January 01, 1970
The construction of Kähler metrics satisfying distinguished curvature conditions is one of the central problems in complex differential geometry. Classical examples include Kähler–Einstein metrics, constant scalar curvature Kähler metrics, extremal Kähler metrics, and positive holomorphic (bi)sectional curvature. In general, the relevant equations are highly nonlinear and depend delicately on the global geometry of the underlying manifold. Obtaining explicit expressions for such metrics is difficult due to the complexity of the curvature conditions. Many constructions proceed by imposing sufficient symmetry so that the curvature equation, originally a nonlinear partial differential equation on the manifold, reduces to an ordinary differential equation for a single profile function satisfying suitable conditions.
One classical approach to this problem was introduced by Klembeck [1], who considered a radially symmetric metric on \(\mathbb{C}^{n}\), and gave conditions under which such metrics are complete and possess positive holomorphic sectional curvature. Building on this radial symmetry, Wu and Zheng [2] later simplified the curvature criteria by reducing the verification to a representative point and introducing an auxiliary function. They applied these criteria to the examples constructed by Klembeck and Cao [3], and also produced a new example that naturally generalizes Klembeck’s metric. Cho and Choi [4] later gave examples of Bergman metrics on \(\mathbb{C}^{n}\) induced by weighted Fock spaces that possess positive holomorphic sectional curvature.
Another important method comes from the Calabi ansatz [5] and its momentum-coordinate formulation. Koiso and Sakane [6] used the momentum map as a coordinate to construct Kähler–Einstein metrics on certain projective bundles. Hwang and Singer [7] formulated the momentum construction for circle-invariant Kähler metrics, in which the Calabi ansatz is expressed in terms of a momentum profile. The theory of Hamiltonian \(2\)-forms developed by Apostolov, Calderbank, Gauduchon, and Tønnesen-Friedman [8]–[11] provides a natural extension of the Calabi ansatz. As observed in this framework, the Calabi ansatz is a special case of a local Kähler metric admitting a Hamiltonian \(2\)-form of order one. The above methods have been widely applied to the study of special Kähler metrics, including Kähler–Einstein metrics, Kähler–Ricci solitons, constant scalar curvature Kähler metrics, and extremal Kähler metrics; see, for example, [12]–[19].
The methods employed in this paper are primarily based on the Delzant–Guillemin–Abreu framework. Delzant [20] classified compact symplectic toric manifolds by their moment polytopes, showing that the moment image determines the equivariant symplectic type. Guillemin [21] gave the canonical formula for toric Kähler metrics, and Calderbank, David, and Gauduchon [22] then provided a simple derivation of Guillemin’s formula. Abreu [23] described compatible toric complex structures on a fixed symplectic toric manifold in terms of symplectic potentials. By Moser’s theorem, varying the compatible Kähler form in a fixed cohomology class while keeping the complex structure fixed is equivalent, up to diffeomorphism, to fixing the symplectic form and varying the compatible complex structure. Thus Abreu’s description shows that the study of Kähler forms may be reformulated as the study of compatible toric complex structures. The above framework also extends to non-compact toric manifolds [24]–[26] and orbifolds [27]–[29].
This framework implies that the problem of constructing Kähler metrics with prescribed curvature can be reduced to the construction of a suitable symplectic potential. This framework has been widely used in the study of Kähler–Einstein metrics and extremal Kähler metrics.
Abreu applied this framework to Calabi’s four-parameter family of \(U(n)\)-invariant extremal Kähler metrics on Hirzebruch manifolds and showed that this family recovers several classical cohomogeneity-one examples in [5], [16], [17], [30]. In a similar spirit, Gauduchon used the momentum profile on \(\mathcal{O}(-\ell)\) to give a unified treatment of invariant scalar-flat Kähler metrics, organizing several previously known complete scalar-flat examples within a single construction [31]. These examples suggest that the Guillemin–Abreu formalism can serve, at least in part, as a unified framework for metric constructions. This unifying aspect is one of the points emphasized in the present paper. For further applications, see [26], [27], [29], [32]–[34].
This paper makes two contributions. First, it develops the use of the Guillemin–Abreu formalism to holomorphic (bi)sectional curvature. Second, it applies the formalism beyond line-bundle examples, to the total spaces of a class of higher-rank vector bundles.
The study of holomorphic (bi)sectional curvature appears to be less developed since most previous works have primarily been concerned with Ricci and scalar curvature [25], [27], [31], [32], [34]. We carry this out for \(\mathbb{C}^{n}\), \(\mathcal{O}(-\ell)\), and \(M_{n,\ell}=\mathbb{P}(\mathcal{O}(-\ell) \oplus \mathrm{1}_{\mathbb{CP}^{n-1}})\). Since positive holomorphic sectional curvature metrics on these spaces have already been studied in [1], [2], [18], [19], our main purpose here is not to give new constructions, but to show that the Guillemin–Abreu formalism provides a unified formulation and at the same time simplifies the corresponding curvature expressions relevant to positive holomorphic (bi)sectional curvature, in much the same way as it does for Kähler–Einstein and extremal Kähler metrics [25], [31]. This unifying and simplifying feature is reflected throughout this paper. In particular, it clarifies the mechanism behind the construction of Duan and Guan [19], and leads to a new result on \(M_{n,\ell}\): when the slope is close to \(1\), the extremal metrics have positive holomorphic sectional curvature.
The second contribution is to apply the Guillemin–Abreu theory to a class of vector bundles and to construct complete scalar-flat Kähler metrics and Ricci-flat Kähler metrics on their total spaces. More precisely, we construct a suitable function \(\Theta_{p,q}(x)\) on \(M=\operatorname{Tot}(\mathcal{O}(-k)\oplus \mathcal{O}(-k)\to \mathbf{\mathbb{CP}}^{n})\), and prove that
Proposition 1. The function \(\Theta_{0,0}\) determines a complete scalar-flat Kähler metric on \(M\). Moreover, when \(2k=n+1\), the resulting metric is Ricci-flat.
Our construction is intended to give an explicit Guillemin–Abreu coordinate realization on \(M=\operatorname{Tot}(\mathcal{O}(-k)\oplus \mathcal{O}(-k)\to \mathbf{\mathbb{CP}}^{n})\), together with a direct curvature computation.
In fact, the argument extends with no essential changes to \(M=\textrm{{Tot}}([\mathcal{O}(-k) ]^{\oplus r} \to \mathbb{CP}^{n}).\) However, to keep computations manageable, in this paper we treat only the case \(r=2\), which already contains the essential features of the general case.
Note that \(c_1(M)=(n+1-2k)\pi^*c_1(\mathcal{O}(1)).\) Thus the equation admits a Ricci-flat solution precisely in the case \(c_1(M)=0\). In the case \(c_1(M)>0\), the obstruction comes from the Bonnet–Myers theorem. For the case of \(c_{1}(M)<0\), see [35], where the existence of a complete Kähler–Einstein metric is established on a neighbourhood of the zero section. Our method does not seem to yield a result of this type. The reason is that the coordinates used in our construction are not directly related to intrinsic coordinates on the underlying manifold. This is also the main obstruction to studying metrics of positive sectional curvature on \(M\). The relevant symmetries are not visible in these coordinates, and therefore the curvature cannot be reduced, as in the case of \(U(n)\)-invariant metrics on \(\mathbb{C}^{n}\), to a computation at a single distinguished point. For constructions of metrics with positive holomorphic sectional curvature on \(M\), see [36].
Our scalar-flat metrics are special cases of the construction in [37] by setting \(\mathscr{F}=\mathscr{F}_{1,n}=\mathbb{CP}^{n}\) and \(q=2\). Results of Ricci-flat metrics are special cases of [7] by setting \(\Lambda=\mathcal{O}(-\frac{n+1}{2})\), \(D=\mathbb{CP}^n\) and \(n=2\). When \(k=n=1\), it is known in the physics literature as the resolved conifold metric [38].
In Section 2, we recall the aspects of Guillemin–Abreu theory needed in this paper and explain how toric Kähler metrics with prescribed curvature can be constructed via suitable symplectic potentials. We also present the construction in a step-by-step form, together with the supporting theorems and sample computations for intermediate verifications, such as completeness of the induced complex structure.
In Section 3, we derive the admissibility conditions for the symplectic potentials on \(\mathbb{C}^n\), \(\mathcal{O}(-\ell)\), and \(M_{n,\ell}\), and compute the corresponding curvature expressions.
In Section 4, we revisit metrics on \(\mathbb{C}^n\), \(\mathcal{O}(-\ell)\), and \(M_{n,\ell}\). For \(\mathbb{C}^n\) and \(\mathcal{O}(-\ell)\), we show how the Guillemin–Abreu formalism extends to the study of holomorphic sectional curvature and recovers the known positivity criteria. For the Hirzebruch manifolds \(M_{n,\ell}\), the curvature expressions lead to the observation that, when the slope is sufficiently close to \(1\), the extremal metrics have positive holomorphic sectional curvature.
In Section 5, we consider \(\textrm{{Tot}}(\mathcal{O}(-k) \oplus \mathcal{O}(-k) \to \mathbb{CP}^{n})\), and explicitly construct scalar-flat metrics on them, identifying in particular the Ricci-flat case.
