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... 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 [56,59,20]. 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. ...

... Formal metric structure and uniqueness. In the Kähler setting, the GIT framework is naturally completed by a formal Riemannian metric, known as the Mabuchi-Semmes-Donaldson metric [56,59,20]. This metric formally gives K α 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 [59] 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). ...

On a compact complex manifold $(M, J)$ endowed with a holomorphic Poisson tensor $\pi_J$ and a deRham class $\alpha\in H^2(M, \mathbb R)$, we study the space of generalized K\"ahler (GK) structures defined by a symplectic form $F\in \alpha$ and whose holomorphic Poisson tensor is $\pi_J$. We define a notion of generalized K\"ahler class of such structures, and use the moment map framework of Boulanger and Goto to extend the Calabi program to GK geometry. We obtain generalizations of the Futaki--Mabuchi extremal vector field and Calabi--Lichnerowicz--Matsushima result 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. 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 K\"ahler 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 and He, we show in some settings that this condition suffices for existence and thus construct new examples.

... In general, this action must also be present as a counterterm to renormalize the divergences that are present in (1.1), in addition to S L . Two-dimensional gravitational actions other than the Liouville or cosmological constant actions can be constructed and have been studied in the mathematical literature, like the Mabuchi and Aubin-Yau actions [2][3][4][5]. These latter functionals involve not only the conformal factor σ but also the Kähler potential φ and do admit generalizations to higher-dimensional Kähler manifolds. ...

... Despite some effort, we could not express the latter in terms of purely local quantities like the conformal factor σ and the Kähler potential. Similar non-local terms also appeared in the scalar case at order m 4 and it seems that in the present fermionic case they already are unavoidable at order m 2 . We plan to return to the case of general topology with a detailed account of the role of the zero-modes in a future publication. ...

... For λ n = 0, however, one can always choose a basis of JHEP11(2021)165 definite chirality eigenfunctions. 4 As is well known, the difference of positive and negative chirality zero-modes of D is called its index. ...

A bstract
We work out the effective gravitational action for 2D massive Euclidean fermions in a small mass expansion. Besides the leading Liouville action, the order m ² gravitational action contains a piece characteristic of the Mabuchi action, much as for 2D massive scalars, but also several non-local terms involving the Green’s functions and Green’s functions at coinciding points on the manifold.

... of smooth functions on X to find a canonical metric in the same cohomology class as ω. Mabuchi [26], Semmes [28], and Donaldson [17] independently found a Riemannian structure on H given by φ, ψ u := 1 Vol(X) X φψω n u for u ∈ H and φ, ψ ∈ T u H = C ∞ (X ). Later, Chen [7] showed that this Riemannian structure makes H a metric space (H, d 2 ). ...

... (5) Most recently, Darvas [10] showed that the space E ψ (X , ω) has a natural complete metric, when ω is Kähler and ψ is a low energy weight as introduced by Guedj-Zeriahi [21]. In the process, Darvas used the geodesics on H introduced by [17,26,28]. ...

In this paper, we show that the low energy spaces in the prescribed singularity case Eψ(X,θ,ϕ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {E}}_{\psi }(X,\theta ,\phi )$$\end{document} have a natural topology which is completely metrizable. This topology is stronger than convergence in capacity.

... We show that this system admits a natural interpretation in terms of a generalized notion of curvature, and that this notion of curvature admits, just as the standard notion, an interpretation in terms of Deligne pairings [14]. We establish the C 0 estimate for this Hessian coupled system, and give an interpretation of this coupled system as a variational problem for an energy functional in an infinite dimensional Riemannian manifold of negative sectional curvature, generalizing the constructions of Donaldson [5], Mabuchi [11], and Semmes [16]. ...

... Similar to [5,11,16], we can recast this as a degenerate Hessian equation on a product manifold of dimension n + 1. First we complexify the time variable t by adding an imaginary part, and assume everything is independent of the imaginary part of t. ...

In this paper, we propose a coupled system of complex Hessian equations which generalizes the equation for constant scalar curvature Kähler (cscK) metrics. We show this system can be realized variationally as the Euler–Lagrange equation of a Hessian version of the Mabuchi K-energy in an infinite dimensional space of k-Hessian potentials, which can be seen as an infinite dimensional Riemannian manifold with negative sectional curvature. Finally, we prove an a priori C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^0$$\end{document}-estimate for this system which depends on the Entropy, which generalizes a fundamental result of Chen and Cheng [1] for cscK metrics.

... As discovered by Semmes [69] and Donaldson [33], the above equation can be understood as a complex Monge-Ampère equation as follows. For each ...

