The concept of RC-positivity and uniform RC-positivity is introduced by Xiaokui Yang to solve a conjecture of Yau on projectivity and rational connectedness of a compact Kähler manifold with positive holomorphic sectional curvature. Some main theorems in Yang’s proof hold under a weaker condition called weak RC-positivity. It is therefore natural to ask if (uniform) weak RC-positivity implies (uniform) RC-positivity. Another motivation for studying this problem is to understand the relation between rational connectedness of \(X\) and (uniform) RC-positivity of the holomorphic tangent bundle \(TX\).
In this paper, we obtain results in this direction. In particular, we show that if a vector bundle \(E\) is uniformly weakly RC-positive, then \(S^kE\otimes \det E\) is uniformly RC-positive for any \(k\geq 0\), and \(S^kE\) is uniformly RC-positive for \(k\) large. We also discuss an approach that might lead to a solution to the question of whether weak RC-positivity of \(E\) implies RC-positivity of \(E\).
In [1], Yang introduces the notion of RC-positivity as a differential geometric counterpart of rational connectedness. RC-positivity plays a crucial role in Yang’s proof of a conjecture of Yau: If a compact Kähler manifold has positive holomorphic sectional curvature, then the manifold is projective and rationally connected. A stronger notion called uniform RC-positivity is introduced by Yang in [2] which also can be used to prove the same conjecture of Yau. For the semipositive case of Yau’s conjecture, see [3]–[5].
Let us recall the definition of RC-positivity and uniform RC-positivity. Let \(E\) be a holomorphic vector bundle of rank \(r\) over a compact complex manifold \(X\) of dimension \(n\). Given a Hermitian metric \(H\) on \(E\), we denote the Chern curvature of \(H\) by \(\Theta^H\), which is an \(\text{End}E\)-valued \((1,1)\)-form. We denote by \(TX\) the holomorphic tangent bundle of \(X\). For a vector \(u\in E_t\) and a tangent vector \(v\in T_tX\) with \(t\in X\), we define the expression \[H(\Theta^H u,u)(v,\bar{v})\] to be \(\sum_{j,k} H(\Theta^H_{j\bar{k}}u,u)v_j \bar{v}_k\) locally where we write the curvature \(\Theta^H=\sum_{j,k}\Theta^H_{j\bar{k}}dt_j\wedge d\bar{t}_k\) and \(v=\sum_j v_j\partial/\partial t_j\).
Definition 1. A Hermitian metric \(H\) on a holomorphic vector bundle \(E\to X\) is called RC-positive if for any \(t\in X\) and any nonzero \(u\in E_t\), there is a nonzero tangent vector \(v\in T_tX\) such that \(H(\Theta^Hu,u)(v,\bar v)>0\). On the other hand, a Hermitian metric \(H\) is called uniformly RC-positive if for any \(t\in X\), there is a nonzero tangent vector \(v\in T_tX\) such that for any nonzero \(u\in E_t\), we have \(H(\Theta^H u,u)(v,\bar v)>0\).
A holomorphic vector bundle \(E\to X\) is called (uniformly) RC-positive if it admits a (uniformly) RC-positive Hermitian metric.
It is clear that uniform RC-positivity implies RC-positivity. To motivate the definition of (uniform) weak RC-positivity, let us consider a Hermitian metric \(H\) on \(E\) and the induced metric \(h\) on the line bundle \(O_{P(E^*)}(1)\) over the projectivized bundle \(P(E^*)\). By a standard computation (for example, see [1]), we know that if \((E,H)\) is RC-positive, then the curvature \(\Theta\) of \(h\) is positive on every fiber and has at least \(r\) positive eigenvalues at every point in \(P(E^*)\). The existence of such a metric \(h\) on \(O_{P(E^*)}(1)\) is called weak RC-positivity of \(E\) ([1]).
Similarly, using the same computation, we see that if \((E,H)\) is uniformly RC-positive, then the curvature \(\Theta\) of the induced metric \(h\) satisfies
\(\Theta\) is positive on every fiber.
