Sharp Inequalities for Determinants of Toeplitz Operators and dbar-Laplacians on Line Bundles

Theorem 1. Let L be an ample line bundle su h that the adjoint line bundle L + K^X is globally generated. Then the absolute maximum of the fun tional F^ω⁰ on H^ω⁰ is attained at any riti al point u. Moreover, any smooth maximizer of F^ω⁰ on H^ω⁰ is unique (up to addition of onstants) modulo the a tion of Aut⁰(X, L). In parti ular, su h a maximizer In the ase when the ample line bundle L = −K^X , so that X is ^a Fano manifold, the spa e H⁰ (X, L + KX) is one-dimensional and hen e L^ω⁰ (u) = − 1 N log R e −(u+ψ0) . Then it is well-known that any riti al point may be identied with ^a Kähler-Einstein metri on X . It should be emphasized that the existen e of riti al points of F^ω⁰ a very di ult issue losely related to onje tures of Yau, Tian, Donaldson and others in Kähler geometry [51, 27, 49℄. Even in the ase L = −K^X there are well-known examples already on omplex surfa es, where riti al points do not exist. Next, assume that (X, L) is K−homogenous, i.e. that X admits ^a transitive a tion by ^a ompa t semi-simple Lie group K, whose a tion on X lifts to L. We will then take ω⁰ as the unique Kähler form in c1(L) whi h is invariant under the a tion of K on X.

Corollary 2. Let L → X be a K−homogenous ample holomorphi line bund le over ^a ompa t omplex manifold X and denote by ω⁰ be the unique K−invariant Kähler metri in c1(L). Then, for any fun tion u in H^ω⁰ $$-\mathcal{L}_{\omega_0}(u) \le -\mathcal{E}_{\omega_0}(u)$$ with equality i the fun tion u is onstant, modulo the a tion of Aut⁰(X, L). Surprisingly, spe ializing to the ase when X is a omplex urve (i.e. n = 1) allows one to take u as any smooth fun tion (whi h is not true in higher dimensions, as shown in [39℄ in the ase when L = −K^X ; see remark 12). More generally, we an then take u to be in the Sobolev space $W^{2,1}(X)$ of all functions u on X such that u and its differential du are square integrable. We will next consider the homegenous case, i.e. when $X = \mathbb{P}^1$ , the complex projective line (i.e. topologically $X = S^2$ , the two-sphere) and hence K = SU(2). In this case any ample line bundle L may be written as $\mathcal{O}(k)$ , where k is a positive integer and $H^0(\mathbb{P}^1, \mathcal{O}(k) + K_{\mathbb{P}^1}) = H^0(\mathbb{P}^1, \mathcal{O}(k-2))$ may be identified with the space of all polynomials of at most degree m := k - 2 on the affine piece $\mathbb{C}$ in $\mathbb{P}^1$ (assuming $k \geq 2$ ). Moreover, if we take $\psi_{0,k}(z) = k \log(1 + z\bar{z})$ as the fixed invariant weight on $\mathcal{O}(k)$ , in the usual trivialization over the affine piece $\mathbb{C}$ , then, under the identification above, the Hermitian product on $H^0(\mathbb{P}^1, \mathcal{O}(k) + K_{\mathbb{P}^1})$ may be written as (1.7) $$\langle p_m, p_m \rangle_{\psi_0 + u} := \int_{\mathbb{C}} \frac{|p_m|^2}{(1 + z\bar{z})^m} e^{-u} \omega_0$$ for $p_m(z)$ a polynomial on $\mathbb{C}$ of degree at most m (compare section 3.3.1). Hence, $$\mathcal{L}_{\omega_{0,k}}(u) := (m+1)\mathcal{L}_m(u) := -\log \det(c_{ij} \int_{\mathbb{C}} \frac{z^i \bar{z}^j}{(1+z\bar{z})^m} e^{-u} \omega_0),$$ where i, j = 0, ..., m and $1/c_{ij} = (m+1)\binom{m}{i}\binom{m}{j}$ .