... Theorem 2.4.1 ([69]). We denote the d and d c -operator on X by d and d c , ...

The aim of this paper is to solve a uniform version of the Yau-Tian-Donaldson conjecture for polarized toric manifolds. Also, we show a combinatorial sufficient condition for uniform relative K-polystability.

... We show that this system admits a natural interpretation in terms of a generalized notion of curvature, and that this notion of curvature admits, just as the standard notion, an interpretation in terms of Deligne pairings [13]. We establish the C 0 estimate for this Hessian coupled system, and give an interpretation of this coupled system as a variational problem for an energy functional in an infinite dimensional Riemannian manifold of negative sectional curvature, generalizing the constructions of Donaldson [4], Mabuchi [10], and Semmes [15]. ...

... Similar to [10,15,4], we can recast this as a degenerate Hessian equation on a product manifold of dimension n + 1. First we complexify the time variable t by adding an imaginary part, and assume everything is independent of the imaginary part of t. ...

In this paper, we propose a coupled system of complex Hessian equations which generalizes the equation for constant scalar curvature K\"ahler (cscK) metrics. We show this system can be realized variationally as the Euler-Lagrange equation of a Hessian version of the Mabuchi K-energy in an infinite dimensional space of $k$-Hessian potentials, which can be seen as an infinite dimensional Riemannian manifold with negative sectional curvature. Finally, we prove an a priori $C^0$-estimate for this system which depends on the Entropy, which generalizes a fundamental result of Chen and Cheng for cscK metrics.

... In 1991, Semmes [82,83] rediscovered this and noticed that the geodesics satisfy a homogeneous complex Monge-Ampére equation det∂∂F (t) = 0, where we regard t as the real part of a complex coordinate z 0 . ...

In this article, we give a further survey of some progress of the applications of group actions in the complex geometry after my earlier survey around 2020, mostly related to my own interests.

... As observed by Donaldson [15] and Semmes [26], the geodesic equation is equivalent to a homogeneous complex Monge-Ampère equation in the product space X × Σ, where Σ ∼ = [0, 1] × S 1 can be embedded as an annulus in C. Notice that any path ϕ(t) of functions on X can be viewed as a function Φ on X × Σ via Φ(·, t, e is ) = ϕ(t). ...

We solve the geodesic equation in the space of K\"ahler metrics under the setting of asymptotically locally Euclidean (ALE) K\"ahler manifolds and we prove global $\mathcal{C}^{1,1}$ regularity of the solution. Then, we relate the solution of the geodesic equation to the uniqueness of scalar-flat ALE metrics. To this end, we study the asymptotic behavior of $\varepsilon$-geodesics at spatial infinity. Under the assumption that the Ricci curvature of a reference ALE K\"ahler metric is non-positive, convexity of the Mabuchi $K$-energy along $\varepsilon$-geodesics. However, we will also prove that on the line bundle $\mathcal{O}(-k)$ over $\mathbb{C}\mathbb{P}^{n-1}$ with $n \geq 2$ and $k \neq n$, no ALE K\"ahler metric can have non-positive (or non-negative) Ricci curvature.

... The origins of the plurisubharmonic geodesics lie in studying Kähler metrics on compact complex manifolds (X, ω). Starting with [49], a notion of geodesics in the spaces of such metrics has been playing a prominent role in Kähler geometry and has found a lot of applications; see, for example, [64], [21], [38], [39], and the bibliography therein. A considerable progress was made then by relating the metrics to quasi-psh functions on compact complex manifolds, see [10], [11], [26]- [34], [41], [62], [63], and many others. ...

We give a short survey on plurisubharmonic interpolation, with focus on possibility of connecting two given plurisubharmonic functions by plurisubharmonic geodesic.

... One can compute the Levi-Civita connection ∇ (·) (·) of this inner-product and the associated geodesic equation. For a thorough discussion of the L 2 Mabuchi-Semmes-Donaldson geometry, as well as its Levi-Civita connection, we refer to the surveys [8, Section 4], [23, Section 3.1], as well as the original papers [17,30,39,40]. Unfortunately smooth geodesics connecting arbitrary u 0 , u 1 ∈ H ω don't exist, but a weak notion of geodesic was studied by Chen [17]. ...

Let \((X,\omega )\) be a Kähler manifold and \(\psi : \mathbb R \rightarrow \mathbb R_+\) be a concave weight. We show that \({\mathcal {H}}_\omega \) admits a natural metric \(d_\psi \) whose completion is the low energy space \({\mathcal {E}}_\psi \), introduced by Guedj–Zeriahi. As \(d_\psi \) is not induced by a Finsler metric, the main difficulty is to show that the triangle inequality holds. We study properties of the resulting complete metric space \(({\mathcal {E}}_\psi ,d_\psi )\).