\(\Theta\) has at least \(r\) positive eigenvalues at every point in \(P(E^*)\).
For any point \(t\in X\), there exists a nonzero tangent vector \(v\in T_tX\), such that \(\Theta(\tilde{v},\bar{\tilde{v}})|_{(t,[\zeta])}>0\) for any lift \(\tilde{v}\) of \(v\) to \(T_{(t,[\zeta])}P(E^*)\).
Following Yang, we call the existence of such a metric \(h\) on \(O_{P(E^*)}(1)\) uniform weak RC-positivity of \(E\). Note that in the third condition, we consider the lifts to the tangent space \(T_{(t,[\zeta])}P(E^*)\) for any point \([\zeta]\) in the fiber \(P(E_t^*)\) not just one point \([\zeta]\). The second condition is implied by the first and the third, so we will omit it later on. Let us summarize the definition.
Definition 2. The bundle \(E\) is called weakly RC-positive if there exists a metric \(h\) on \(O_{P(E^*)}(1)\) with properties a and b. The bundle \(E\) is called uniformly weakly RC-positive if there exists a metric \(h\) on \(O_{P(E^*)}(1)\) with properties a and c.
In Yang’s solution to Yau’s conjecture, two main theorems [1], although formulated in terms of RC-positivity, hold under weak RC-positivity. So, it is natural to ask if weak RC-positivity of \(E\) implies RC-positivity of \(E\) ([1] and [6]). This question has the same flavor as a conjecture of Griffiths [7]: If \(E\) is ample, then \(E\) is Griffiths positive. For the developments on the Griffiths conjecture, see [8]–[25].
In this paper, we make some progress in this direction. In particular, we prove the following theorem regarding uniform RC-positivity.
Theorem 1. If \(E\) is uniformly weakly RC-positive over a compact Kähler manifold \(X\), then \(S^kE\otimes \det E\) is uniformly RC-positive for any \(k\geq 0\), and \(S^kE\) is uniformly RC-positive for \(k\) large.
Another motivation for establishing Theorem 1 is to understand the relation between rational connectedness of \(X\) and (uniform) RC-positivity of the holomorphic tangent bundle \(TX\). According to Yang [1] and [2], for a compact Kähler manifold \(X^n\), if one of the following is true, then \(X\) is projective and rationally connected.
The holomorphic tangent bundle \(TX\) is uniformly RC-positive.
The exterior power \(\wedge^p TX\) is RC-positive for \(1\leq p\leq n\).
One can ask if the converse is true ([2]). A partial converse is proved in [2]: if \(X\) is projective and rationally connected, then the line bundle \(O_{\wedge^p TX}(-1)\) is RC-positive for \(1\leq p \leq n\). So, Theorem 1 can be viewed as a step towards this converse problem: constructing uniformly RC-positive Hermitian metrics out of metrics on the line bundle \(O_{P(E^*)}(1)\).
We also prove a lemma (Lemma 2 in Section 4) and discuss how a variant of this lemma might lead to a solution to the original question of Yang, namely, weak RC-positivity of \(E\) implying RC positivity of \(E\).