Our notation is close to that used by Gauduchon; see [31]. The letter \(x\) may denote a point \(x=(x_1,\ldots,x_N)\in\mathbb{R}^N.\) When no confusion can arise, \(x\) will also denote a sum of some components, with the summation range specified in context. For functions \(F(x)\), \(G(x)\) and \(H(y)\), we let \(F''(x)=\frac{1}{\Theta(x)}\) and use the notation \(G_{ij}=\frac{\partial^2 G}{\partial x_i\partial x_j}\) and \(H_{ij}=\frac{\partial^2 H}{\partial y_i\partial y_j}.\)
To keep our other notations as close as possible to that used in the references, we allow a mild abuse of notation and occasionally use the same symbol for related quantities. The intended meaning will always be clear from the context.
In this section, we explain how the Guillemin-Abreu theory is applied to the construction of Kähler metrics with prescribed curvature. We first present the overall strategy, and then discuss each step in detail, together with the relevant results ensuring that the construction is well defined.
Given a Kähler toric manifold \((M,J_M,\omega,\tau,\mu)\), we compute its corresponding Delzant set \(P=\mu(M)\).
By Delzant’s construction and Guillemin’s result [20], \(P\) determines a symplectic toric manifold \((M_P,\omega_P,\tau_P)\), uniquely up to equivariant symplectomorphism. The Guillemin–Abreu construction [21], [25] then provides a compatible toric complex structure \(J_P\) on this model, giving a Kähler toric manifold \((M_P,J_P,\omega_P,\tau_P).\)
By Abreu’s description, the compatible toric complex structures \(J\) on \((M_P,\omega_P,\tau_P)\) are parametrized by suitable symplectic potentials \(G = G^{\mathrm{can}} + h\) on \(P\).
The prescribed curvature condition can be translated into an equation for that potential. The construction therefore reduces to solving the resulting equation for the unknown function \(h\).
One can prove that the underlying toric complex manifold is unchanged. More precisely, there are equivariant biholomorphisms \(\varphi_J:(M_P,J_P,\tau_P) \to (M_P,J,\tau_P)\).
The above results mean that \((M_P, \omega_P,J)\) is equivariantly Kähler isomorphic to \((M_P,\omega_J,J_P)\), where \(\omega_J=(\varphi_J)^* \omega_P\) and \([\omega_J]=[\omega_P] \in H^2(M_P)\). Again, we can pull back \(\omega_J\) on \(M_P\) to obtain a new Kähler form on the original manifold \(M\). It is again an equivariantly Kähler isomorphism.
The validity of this procedure is based on the following standard results.
Definition 1. ([25]) A toric symplectic manifold is a connected symplectic manifold \((M^{2n},\omega)\), equipped with an effective Hamiltonian action of the \(n\)-torus \(\tau:\mathbb{T}^n \cong \mathbb{R}^n/2\pi\mathbb{Z}^n \hookrightarrow \operatorname{Ham}(M,\omega),\) such that the corresponding moment map \(\mu:M\to \mathbb{R}^n,\) which is well-defined up to a constant, is proper onto its convex image \(P=\mu(M)\subset \mathbb{R}^n.\) Note that the requirement that the moment map be “proper onto its convex image” is automatic for compact manifolds.
Definition 2. ([25]) A convex polyhedral set \(P\) in \(\mathbb{R}^n\) is called simple and integral if
there are \(n\) edges meeting at each vertex \(p\).
the edges meeting at the vertex \(p\) are rational, i.e., each edge is of the form \(p+t v_i, 0\leq t\leq \infty,\) where \(v_i\in \mathbb{Z}^n\).
the \(v_1,\ldots,v_n\) in \(\textrm{(ii)}\) can be chosen to be a \(\mathbb{Z}\)-basis of the lattice \(\mathbb{Z}^n\).
A facet is a face of \(P\) of codimension one.
A Delzant set is a simple and integral convex polyhedral set \(P\subset \mathbb{R}^n\). A Delzant polytope is a compact Delzant set.
Two Delzant sets are said to be isomorphic if one can be mapped to the other by a translation.
We show that the manifolds considered in this paper are toric symplectic manifolds. It suffices to treat the properness of the moment map in a representative example; the remaining conditions are immediate from the construction. Consider \(\operatorname{Tot}(\mathcal{O}(-1)\oplus \mathcal{O}(-1)\to \mathbb{CP}^1)\) in [39], the moment map is given by \(\mu_a: \mu^{-1}_N(a)/U(1)\to \mathbb{R}^3,\;[z_1:z_2:z_3:z_4]\mapsto (y_1,y_2,y_3)=\frac{1}{2}(|z_1|^2,|z_3|^2,|z_4|^2)\).
If \(K\) is a compact subset of the Delzant set \(P\), then \(\widetilde{K}=\{ z \in \mu^{-1}_N(a)\mid \mu_a([z])\in K\}\) is a closed and bounded subset of \(\mathbb{C}^4\), and thus is compact. Then \(\mu_a^{-1}(K)=\widetilde{K}/U(1)\) is compact. Hence \(\mu_a\) is proper.
We now explain how to associate a Delzant set to a symplectic toric manifold. We describe the following procedure and then illustrate it with examples.
Consider the standard action of the complex torus \(\mathbb{T}_{\mathbb{C}}^d=(\mathbb{C}^*)^d\) on \(\mathbb{C}^d\) given by \((\tilde{t},z) \longmapsto \tilde{t}\cdot z =(\tilde{t}_1z_1,\ldots,\tilde{t}_dz_d)\) as in [39].
Let \(N_{\mathbb{C}}=(\mathbb{C}^*)^{d-n}\) and fix any injective group homomorphism \(\rho_{\mathbb{C}}:N_{\mathbb{C}} \longrightarrow \mathbb{T}_{\mathbb{C}}^d\) as in [39]. The map \(\rho_{\mathbb{C}}\) is of the form \(\rho_{\mathbb{C}}(\lambda_1,\ldots,\lambda_{d-n}) = \left( \prod_{\ell=1}^{d-n}\lambda_\ell^{Q^\ell_1}, \ldots, \prod_{\ell=1}^{d-n}\lambda_\ell^{Q^\ell_d} \right),\) where \(Q^\ell_k\) are integers [39].
Following the method used in [39], we construct a matrix \(B\). Then [39] gives the Delzant set \(\Delta = \left\{ \xi\in \mathbb{R}^n \;\middle|\; L(\xi)_j=\langle v^j,\xi\rangle-\kappa_j\geq 0 \right\},\) where \(v^j\), \(j=1,\ldots,d\), are the column vectors of \(B\). Here \(\kappa=(\kappa_1,\ldots,\kappa_d)\in\mathbb{R}^d\) denotes the constant term in the standard \(\mathbb{T}^d\)-moment map and is chosen so that \(Q^{\mathsf T}\kappa=-a\), where \(a\) is a regular value of the moment map. By choosing a suitable \(a\), we can take \(\kappa=(0,\ldots,0,1)\).
Example 1. The \(\mathbb{C}^*\)-action \(\lambda\cdot(z_1,z_{2},z_{3}) = (\lambda z_1,\lambda z_{2},\lambda^{-\ell}z_{3})\) on the open set \(U=(\mathbb{C}^{2}\setminus\{0\})\times \mathbb{C} \subset \mathbb{C}^{3}\) gives \(U/\mathbb{C}^* \cong \operatorname{Tot}(\mathcal{O}(-\ell)\to \mathbb{C}P^1).\)
In the notation introduced above, this action is induced by \(\rho_{\mathbb{C}}:\mathbb{C}^*\rightarrow(\mathbb{C}^*)^{3}, \rho_{\mathbb{C}}(\lambda) = (\lambda,\lambda,\lambda^{-\ell}),\) and thus \(Q= \begin{pmatrix} 1 & 1 & -\ell \end{pmatrix}^{\mathsf T}.\) Following the method used in [39], we choose \(B= \begin{pmatrix} -1 & 1 & 0\\ \ell & 0 & 1 \end{pmatrix}.\)
As explained in [25], the Delzant set used for the Kähler metric constructions can be obtained from the standard Delzant set by applying a linear transformation in \(\mathrm{GL}(n,\mathbb{R})\). Following Abreu’s convention, we still call the transformed moment image the Delzant set. Here we set the transformation \(T=\begin{pmatrix} -\frac{1}{\ell} & 1 \\ \frac{1}{\ell} & 0 \end{pmatrix}\), then we get \(B'=(T^{-1})^{\mathsf T} B \begin{pmatrix} \ell & 0 & 1\\ 0 & \ell & 1 \end{pmatrix}.\) The Delzant set is \(\{(x_1,x_2) \in \mathbb{R}^2| \ell x_1 \geq 0,\ell x_2\geq 0,x_1+x_2\geq 1\}\).