... where the wedge power is interpreted in Bedford-Taylor sense [1]. For smooth geodesic segments in (H, d 2 ), Semmes [61] and Donaldson [33] have made similar observations before. The uniqueness of the solution of (2.12) is assured by [38,Lemma 5.25]. ...

From the work of Phong and Sturm in 2007, for a polarised projective manifold and an ample test configuration, one can associate the geodesic ray of plurisubharmonic metrics on the polarising line bundle using the solution of the Monge-Amp\`ere equation on an equivariant resolution of singularities of the test configuration. We prove that the Mabuchi 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 the infinity of the space of positive metrics, viewed - as it is usually done for spaces of negative curvature - through geodesic rays.

... The analogous L 2 type metric was introduced by Mabuchi [Ma87] (independently discovered by Semmes [Se92] and Donaldson [Do99]). This Riemannian metric was studied in detail by Chen [Ch00], and for more historical details we refer to [Da19,Chapter 3]. ...

In previous work, Darvas-George-Smith obtained inequalities between the large scale asymptotic of the $J$ functional with respect to the $d_1$ metric on the space of toric K\"ahler metrics/rays. In this work we prove sharpness of these inequalities on all toric K\"ahler manifolds, and study the extremizing potentials/rays. On general K\"ahler manifolds we show that existence of radial extremizers is equivalent with the existence of plurisupported currents, as introduced and studied by McCleerey.

... First we recall a few basic facts in Kähler geometry, cf. [27], [13], [10] and [24]. Consider a subgeodesic ray in the space of Kähler potentials on CP 1 . ...

The aim of this paper is to study the residual Monge-Amp\`ere mass of a plurisubharmonic function with isolated singularity at the origin in $\mathbb{C}^2$. We proved that the residual mass is zero if its Lelong number is zero at the origin, provided that it is $S^1$-invariant and radially regular. This result answers the zero mass conjecture raised by Guedj and Rashkovskii in this special case.

... The Mabuchi metric (defined in [Mab87] and studied independently also in the early references [Sem92,Don99]) plays an important role in recent advances in Kähler geometry. A path of Kähler metrics ...

In this paper we use techniques of geometric quantization to give a geometric interpretation of the Peter--Weyl theorem. We present a novel approach to half-form corrected geometric quantization in a specific type of non-K\"ahler polarizations and study one important class of examples, namely cotangent bundles of compact semi-simple groups $K$. Our main results state that this canonically defined polarization occurs in the geodesic boundary of the space of $K\times K$-invariant K\"ahler polarizations equipped with Mabuchi's metric, and that its half-form corrected quantization is isomorphic to the K\"ahler case. An important role is played by invariance of the limit polarization under a torus action. Unitary parallel transport on the bundle of quantum states along a specific Mabuchi geodesic, given by the coherent state transform of Hall, relates the non-commutative Fourier transform for $K$ with the Borel--Weil description of irreducible representations of $K$.

... As observed by Donaldson in [6], these can be regarded as a family of Kähler metrics within a fixed Kähler class [ω] on a fixed complex manifold (M, J). The space H ω of these Kähler metrics was studied by Semmes in [31] and Donaldson in [6]. In particular, H ω is an infinite dimensional symmetric space of nonpositive curvature. ...

Let $(M, \omega, J)$ be a K\"ahler manifold, equipped with an effective Hamiltonian torus action $\rho: T \rightarrow \mathrm{Diff}(M, \omega, J)$ by isometries with moment map $\mu: M \rightarrow \mathfrak{t}^{*}$. We first construct a singular mixed polarization $\mathcal{P}_{\mathrm{mix}}$ on $M$. Second, we construct a one-parameter family of complex structures $J_{t}$ on $M$ which are compatible with $\omega$. Furthermore, the path of corresponding K\"ahler metrics $g_{t}$ is a complete geodesic ray in the space of K\"ahler metrics of $M$, when $M$ is compact. Finally, we show that the corresponding family of K\"ahler polarizations $\mathcal{P}_{t}$ associated to $J_{t}$ converges to $\mathcal{P}_{\mathrm{mix}}$ as $t \rightarrow \infty$.

... On the other hand, it was discovered independently by Semmes [45] and Donaldson [20] that the geodesic equation in the space of Kähler potentials on a compact Kähler manifold is equivalent to the homogeneous Monge-Ampère equation on a compact Kähler manifold with boundary, the product of the Kähler manifold and an annulus in the complex plane. In general the solution is at most C 1,1 -smooth, due to an example of Gamelin Let h : R + → (0, ∞) be an increasing function such that ...