Corollary 3. Let u be a function in the Sobolev space $W^{2,1}(S^2)$ on the two-sphere $S^2$ and denote by $\omega_0$ the volume form corresponding to the metric on $S^2$ with constant curvature and volume one. Then $$-\mathcal{L}_m(u) \le -(m+1) \int_{S^2} u\omega_0 + (\frac{m+1}{m+2}) \frac{1}{2} \int_{S^2} du \wedge du^c$$ with equality iff there exists a Möbius transformation M of $S^2$ such that $\omega_u = M^*\omega_0$ . The case when m=0, so that $-\mathcal{L}_m(u)=\log\int_{\mathbb{C}}e^{-u}\omega_0$ , is precisely the celebrated Moser-Trudinger-Onofri inequality 1.1. The reduction of the proof of Corollary 3 to Corollary 2, is based on properties of the projection operator $P_{\omega}$ (formula 1.8). 1.2.1. Application to determinants of $\bar{\partial}$ -Laplace operators and Analytic Torsion. Consider again the case when $X = \mathbb{P}^1$ is the complex projective line equipped with the standard Kähler form $\omega_0$ . Any function u corresponds to a metric on $\mathcal{O}(m)$ with weight $m\psi_0 + u$ , where m is a fixed nonnegative integer. Hence, the pair $(\omega_0, u)$ induces natural Hilbert norms on the space $\Omega^{0,q}(\mathcal{O}(m))$ of smooth (0,q)-forms with values in $\mathcal{O}(m)$ , where q=0,1. Denote by $\Delta_{\bar{\partial}_u}^{(m)}$ the corresponding $\bar{\partial}$ -Laplace (Dolbeault) operator acting on the space $\Omega^0(\mathcal{O}(m))$ , i.e. $\Delta_{\bar{\partial}_u}^{(m)} = \bar{\partial}^\bar{\partial}$ , where $\bar{\partial}$ is the formal adjoint of the $\bar{\partial}$ -operator $$\bar{\partial}: \Omega^{0,0}(\mathcal{O}(m)) \to \Omega^{0,1}(\mathcal{O}(m))$$ Note that $\bar{\partial}$ may be expressed in terms of the adjoint $\bar{\partial}^{,0}$ induced by u = 0 as $$\bar{\partial}^ = e^u \bar{\partial}^{,0} e^{-u}$$ The zeta function regularized determinant of the operator obtained by restricting $\Delta_{\bar{\partial}_u}^{(m)}$ to the orthogonal complement of its kernel will be denoted by $\det \Delta_{\bar{\partial}_{-}}^{(m)}$ (compare [14]). Given the result in the previous corollary, the anomaly formula (i.e. a family Riemann-Roch-Grothendieck theorem) of Bismut-Gillet-Soulé [14] now implies the following positive solution of Fang's conjecture