Example 2. The \(\mathbb{C}^*\)-action \(\lambda\cdot(z_1,z_{2},z_{3},z_{4}) = (\lambda z_1,\lambda z_{2},\lambda^{-\ell}z_{3},\lambda^{-\ell}z_{4})\) on the open set \(U=(\mathbb{C}^{2}\setminus\{0\})\times \mathbb{C}^2 \subset \mathbb{C}^{4}\) gives \(U/\mathbb{C}^* \cong \operatorname{Tot}(\mathcal{O}(-\ell)\oplus \mathcal{O}(-\ell) \to \mathbb{CP}^1).\)
Similarly, this action is induced by \(\rho_{\mathbb{C}}:\mathbb{C}^*\rightarrow(\mathbb{C}^*)^{4}, \rho_{\mathbb{C}}(\lambda) = (\lambda,\lambda,\lambda^{-\ell},\lambda^{-\ell}),\) and thus \(Q= \begin{pmatrix} 1 & 1 & -\ell & -\ell \end{pmatrix}^{\mathsf T}.\) Again, we choose \(B= \begin{pmatrix} -1 & 1 & 0 & 0 \\ \ell & 0 & 1 & 0\\ \ell & 0 & 0 & 1\\ \end{pmatrix}.\) Similarly, we set \(T=\begin{pmatrix} -\frac{1}{\ell} & 1 & 1 \\ \frac{1}{\ell} & 0 & 0\\ 0 & 1 & 0 \\ \end{pmatrix}\) and have \(B'=(T^{-1})^{\mathsf T}B= \begin{pmatrix} \ell & 0 & 0 & 1\\ 0 & \ell & 0 & 1\\ 0 & 0 & 1 & -1 \end{pmatrix}.\) The Delzant set is \(\{(x_1,x_2,x_3) \in \mathbb{R}^3| \ell x_1\geq 0,\ell x_2\geq 0,x_3\geq 0,x_1+x_2-x_3\geq 1\}\).
The following results give the compact and noncompact versions of Delzant’s classification, together with the corresponding admissibility criteria for symplectic potentials.
Theorem 2 ([23]). Let \((M,\omega,\tau)\) be a compact, connected, \(2n\)-dimensional Hamiltonian \(\mathbb{T}^{n}\)-space, on which the action of \(\mathbb{T}^{n}\) is effective with moment map \(\phi : M \to \mathbb{R}^{n}\). Then the image \(P\) of \(\phi\) is a Delzant polytope, and \((M,\omega,\tau)\) is isomorphic as a Hamiltonian \(\mathbb{T}^{n}\)-space to \((M_P,\omega_P,\tau_P)\).
Theorem 3 ([25]). Let \((M,\omega,\tau)\) be a toric symplectic manifold, with a moment map \(\mu:M\to \mathbb{R}^n .\) Then \(P=\mu(M)\) is a Delzant set.
Two toric symplectic manifolds are equivariant symplectomorphic, with respect to a fixed torus acting on both, if and only if their associated Delzant sets are isomorphic. Moreover, every Delzant set arises from some toric symplectic manifold.
Theorem 4 ([23]). Let \((M_P, \omega_P, \tau_P)\) be the toric symplectic manifold associated to a Delzant polytope \(P \subset \mathbb{R}^n\), and \(J\) any compatible toric complex structure. Then \(J\) is determined by a “potential” \(G \in C^\infty(P^\circ)\) of the form \[\label{eq:2469} G = G^{\mathrm{can}} + h\qquad{(1)}\] where \(G^{\mathrm{can}}\) is given by \(G^{\mathrm{can}}(x)=\frac{1}{2}\sum_{r=1}^{d}l_{r}(x)\log l_{r}(x)\), \(h\) is smooth on the whole \(P\), and the matrix \((G_{ij}) = \mathrm{Hess}_x(G)\) is positive definite on \(P^\circ\) and has determinant of the form \[\begin{align} \det(G_{ij}) = \left[ \delta(x) \cdot \prod_{r=1}^d l_r(x) \right]^{-1} \end{align}\] with \(\delta\) being a smooth and strictly positive function on the whole \(P\).
Conversely, any such \(G\) determines a compatible toric complex structure \(J\) on \((M_P, \omega_P)\), which in the \((x,y)\) symplectic coordinates of \(M_P^\circ \cong P^\circ \times \mathbb{T}^n\) has the form \[J = \begin{bmatrix} 0 & -(G_{ij})^{-1} \\ (G_{ij}) & 0 \end{bmatrix}.\]
For the non-compact case, an analogous statement holds [25]. Complete toric compatible complex structures are described by symplectic potentials of the form \(G = G^{\mathrm{can}} + h ,\) satisfying the same conditions as in the compact case. Conversely, any such potential defines a toric compatible complex structure, which is not necessarily complete. Here completeness of a toric compatible complex structure is understood in the following sense.
Definition 3 ([25]). A toric compatible complex structure \(J\) on a symplectic toric manifold \(M\) is said to be complete if the \(J\)-holomorphic vector fields \(JY_1,\ldots,JY_n\) are complete, where \(Y_1,\ldots,Y_n\) are the Hamiltonian vector fields generating the torus action.
Let \(G=G^{\mathrm{can}}+h\) be a candidate symplectic potential. To show that it defines an admissible toric Kähler metric, we shall use the following admissibility criterion, denoted by \((\star)\):
The function \(h\) is smooth on \(P\).
The Hessian matrix \((G_{ij})\) is positive definite on \(P^\circ\).
The function \(\delta=\bigl( \det (G_{ij})\prod_{r=1}^d \ell_r(x)\bigr)^{-1}\) is smooth and strictly positive on \(P\).
In the non-compact case, we must also check the induced toric complex structure is complete. In the compact case, this condition is automatic.
We say that \(h\) satisfies \((\star)\) if the corresponding potential \(G=G^{\mathrm{can}}+h\) satisfies \((\star)\).
We next explain why the complex structure \(J\) obtained from the preceding theorem is biholomorphic to the original complex structure \(J_P\).
In the compact case, this follows from the following result.
Proposition 5 ([23]). Let \((M,\omega,J_M,\tau)\) be a compact Kähler toric manifold, with moment polytope \(P\subset\mathbb{R}^{n}\), and \(J\) is any toric compatible complex structure, we have that:
\((M,\omega,\tau)\) is equivariantly symplectomorphic to \((M_P,\omega_P,\tau_P)\).
\((M,J_M,\tau)\) is equivariantly biholomorphic to \((M_P,J_P,\tau_P)\).
\((M_P,J,\tau_P)\) is equivariantly biholomorphic to \((M_P,J_P,\tau_P)\).
In the non-compact case, the analogue of (ii) follows from [40]. For (iii), if the toric compatible complex structure \(J\) is complete, then \((M_P,J,\tau_P)\cong (M_P,J_P,\tau_P)\) equivariantly biholomorphically. This conclusion is used in [26], and the proof is the same as in Appendix A of [23]. Note that there is no immediate relation between completeness of a toric compatible complex structure and completeness of the associated toric Kähler metric [26].
The completeness of a toric compatible complex structure can be studied as follows. If \(J_u\) is determined by \(u\), set \(\xi=\nabla u:P^\circ\to \mathbb{R}^n\), then the \(J\)-holomorphic coordinates are given by \((\xi,\theta)=\left(\frac{\partial u}{\partial x},\theta\right),\) \(J\) is complete precisely when the map \(\xi\) is onto. The above argument is the same as in the proof of [26].
Example 3. Consider the standard toric Kähler manifold \(\mathbb{C}^n\), whose Delzant set is \(P=\mathbb{R}_{\geq 0}^n\). Here \(x\) denotes \(\sum_{i=1}^n x_i\). Let \[G(x) = \frac{1}{2}\sum_{i=1}^n x_i\log x_i -\frac{1}{2} x\log x +F(x) .\] Assuming \(\Theta(x)>0\), the function \(F'\) is strictly increasing. The induced toric complex structure \(J_G\) is complete if and only if \(\nabla G(P^\circ)=\mathbb{R}^n .\)
For \(y=(y_1,\ldots,y_n)\in\mathbb{R}^n\), the inverse of \(\nabla G\), whenever defined, is given by \[x_i = (F')^{-1}\left( \frac{1}{2}\log\sum_{j=1}^n e^{2y_j} \right) \frac{e^{2y_i}}{\sum_{j=1}^n e^{2y_j}} .\] Indeed, the function \(\frac{1}{2}\log \sum_{j=1}^n e^{2y_j}\) ranges over all of \(\mathbb{R}\): it takes the value \(\frac{1}{2}\log(n e^{2c})\) when \(y_1=\cdots=y_n=c\).
Hence \(\nabla G(P^\circ)=\mathbb{R}^n\) is equivalent to \(\lim_{x\to 0^+}F'(x)=-\infty\), and \(\lim_{x\to+\infty}F'(x)=+\infty.\) Since \(F''(x)=1/\Theta(x)\), this is equivalent to \(\int_0^1 \frac{dx}{\Theta(x)}=+\infty\), and \(\int_1^{+\infty}\frac{dx}{\Theta(x)}=+\infty .\)
Example 4. Consider the Delzant set \(P \subset \mathbb{R}^{n+2}\) defined by \(l_i=kx_i,\;i=1,\cdots,n+1,\; l_{n+2}=x_{n+2},\; l_0=x-x_{n+2}-1\), where we set \(x=\sum_{i=1}^{n+1}x_i\). We consider the symplectic potential \[G= \frac{1}{2}\sum_{i=0}^{n+2}l_i\log l_i +\frac{k+1}{2}x -\frac{1}{2}(x-1)\log(x-1) -\frac{1}{2}kx\log x +F(x).\]
Assuming \(\Theta(x)>0\), the function \(F'\) is strictly increasing. The induced toric complex structure \(J_G\) is complete if and only if \(\nabla G(P^\circ)=\mathbb{R}^{n+2}.\) Let \(y=(y_1,\ldots,y_{n+2})\in\mathbb{R}^{n+2}\), and set \(S=\sum_{i=1}^{n+1}e^{2y_i/k}\) and \(T=\frac{k}{2}\log S +\frac{1}{2}\log\bigl(1+e^{2y_{n+2}}\bigr) -\frac{k}{2}\log k-\frac{k+1}{2}\). Since adding a linear function to \(F\) does not affect \(F''\), this constant is irrelevant for the completeness criterion.