We study weak quasi-plurisubharmonic solutions to the Dirichlet problem for the complex Monge–Ampère equation on a general Hermitian manifold with non-empty boundary. We prove optimal subsolution theorems: for bounded and Hölder continuous quasi-plurisubharmonic functions. The continuity of the solution is proved for measures that are well dominated by capacity, for example measures with Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}, p>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1$$\end{document} densities, or moderate measures in the sense of Dinh–Nguyen–Sibony.

... That is, t → ϕ(t) is a geodesic precisely when t → Lϕ Θ (t) is affine in t. These facts go back to Mabuchi, Semmes, and Donaldson [19,Section 1], [13,Section 3.1], [8,Section 6]. ...

The Chebyshev potential of a K\"ahler potential on a projective variety, introduced by Witt Nystr\"om, is a convex function defined on the Okounkov body. It is a generalization of the symplectic potential of a torus-invariant K\"ahler potential on a toric variety, introduced by Guillemin, that is a convex function on the Delzant polytope. A folklore conjecture asserts that a curve of Chebyshev potentials associated to a curve in the space of K\"ahler potentials is linear in the time variable if and only if the latter curve is a geodesic in the Mabuchi metric. This is classically true in the special toric setting, and in general Witt Nystr\"om established the sufficiency. The goal of this article is to disprove this conjecture. More generally, we characterize the Fubini--Study geodesics for which the conjecture is true on projective space. The proof involves explicitly solving the Monge--Amp\`ere equation describing geodesics on the subspace of Fubini--Study metrics and computing their Chebyshev potentials.

... We endow the infinite-dimensional space H with a Riemannian metric named after Mabuchi [19,25,14]: at (φ, v) ∈ T H ≃ H × C 1,1 (M, R), the squared norm of v is ...

We prove the convergence of quantized Bergman geodesics to the Mabuchi geodesics for the initial value problem, in the case of real-analytic initial data and in short time. This partially solves a conjecture of Y. Rubinstein and the last author. We also prove that the boundary value problem generically admits no solution in real-analytic regularity.

... Let h L 0 , h L 1 be two smooth positive metrics on L. A geodesic equation (with respect to L 2metric) for a smooth curve h L t connecting h L 0 and h L 1 was found by Mabuchi [40]. Later, Semmes [50] and Donaldson [24] independently discovered that this equation can be understood as a complex Monge-Ampère equation. ...

The main goal of this article is to study for a projective manifold and an ample line bundle over it the relation between metric and algebraic structures on the associated section ring. More precisely, we prove that once the kernel is factored out, the multiplication operator of the section ring becomes an approximate isometry (up to normalization) with respect to the $L^2$-norms and the induced Hermitian tensor product norm. We also show that the analogous result holds for the $L^1$ and $L^{\infty}$-norms if instead of the Hermitian tensor product norm, we consider the projective and injective tensor norms induced by $L^1$ and $L^{\infty}$-norms respectively. Then we show that $L^2$-norms associated with continuous plurisubharmonic metrics are actually characterized by the multiplicativity properties of this type.

... Thus it should be naturally related to the existence problem of geodesics on the space of Kähler metrics (cf. [23,[36][37][38]41] ...

We establish an asymptotic version of Bismut’s local family index theorem for the Bergman kernel. The key idea is to use the superconnection as in the local family index theorem.

... Here the subscript of the ∂∂ operator indicates that it is to be taken with respect to the variables τ, x jointly. If u τ does not depend on the imaginary part of τ , this means that u t = u τ is a geodesic for the Riemannian structure on the Mabuchi space, see [16], [10] For a complex geodesic i∂∂ τ x u τ is a (1, 1)-form on U × X, where U is an open set in C, which we usually take to be a strip, if u τ is independent of s. We also note that i∂∂ τ x u τ + ω ≥ 0 on U × X implies that u τ is subharmonic in τ for x fixed, so u t is a convex function of t if u τ is independent of s. ...

We give some remarks on geodesics in the space of K\"ahler metrics that are defined for all time. Such curves are conjecturally induced by holomorphic vector fields, and we show that this is indeed so for regular geodesics, whereas the question for generalized geodesics is still open (as far as we know). We also give a result about the derivative of such geodesics which implies a variant of a theorem of Atiyah and Guillemin-Sternberg on convexity of the image of certain moment maps.