For the proof of Theorem 1, instead of the fibration \(p:P(E^*)\to X\), we will work on a more general fibration and prove a general theorem which contains Theorem 1 as a special case. We consider a proper holomorphic surjection \(p:\mathcal{X}^{n+m}\to Y^m\) between two complex manifolds with \(\mathcal{X}\) Kähler, \(Y\) compact, and the differential \(dp\) surjective at every point. We denote the fibers \(p^{-1}(t)\) by \(\mathcal{X}_t\) for \(t\in Y\). Let \((L,h)\) be a Hermitian line bundle over \(\mathcal{X}\). Let \[V_t=H^0(\mathcal{X}_t, L|_{\mathcal{X}_t}\otimes K_{\mathcal{X}_t}).\] We assume that \(\dim V_t\) is independent of \(t\in Y\). So, the direct image of the sheaf of sections of \(L\otimes K_{\mathcal{X}/Y}\) is locally free by Grauert’s direct image theorem, where \(K_{\mathcal{X}/Y}\) is the relative canonical bundle. We denote by \(V\) the associated vector bundle over \(Y\). There is a naturally defined Hermitian metric \(H\) on \(V\). For \(u\) in \(V_t\) with \(t\in Y\), \[\label{metric} H(u,u):=\int_{\mathcal{X}_t}h(u,u).\tag{1}\] We extend the metric \(h\) to act on sections \(u\) of \(L|_{\mathcal{X}_t}\otimes K_{\mathcal{X}_t}\) so that \(h(u,u)\) is an \((n,n)\)-form on \(\mathcal{X}_t\). In terms of local coordinates, if \(u=u'\otimes e\) with \(u'\) an \((n,0)\)-form and \(e\) a frame of \(L|_{\mathcal{X}_t}\), then \(h(u,u)=c_n u' \wedge \overline{u'} h(e,e)\) where \(c_n=i^{n^2}\). Under this more general fibration, we can show
Theorem 2. If the curvature \(\Theta\) of \(h\) is positive on every fiber, and for any point \(t\in Y\), there exists a nonzero tangent vector \(v\in T_tY\), such that \(\Theta(\tilde{v},\bar{\tilde{v}})|_{(t,z)}>0\) for any lift \(\tilde{v}\) of \(v\) to \(T_{(t,z)}\mathcal{X}\), then the Hermitian bundle \((V,H)\) is uniformly RC-positive.
Actually, the precise statement we prove in Theorem 2 is: For a fixed point \(t_0 \in Y\), if the curvature \(\Theta\) of \(h\) is positive on the fiber \(\mathcal{X}_{t_0}\) , and there exists a nonzero tangent vector \(v\in T_{t_0}Y\), such that \(\Theta(\tilde{v},\bar{\tilde{v}})|_{(t_0,z)}>0\) for any lift \(\tilde{v}\) of \(v\) to \(T_{(t_0,z)}\mathcal{X}\), then the Hermitian bundle \((V,H)\) is uniformly RC-positive at \(t_0\).
Now, we consider the fibration \(p:P(E^*)\to X\), and we assume \(X\) is Kähler to make sure \(P(E^*)\) is Kähler (see [25]). Therefore, by Theorem 2, we have Theorem 1. Indeed, the vector bundle \(V\) in Theorem 2 is associated with the direct image of \(L\otimes K_{\mathcal{X}/Y}\). In the present situation, the relative canonical bundle \(K_{P(E^*)/X}\) is isomorphic to \(O_{P(E^*)}(-r)\otimes p^*\det E\). If we choose \(O_{P(E^*)}(r+k)\) for the line bundle \(L\), then \(V\) is \(S^k\otimes \det E\). On the other hand, if we choose \(O_{P(E^*)}(k)\otimes K^{-1}_{P(E^*)/X}\) for \(L\), then \(V\) is \(S^kE\) (we use an arbitrary metric \(g\) on \(K^{-1}_{P(E^*)/X}\), and the effect of \(g\) can be absorbed by taking \(k\) large).
The proof of Theorem 2 is an adaptation of [25], but we still include the details for completeness (the original argument is due to Berndtsson in [10] and [26]. See also [27]).
This paper is organized as follows. In Section 2, we give a local expression for the assumption (uniform weak RC-positivity) in Theorem 2 which will be used in the proof of the main theorem. In Section 3, we prove Theorem 2. In Section 4, we discuss a characterization of weak RC-positivity and its possible application.
I would like to thank Shin-ichi Matsumura for bringing to my attention the question of RC-positivity and weak RC-positivity. I am grateful to László Lempert, Siarhei Finski, and Xiaokui Yang for their interest in the paper. Thanks are also due to the Erdős Center, Budapest for the support.
Let \(p:\mathcal{X}^{m+n}\to Y^m\) be the fibration in the introduction, and \((L,h)\to \mathcal{X}\) be the Hermitian line bundle. In this section, we will explain the meaning of the assumption (uniform weak RC-positivity) in Theorem 2.
