve D. We also show that this conjecture can be reduced to a fixed point problem.
... The concept of plurisubharmonic geodesics was first introduced in Mabuchi's seminal work on constant scalar curvature Kähler metrics [45]. Subsequently, Semmes [52] and Donaldson [31] independently showed that Mabuchi geodesics can be understood as solutions to certain degenerate homogeneous complex Monge-Ampère equations. This concept has been further refined to describe geodesics as the upper envelopes of subgeodesics, a perspective that Berman-Berndtsson [11], Abja [1], Abja-Dinew [2,3], and Rashkovskii [48] adapted to the local setting. ...
This study examines geodesics and plurisubharmonic envelopes within the Cegrell classes on bounded hyperconvex domains in . We establish that solutions possessing comparable singularities to the complex Monge–Ampère equation are identical, affirmatively addressing a longstanding open question raised by Cegrell. This achievement furnishes the most general form of the Bedford–Taylor comparison principle within the Cegrell classes. Building on this foundational result, we explore plurisubharmonic geodesics, broadening the criteria for geodesic connectivity among plurisubharmonic functions with connectable boundary values. Our investigation also delves into the notion of rooftop envelopes, revealing that the rooftop equality condition and the idempotency conjecture are valid under substantially weaker conditions than previously established, a finding made possible by our proven uniqueness result. The paper concludes by discussing the core open problems within the Cegrell classes related to the complex Monge–Ampère equation.
... As a concluding remark, we would like to mention some related works concerning the equation (1.1) in the setting when B has a boundary. First, when B is an annuli in C, weak solutions to the Dirichlet problem associated with (1.1) correspond to the Mabuchi geodesics in the space of all Kähler potentials, [60] [26], and they always exist, see [14] (cf. also [71] for a related result on pseudoconvex domains in C m ). ...
For a polarized family of complex projective manifolds, we study the Hermitian Yang-Mills functionals on the sequence of vector bundles over the base of the family associated with direct image sheaves of the tensor powers of the polarization. We make a connection between the asymptotic minimization of these functionals, for big tensor powers of the polarization, and the minimization of the so-called Wess-Zumino-Witten functional defined on the space of all relatively K\"ahler -forms on the fibration. We establish the sharp lower bounds on the latter functional in terms of the limiting Harder-Narasimhan measure, which is a certain algebraic invariant of the family. As an application, in a fibered setting, we prove an asymptotic converse to the Andreotti-Grauert theorem conjectured by Demailly.
Consider a compact Kähler manifold ( X , ω ) (X,\omega ) and the space E ( X , ω ) = E \mathcal {E}(X,\omega )\!= \mathcal {E} of ω \omega –plurisubharmonic functions of full Monge–Ampère mass on it. We introduce a quantity ρ [ u , v ] \rho [u,v] to measure the distance between u , v ∈ E u, v\in \mathcal {E} ; ρ [ u , v ] \rho [u,v] is not a number but rather a decreasing function on a certain interval ( 0 , V ) ⊂ R (0,V)\subset \mathbb {R} . We explore properties of ρ [ u , v ] \rho [u,v] , and using them we study Lagrangians and associated energy spaces of ω \omega –plurisubharmonic functions. Many results here generalize Darvas’s findings about his metrics d χ d_\chi .
Our interest is the behavior of weak geodesics between two plurisubharmonic functions on pseudoconvex domains. We characterize the convergence condition along the geodesic between toric psh functions with a pole at origin on a unit ball in by means of Lelong numbers.
In this paper, we report a "new" continuity path which links the constant scalar curvature equation to a second order elliptic equation. This is largely an expository article where we describes various aspects of geometry and analysis associated with path.