... For ψ ≥ 0, the Dirichlet problem for complex Monge-Ampère equation becomes degenerate, which is much more complicated. In [7], Chen solved the Dirichlet problem for homogeneous complex Monge-Ampère equation on M = X × A and proved the existence of C 1,α -regularity (weak) geodesics in the space of Kähler potentials [11,32,35], where A = S 1 × [0, 1] and X is a closed Kähler manifold. The Dirichlet problem for degenerate complex Monge-Ampère equation was further complemented by Błocki [2], Phong-Sturm [34] and Boucksom [3]. ...

We derive the solvability and regularity of the Dirichlet problem for fully non-linear elliptic equations possibly with degenerate right-hand side on Hermitian manifolds, through establishing a quantitative version of boundary estimate under a subsolution assumption. In addition, we construct the subsolution when the background manifold is a product of a closed Hermitian manifold with a compact Riemann surface with boundary.

... plays some important roles in complex geometry and complex analysis (cf. [4,18,26,49,63]). See also [42,58,62]. ...

In this paper we propose new insights and ideas to set up quantitative boundary estimates for solutions to Dirichlet problem of a class of fully non-linear elliptic equations on compact Hermitian manifolds with real analytic Levi flat boundary. With the quantitative boundary estimates at hand, we can establish the gradient estimate and give a unified approach to investigate the existence and regularity of solutions of Dirichlet problem with sufficiently smooth boundary data, which include the geodesic equation in the space of K\"ahler metrics as a special case. Our method can also be applied to Dirichlet problem for analogous fully non-linear elliptic equations on a compact Riemannian manifold with concave boundary.

... However, since what tradition calls weak geodesics turned out to be rather more important than smooth solutions of the geodesic equation ∇ψψ = 0, they earned the right to a short name, geodesic, and this is the name we will be using. Over the years geodesics in various subspaces of E have been found to be subject to conservation laws [S,Corollary 3.19], [Be2,Proposition 2.2], [Da1], [DNL,Theorem 4.4]. The conserved quantities can be expressed in terms of the velocity of the geodesic and its decreasing rearrangement. ...

Consider a compact K\"ahler manifold $(X,\omega)$ and the space $\cal E(X,\omega)=\cal E$ of $\omega$--plurisubharmonic functions of full Monge--Amp\`ere mass on it. We introduce a quantity $\rho[u,v]$ to measure the distance between $u, v\in\cal E$; $\rho[u,v]$ is not a number but rather a decreasing function on a certain interval $(0,V)\subset\mathbb R$. We explore properties of $\rho[u,v]$, and using them we study Lagrangians and associated energy spaces of $\omega$--plurisubharmonic functions. Many results here generalize Darvas's findings about his metrics $d_\chi$.

... The right-hand side is an increasing function of A and by (32) we know A ≥ 1 n+1 . This means the right hand side is minimized at this value, so Springer Nature journal content, brought to you courtesy of Springer Nature Customer Service Center GmbH ("Springer Nature"). ...

We obtain sharp inequalities between the large scale asymptotic of the J functional with respect to the d1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_1$$\end{document} metric on the space of Kähler metrics. Applications regarding the initial value problem for geodesic rays are presented.

... On the other hand, it was discovered independently by Semmes [Se92] and Donaldson [Do99] that the geodesic equation in the space of Kähler potentials on a compact Kähler manifold is equivalent to the homogeneous Monge-Ampère equation on a compact Kähler manifold with boundary, the product of the Kähler manifold and an annulus in the complex plane. In general the solution is at most C 1,1 -smooth, due to an example of Gamelin and Sibony [GS80]. ...

We study weak quasi-plurisubharmonic solutions to the Dirichlet problem for the complex Monge-Am\`ere equation on a general Hermitian manifold with non-empty boundary. We prove optimal subsolution theorems: for bounded and H\"older continuous quasi-plurisubharmonic functions. The continuity of the solution is proved for measures that well dominated by capacity, for example measures with $L^p$, $p>1$ densities, or moderate measures in the sense of Dinh-Nguyen-Sibony.

... Motivated by the study of canonical metrics on X, in [Mab87] Mabuchi introduced a Riemannian structure on the space of Kähler potentials, giving rise to the notion of geodesics connecting two elements in H. As discovered later by Semmes [Sem92] and Donaldson [Don99], the geodesic equation can be formulated as a degenerate homogeneous complex Monge-Ampère equation. ...

We prove a geodesic distance formula for quasi-psh functions with finite entropy, extending results by Chen and Darvas. We work with big and nef cohomology classes: a key result we establish is the convexity of the $K$-energy in this general setting. We then study Monge-Amp\`ere measures on contact sets, generalizing a recent result by the first author and Trapani.