Corollary 4. Given the line bundle $\mathcal{O}(m) \to \mathbb{P}^1$ , the corresponding func- $$u \mapsto \det \Delta_{\bar{\partial}_u}^{(m)}$$ $u\mapsto \det \Delta^{(m)}_{\bar{\partial}_u}$ on the space of all smooth functions u on $\mathbb{P}^1$ attains its maximum precisely for u a constant function. In fact, the proof of the previous Corollary, will give the stronger statement that the inequality 1.2 for $\det \Delta_{\bar{\partial}_u}$ stated in the introduction holds and that this latter inequality is equivalent to Corollary 3. Note that a direct consequence of the previous corollary is the following reponse to a variant of Kac's classical question "Can one hear the shape of a drum?" [37]: if the $\partial$ -Laplacian on some power $\mathcal{O}(m)$ induced by a smooth metric h on $\mathcal{O}(1) \to \mathbb{P}^1$ has the same spectrum (including multiplicities) as the $\bar{\partial}$ -Laplacian induced by the standard SU(2) invariant metric $h_0$ , then $h = Ch_0$ for a positive number C. Finally it should be pointed out that in the general case of an ample line bundle L Theorem 1 yields a bound on the twisted Ray-Singer analytic torsion (see for example [14]) associated to a semi-positively curved metric on L in terms of the corresponding Quillen metric and the functional $\mathcal{E}$ . This is a direct consquence of the fact that $L + K_X$ is ample, so that the higher cohomology groups $H^q(X, L + K_X)$ , $q \geq 1$ , vanish, combined with the anomaly formula of Bismut-Gillet-Soulé [14]. For the sake of brevity the details are omitted. 1.3. Further relations to previous results. In this case when L= $-K_X$ the first statement of Theorem 1 is a result of Ding-Tian[25] and the "uniqueness" of critical points (i.e. Kähler-Einstein metrics in this case) was proved earlier by Bando-Mabuchi [4]. See [10] for a generalization of this latter result to funtions of "finite energy", in the case when $\operatorname{Aut}_0(X,L)$ is discrete (compare remark 5). The extremal property of the critical points in Theorem 1 can also be seen as an analog of a result of Donaldson (Theorem 2 in [28]) who furthermore assumed that $Aut_0(X, L)$ is discrete. In this latter setting the role of the space $H^0(X, L+K_X)$ is played by $H^0(X, L)$ equipped with the scalar products induced by the weight $\psi_0 + u$ and the integration measure $(\omega_u)^n/n!$ Note however that in Donaldson's setting the functional corresponding to $\mathcal{F}_{\omega_0}$ is minimized on its critical points (compare section 5.1 and the discussion in section 5 in [13]). In the terminilogy of [28] these latter critical points correpond to balanced metrics. Donaldson used his result, combined with the deep convergence results in [27] for balanced metrics, in the limit when L is replaced by a large tensor power, to prove a lower bound on Mabuchi's K-energy functional. It will be shown in section 5 how to deduce this latter result more directly from Theorem 1 above. It should also be pointed out that the inequality proved by Donaldson corresponds to a lower bound on $\mathcal{F}_{\omega_0}(u)$ in the present setting, which however will depend on u through its volume form $(\omega_u)^n/n!$ (see the end of section 5.1). 1.4. Concerning the proof of Theorem 1. The proof of Theorem 1 relies on the recent work [13] of Berndtsson combined with some global pluripotential theory developed in [9, 11] (see also [10] for the case $L = -K_X$ ). On one hand [13] gives that $\mathcal{F}_{\omega_0}$ is "geodesically" convex wrt the Riemann metric on the space $\mathcal{H}_{\omega_0}$ introduced by Mabuchi [43]. In turn, this fact is used to show that any critical point maximizes $\mathcal{F}_{\omega_0}$ on $\mathcal{H}_{\omega_0}$ , using the existence of (generalized) $C^0$ -geodesics in the closure $\overline{\mathcal{H}}_{\omega_0}$ . On the other hand, a main point in the proof of the "uniqueness" of critical points is to show that there are no smooth extremal points of $\mathcal{F}_{\omega_0}$ in the "boundary" of $\mathcal{H}_{\omega_0}$ , i.e. in $\overline{\mathcal{H}}_{\omega_0} - \mathcal{H}_{\omega_0}$ . Following [9, 10] this is shown by extending $\mathcal{F}_{\omega_0}$ to a (Gâteaux) differentiable function on all of $C^0(X)$ , by replacing $\mathcal{E}_{\omega_0}$ with the composed map $\mathcal{E}_{\omega_0} \circ P_{\omega_0}$ , where $P_{\omega_0}$ is the following (non-linear) projection operator from $C^0(X)$ onto $\mathcal{C}^0(X) \cap \overline{\mathcal{H}}_{\omega_0}$ : (1.8) $$P_{\omega_0}[u](x) = \sup \{v(x) : v \in \mathcal{H}_{\omega_0}, \ v \le u\}$$ Remark 5. Consider the setting of Theorem 1 and assume that there exists a (smooth) critical point, which we may assume is given by 0. Then the inequality furnised by the theorem, i.e. $$\mathcal{F}_{\omega_0}(u) := \mathcal{E}_{\omega_0}(u) - \mathcal{L}_{\omega_0}(u) \le 0$$ actually holds for all u in $\mathcal{E}^1(X, \omega_0)$ , i.e. for al u in the convex set of all u in $\overline{\mathcal{H}}_{\omega_0}$ with finite energy; $\mathcal{E}(u) > -\infty$ , where $$\mathcal{E}(u) := \inf_{u' \ge u} \mathcal{E}(u')$$ when u' ranges over all elements in $\mathcal{H}_{\omega_0}$ such that $u' \geq u$ . Equivalently, $\int_X (\omega_u)^n = \operatorname{Vol}(L)$ and $-\int_X u(\omega_u)^n < \infty$ in terms of non-pluripolar products (see [10] and references therein). The inequality on all of $\mathcal{E}^1(X,\omega_0)$ is simply obtained by writing u as a decreasing limit of elements in $\mathcal{H}_{\omega_0}$ and using the continuity of $\mathcal{E}$ and $\mathcal{L}_{\omega_0}$ under such limits [10] (note that $e^{-u}$ is integrable if $\mathcal{E}(u) > -\infty$ [10]). Moreover, in the case when $\operatorname{Aut}_0(X, L)$ is discrete it can be shown that any maximizer of $\mathcal{F}_{\omega_0}$ on $\mathcal{E}^1(X, \omega_0)$ , is in fact equal to a constant. The proof is a simple adaptation of the argument in [10] concerning the case $L = -K_X$ . It would be interesting to know if the general uniqueness statement in Theorem 1 also remains true in the larger class $\mathcal{E}^1(X, \omega_0)$ ? It is a pleasure to thank Bo Berntdsson for illuminating discussions on the topic of the present paper, in particular in connection to [13]. It is an equal pleasure to thank Sébastien Boucksom, Vincent Guedj and Ahmed Zeriahi for discussions and stimulation coming from the colaboration [10]. The author is also grateful to Yanir Rubinstein and Bálint Virág for helpful comments on a preliminary version of this paper. Organization. In section 2 preliminaires for the proofs of the main results appearing in section 3 are given. The proof of the uniqueness statement in the main theorem relies on higher order regularity for "geodesics" defined by inhomogenous Monge-Ampère equations. An alternative proof based on considerably more elementary regularity results is given in section 3.6. In section 3.5 applications to Arithmetic (Arakelov) geometry are briefly indicated. In section 4 some of the previous results are interpreted in terms of SU(2)—invariant determinantal random point process on $S^2$ . Finally, in section 5 the limit when the line bundle L is replaced by a large tensor power is studied and a new proof of the lower bound on Mabuchi's K—energy for a polarized projective manifold is given. Relations to Donaldson's work are also discussed. In the appendix some formulas involving Bergman kernels are recalled and a "Bergman kernel proof" of Theorem 9 is given.

Theorem 7. (Chen) Assume that the boundary data in the Dirichlet problem 2.1 for the Monge-Ampère operator on M is smooth on $\partial M$ . Then $U \in \mathcal{C}^{1,1}_{\mathbb{C}}(M)$ . More precisely, the mixed second order complex derivatives of U are uniformly bounded, i.e. there is a positive constant C such that $$0 \le (dd^c U + \pi_X^ \omega_0) \le C(\pi_X^ \omega_0 + \pi_A^* \omega_A)$$ where $\omega_A$ is the Eucledian metric on A. In the statement above we have used the (non-standard) notation $\mathcal{C}_{\mathbb{C}}^{1,1}(M)$ for the set of all functions U such that, locally, the current $dd^cU$ has coefficients in $L^{\infty}$ . Such a U is called $almost \, \mathcal{C}^{1,1}$ in [16]. Note that if $U \in \mathcal{H}_{\pi_X^\omega_0}(M)$ then this is equivalent to U having a bounded Laplacian $\Delta_M U$ , where $\Delta_M$ is the Laplacian on M wrt the Kähler metric $\pi_X^\omega_0 + \pi_A^*\omega_A$ on M. As will be explained in section 3.6 the proof of the uniqueness statement in Theorem 1 may actually be obtained by only using the bounds on the derivatives of $u_t$ on X for t fixed. As shown very recently in [11] such bounds may be obtained by working directly with the envelope 2.2.