Solving \(\nabla G=y\), one obtains \[x_i = (F')^{-1}(T) \frac{e^{2y_i/k}}{S(y)}, i=1,\ldots,n+1, \textrm{ and } x_{n+2} = \bigl((F')^{-1}(T)-1\bigr) \frac{e^{2y_{n+2}}}{1+e^{2y_{n+2}}}.\]
Since the function \(\frac{k}{2}\log S(y) +\frac{1}{2}\log\bigl(1+e^{2y_{n+2}}\bigr)\) ranges over all of \(\mathbb{R}\), we have \(\nabla G(P^\circ)=\mathbb{R}^{n+2}\) is equivalent to \(F'((1,+\infty))=\mathbb{R},\) or equivalently, \(\lim_{x\to1^+}F'(x)=-\infty\), and \(\lim_{x\to+\infty}F'(x)=+\infty.\) This is equivalent to \(\int_1^2\frac{dx}{\Theta(x)}=+\infty\), and \(\int_2^{+\infty}\frac{dx}{\Theta(x)}=+\infty.\)
The remaining problem is to choose an admissible symplectic potential whose associated Kähler metric has the desired curvature property. We therefore now study the curvature formulas. Note that the following formulas hold for an arbitrary admissible symplectic potential \(G\), and are not restricted to the canonical potential.
Suppose \((M,\omega,J,\tau,\mu)\) is a Kähler toric manifold and its Delzant set \(\Delta \subset \mathbb{R}^{n}\) is defined by \(l_{i}(x),\;i=1,\dots, d\), then the standard symplectic potential function \(G\) on \(\Delta^\circ\) and the corresponding Kähler potential \(H\) on \(\mu^{-1}(\Delta^\circ)\) are given by \[\begin{align} G =\frac{1}{2} \sum\limits_{i=1}^{d} l_{i}(x)\log l_{i}(x), H(y)=\sum^{n}_{i=1}x_{i}\frac{\partial G}{\partial x_{i}}-G(x) \textrm{ where } y_{i}=\frac{\partial G}{\partial x_{i}} \end{align}\]
Moreover, the following propositions hold on \(M^\circ\), the open dense subset of \(M\) where the \(T^{n}\) action is free. We use the complex coordinates \(u_i = y_i + \sqrt{-1}\theta_i\) via the identification of \(\mu^{-1}(\overset{\circ}{\Delta}) \cong \mathbb{T}_{\mathbb{C}}^n\), and the action-angle coordinates \((x_i, \theta_i)\) via \(\mu^{-1}(\Delta^\circ) \cong \Delta^\circ \times \mathbb{T}^n\).
Proposition 6 ([[39]][23]). The Kähler form \(\omega\), the compatible toric complex structure \(J\) and the Riemannian metric compatible with \(\omega\) and the complex structure \(J\) are \[\begin{align} J = \begin{bmatrix} 0 & -(G_{ij})^{-1} \\ (G_{ij}) & 0 \end{bmatrix}, \end{align}\] \[\omega = \sum_{i=1}^n dx_i \wedge d\theta_i = \sum_{i,j=1}^n \frac{\partial^2 H}{\partial y_i \partial y_j} dy_i \wedge d\theta_j = \frac{\sqrt{-1}}{2} \sum_{i,j=1}^n \frac{\partial^2 H}{\partial y_i \partial y_j} du_i \wedge d\bar{u}_j,\]
\[g = \sum_{i,j=1}^n \frac{\partial^2 H}{\partial y_i \partial y_j} (dy_i dy_j + d\theta_i d\theta_j) = \sum_{i,j=1}^n \frac{\partial^2 G}{\partial x_i \partial x_j} dx_i dx_j + \sum_{i,j=1}^n \frac{\partial^2 H}{\partial y_i \partial y_j} d\theta_i d\theta_j.\]
Proposition 7. The curvature tensor components, the Ricci form, and the scalar curvature are given by the following formulas. \[\label{eq:jiemian} R_{i \bar{j} k \bar{l}}= -H_{al}( \frac{\partial H_{bk}}{\partial x_{a}}\frac{\partial H_{ij}}{\partial x_{b}}+H_{bk}\frac{\partial^{2} H_{ij}}{\partial x_{a}\partial x_{b}})+G_{pq}H_{ak}\frac{\partial H_{iq}}{\partial x_{a}}H_{bl}\frac{\partial H_{pj}}{\partial x_{b}},\qquad{(2)}\] \[\label{eq:ric}\rho=-\frac{\partial^2 H_{mj}}{\partial x_m \partial x_k} H_{ki}du_{i} \wedge d\bar{u}_{j} \textrm{ and }s=- \sum^{n}_{i,j=1}\frac{\partial H_{ij}}{\partial x_{i}\partial x_{j}}.\qquad{(3)}\]
Proof. The formulas for the scalar curvature and the Ricci form follow from [23] and [31]. We compute \[\begin{align} R_{i \bar{j} k \bar{l}} &= - \frac{\partial^2 H_{ij}}{\partial y_k \partial y_l} + G_{pq} \frac{\partial H_{iq}}{\partial y_k} \frac{\partial H_{pj}}{\partial y_l} = - \frac{\partial}{\partial y_l} \left( \frac{\partial}{\partial y_k} H_{ij} \right) + G_{pq} \frac{\partial x_m}{\partial y_k} \frac{\partial}{\partial x_m} H_{iq} \frac{\partial x_n}{\partial y_l} \frac{\partial}{\partial x_n} H_{pj} \\ &= - \frac{\partial x_a}{\partial y_l} \frac{\partial}{\partial x_a} \left( \frac{\partial x_b}{\partial y_k} \frac{\partial}{\partial x_b} H_{ij} \right) + G_{pq} H_{mk} \frac{\partial H_{iq}}{\partial x_m} H_{nl} \frac{\partial H_{pj}}{\partial x_n} \\ &= - H_{al} \frac{\partial}{\partial x_a} \left( H_{bk} \frac{\partial H_{ij}}{\partial x_b} \right) + G_{pq} H_{mk} \frac{\partial H_{iq}}{\partial x_m} H_{nl} \frac{\partial H_{pj}}{\partial x_n} \\ &= - H_{al} \left( \frac{\partial H_{bk}}{\partial x_a} \frac{\partial H_{ij}}{\partial x_b} + H_{bk}\frac{\partial^2 H_{ij}}{\partial x_a \partial x_b} \right) + G_{pq} H_{ak} \frac{\partial H_{iq}}{\partial x_a} H_{bl} \frac{\partial H_{pj}}{\partial x_b}. \end{align}\] ◻
We briefly summarize the construction. Suppose, for example, that one aims to construct a constant scalar curvature Kähler metric on \(M\). We first pass to the Delzant set \(P\). The data of \(P\) determine the canonical symplectic potential \(G^{\mathrm{can}}\). Set \(G=G^{\mathrm{can}}+h.\) The scalar curvature \(s\) can be computed explicitly from \(G\). Hence the constant scalar curvature problem is reduced to solving \(s=\textrm{constant},\) for the unknown \(h\) satisfying \((\star)\).
The purpose of this section is to derive explicit formulas for the (bi)sectional curvature of \(\mathbb{C}^n\), \(\mathcal{O}(-\ell)\), and \(M_{n,\ell}\), and to give criteria for the positivity of (bi)sectional curvature. The point is that, in each case, after a suitable choice of \(h\), the symplectic potential \(G=G^{\mathrm{can}}+h\) can be written in the same form. For this ansatz, checking \((\star)\) for \(h\) reduces to checking the corresponding conditions on \(\Theta(x)\).
For convenience, we introduce the following temporary notation. Set \(x=\sum_{i=1}^n x_i\). Define \(f_0=x\), \(\tilde{f}_0=\ell x\), \(f_i=\ell x_i\), \(g_1=x-\lambda\), \(g_2=x-a\), \(g_3=b-x\) and \(\eta(\cdot)=\frac{1}{2}(\cdot)\log (\cdot)\).
The corresponding Delzant sets of \(\mathbb{C}^{n}\), \(\mathcal{O}(-\ell)\), and \(M_{n,\ell}\) are \[P_1=\{x_i\geq 0\},\;P_2=\{f_i \geq 0,~g_1\geq 0 \},\;P_3= \{ f_i\geq 0, g_2\geq 0, g_3\geq 0 \},\] respectively, where the indexed inequalities are understood to hold for \(1\leq i \leq n\). The Delzant set of \(M_{n,\ell}\) can also be obtained by a translation by the standard one [25]. We then define \[h_1=-\eta(f_0)+F(x),\;h_2=-\eta(\tilde{f}_0)-\eta(g_1)+F(x),\;h_3=-\eta(\tilde{f}_0)-\eta(g_2)-\eta(g_3)+F(x).\]
For the three Delzant sets under consideration, the admissible symplectic potentials \(G_i=G_i^{\mathrm{can}}+h_i\) have the same form, with the case of \(\mathbb{C}^n\) obtained by setting \(\ell=1\): \[\begin{align} G(x)=\frac{1}{2}\sum^{n}_{i=1}\ell x_{i}\log(\ell x_{i})-\frac{1}{2}\ell x\log(\ell x)+F(x), \end{align}\]
A direct computation gives \(\textrm{det}(G_{ij})=\ell^{n-1} x(2^{n-1}\Theta(x)\prod{x_{i}})^{-1}\). Moreover \((G_{ij})\) is positive definite on \(\mathbb{R}_{> 0}^n\) if and only if \(\Theta(x)>0\) when \(x>0\). By [19], the metrics are complete if \(\int_a^{\infty}\frac{dx}{\sqrt{\Theta(x)}}=+\infty\) for some suitable \(a>0\).