... For 1 ≤ m < n a problem arise, since if C n ∋ z → u(z) is m-subharmonic, then C n × C ∋ (z, λ) → u(z) need not to be (m+1)-subharmonic, it is only m-subharmonic. For this reason, we can not directly use Semmes' method [55] to construct geodesics. This leads us to consider the subspace, E 1,m , of (E 1,m , d) containing those functions that are also (m + 1)-subharmonic. ...

With inspiration from the K\"ahler geometry, we introduce a metric structure on the energy class, $\mathcal{E}_{1,m}$, of $m$-subharmonic functions with bounded energy and show that it is complete. After studying how the metric convergence relates to the accepted convergences in this Caffarelli-Nirenberg-Spruck model, we end by constructing geodesics in a subspace of our complete metric space.

... This endows the configuration space of the system with the structure of an effective phase space whose Kähler quantization determines the lowest Landau level (LLL). Natural families of deformations of the physical metric are provided by paths which are geodesics relative to the Mabuchi metric on the space of Kähler structures [12,13]. The latter are generated by imaginary time Hamiltonian flows [14]. ...

We study the change of the Laughlin states under large deformations of the geometry of the sphere and the plane, associated with Mabuchi geodesics on the space of metrics with Hamiltonian $S^1$-symmetry. For geodesics associated with the square of the symmetry generator, as the geodesic time goes to infinity, the geometry of the sphere becomes that of a thin cigar collapsing to a line and the Laughlin states become concentrated on a discrete set of $S^1$--orbits, corresponding to Bohr-Sommerfeld orbits of geometric quantization. The lifting of the Mabuchi geodesics to the bundle of quantum states, to which the Laughlin states belong, is achieved via generalized coherent state transforms, which correspond to the KZ parallel transport of Chern-Simons theory.

The aim of this paper is to study the residual Monge–Ampère mass of a plurisubharmonic function with isolated singularity at the origin in C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {C}}}^2$$\end{document}. We prove that the residual mass is zero if its Lelong number is zero at the origin, provided that it is S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^1$$\end{document}-invariant. This result answers the zero mass conjecture raised by Guedj and Rashkovskii in this special case. More generally, we obtain an estimate on the residual mass by the maximal directional Lelong number and Lelong number at the origin.

A bstract
We explore the effective gravitational action for two-dimensional massive Euclidean Majorana fermions in a small mass expansion, continuing and completing the study initiated in a previous paper [1]. We perform a detailed analysis of local zeta functions, heat kernels, and Green’s functions of the Dirac operator on arbitrary Riemann surfaces. We obtain the full expansion of the effective gravitational action to all orders in m ² . For genus one and larger, this requires the understanding of the role of the zero-modes of the (massless) Dirac operator which is worked out.
Besides the Liouville action, at order m ⁰ , which only involves the background metric and the conformal factor σ , the various contributions to the effective gravitational action at higher orders in m ² can be expressed in terms of integrals of the renormalized Green’s function at coinciding points of the squared (massless) Dirac operator, as well as of higher Green’s functions. In particular, at order m ² , these contributions can be re-written as a term ∫ e 2 σ σ characteristic of the Mabuchi action, much as for 2D massive scalars, as well as several other terms that are multi-local in the conformal factor σ and involve the Green’s functions of the massless Dirac operator and the renormalized Green’s function, but for the background metric only, and certain area-like parameters related to the zero-modes.

We explore the effective gravitational action for two-dimensional massive Euclidean Majorana fermions in a small mass expansion, continuing and completing the study initiated in a previous paper. We perform a detailed analysis of local zeta functions, heat kernels, and Green's functions of the Dirac operator on arbitrary Riemann surfaces. We obtain the full expansion of the effective gravitational action to all orders in $m^2$. For genus one and larger, this requires the understanding of the role of the zero-modes of the (massless) Dirac operator which is worked out. Besides the Liouville action, at order $m^0$, which only involves the background metric and the conformal factor $\sigma$, the various contributions to the effective gravitational action at higher orders in $m^2$ can be expressed in terms of integrals of the renormalized Green's function at coinciding points of the squared (massless) Dirac operator, as well as of higher Green's functions. In particular, at order $m^2$, these contributions can be re-written as a term $\int e^{2\sigma}\, \sigma$ characteristic of the Mabuchi action, much as for 2D massive scalars, as well as several other terms that are multi-local in the conformal factor $\sigma$ and involve the Green's functions of the massless Dirac operator and the renormalized Green's function, but for the background metric only.