Assume that the Hermitian metric \(h\) on the line bundle \(L\) has its curvature \(\Theta\) positive on every fiber. For such an \(h\), we can decompose its curvature \(\Theta=\Theta_{\mathcal{H}}+\Theta_\mathcal{V}\) where \(\Theta_\mathcal{H}\) is the horizontal component and \(\Theta_\mathcal{V}\) is the vertical component. The horizontal component \(\Theta_\mathcal{H}\) can be viewed as an element in \(C^{\infty}(\mathcal{X}, p^*(\wedge^{1,1}T^*Y))\).
We denote by \((t_1,\ldots,t_m)\) the local coordinates in \(Y\) and by \((t_1,\ldots,t_m,z_1,\ldots, z_n)\) the local coordinates in \(\mathcal{X}\), and assume that \(h=e^{-\phi}\) locally. We write \(\phi_{i\bar{j}}=\phi_{t_i\bar{t}_j}\), \(\phi_{\lambda\bar{\mu}}=\phi_{z_\lambda \bar{z}_\mu}\), etc. Since \(\Theta\) is positive on each fiber, the matrix \((\phi_{\lambda\bar{\mu}})\) is positive definite, and we denote its inverse by \((\phi^{\lambda\bar{\mu}})\). The horizontal component \(\Theta_\mathcal{H}\) and the vertical component \(\Theta_\mathcal{V}\) have the local expressions \[\begin{align} &\Theta_\mathcal{H}=\sum_{i,j}( \phi_{i\bar{j}}-\sum_{\lambda,\mu} \phi_{i\bar{\mu}}\phi^{\lambda \bar{\mu}}\phi_{\lambda \bar{j}})dt_i\wedge d\bar{t}_j\tag{2}\\ &\Theta_\mathcal{V}=\sum_{\lambda,\mu}\phi_{\lambda\bar{\mu}}\delta z_\lambda\wedge \delta \bar{z}_{\mu}\tag{3} \end{align}\] where \(\delta z_\lambda=dz_\lambda+\sum_{i,\mu}\phi^{\lambda\bar{\mu}}\phi_{i\bar{\mu}}dt_i\). That (2 ) and (3 ) are independent of local coordinates is discussed in [28], and the horizontal component \(\Theta_\mathcal{H}\) is called the geodesic curvature there.
Recall the assumption in Theorem 2: the curvature \(\Theta\) of \(h\) is positive on every fiber, and for any point \(t\in Y\), there exists a nonzero tangent vector \(v\in T_tY\), such that \(\Theta(\tilde{v},\bar{\tilde{v}})|_{(t,z)}>0\) for any lift \(\tilde{v}\) of \(v\) to \(T_{(t,z)}\mathcal{X}\). We may assume the tangent vector \(v\) is \(\partial/\partial t_1\) in the local coordinates above, and we denote the lift by \(\tilde{v}=\partial/\partial t_1+\sum_\lambda a_\lambda \partial/\partial z_\lambda\) with \(a_\lambda\in\mathbb{C}\). We then have \[\begin{align} \Theta(\tilde{v},\bar{\tilde{v}})&=\phi_{1\bar{1}}+\sum_\mu \phi_{1\bar{\mu}}\overline{a_\mu}+\sum_\lambda \phi_{\lambda\bar{1}}a_\lambda+\sum_{\lambda,\mu}\phi_{\lambda\bar{\mu}}a_\lambda\overline{a_\mu}\\&=\phi_{1\bar{1}}-\sum_{\lambda,\mu}\phi_{1\bar{\mu}}\phi^{\lambda\bar{\mu}}\phi_{\lambda\bar{1}}+\sum_{\lambda,\mu}\phi_{1\bar{\mu}}\phi^{\lambda\bar{\mu}}\phi_{\lambda\bar{1}}+\sum_\mu \phi_{1\bar{\mu}}\overline{a_\mu}+\sum_\lambda \phi_{\lambda\bar{1}}a_\lambda+\sum_{\lambda,\mu}\phi_{\lambda\bar{\mu}}a_\lambda\overline{a_\mu}\\&=\phi_{1\bar{1}}-\sum_{\lambda,\mu}\phi_{1\bar{\mu}}\phi^{\lambda\bar{\mu}}\phi_{\lambda\bar{1}}+\|\sqrt{A^{-1}}\phi_1+\sqrt{A}\bar{a}\|^2, \end{align}\] where in the last equality we denote the matrix \((\phi_{\lambda\bar{\mu}})\) by \(A\), the column vector \((\phi_{\lambda\bar{1}})\) by \(\phi_1\), and the column vector \((a_\lambda)\) by \(a\). Let us summarize the computation as a lemma.