By \((\star)\), for \(\mathbb{C}^n\), we require \(\Theta(0)=0,\; \Theta'(0)=2\), \(\Theta(x)>0\) for \(x>0\), and \(\int_{0}^1\frac{dx}{\Theta(x)}=+\infty\) and \(\int^{+\infty}_1 \frac{dx}{\Theta(x)}=+\infty\).
For \(\mathcal{O}(-\ell)\), we require \(\Theta(\lambda)=0,\; \Theta'(\lambda)=2\), \(\Theta(x)>0\) for \(x>\lambda\), and \(\int_{\lambda}^{\lambda+1}\frac{dx}{\Theta(x)}=+\infty\) and \(\int^{+\infty}_{\lambda+1} \frac{dx}{\Theta(x)}=+\infty\).
For \(M_{n,\ell}\), we require \(\Theta(a)=0,\;\Theta'(a)=2,\;\Theta(b)=0, \;\Theta'(b)=-2\), and \(\Theta(x)>0\) for \(a<x<b\).
These are the usual endpoint conditions for the momentum profile; compare [31].
The conditions \(\Theta(a)=0\) and \(\Theta'(a)=2\) imply that, near \(x=a\) and on the side \(x>a\), \[F''(x)=\frac{1}{2(x-a)}+\varphi(x), \textrm{ i.e., } F(x)=\frac{1}{2}(x-a)\log(x-a)+\psi(x),\] where \(\varphi\) and \(\psi\) are smooth near \(x=a\). The boundary conditions therefore have a natural interpretation in the Guillemin–Abreu theory. They correspond to adding facets to the polytope such that it becomes the one associated with the standard potential.
By symmetry, global positivity of the holomorphic (bi)sectional curvature reduces to checking positivity at a suitably chosen point. For instance, consider a radially symmetric metric on \(\mathbb{C}^n\), the \(U(n)\)-symmetry allows one to reduce the curvature calculation to points of the form \((z_1,0,\ldots,0)\).
The metrics considered on \(\mathbb{C}^n, \mathcal{O}(-\ell)\) and \(M_{n,\ell}\) all have \(U(n)\)-symmetry, the corresponding representative point, according to Gauduchon’s result [31], in our coordinates is \((x_1,0,\ldots,0)\) where \(x_1>0\). Since the curvature formulas are computed on \(P^\circ\), which does not contain boundary points, we use the normalized curvature components. We define \[\begin{align} \widetilde{R}_{i \bar{j} k \bar{l}} =\lim_{(x_2,\ldots,x_n)\to (0,\ldots,0)} \frac{ R_{i \bar{j} k \bar{l}} }{ \sqrt{ g_{i \bar{i}} g_{j \bar{j}} g_{k \bar{k}} g_{l \bar{l}} } }. \end{align}\] The positivity of the holomorphic (bi)sectional curvature can be checked at points of the form \((x_1,0,\ldots,0)\), where \(x_1>0\). For simplicity, we shall still denote these normalized components by \(R_{i \bar{j} k \bar{l}}\).
In our construction, the symplectic potentials for the three classes of manifolds take the same form. It is therefore enough to carry out the computation once. The computation is most efficient using symbolic computation software. A direct symbolic computation with \(n\) treated as a formal parameter is difficult to implement. We therefore carry out the calculation for \(n\le 10\). The ansatz and the resulting expressions have a regular structure, it is enough for the reader to verify the identities below in the cases \(n\le 10\).
The results are as follows. Notice that the case \(\ell=1\) recovers the \(\mathbb{C}^n\) case. The results are consistent with [19] and [18].
Proposition 8. At the representative point \((x_1,0,\ldots,0)\) where \(x_1>0\), the nonzero normalized curvature components associated with \(G(x)=\frac{1}{2}\sum^{n}_{i=1}\ell x_{i}\log(\ell x_{i})-\frac{1}{2}\ell x \log(\ell x)+F(x)\) are: \[\begin{gather} A=R_{1 \bar{1} 1 \bar{1}}=- \Theta''(x),\\[1ex] B=R_{i \bar{i} 1 \bar{1}}=R_{i \bar{1} 1 \bar{i}}=R_{1 \bar{i} i \bar{1}}=R_{1 \bar{1} i \bar{i}}=\frac{\Theta(x)-x\Theta'(x)}{x^2},\;i \geq 2,\\[1ex] C=R_{i \bar{i} i \bar{i}}= \frac{-2\ell \Theta(x)+4x}{\ell x^2} ,\; i \geq 2,\\[1ex] \frac{C}{2}=R_{i \bar{j} j \bar{i}}=R_{i \bar{i} j \bar{j}}= \frac{-\ell \Theta(x)+2x}{\ell x^2} ,\;2\leq i,j\leq n,\;i\neq j.\end{gather}\]
For \(X=x_{i}\frac{\partial}{\partial u_{i}}\) and \(Y=y_{i}\frac{\partial}{\partial u_{i}}\), \[\begin{align} R_{X\bar{X}Y\bar{Y}} &= R_{i\bar{j}k\bar{l}} x_i \bar{x}_j y_k \bar{y}_l = |x_1|^2 y_k \bar{y}_l R_{1\bar{1}k\bar{l}} + x_1 \bar{x}_i y_i \bar{y}_1 R_{1\bar{i}i\bar{1}} + \sum_{i \geq 2} x_i \bar{x}_j y_k \bar{y}_l R_{i\bar{j}k\bar{l}} \\ &= |x_1|^2 |y_1|^2 A + \left( \sum_{i \geq 2} |x_1 y_i + x_i y_1|^2 \right) B + \left( \sum_{i \geq 2} |x_i|^2 |y_i|^2 \right) C + \left( \sum_{i,j \geq 2, i > j} |x_i y_j + x_j y_i|^2 \right) \frac{C}{2}. \end{align}\] \[\begin{align} R_{X\bar{X}X\bar{X}}&= |x_1|^4 A + 4 \left( \sum_{i \geq 2} |x_1 x_i|^2 \right) B + \left( \sum_{i \geq 2} |x_i|^4 \right) C +\left( \sum_{i,j \geq 2, i > j} |x_i x_j |^2 \right) 2C \\ &= |x_1|^4 A + 4 |x_1|^2\left( \sum_{i \geq 2} | x_i|^2 \right) B + \left( \sum_{i \geq 2} |x_i|^2 \right)^2 C. \end{align}\]
For the metrics under consideration, positivity of the holomorphic bisectional curvature is equivalent to \(A>0,B>0\), and \(C>0\). Similarly, positivity of the holomorphic sectional curvature is equivalent to \(A>0,C>0\), and \(D:=2B+\sqrt{AC}>0.\)
In this section, we review several examples of Kähler toric manifolds admitting metrics with positive holomorphic (bi)sectional curvature.
The following criterion is parallel to the positivity criteria obtained by Duan–Guan and by Wu–Zheng for \(U(n)\)-invariant Kähler metrics on \(\mathbb{C}^n\); see [19] and [2].
Theorem 9. On \(\mathbb{C}^n\), the metrics under consideration have positive holomorphic bisectional curvature if and only if the following conditions hold: \[\begin{align} \Theta(0)=0, ~\Theta'(0)=2, \;\Theta(x)>0 ,~ \Theta''(x) < 0 ,~x \in (0,+ \infty). \end{align}\] In particular, the completeness conditions for the induced toric complex structure are automatic under these assumptions by \(\Theta(x)\le 2x,~x \in (0,+ \infty)\).
Wu–Zheng use curvature criteria to examine explicit examples of \(U(n)\)-invariant Kähler metrics on \(\mathbb{C}^n\). We use the same strategy here. The point is not to give a new construction of positively curved metrics on \(\mathbb{C}^n\), but rather to illustrate how the Guillemin–Abreu method leads to curvature criteria that can be applied directly to examples. Here we set \(\rho=r^2=|z|^2\).
Example 5 ([1]). Klembeck introduced a metric on \(\mathbb{C}^{n}\) with positive holomorphic bisectional curvature whose potential function is \(p(\rho)=\int^{\rho}_{0}\frac{\log(1+w)}{w}dw\), and \(\Theta(x)=1-e^{-2x}\).
Example 6 ([2]). A natural perturbation of Klembeck’s metric is the following: \(p(\rho)=\int^{\rho}_{0}\frac{(1+w)^{1-c}-1}{(1-c)w}dw\) where \(0<c<1\), and \(\Theta(x)=u^d-u^{d-1}\) where \(0<d=1-c<1, u=(2dx+1)^{1/d}\).
Example 7 ([3]). The explicit expression of Cao’s cigar was given in [2], defined as \(\rho f=H^{-1}(\rho)\) where \(H(x)=\sqrt[n]{P(x)e^{x}-P(0)}\), \(P(x)\) is the unique polynomial satisfying \(P+P'=nx^{n-1}\). The corresponding \(\Theta(x)\) is given by \(\Theta(x)=\frac{2 H(2x)}{(H(2x))'}=\frac{e^{-2x}}{(2x)^{n-1}}H(2x)^n\) with \(\Theta''(x)=\frac{-e^{-2x}L(x)}{x^{n+1}}\) where \(L(0)=L'(0)=L''(0)=0\), and \(L'''(x)=8n(n-1)e^{2x}x^{n-2}\), \(n \geq 2\).