Theorem 8. Assume that the boundary data in the Dirichlet problem 2.1 for the Monge-Ampère operator on M is in $C^{1,1}(\partial M)$ . Then $u_t \in C^{1,1}_{\mathbb{C}}(X)$ . More precisely, the mixed second order complex derivatives of $u_t$ on X are uniformly bounded, i.e. there is a positive constant C such that $$0 \le (dd^c u_t + \omega_0) \le C\omega_0$$ on X. One of the virtues of this latter approach is that the proof is remarkly simple when X is homogenous. 2.2. The functional $\mathcal{L}_{\omega_0}$ . First note that the functional $\mathcal{L}_{\omega_0}(u)$ defined by formula 1.5 is increasing on $\mathcal{C}^0(X)$ , wrt the usual order relation. This is an immediate consequence of the basic geometric interpretation in [9] of $\mathcal{L}_{\omega_0}(u)$ as propopertional to the logarithmic volume of the unit-ball in the Hilbert space $H^0(X, L + K_X)$ equipped with the Hermitian product induced by the weight $\psi_0 + u$ . Alternatively, it follows from formula 2.3 below which shows that the differential of the functional $\mathcal{L}_{\omega_0}$ on $\mathcal{C}^0(X)$ may be represented by the positive measure $\beta_u$ . Integrating $\beta_u$ along a line segment in $\mathcal{C}^0(X)$ equipped with its affine structure then shows that $\mathcal{L}_{\omega_0}(u)$ is increasing. The differential of the functional $\mathcal{L}_{\omega_0}$ on $\mathcal{C}^0(X)$ is given by $$(2.3) (d\mathcal{L}_{\omega_0})_u = \beta_u,$$ in the sense that given any smooth function v we have that $$d(\mathcal{L}_{\omega_0}(u+tv))/dt_{t=0} = \int_X \beta_u v,$$ where $\beta_u$ is the Bergman measure associated to u. This latter measure is the positive measure on X defined as (2.4) $$\beta_u = (i^{n^2} \frac{1}{N} \sum_{i=1}^N s_i \wedge \bar{s}_i e^{-\psi_0}) e^{-u}$$ in terms of any given orthonormal base $(s_i)$ in the Hilbert space $H^0(X, L + K_X)$ equipped with the Hermitian product induced by the weight $\psi_0 + u$ (compare section 6.1). In particular this means that $\beta_u$ may be represented as $e^{-u}$ times a strictly positive smooth measure on X if $L + K_X$ is globally generated. The proof of formula 2.3 follows more or less directly from the definition (see [13] for a geometric argument). The following theorem, which is direct consequence of a result of Berndtsson about the curvature of direct image bundles [13], considers the second derivatives of $\mathcal{L}_{\omega_0}$ along a psh path. As a courtesy to the reader a proof of the theorem, using Bergman kernels, is given in the appendix.

Theorem 9. (Berndtsson) Let $u_t$ be a continuous psh path in $\mathcal{H}_{\omega_0}$ . Then the function $t \mapsto \mathcal{L}_{\omega_0}(u_t)$ is convex. Moreover, if $\mathcal{L}_{\omega_0}(u_t)$ is affine and $u_t$ is a smooth psh path with $\omega_{u_t} > 0$ on X for all t, then there is an automorphism $S_1$ of (X, L), homotopic to the identity, such that $u_1 - u_0 = S_1^* \psi_0 - \psi_0$ . The convexity statement in [13] assumed in fact that $u_t$ be smooth. However, by uniform approximation the convexity statement above in fact holds for any continuous psh path in $C^0(X)$ . Indeed, if $u_t$ is such a path, then there exists, for example by Richbergs's approximation theorem [22], a sequence $U^j$ converging uniformly towards U on M such that $dd^cU^j + \pi^*\omega_0 > 0$ . Applying the theorem above to each $U^j$ and letting j tend to infinity then gives that $f(t) := \mathcal{L}_{\omega_0}(u_t)$ is a uniform limit of convex functions and hence convex, proving the claim. However, for the uniqueness statement the argument in [13] seems to require that $\omega_{u_t}$ be reasonably smooth in (t, x). Moreover, the assumption that $\omega_t > 0$ is crucial to be able to define the vector fields $V_t$ that integrate to the automorphism $S_1$ (see formula 3.6). 2.3. The functional $\mathcal{E}_{\omega_0}$ . First recall the following well-known formula for the differential of the energy functional $\mathcal{E}_{\omega_0}$ defined by formula 1.3: $$(2.5) (d\mathcal{E}_{\omega_0})_u = \omega_u^n / n!$$ The following generalization from [9] of the previous formula to the functional $\mathcal{E}_{\omega_0} \circ P_{\omega_0}$ , where $P_{\omega_0}$ is the non-linear projection 1.8, will be crucial for the proof of Theorem 1:

Theorem 10. The functional $\mathcal{E}_{\omega_0} \circ P_{\omega_0}$ is Gâteaux differentiable on $\mathcal{C}^0(X)$ . Its differential at the point u is represented by the measure $\omega_{P_{\omega_0}u}^n/n!$ , i.e. given $u, v \in \mathcal{C}^0(X)$ the function $\mathcal{E}_{\omega_0}P_{\omega_0}(u+tv)$ is differentiable on $\mathbb{R}_t$ and (2.6) $$d\mathcal{E}_{\omega_0} P_{\omega_0}(u+tv)/dt_{t=0} = \int_X v \omega_{P_{\omega_0} u}^n / n!$$ As for the second derivatives of $\mathcal{E}_{\omega_0}$ we have the following Proposition which is well-known (at least in the smooth case):

Proposition 11. The following properties of $\mathcal{E}_{\omega_0}$ hold: - The functional $\mathcal{E}_{\omega_0}$ on $\overline{\mathcal{H}}_{\omega_0} \cap \mathcal{C}^0(X)$ is concave wrt the affine structure on $\mathcal{C}^0(X)$ . - Let $u_t$ be a $\mathcal{C}^0$ geodesic in $\overline{\mathcal{H}}_{\omega_0}$ connecting $u_0$ and $u_1$ . Then the functional $t \mapsto \mathcal{E}_{\omega_0}(u_t)$ is affine and continuous on [0,1].

Lemma 16. The right tangent vector $v_t$ of $u_t$ at t is uniformly bounded on M.

Proposition 17. Let $u_0$ be a critical point of $\mathcal{F}_{\omega_0}$ on $\overline{\mathcal{H}}_{\omega_0} \cap \mathcal{C}^{1,1}(X)$ , $u_1$ an arbitrary element in $\overline{\mathcal{H}}_{\omega_0} \cap \mathcal{C}^{1,1}(X)$ and $u_t$ the geodesic connecting $u_0$ and $u_1$ . If $\mathcal{L}_{\omega_0}(u_t)$ is affine, then there is an automorphism $S_1$ of (X, L), homotopic to the identity, such that $u_1 - u_0 = S_1^* \psi_0 - \psi_0$ . Proof. Step 1: $u_t \in \mathcal{C}^{\infty}(X)$ . First note that by Theorem 8 $u_t \in \mathcal{C}^{1,1}_{\mathbb{C}}(X)$ . Moreover, as shown in the beginning of section 3.1 it follows under the assumptions above that, for any t, the function $u_t$ satisfies the Euler-Lagrange equations 1.6 on X. Hence, just as in section 3.1 Blocki's complex version of the regularity result of Trudinger, now applied to local patches of $\{t\} \times X$ immediately gives that $u_t \in \mathcal{C}^{\infty}(X)$ (when n = 1 this follows from basic linear elliptic theory). Step 2: $\Delta_X v_t \in L^{\infty}(X)$ uniformly wrt t. Differentiating the Euler-Lagrange equation wrt t from the right gives (3.14) $$ndd^{c}v_{t} \wedge (\omega_{t})^{n-1} = \frac{d\beta_{u_{t}}(x)}{dt} = :R[v_{t}],$$ in the sense of currents. Of course, this would follow immediately from the chain rule if $u_t$ were smooth in (t, x). In the present case it is proved in lemma 18 below. Moreover, lemma 24 in the appendix implies the bound (3.15) $$||R[v_t]/(\omega_0)^n||_{L^{\infty}(X)} \le C ||v_t||_{L^{\infty}(X)}$$ To see this, just note that $$R[v] \le 2 \|v\|_{L^{\infty}(X)} \int_{X} |K(x,y)|^{2} e^{-(\psi(x)+\psi(y))} = 2 \|v\|_{L^{\infty}(X)} \beta_{u},$$ using the well-known "reproducing property" of the Bergman kernel (formula 6.3 in the appendix). By formula 6.5 in the appendix this proves the inequality 3.15. Now, since $\omega_t > \delta \omega_0$ , formula 3.14 gives that the distribution $\Delta_{\omega_t} v_t$ , where $\Delta_{\omega_t}$ is the Laplacian on X wrt the metric $\omega_t := \omega_{u_t}$ , is in $L^{\infty}(X)$ uniformly wrt t and $$\|\Delta_{\omega_t} v_t\|_{L^{\infty}(X)} \le C \|v_t\|_{L^{\infty}(X)} \le C',$$ by lemma 16. Step 3: $\Delta_M u \in W^{1,p}(M)$ for any $p \geq 1$ . First observe that by step 1 (3.16) $$\partial_z(\partial_{z_i}\partial_{\bar{z}_j}u) \in L^{\infty}(X),$$ uniformly wrt t. Also note that (3.17) $$\partial_t(\partial_{z_i}\partial_{\bar{z}_j}u) \in L^{\infty}(X),$$ uniformly wrt t. Indeed, $\partial_t(\partial_{z_i}\partial_{\bar{z}_j}u) = \partial_{z_i}(\partial_{\bar{z}_j}\partial_t u) = (\partial_{z_i}\partial_{\bar{z}_j})v_t \in L^p(X)$ , uniformly wrt t, for any p > 1, by step 2 and local elliptic estimates for $\Delta_X$ . Next, we will use that the following identity proved in lemma 19 below (3.18) $$\partial_t \partial_{\bar{t}} u = |V_t|_{ut}^2 = |\partial_{\bar{z}} v_t|_{ut}^2,$$ where $|V_t|_{\omega_t}^2$ denotes the point-wise norm of $V_t$ wrt the metric $\omega_t$ (where we have used that $\omega_t > 0$ ). First we have (3.19) $$\partial_z(\partial_t \partial_{\bar{t}} u) = \partial_z \left| \partial_{\bar{z}} v_t \right|_{\omega_t}^2 \in L^p(X),$$ uniformly wrt t, for any p > 1 using Step 1 and Step 2 combined with local elliptic estimates on X for $\Delta_X$ . Next (3.20) $$\partial_t(\partial_t\partial_{\bar{t}}u) \in L^p(X),$$ uniformly wrt t. Indeed, $\partial_t(\partial_t\partial_{\bar{t}}u) = \partial_t |\partial_{\bar{z}}\partial_t u|_{\omega_t}^2$ and since locally $\partial_t\omega_t =$ $\partial_t(\partial_z\partial_{\bar{z}}u)$ 3.20 follows from 3.17 and 3.19 combined with Leibniz product rule. All in all this proves Step 3. Now by Step 3 and elliptic estimates for the Laplacian we have $u \in$ $W^{3,p}(M)$ . In particular, u is locally in $\mathcal{C}^2(M)$ . As a consequence the proof of Theorem 2.6 in [13] immediately gives that $V_t$ is a holomorphic vector field on X for any t. Finally, we will recall a slight variant of the argument in [13] which shows that $\partial_t V_t = 0$ for $V_t$ seen as a distribution on the interiour of M. To simplify the notation we assume that n=1, but modulo the change to matrix notation the case n > 1 is the same. First we write 3.6 in the form $$(3.21) \omega V = \partial_{\bar{z}} \partial_t u,$$ where we have identified V and $\omega$ with elements in $L^p(M)$ for p >> 1. By Leibniz rule $$\partial_{\bar{t}}(V\omega) = (\partial_{\bar{t}}V)\omega + V(\partial_{\bar{t}}\omega)$$ Next, observe that $$\partial_{\bar{t}}\omega = \partial_{\bar{t}}(\partial_z\partial_{\bar{z}}u) = \partial_{\bar{z}}(\partial_{\bar{t}}\partial_z u) = \partial_{\bar{z}}(\omega\bar{V}),$$ using 3.21 in the last step. Hence, since, as shown above, $\partial_{\bar{z}}V=0$ , the two previous equations together give $$\partial_{\bar{t}}(V\omega) = (\partial_{\bar{t}}V)\omega + \partial_{\bar{z}}(V\omega\bar{V}) = \partial_{\bar{z}}(\partial_{\bar{t}}\partial_t u),$$ also using 3.21 in the last step and commuting $\partial_{\bar{z}}$ and $\partial_{\bar{t}}$ . Since, $V\omega\bar{V}=$ $|V|_{\omega}^2$ it follows by 3.18 that $(\partial_{\bar{t}}V)\omega=0$ . But since, $\omega>0$ and $(\partial_{\bar{t}}V)$ is in $L^p(M)$ for all p>1 this forces $(\partial_{\bar{t}}V)=0$ a.e. on M. In particular, $(\partial_{\bar{t}}V)=0$ as a distribution on M. Hence, it follows that the distribution $V_t$ is in the null-space of the $\bar{\partial}$ -operator on M. By local elliptic theory it follows that $V_t$ is smooth and hence holomorphic in the interiour of M. Finally, the automorphism $S_1$ is obtained precisely as in the end of section 3.1.