We derive the geodesic equation for relatively Kähler metrics on fibrations and prove that any two such metrics with fibrewise constant scalar curvature are joined by a unique smooth geodesic. We then show convexity of the log‐norm functional for this setting along geodesics, which yields simple proofs of Dervan and Sektnan's uniqueness result for optimal symplectic connections and a boundedness result for the log‐norm functional. Next, we associate to a fibration degeneration a unique geodesic ray defined on a dense open subset. Calculating the limiting slope of the log‐norm functional along a globally defined smooth geodesic ray, we prove that fibrations admitting optimal symplectic connections are polystable with respect to a large class of fibration degenerations that are smooth over the base. We give examples of such degenerations in the case of projectivised vector bundles and isotrivial fibrations.

In (Donaldson in J Differ Geom 70(3):453–472, 2005), it was asked whether the lower bound of the Calabi functional is achieved by a sequence of the normalized Donaldson–Futaki invariants. We answer to the question for the Ricci curvature formalism, in place of the scalar curvature. Our principle is that the stability indicator is optimized by the multiplier ideal sheaves of certain weak geodesic ray asymptotic to the geometric flow. We actually obtain the results for the two cases: the inverse Monge–Ampère flow and the Kähler–Ricci flow.

This paper surveys the role of moment maps in Kähler geometry. The first section discusses the Ricci form as a moment map and then moves on to moment map interpretations of the Kähler–Einstein condition and the scalar curvature (Quillen–Fujiki–Donaldson). The second section examines the ramifications of these results for various Teichmüller spaces and their Weil–Petersson symplectic forms and explains how these arise naturally from the construction of symplectic quotients. The third section discusses a symplectic form introduced by Donaldson on the space of Fano complex structures.

We study the dependence of the Laughlin states on the geometry of the sphere and the plane, for one-parameter Mabuchi geodesic families of curved metrics with Hamiltonian S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^1$$\end{document}-symmetry. For geodesics associated with convex functions of the symmetry generator, as the geodesic time goes to infinity, the geometry of the sphere becomes that of a thin cigar collapsing to a line and the Laughlin states become concentrated on a discrete set of S1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S^1$$\end{document}-orbits, corresponding to Bohr–Sommerfeld orbits of geometric quantization.

Given a polarized projective variety ( X , L ) {(X,L)} over any non-Archimedean field, assuming continuity of envelopes, we define a metric on the space of finite-energy metrics on L , related to a construction of Darvas in the complex setting. We show that this makes finite-energy metrics on L into a geodesic metric space, where geodesics are given as maximal psh segments. Given two continuous psh metrics, we show that the maximal segment joining them is furthermore continuous. Our results hold in particular in all situations relevant to the study of degenerations and K-stability in complex geometry.

We raise our cups to Urban Cegrell, gone but not forgotten, gone but ever here. Until we meet again in Valhalla!
With inspiration from the Kähler geometry, we introduce a metric structure on the energy class, $\mathcal {E}_{1,m}$, of $m$-subharmonic functions with bounded energy and show that it is complete. After studying how the metric convergence relates to the accepted convergences in this Caffarelli–Nirenberg–Spruck model, we end by constructing geodesics in a subspace of our complete metric space.

We develop an algebro-geometric formulation for neural networks in machine learning using the moduli space of framed quiver representations. We find natural Hermitian metrics on the universal bundles over the moduli whose expressions are independent of dimension vector, and show that their Ricci curvatures give a Kähler metric on the moduli. Moreover, we use toric moment maps to construct activation functions and prove the universal approximation theorem for the softmax function (also known as Boltzmann distribution) using toric geometry of the complex projective space.

We prove a geodesic distance formula for quasi-psh functions with finite entropy, extending results by Chen and Darvas. We work with big and nef cohomology classes: a key result we establish is the convexity of the K-energy in this general setting. We then study Monge—Ampère measures on contact sets, generalizing a recent result by the first author and Trapani.

In previous work, Darvas, George and Smith obtained inequalities between the large scale asymptotic of the J functional with respect to the d1 metric on the space of toric Kähler metrics/rays. In this work we prove sharpness of these inequalities on all toric Kähler manifolds, and study the extremizing potentials/rays. On general Kähler manifolds we show that existence of radial extremizers is equivalent with the existence of plurisupported currents, as introduced and studied by McCleerey.

Following Kobayashi, we consider Griffiths negative complex Finsler bundles, naturally leading us to introduce Griffiths extremal Finsler metrics. As we point out, this notion is closely related to the theory of interpolation of norms, and is characterized by an equation of complex Monge–Ampère type, whose corresponding Dirichlet problem we solve. As applications, we prove that Griffiths extremal Finsler metrics quantize solutions to a natural PDE in Kähler geometry, related to the construction of flat maps for the Mabuchi metric.