We study the existence of extremal K\"ahler metrics on K\"ahler manifolds. After introducing a notion of relative K-stability for K\"ahler manifolds, we prove that K\"ahler manifolds admitting extremal K\"ahler metrics are relatively K-stable. Along the way, we prove a general lower bound on the Calabi functional involving test configurations and their associated numerical invariants, answering a question of Donaldson. When the K\"ahler manifold is projective, our definition of relative K-stability is stronger than the usual definition given by Sz\'ekelyhidi. In particular our result strengthens the known results in the projective case (even for constant scalar curvature K\"ahler metrics), and rules out a well known counterexample to the "na\"ive" version of the Yau-Tian-Donaldson conjecture in this setting.
Complex Legendre duality is a generalization of Legendre transformation from Euclidean spaces to Kahler manifolds, that Berndtsson and collaborators have recently constructed. It is a local isometry of the space of Kahler potentials. We show that the fixed point of such a transformation must correspond to a real analytic Kahler metric.
We study the gravitational action induced by coupling two-dimensional non-conformal, massive matter to gravity on a compact Riemann surface. We express this gravitational action in terms of finite and well-defined quantities for any value of the mass. A small-mass expansion gives back the Liouville action in the massless limit, the Mabuchi and Aubin-Yau actions to first order, as well as an infinite series of higher-order contributions written in terms of purely geometric quantities.
We study the gravitational action induced by coupling two-dimensional non-conformal, massive matter to gravity on a Riemann surface with boundaries. A small-mass expansion gives back the Liouville action in the massless limit, while the first-order mass correction allows us to identify what should be the appropriate generalization of the Mabuchi action on a Riemann surface with boundaries. We provide a detailed study for the example of the cylinder. Contrary to the case of manifolds without boundary, we find that the gravitational Lagrangian explicitly depends on the space-point, via the geodesic distances to the boundaries, as well as on the modular parameter of the cylinder, through an elliptic theta-function.
Let ( X , ω ) (X,\omega ) be a compact Kähler manifold and θ \theta be a smooth closed real ( 1 , 1 ) (1,1) -form that represents a big cohomology class. In this paper, we show that for p ≥ 1 p\geq 1 , the high energy space E p ( X , θ ) \mathcal {E}^{p}(X,\theta ) can be endowed with a metric d p d_{p} that makes ( E p ( X , θ ) , d p ) (\mathcal {E}^{p}(X,\theta ),d_{p}) a complete geodesic metric space. The weak geodesics in E p ( X , θ ) \mathcal {E}^{p}(X,\theta ) are the metric geodesic for ( E p ( X , θ ) , d p ) (\mathcal {E}^{p}(X,\theta ), d_{p}) . Moreover, for p > 1 p > 1 , the geodesic metric space ( E p ( X , θ ) , d p ) (\mathcal {E}^{p}(X,\theta ), d_{p}) is uniformly convex.
The main goal of this article is to describe a relation between the asymptotic properties of filtrations on section rings and the geometry at infinity of the space of Kähler potentials. More precisely, for a polarized projective manifold and an ample test configuration, Phong and Sturm associated a geodesic ray of plurisubharmonic metrics on the polarizing line bundle. On the other hand, for the same data, Witt Nyström associated a filtration on the section ring of the polarized manifold. In this article, we establish a folklore conjecture that the pluripotential chordal distance between the geodesic rays associated with two ample test configurations coincides with the spectral distance between the associated filtrations on the section ring. This gives an algebraic description of the boundary at infinity of the space of positive metrics, viewed – as it is usually done for spaces of negative curvature – through geodesic rays.
We prove that Calabi–Yau metrics on compact Calabi–Yau manifolds whose Kähler classes shrink the fibers of a holomorphic fibration have a priori estimates of all orders away from the singular fibers. To this end, we prove an asymptotic expansion of these metrics in terms of powers of the fiber diameter, with ‐order remainders that satisfy uniform ‐estimates with respect to a collapsing family of background metrics. The constants in these estimates are uniform not only in the sense that they are independent of the fiber diameter, but also in the sense that they only depend on the constant in the estimate for known from previous work of the second‐named author. For , the new estimates are proved by blowup and contradiction, and each additional term of the expansion arises as the obstruction to proving a uniform bound on one additional derivative of the remainder.
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