Lemma 18. Under the assumptions in the previous proposition the following holds: $$\frac{d}{dt} \int_{X} (\omega_t)^n f = \int_{X} n v_t \wedge (\omega_t)^{n-1} \wedge dd^c f,$$ where f is a given smooth function on X.

Lemma 19. Under the assumptions in the previous proposition the following holds: $\partial_t \partial_{\bar{t}} u \in L^{\infty}(X)$ uniformly in t and $$\partial_t \partial_{\bar{t}} u = \left| \bar{\partial}_X \partial_t u \right|_{\omega_{nx}}^2.$$ Proof. By assumption the Monge-Ampère measure $(dd^cU + \pi_X^*\omega_0)^{n+1}$ vanishes on M. Moreover, by Step 1 in the proposition above $\Delta_X u_t \in C^\infty(X)$ for any t with bounds on the Sobolev norms which are uniform wrt t. Combining this latter fact with lemma 16 gives that U is Lipschitz on M. Finally, as shown in Step 2 in the proof of proposition above $\Delta_X \partial_t u_t \in L^\infty(X)$ uniformly wrt t. We will next show that these properties are enough to prove the lemma. As the statement is local we may as well consider the restriction of u := U to an open set biholomorphic to a domain in $\mathbb{C}^{n+1} = \mathbb{C}_t \times \mathbb{C}_z^n$ . Denote by $u^{\epsilon}$ the local smooth function obtain as the convolution of u with a fixed local compactly supported smooth family of approximations of the identity. Expanding gives $$(3.25) \qquad (dd^c U + \pi_X^* \omega_0)^{n+1} = (\partial_t \partial_{\bar{t}} u^{\epsilon} - |\partial_{\bar{z}} \partial_t u^{\epsilon}|_{\omega_{,\epsilon}}^2) (\omega_{u^{\epsilon}})^n \wedge dt \wedge d\bar{t}.$$ Now since, by assumption, $|\partial_{\bar{z}}\partial_t u^{\epsilon}|^2_{\omega_u^{\epsilon}} \leq C$ the second term tends to $|\partial_{\bar{z}}\partial_t u|^2_{\omega_u})(\omega_u)^n \wedge dt \wedge d\bar{t}$ weakly when $\epsilon \to 0$ . Moreover, by assumption $u^{\epsilon} \to u$ uniformly locally and since the Monge-Ampère operator is continous, as a measure, under uniform limits of psh functions [23] it will now be enough to prove that $$(3.26) \qquad (\partial_t \partial_{\bar{t}} u^{\epsilon})(\omega_{u^{\epsilon}})^n \wedge dt \wedge d\bar{t} \to (\partial_t \partial_{\bar{t}} u)(\omega_u)^n \wedge dt \wedge d\bar{t}$$ weakly, where the right hand sice is well-defined since $\partial_t \partial_{\bar{t}} u_t$ defines a positive measure on $\mathbb{C}^{n+1}$ and $(\omega_{u_t})^n/\omega_0^n$ is continous on $\mathbb{C}^{n+1}$ . To this end fix a test funtion f i.e. a smooth and compactly supported function on $\mathbb{C}^{n+1}$ . Then, with $\int$ denoting the integral over $\mathbb{C}^{n+1}$ , $$\int f(\omega_{u^{\epsilon}})^n (\partial_t \partial_{\bar{t}} u^{\epsilon}) \wedge dt \wedge d\bar{t} =: \int g_{\epsilon} (\partial_t \partial_{\bar{t}} u^{\epsilon}) = -\int (\partial_t g_{\epsilon}) (\partial_{\bar{t}} u^{\epsilon})$$ By assumption $(\partial_t g_{\epsilon})$ and $(\partial_{\bar{t}} u^{\epsilon})$ tend to $(\partial_{\bar{t}} u)$ and $(\partial_{\bar{t}} u)$ , respectively in $L^p(X)$ for any p > 1, uniformly wrt t (more precisily by the assumption on $\Delta_X u_t$ and the fact that u is Lipschitz). Hence, by Hölders's inequality $$\int g_{\epsilon}(\partial_t \partial_{\bar{t}} u^{\epsilon}) \to -\int (\partial_t g)(\partial_{\bar{t}} u).$$ Finally, since $(\partial_t g) \in L^{\infty}(X)$ uniformly wrt t (by the assumption on $\Delta_X u_t$ ) and since $\partial_t \partial_{\bar{t}} u$ defines a positive measure, Leibniz rule combined with the dominated convergence theorem gives (by a simple argument using a regularization of g) $$-\int (\partial_t g)(\partial_{\bar{t}} u) = \int g(\partial_t \partial_{\bar{t}} u)$$ This proves 3.26 and hence finishes the proof of the lemma.

Theorem 20. The following upper bound on the moment generating function of the fluctuation of a linear statistic in the point process 4.4 with N-particles on $S^2$ holds (4.5) $$\log \mathbb{E}_{N}(e^{t(\widetilde{u(x_{1})} + \widetilde{u(x_{2})} + \dots + \widetilde{u(x_{N})})}) \leq \frac{N}{N+1} \frac{t^{2}}{2} \|du\|^{2}$$ for any $t \in \mathbb{R}$ with equality iff $\omega_0 - tdd^c u$ is the pull-back of $\omega_0$ under a conformal transformation of $S^2$ .