We first observe that, within the present ansatz, the metrics on \(\mathcal{O}(-\ell)\) do not have positive holomorphic bisectional curvature. Indeed, positive holomorphic bisectional curvature requires \(\Theta(x)-x\Theta'(x)>0.\) Taking the limit \(x\to\lambda^+\) in the two inequalities gives \(0\leq \Theta(\lambda)-\lambda\Theta'(\lambda)=-2\lambda.\) This contradicts \(\lambda>0\).
We now determine when the metric has positive holomorphic sectional curvature. Equivalently, we study when the following expression is positive everywhere. This follows the approach of Duan and Guan [19]. \[\begin{align} \mathcal{F}&=R_{X\bar{X}X\bar{X}} =|x_{1}|^4A+4B|x_1|^2 \sum^{n}_{i \geq 2}|x_{i}|^2 +C(\sum^{n}_{i \geq 2}|x_{i}|^2)^2 =As^2+4Bs(1-s)+C(1-s)^2 \\ &=-s^2\Theta''(x)+\frac{4(\Theta(x)-x\Theta'(x))}{x^2}s(1-s)+\frac{-2 \ell \Theta(x)+4x}{\ell x^2}(1-s)^2. \end{align}\]
Suppose a positive function \(\Theta_0(x)\) defined on \((0,+ \infty)\) satisfies \(\Theta_0''(x)<0\), \(\Theta_0(0)=0\), \(\Theta_0'(0)=2\) and thus for \(x \in (0,+ \infty)\), \[-s^2\Theta_0''(x)+\frac{4(\Theta_0(x)-x\Theta_0'(x))}{x^2}s(1-s)+\frac{-2\Theta_0(x)+4x}{x^2}(1-s)^2>0 .\]
Define \(\Theta(x)=\Theta_0(x-\lambda)\), then we have, for \(x \in (\lambda,+ \infty)\) \[\mathcal{G}=-s^2\Theta''(x)+\frac{4(\Theta(x)-(x-\lambda)\Theta'(x))}{(x-\lambda)^2}s(1-s)+\frac{-2\Theta(x)+4(x-\lambda)}{(x-\lambda)^2}(1-s)^2>0.\]
In particular, \[\begin{gather}\mathcal{G}_1=-s^2\Theta''(x)\geq 0, \\ \mathcal{G}_2=\frac{4(\Theta(x)-(x-\lambda)\Theta'(x))}{(x-\lambda)^2}s(1-s)\geq 0, \\ \mathcal{G}_3=\frac{-2\Theta(x)+4(x-\lambda)}{(x-\lambda)^2}(1-s)^2\geq 0.\end{gather}\]
To ensure that \(\mathcal{F}\) is positive, it suffices that the following expression is positive, \[\begin{align}\mathcal{H}&=\ell x^2\mathcal{F} - \ell(x - \lambda)^2(\mathcal{G}_1 +\mathcal{G}_2 )- (x - \lambda)^2 \mathcal{G}_3\\ &=2(1 - \ell)(1 - s)^2 \Theta(x) + \lambda \left( 4(1 - s)^2 + 4\ell(s-1)s\, \Theta'(x) \right)+ \lambda \ell s^2 (\lambda - 2x) \Theta''(x). \end{align}\]
By computing the discriminant, we have \[\begin{align} -2 \lambda \ell \left(\Theta'(x)\right)^2 + (\lambda - 2x)\left(2\lambda + \Theta(x) - \ell \Theta(x)\right) \Theta''(x) >0. \end{align}\]
Using the relation \(0 \leq \Theta'(x)\leq 2\), we obtain a sufficient condition: \[\begin{align} -4 \lambda \ell \Theta'(x) + (\lambda - 2x)\left(2\lambda + \Theta(x) - \ell \Theta(x)\right) \Theta''(x) >0 . \end{align}\]
Proposition 10. For \(\lambda>0,~ \epsilon \in (0,2)\), suppose \(\Theta(x)\) is smooth on \([\lambda,+\infty)\), positive on \((\lambda,+\infty)\) with \(\Theta(\lambda)=0,~ \Theta'(\lambda)=2,~ \Theta''(x)<0\). If, in addition, the following holds: \[\begin{align} (\lambda - 2x) \Theta''(x) -2 \Theta'(x)>0 \text{ when } \ell=1, \end{align}\]\[\begin{align} \epsilon(\lambda - 2x) \Theta''(x)-4 \ell \Theta'(x) >0 \text{ and } \Theta(x)<\frac{2-\epsilon}{\ell-1}\lambda \text{ when } \ell>1, \end{align}\]
then \(\Theta(x)\) induces a Kähler metric with positive holomorphic sectional curvature on \(\mathcal{O}(- \ell)\). The completeness conditions follow immediately from \(\Theta(x)\leq 2(x-\lambda)\).
The preceding method does not apply directly to the manifold \(M_{n,\ell}\), since the corresponding function \(\Theta(x)\) vanishes at both endpoints of an interval, which cannot be satisfied by a translation of the profiles \(\Theta(x)\) associated with \(\mathbb{C}^n\) or \(\mathcal{O}(-\ell)\).
By analyzing the positivity inequalities \(A>0,\; C>0,\; D>0,\) Yang and Zheng construct a Kähler metric with positive holomorphic sectional curvature on \(M_{n,\ell}\) [18]. Their strategy is also based on reducing the curvature positivity problem to a system of inequalities for a single function. Compared with their curvature expressions [18], the formulas obtained from the present Guillemin–Abreu theory are more tractable. Hence one could in principle carry out a similar construction in our setting and obtain the corresponding positively curved metrics.
However, this would not produce a genuinely new construction. As explained in the Introduction, our purpose here is instead to illustrate the convenience of the Guillemin–Abreu approach: a single curvature computation applies simultaneously to the three classes of manifolds under consideration.
As an application of the curvature formulas, we give a new observation concerning extremal metrics on \(M_{n,\ell}\). It follows from Gauduchon’s formulas that the metric is extremal on \(M_{n,\ell}\) if and only if \(\Theta(x)\) is of the following form; see [31] \[\begin{align} \Theta(x)=\frac{c_{0}x^{n+2}+c_{1}x^{n+1}+\frac{2}{\ell}x^n+c_{2}x+c_{3}}{x^{n-1}},\;x\in [a,b]. \end{align}\] where \(c_0,c_1,c_2,c_3\) are determined by the boundary conditions \(\Theta(a)=0, \;\Theta'(a)=2,\;\Theta(b)=0,\; \Theta'(b)=-2\). We claim that the above extremal metric on \(M_{n,\ell}\) has positive holomorphic sectional curvature if the slope is close to 1.
Proposition 11. For \(\ell \geq 1\), there exists \(\epsilon > 0\) such that if \(0<k-1<\epsilon\) where \(k=\frac{b}{a}\), then \[\begin{align} \Theta(x)=\frac{c_{0}x^{n+2}+c_{1}x^{n+1}+\frac{2}{\ell}x^n+c_{2}x+c_{3}}{x^{n-1}} \end{align}\]
induces a Kähler metric with positive holomorphic sectional curvature on \(M_{n,\ell}\) where \(c_{0},c_{1},c_{2},c_{3}\) are constants determined by the extremal Kähler metric conditions.