Lemma 1. \(\Theta(\tilde{v},\bar{\tilde{v}})>0\) for any lift \(\tilde{v}\) if and only if \(\phi_{1\bar{1}}-\sum_{\lambda,\mu}\phi_{1\bar{\mu}}\phi^{\lambda\bar{\mu}}\phi_{\lambda\bar{1}}>0\), which is also equivalent to \(\Theta_{\mathcal{H}}(\tilde{v},\bar{\tilde{v}})>0\).
Recall the setup: \(p:\mathcal{X}^{m+n}\to Y^m\) is a proper holomorphic submersion with \(\mathcal{X}\) Kähler and \(Y\) compact, \((L,h)\to \mathcal{X}\) is a Hermitian line bundle, and \(V\to Y\) is a holomorphic vector bundle associated with the direct image \(p_*(L\otimes K_{\mathcal{X}/Y})\). The fibers of \(V\) are \(V_t=H^0(\mathcal{X}_t, L|_{\mathcal{X}_t}\otimes K_{\mathcal{X}_t}).\) The bundle \(V\) carries a Hermitian metric \(H(u,u)=\int_{\mathcal{X}_t}h(u,u)\).
A smooth local section \(u\) of the bundle \(V\) is represented by a smooth \((n,0)\)-from with values in \(L\) over \(p^{-1}(W)\) for some open set \(W\) in \(Y\) such that the restriction to each fiber is holomorphic. Any representative of \(u\) is denoted by \(\mathbf{u}\). We use \((t_1,\dots, t_m)\) for local coordinates in \(Y\). In general, we have \[\bar{\partial}\mathbf{u}=\sum_j d\bar{t}_j\wedge \nu_j+ \sum_j \eta_j\wedge dt_j\] where \(\eta_j\) are of bidegree \((n-1,1)\) and \(\nu_j\) are of bidegree \((n,0)\) whose restrictions to fibers are holomorphic (thus \(\nu_j\) define sections of the bundle \(V\)). The Hermitian holomorphic vector bundle \((V, H)\) admits the Chern connection \(D=D'+D{''}\). The \((0,1)\)-part \(D^{''}\) is given by \(D^{''}u=\sum_j \nu_j d\bar{t}_j\). Therefore, a section \(u\) of \(V\) is holomorphic if and only if \(\bar{\partial}\mathbf{u}= \sum \eta_j\wedge dt_j\).
For the \((1,0)\)-part \(D'\), we consider some local frame \(e\) of \(L\) and write \(\mathbf{u}=u'\otimes e\) with \(u'\) an \((n,0)\)-form. Denote \(h(e,e)=e^{-\phi}\) and define \[\label{phi} \partial^\phi \mathbf{u}:=(\partial^\phi u')\otimes e = e^\phi \partial(e^{-\phi}{u'})\otimes e\tag{4}\] which is an \((n+1,0)\)-form with values in \(L\). It is straightforward to check that (4 ) is independent of the choice of the frame \(e\). We can write \(\partial^\phi \mathbf{u}=\sum_j dt_j\wedge \mu_j\) where \(\mu_j\) are of bidegree \((n,0)\). If we denote by \(P(\mu_j)\) the orthogonal projection of \(\mu_j\) on the space of holomorphic forms on each fiber, then \(D'u=\sum_j P(\mu_j)dt_j\).