Corollary 21. In the setting of the previous theorem the following large deviation bound holds: for any given positive number $\lambda$ : $$Prob_N \{ \frac{1}{N} (u(x_1) + \dots + u(x_N)) > \lambda \} \le e^{-\frac{N^2 \lambda^2}{2||du||^2} \frac{N+1}{N}}$$ if the linear statistic 4.2 is centered, i.e. if its expected value vanishes. Proof. The proof of this consequence of the previous theorem is a standard application of Markov's inequality: for any given t > 0 we have $$Prob\{Y > 1\} = Prob\{e^{tY} > e^t\} \le e^{-t}\mathbb{E}(e^{tY}),$$ where in our case $Y = \frac{1}{N\lambda}(u(x_1) + ... + u(x_N))$ . By the previous theorem the rhs above is bounded by $e^{-t+ct^2}$ for $c = \frac{N}{N+1}\frac{1}{2}\left\|d(\frac{1}{N\lambda}u)\right\|^2$ . Setting t = 1/2c (i.e. optimizing over t) finally proves the corollary. Note that effective bounds as above are usually called Chernoff bounds in the classical probabilistic setting where the role of the linear statistic is played by a random variable Y of the form $Y = \frac{1}{N}(Y_1 + ... + Y_N)$ , where $Y_i$ are independent random variables with identical symmetric distribution. The bound in the previous corollary should be compared with the general non-effective bound (4.7) $$\operatorname{Prob}_{N_k} \left\{ \frac{1}{N} (u(x_1) + \dots + u(x_N)) > \lambda \right\} \le C e^{-N^2/C},$$ where C is a non-explicit constant, implied by the large deviation principle proved in [8] for determinantal point process in the general line bundle setting (compare the beginning of this section). Note also that the bound 4.7 is essentially contained in the analysis in [52], since $X = \mathbb{P}^1$ in this case. In the large N-limit the inequality in the previous theorem is also closely related to a $Central\ Limit\ Theorem\ (CLT)$ for the linear statistic 4.2. Indeed, when N tends to infinity it can be shown that the inequality 4.5 becomes an asymptotic equality, i.e. (4.8) $$\lim_{N \to \infty} \log \mathbb{E}_N(e^{-t(u(\tilde{x}_i) + \dots + u(\tilde{x}_N))}) = \frac{t^2}{2} \int_{S^2} du \wedge du^c$$ for any $t \in \mathbb{R}$ . In turn, by basic probability theory, this latter fact can be shown to be equivalent to the following CLT: $$\widetilde{u(x_1)} + \widetilde{u(x_2)} + \dots + \widetilde{u(x_N)} \to \mathcal{N}(0, \frac{1}{2} \|du\|^2),$$ in distribution, when $N \to \infty$ , where $\mathcal{N}(0, \frac{1}{2} ||du||^2)$ is the centered normal variable with variance $\frac{1}{2} ||du||^2$ . See [47] for combinatorial proofs of this CLT on the sphere and [7] for general results in the line bundle setting, using Bergman kernel asymptotics. It is also interesting to compare with the case of unitary random matrices, where the the role of the asymptotics 4.8 is played by $Szeg\ddot{o}$ 's strong limit theorem [24]. See also [36] for the case of Hermitian random matrices and [2] for normal random matrices. Loosely, speaking the CLT theorem above may also be formulated as the statement that the the potential of the fluctuations of the empirical measure 4.1 on $S^2$ converges in distribution to the Gaussian free field on $S^2$ (see the introduction in [47] and references therein). Remark 22. Consider the probability measure on $gl(N, \mathbb{C})$ obtained by declaring the complex entries of an $N \times N$ matrix to be i.i.d complex Gaussians. Let $\Phi_N$ be the map defined by $$\Phi_N: (G_1, G_2) \mapsto (z_1, ..., z_N)/S_N,$$ where the $z_i$ is are the N zeroes in $\mathbb{C}$ (taking multiplicities into account) of $\det(G_1 - zG_2)$ , i.e. the eigen values of the matrix $G_2(G_1)^{-1}$ , when $G_1$ is invertible. A remarkable result in [38] says that the push-forward under $\Phi_N$ of the product probability measure on $gl(N, \mathbb{C}) \times gl(N, \mathbb{C})$ is precisely the random point process on $S^2$ with N particles defined by the density 4.4 (under stereographic projection).

Theorem 23. Assume that the Kähler metric $\omega_u$ has constant scalar curvature. Then u minimizes Mabuchi's K-energy $\mathcal{M}_{\omega_0}$ on $\mathcal{H}_{\omega_0}$ .

Lemma 24. Let $u_t$ be a family of continuous functions on X such that the right derivative $v_t := \frac{du_t}{dt}_+$ exists and is uniformly bounded on $[0,1] \times X$ . Then (6.6) $$R[v](x) := \frac{d\beta_{u_t}(x)}{dt}_{t=0+} = \int_{X_y} |K_u(x,y)|^2 e^{-(\psi_0(x) + \psi_0(y))} v_0(y) - \beta_{u_t}(x) v_0(x)$$

Definition 6. A continuous path in $\overline{\mathcal{H}}_{\omega_0} \cap C^0(X)$ $u_t$ will be called a $C^0$ -geodesic connecting $u_0$ and $u_1$ if $U(w,x) := u_t(x)$ , where $t = \log |w|$ , is continuous on $$M := \{1 \le |w| \le e\} \times X := A \times X$$ with $dd^c U + \pi_X^* \omega_0 \ge 0$ and $$(2.1) (dd^c U + \pi_X^* \omega_0)^{n+1} = 0$$ in the interiour of M in the sense of pluripotential theory [33, 23], where $\pi_X$ denotes the projection from M to X. As shown in [11, 10] U(w, x) exists and is uniquely defined as the extension from $\partial M$ obtained as the upper envelope $$(2.2) U(w,x) = \sup \left\{ V(w,x) : V \in \mathcal{H}_{\pi_X^*\omega_0}(M), \ V \le U \text{ on } \partial M \right\},$$ where $\mathcal{H}_{\pi_X^\omega_0}(M)$ denotes the set of all smooth functions V on M such that $dd^cU + \pi_X^\omega_0 > 0$ . If $u_t$ is such that $dd^cU + \pi_X^*\omega_0 \geq 0$ then $u_t$ will be called a $psh\ path$ (or a subgeodesic). In local computations we will often make the identification $u_t(x) = U(w,x)$ extending t to a complex variable. Then $u_t(x)$ is independent of the imaginary part of t and is hence convex wrt real t. In the proof of the uniqueness part of Theorem 1 we will have great use for the following regularity result for geodesics in $\overline{\mathcal{H}}_{\omega_0}$ , shown by Chen [19]. See also [16] for a detailed analysis of the proof and some refinements. The proof uses the method of continuity combined with very precise a priori estimates on the perturbed Monge-Ampère equations.