Proof. In this proposition only, set \(k=\frac{b}{a}\). The computations in the proof are most conveniently carried out by symbolic computation. The extremal metric on \(M_{n,\ell}\) is given by the following expressions, \[c_{0}=- \frac{2}{a^2 (k - 1) \ell} \frac{ \displaystyle \sum_{i=0}^{n - 2} \binom{i + 2}{2} (1 + \ell) k^i + \sum_{i = n - 1}^{2n - 3} \binom{2n-1-i}{2} (\ell-1) k^i }{ \displaystyle \sum_{i=0}^{n-1} \binom{i + 3}{3} k^i + \sum_{i = n}^{2n - 2} \binom{2n+1-i}{3} k^i },\] \[c_{1}=\frac{2 \left( (2 + \ell) + \displaystyle \sum_{i = 1}^{n - 2} \binom{i + 2}{2} (2 + \ell) k^i + \left( \binom{n}{2} - 1 \right) \ell k^{n - 1} + \displaystyle \sum_{i = n}^{2n - 2} \binom{2n - i}{2} (\ell - 2) k^i \right)}{ a (k - 1) \ell \left( \displaystyle \sum_{i = 0}^{n - 1} \binom{i + 3}{3} k^i + \displaystyle \sum_{i = n}^{2n - 2} \binom{2n+1-i}{3} k^i \right) },\]
\[c_{2}=\frac{a^n c_1 \left(1 - k^{n + 1} \right) + a^{n + 1} c_0 \left(1 - k^{n + 2} \right)}{k - 1} + \frac{2 a^{n - 1} \left(1 - k^n \right)}{(k - 1) \, \ell},\]
\[c_{3}=\frac{ a^{n + 1} c_1 \left( -k + k^{n + 1} \right) + a^{n + 2} c_0 \left( -k + k^{n + 2} \right) }{k - 1} + \frac{2 a^n \left( -k + k^n \right)}{\ell (k - 1)}.\]
Note that \(c_0, c_1, c_2,\) and \(c_3\) have first order singularities at \(k = 1\). Set \(\widetilde{c}_0=(k-1)c_0\), \(\widetilde{c}_1=(k-1)c_1\), \(\widetilde{c}_2=(k-1)c_2\), \(\widetilde{c}_3=(k-1)c_3\). A direct computation gives
\[\begin{gather} \lim_{k \to 1} \widetilde{c}_0= \frac{-8 (n-1)}{a^2(n + 1) (n + 2) },~\; \lim_{k \to 1} \widetilde{c}_1= \frac{4(2 n - 3)}{an (n + 1)},\; \lim_{k \to 1} \widetilde{c}_2= \frac{-4a^{n-1}(n-3)}{n(n+1)},\\[1ex] \lim_{k \to 1} \widetilde{c}_3= \frac{4 a^n(n-4)}{(n+1)(n+2)},~\; \lim_{k \to 1} (\widetilde{c}_0)'_{k}= \frac{2(n-1)(n+2 \ell)}{a^2 \ell (n + 1) (n + 2) },~\; \lim_{k \to 1} (\widetilde{c}_1)'_{k}= \frac{4-4n}{a(\ell+n \ell)},\\[1ex] \lim_{k \to 1} (\widetilde{c}_2)'_{k}= \frac{-2a^{n-1}(2+\ell(n-3))}{\ell(n+1)},~\;\lim_{k \to 1} (\widetilde{c}_3)'_{k}= \frac{2 a^n(2(n-1)+\ell(n^2-3n-4))}{\ell (n+1)(n+2)}. \end{gather}\]
We now proceed to verify the positivity of \(A\), \(C\), and \(D\). Set \(x=at\), \[\begin{gather} A=\frac{ -c_3 (-1 + n) n - a t \left( c_2 (2 - 3n + n^2) + 2 (a t)^n (c_1 + 3 c_0 a t) \right)}{(a t)^{1 + n}}. \\[1ex] C=-2 \frac{ c_3 + a t \left( c_2 + (a t)^n (c_1 + c_0 a t) \right) }{ (a t)^{1 + n} },\;D=2B+\sqrt{AC}=P+\sqrt{Q}, \end{gather}\]
where \[\begin{gather} P=2 (a t)^{-1 - n} \left( c_3 n - a t \left( c_2 - c_2 n + (a t)^n (c_1 + 2 c_0 a t) \right) \right), \\[1ex] Q= 2 (a t)^{-2(1 + n)} \left( c_3 + a t \left( c_2 + (a t)^n (c_1 + c_0 a t) \right) \right) \\ \;\;\; \cdot \left( c_3 (n - 1) n + a t \left( c_2 (2 - 3n + n^2) + 2 (a t)^n (c_1 + 3 c_0 a t)\right) \right). \end{gather}\]
Define \(\widetilde{A}=(k-1)A\), \(\widetilde{C}=(k-1)C\), \(\widetilde{P}=(k-1)P\), \(\widetilde{Q}=(k-1)^2Q\), then \[\begin{gather} \lim_{(k,t)\to(1,1)} \widetilde{A}= \frac{4}{a}>0 , \lim_{(k,t)\to(1,1)} \widetilde{C}= 0,\lim_{(k,t)\to(1,1)} \widetilde{C}'_{t}= 0,\lim_{(k,t)\to(1,1)} \widetilde{C}'_{k}= \frac{4}{a\ell} ,\\[1ex] \lim_{(k,t)\to(1,1)}\widetilde{C}''_{tt}= \frac{8}{a},\lim_{(k,t)\to(1,1)}\widetilde{C}''_{kt}= \frac{-4(\ell+1)}{a \ell} , \lim_{(k,t)\to(1,1)} \widetilde{P}= \widetilde{Q}= \widetilde{Q}'_{t}=0,\\[1ex]\lim_{(k,t)\to(1,1)} \widetilde{P}'_{t}= \frac{8}{a}, \lim_{(k,t)\to(1,1)} \widetilde{Q}''_{tt}= \frac{32}{a^2} , \lim_{(k,t)\to(1,1)}\widetilde{Q}'_{k}= \frac{16}{a^2\ell},\lim_{(k,t)\to(1,1)}\widetilde{P}'_{k}= -\frac{4}{a}. \end{gather}\] After a translation, the remaining discussion reduces to the following proposition. ◻
Proposition 12. Let \(P,Q,C\in C^2(U)\), where \(U\) is a neighborhood of \((0,0)\). Assume \(C(0,0)=C'_t(0,0)=P(0,0)=Q(0,0)=Q'_t(0,0)=0\). Set \(\alpha_1= C'_k(0,0),\; \alpha_2=C''_{tt}(0,0),\;\alpha_3=P'_t(0,0),\; \alpha_4=Q'_k(0,0),\;\alpha_5=Q''_{tt}(0,0).\) If all \(\alpha_i\) are positive, then there exists \(\varepsilon>0\) such that \(C(k,t)>0\), \(Q(k,t)>0\) and \(P(k,t)+\sqrt{Q(k,t)}>0\) for all \(0<k<\varepsilon,\;0<t<\varepsilon.\)
Proof. In what follows, \(k,t\in(0,\varepsilon)\), and we shrink \(\varepsilon\) when necessary to adjust their range. We first prove \(C>0\). For \(\varepsilon\) small enough, we may assume that \(C'_k(k,0)\ge \frac{\alpha_1}{2}, C''_{tt}(k,t)\ge \frac{\alpha_2}{2}\), and there exists \(M_0>0\) such that \(|C'_t(k,0)|\le M_0 k\).
Hence, choose \(\varepsilon>0\) so that \(M_0\varepsilon\le\alpha_1/4\), for small \(\epsilon\), \[\begin{align} C(k,t) &=C(k,0)+tC'_t(k,0) +t^2\int_0^1(1-\theta)C''_{tt}(k,\theta t)\,d\theta \\ &\ge \frac{\alpha_1}{2}k-M_0kt+\frac{\alpha_2}{4}t^2 \ge \frac{\alpha_1}{4}k+\frac{\alpha_2}{4}t^2>0. \end{align}\]
We now prove \(P+\sqrt{Q}>0\). For \(\varepsilon\) small enough, we may assume that \(P'_t(0,t)\ge \frac{\alpha_3}{2}, Q'_k(k,t)\ge \frac{\alpha_4}{2}, Q''_{tt}(0,t)\ge \frac{\alpha_5}{2}\) on \(U\). Also choose \(M_1>0\) such that \(|P'_k(k,t)|\le M_1\). Then \[P(k,t)=P(0,t)+\int_0^k P'_k(\theta,t)\,d\theta \ge \frac{\alpha_3}{2}t-M_1 k,\] and \[Q(k,t) = Q(0,t)+\int_0^k Q'_k(\theta,t)\,d\theta\geq t^2\int_0^1(1-\theta)Q''_{tt}(0,\theta t)\,d\theta+\frac{\alpha_4}{2}k\ge \frac{\alpha_5}{4}t^2+\frac{\alpha_4}{2}k>0.\] In particular, \[\sqrt{Q(k,t)}\ge \sqrt{\frac{\alpha_4}{2}}\sqrt{k}.\] Taking \(\varepsilon>0\) smaller so that \(M_1 k\le \frac{1}{2}\sqrt{\frac{\alpha_4}{2}}\sqrt{k}\) for \(0<k<\varepsilon\), we obtain \[P(k,t)+\sqrt{Q(k,t)} \ge \frac{\alpha_3}{2}t-M_1 k+\sqrt{\frac{\alpha_4}{2}}\sqrt{k} > 0.\] ◻
The construction developed in this section applies more generally to \(\textrm{{Tot}}([\mathcal{O}(-k) ]^{\oplus r} \to \mathbb{CP}^{n})\). However, to keep the computations manageable, we restrict ourselves to the case \(r=2\). Again, the following computation is most conveniently verified by symbolic computation software. Similarly, in view of the regular structure of the ansatz and of the resulting expressions, it is enough for the reader to verify the identities below in the cases \(n\le 10\).
Consider \(\mathbb{C}^{*}\) acting on \(\mathbb{C}^{n+3}\) by \[\begin{align} \lambda \cdot (z_1,z_2,\cdots ,z_{n+3})=(\lambda z_1,\lambda z_2,\cdots,\lambda z_{n+1},\lambda^{-k} z_{n+2},\lambda^{-k} z_{n+3}). \end{align}\]
The above action on \(U=(\mathbb{C}^{n+1}\setminus\{0\})\times \mathbb{C}^2 \subset \mathbb{C}^{n+3}\) determines \(M=\textrm{Tot}(\mathcal{O}(-k) \oplus \mathcal{O}(-k) \to \mathbb{CP}^{n})\).