We study and develop pluripotential theory over a non-Archimedean field, in itself, and through its interactions with complex geometry. Its main objects are plurisubharmonic (or psh) metrics on a line bundle L over a variety X over a non-Archimedan field. The global theory of such psh metrics has recently been developed by Boucksom-Eriksson-Favre-Jonsson et al. The most well known case is that of a field K endowed with the trivial absolute value; in this thesis, we will focus on fields endowed with nontrivial absolute values.We first look into the image of the asymptotic Fubini-Study operator over general non-Archiemdean fields, which allows us to approximate plurisubharmonic (psh) metrics on an ample line bundle L using norms acting on the sections of the tensor powers of L. Then, extending an important construction from complex geometry to the non-Archimedean world, we show that there exist plurisubharmonic geodesics in spaces of finite-energy psh metrics on an ample line bundle, and study their regularity properties with respect to the regularity of their endpoints. Finally, we consider an analytic degeneration of complex varieties, X, fibred over the unit disc, which we identify with a variety X_K over the non-Archimedean field of complex Laurent series. Given a relatively ample line bundle on X, we construct the geodesic metric space of relatively maximal finite-energy metrics on L. We show that the space of non-Archimedean finite-energy metrics on L_K (having once again identified L with a variety over the field K) embeds isometrically and geodesically into the former, allowing us to deduce convexity of some non-Archimedean versions of various functionals related to K-stability.

We give some remarks on geodesics in the space of Kähler metrics that are defined for all time. Such curves are conjecturally induced by holomorphic vector fields, and we show that this is indeed so for regular geodesics, whereas the question for generalized geodesics is still open (as far as we know). We also give a result about the derivative of such geodesics which implies a variant of a theorem of Atiyah and Guillemin-Sternberg on convexity of the image of certain moment maps.

We give a criterion for the projectivisation of a reflexive sheaf on a klt space to be induced by a projective representation of the fundamental group of the smooth locus. This criterion is then applied to give a characterisation of finite quotients of projective spaces and Abelian varieties by Q \mathbb {Q} -Chern class (in)equalities and a suitable stability condition. This stability condition is formulated in terms of a naturally defined extension of the tangent sheaf by the structure sheaf. We further examine cases in which this stability condition is satisfied, comparing it to K-semistability and related notions.

Given \((X,\omega )\) compact Kähler manifold and \(\psi \in \mathcal {M}^{+}\subset PSH(X,\omega )\) a model type envelope with non-zero mass, i.e. a fixed potential determining a singularity type such that \(\int _{X}(\omega +dd^{c}\psi )^{n}>0\), we prove that the \(\psi -\)relative finite energy class \(\mathcal {E}^{1}(X,\omega ,\psi )\) becomes a complete metric space if endowed with a distance d which generalizes the well-known \(d_{1}\) distance on the space of Kähler potentials. Moreover, for \(\mathcal {A}\subset \mathcal {M}^{+}\) totally ordered, we equip the set \(X_{\mathcal {A}}:=\bigsqcup _{\psi \in \overline{\mathcal {A}}}\mathcal {E}^{1}(X,\omega ,\psi )\) with a natural distance \(d_{\mathcal {A}}\) which coincides with d on \(\mathcal {E}^{1}(X,\omega ,\psi )\) for any \(\psi \in \overline{\mathcal {A}}\). We show that \(\big (X_{\mathcal {A}},d_{\mathcal {A}}\big )\) is a complete metric space. As a consequence, assuming \(\psi _{k}\searrow \psi \) and \(\psi _{k},\psi \in \mathcal {M}^{+}\), we also prove that \(\big (\mathcal {E}^{1}(X,\omega ,\psi _{k}),d\big )\) converges in a Gromov-Hausdorff sense to \(\big (\mathcal {E}^{1}(X,\omega ,\psi ),d\big )\) and that there exists a direct system \(\Big \langle \big (\mathcal {E}^{1}(X,\omega ,\psi _{k}),d\big ),P_{k,j}\Big \rangle \) in the category of metric spaces whose direct limit is dense into \(\big (\mathcal {E}^{1}(X,\omega ,\psi ),d\big )\).

Let $(X,\omega)$ be a K\"ahler manifold and $\psi: \Bbb R \to \Bbb R_+$ be a concave weight. We show that $\mathcal H_\omega$ admits a natural metric $d_\psi$ whose completion is the low energy space $\mathcal E_\psi$, introduced by Guedj-Zeriahi. As $d_\psi$ is not induced by a Finsler metric, the main difficulty is to show that the triangle inequality holds. We study properties of the resulting complete metric space $(\mathcal E_\psi,d_\psi)$.

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