As computed in Section 2, the Delzant set is given by \[\begin{align} \{(x_1,x_2,\cdots,x_{n+2}) \in \mathbb{R}^{n+2} | l_{i}=kx_i\geq 0,l_{n+2}=x_{n+2}\geq 0,l_{0}=\sum^{n+1}_{i=1}x_i-x_{n+2}-1\geq 0\}. \end{align}\]
Define \(x=\sum^{n+1}_{i=1}x_i\) and set \(h(x)=\frac{k+1}{2}x-\frac{1}{2}(x-1)\log(x-1)-\frac{1}{2}kx\log(x)+F(x)\), we choose the symplectic potential as \[\begin{align} G=\frac{1}{2}\sum^{n+2}_{i=0}l_{i}\log(l_{i})+\frac{k+1}{2}x-\frac{1}{2}(x-1)\log(x-1)-\frac{1}{2}kx\log(x)+F(x). \end{align}\] \[\label{eq:g} (G_{ij}) = \begin{pmatrix} \mathbf{\Lambda} + c_0 \cdot \mathbf{E}_{n+1 \times n+1} & -\frac{1}{2l_0} \cdot \mathbf{E}_{n+1 \times 1} \\[2ex] -\frac{1}{2l_0} \cdot \mathbf{E}_{1 \times n+1} & \frac{1}{2x_{n+2}} + \frac{1}{2l_0} \end{pmatrix}.\tag{1}\]
where \(\mathbf{\Lambda} =k \operatorname{diag}\left( \frac{1}{2x_1}, \dots, \frac{1}{2x_{n+1}} \right)\), \(c_0=\frac{1}{2l_{0}}-\frac{k}{2x}-\frac{1}{2(x-1)}+\frac{1}{\Theta(x)}\) and \(\mathbf{E}_{m \times n}\) denotes the \(m \times n\) matrix with all entries equal to 1.
Let \(\tilde{x}_r=\sum_{i=1}^{r}x_i\), then the first \(n+1\) leading principal minors and the determinant of the matrix are given by \[\begin{align} D_r &= \left( \prod_{i=1}^{r} \frac{k}{2x_i} \right) \left( 1 + c_0 \sum_{i=1}^{r} \frac{2x_i}{k} \right)=\left( \prod_{i=1}^{r} \frac{k}{2x_i} \right)\left(1- \frac{\tilde{x}_r}{x}+\frac{\tilde{x}_r}{kl_0}-\frac{\tilde{x}_r}{k(x-1)}+\frac{2\tilde{x}_r}{k\Theta(x)} \right). \end{align}\] \[\begin{align} \det(G_{ij})&=\frac{k^{n}x(x-1)}{2^{n+1}\prod^{n+2}_{i=1}x_{i}(x-x_{n+2}-1)\Theta(x)}=\left(\frac{2^{n+1}\Theta(x)}{k^{2n+1}x(x-1)} \cdot \prod^{n+2}_{i=0}l_{i}\right)^{-1}. \end{align}\]
By \((\star)\), \(\Theta(x)\) must satisfy the following conditions: \(\Theta(1)=0\), \(\Theta'(1)=2\), \(\Theta(x)\) is positive and smooth on \(( 1,+ \infty)\), and \(\int_2^{+\infty}\frac{dx}{\Theta(x)}=+\infty\).
We claim that the metric above is complete if and only if \(\int_2^{+\infty}\frac{dx}{\sqrt{\Theta(x)}}=+\infty\). Recall the metric is \(g= \sum_{i,j=1}^{n+2} G_{ij} dx_i dx_j + \sum_{i,j=1}^{n+2} H_{ij} d\theta_i d\theta_j\). Since the metric extends smoothly across all finite facets, it remains only to examine curves escaping to the end \(x=\infty\). \[\begin{align} \sum_{i,j=1}^{n+2} G_{ij}\,dx_i dx_j &= \frac{k}{2}\sum_{i=1}^{n+1}\frac{dx_i^2}{x_i} +c_0\,dx^2 -\frac{1}{l_0}\,dx\,dx_{n+2} +\left(\frac{1}{2x_{n+2}}+\frac{1}{2l_0}\right)dx_{n+2}^2 \\[1ex] &= \frac{1}{\Theta(x)}\,dx^2 +\frac{k}{2}\left( \sum_{i=1}^{n+1}\frac{dx_i^2}{x_i} -\frac{dx^2}{x} \right) +\frac{1}{2}\left( \frac{dx_{n+2}^2}{x_{n+2}} +\frac{(dx-dx_{n+2})^2}{l_0} -\frac{dx^2}{x-1} \right) \\[1ex] &\geq \frac{dx^2}{\Theta(x)}+\frac{k}{2}\left( \frac{dx^2}{x} -\frac{dx^2}{x} \right)+\frac{1}{2}\left( \frac{(dx_{n+2}+dx-dx_{n+2})^2}{x_{n+2}+l_0} -\frac{dx^2}{x-1} \right)\geq \frac{dx^2}{\Theta(x)}. \end{align}\]
For any curve \(\gamma(t)\) escaping to \(x=\infty\), \(L(\gamma)\ge \int \frac{|\dot{x}(t)|}{\sqrt{\Theta(x(t))}}\,dt.\) We define a curve \(\gamma_0(t)= \left( \frac{t}{n+1},\ldots,\frac{t}{n+1},\frac{t-1}{2} \right),\; t \in (2,+\infty)\). Along this curve, we have \(x_1=\cdots=x_{n+1}=\frac{t}{n+1}, x_{n+2}=\frac{t-1}{2}\), then \(|\dot{\gamma}(t)|_g^2=\frac{1}{\Theta(x)}\). This proves the claim.
In the present case, the symmetry of the metric is not sufficiently transparent, so we cannot reduce the computation to a special representative point as in the preceding examples. Nevertheless, one can still study the Kähler–Einstein metrics and the extremal metrics on \(M\) by computing the Ricci curvature and the scalar curvature by using Proposition 7. Since the resulting expressions are rather lengthy, we record only the scalar curvature formula here. This computation is best carried out with symbolic computation software.
Proposition 13. Set \(m=n+1\). The scalar curvature is given by \[s=\frac{n(n - 1 - m x) k \Theta(x) + x(2nm(x-1)+4kx + 2( n -m x) k \Theta'(x) - kx(x-1 ) \Theta''(x))}{kx^2(x - 1)},\]
Recall that the Euler-Lagrange equation defining an extremal Kähler metric can be shown to be equivalent to \(\frac{\partial s}{\partial x_{i}}\equiv \textrm{constant}\), i.e., \(s=px+q\) for constants \(p,q\).
Solving the equations \(s=px+q\) and \(\Theta(1)=0, ~ \Theta'(1)=2\), we obtain \[\label{eq:jizhi} \Theta_{p,q}(x)=\frac{p_1-p_2x-p_3x^{n+1}+p_4x^{n+2}+p_5x^{n+3}-p_6x^{n+4}}{kx^n (1 + n) (2 + n) (3 + n) (4 + n) (x-1)}\tag{2}\]
where \[\begin{align} p_1 &=(1 + n) (-2 n (12 + 7 n + n^2) + k (4 n^2 + n (28 + p + q) + 2 (24 + p + 2 q))) , \\[1ex] p_2 &= (4 + n) (-2 (6 + 11 n + 6 n^2 + n^3) + k (24 + 4 n^2 + p + 3 q + n (20 + p + q))) , \\[1ex] p_3 &= 2 (24 + 50 n + 35 n^2 + 10 n^3 + n^4), \\[1ex] p_4 &=(12 + 7 n + n^2) (2 n (1 + n) + k (4 + q)), \\[1ex] p_5 &=k (4 + 5 n + n^2) (p - q), \\[1ex] p_6 &= k (2 + 3 n + n^2) p. \end{align}\]
Proposition 14. When \(p,q \leq 0\), \(\Theta_{p,q}\) is positive on \((1,+\infty)\).
Proof. Define \(f(x)=p_1-p_2x-p_3x^{n+1}+p_4x^{n+2}+p_5x^{n+3}-p_6x^{n+4}\), then \[f'(x)=-p_2-(n+1)p_3x^{n}+(n+2)p_4x^{n+1}+(n+3)p_5x^{n+2}-(n+4)p_6x^{n+3},\] \[f''(x)=-n(n+1)p_3x^{n-1}+(n+1)(n+2)p_4x^{n}+(n+2)(n+3)p_5x^{n+1}-(n+3)(n+4)p_6x^{n+2},\] \[g(x):=-n(n+1)p_3+(n+1)(n+2)p_4x+(n+2)(n+3)p_5x^{2}-(n+3)(n+4)p_6x^{3},\] \[g'(x)=(n+1)(n+2)p_4+2(n+2)(n+3)p_5x-3(n+3)(n+4)p_6x^{2},\] \[g''(x)=2(n+2)(n+3)p_5-6(n+3)(n+4)p_6x.\]
The positivity of \(\Theta(x)\) can be deduced from the following: \[g'''(x)=-6(n+3)(n+4)p_6=-6(n+3)(n+4)k (2 + 3 n + n^2) p \geq 0,\] \[g''(1)=-2 k (24 + 50 n + 35 n^2 + 10 n^3 + n^4) (2 p + q) \geq 0,\] \[g'(1)=(24 + 50 n + 35 n^2 + 10 n^3 + n^4) (2 n (1 + n) - k (-4 + p + q)) \geq 0,\] \[g(1)=4 k (24 + 50 n + 35 n^2 + 10 n^3 + n^4) >0,\] \[f'(1)=0,\; f(1)=0.\] ◻
The completeness of both the induced complex structure and the metric forces that \(\Theta_{p,q}\) grows at most linearly at infinity, i.e., \(p_5=p_6=0\), that is \(p=q=0\).
We now turn to the study of Kähler–Einstein metrics. Substituting \(\Theta_{0,q}\) into the Kähler–Einstein equation \(\operatorname{Ric}_{ij}=cH_{ij}\), we obtain \(q=\frac{2}{k}(n+2)(n+1)-4(n+2)\) and \(c=\frac{2}{k}(n+1)-4\). Thus \(2k=n+1\).
Proposition 15. The function \(\Theta_{0,0}\) determines a complete scalar-flat Kähler metric on \(M\). Moreover, when \(2k=n+1\), the resulting metric is Ricci-flat.