Next, we are going to prove \((V,H)\) is uniformly RC-positive. Fix a point \(t_0\in Y\). By the assumption in Theorem 2, there exists a nonzero tangent vector \(v_0\in T_{t_0}Y\) such that for any lift \(\tilde{v}\) of \(v_0\) to \(T_{(t_0,z)}\mathcal{X}\), we have \(\Theta(\tilde{v},\bar{\tilde{v}})>0\). So, the goal is to show that the curvature \(\Theta^V\) of \((V,H)\) satisfies \(H(\Theta^V u_0,u_0)(v_0,\bar{v}_0)>0\) for any nonzero vector \(u_0\) in \(V_{t_0}\).
We choose a coordinate system \((t_1,\ldots,t_m)\) around the point \(t_0\) in \(Y\) such that \(v_0=\partial/\partial t_1\) at \(t_0\). Consider a fixed \(u_0\neq 0\) in \(V_{t_0}\). A standard argument allows us to extend \(u_0\) to a local holomorphic section \(u\) of \(V\) such that \(D'u=0\) at \(t_0\) and \(u(t_0)=u_0\neq0\). A straightforward computation gives \[\label{standard} \partial \bar{\partial} H(u,u)=- H(\Theta^V u,u) \text{ at } t_0.\tag{5}\] On the other hand, if we let \(\mathbf{u}\) be a representative of \(u\) and write \(\mathbf{u}=u'\otimes e\) with \(u'\) an \((n,0)\)-form and \(e\) some local frame of \(L\), then \[H(u,u)=p_*(c_n u'\wedge \overline{u'} e^{-\phi})\] where \(e^{-\phi}=h(e,e)\) and \(c_n=i^{n^2}\). According to [10], we can choose a representative \(\mathbf{u}\) such that in \(\bar{\partial}\mathbf{u}= \sum \eta_j\wedge dt_j\), the \(\eta_j\) is primitive on \(\mathcal{X}_{t_0}\). Moreover, \(\partial^\phi u'=0\) at \(t_0\). After using such a representative, we obtain \[\label{4464} \partial\bar{\partial}H(u,u) = -c_n p_* ( u'\wedge\overline{u'}\wedge \partial\bar{\partial} \phi e^{-\phi}) + (-1)^n c_np_* (\bar{\partial}u'\wedge \overline{\bar{\partial}u'}e^{-\phi}) \text{ at } t_0.\tag{6}\] We apply the above \((1,1)\)-form to the tangent vector \(v_0=\partial/\partial t_1\) and get \[\label{4466} \partial\bar{\partial}H(u,u)(v_0,\bar{v}_0) = -c_n p_* ( u'\wedge\overline{u'}\wedge \partial\bar{\partial} \phi e^{-\phi})(v_0,\bar{v}_0) + (-1)^n c_np_* (\bar{\partial}u'\wedge \overline{\bar{\partial}u'}e^{-\phi}) (v_0,\bar{v}_0)\text{ at } t_0.\tag{7}\] Because \(\bar{\partial}\mathbf{u}= \sum \eta_j\wedge dt_j\) and \(\mathbf{u}=u'\otimes e\), we see \(\sum \eta_j\wedge dt_j=\bar{\partial}\mathbf{u}=\bar{\partial}u'\otimes e\). If we write \(\eta_j=\eta_j'\otimes e\), then \(\bar{\partial}u'=\sum \eta_j'\wedge dt_j\). So the last term in (7 ) is equal to \[\label{4467} (-1)^n c_n\int_{\mathcal{X}_{t_0}}(-1)^{n}\sum \eta'_j\wedge \overline{\eta'}_k \wedge dt_j\wedge d\bar{t}_k e^{-\phi}(v_0,\bar{v}_0)= c_n \int_{\mathcal{X}_{t_0}} \eta'_1\wedge \overline{\eta'}_1 e^{-\phi}\leq 0;\tag{8}\] the last inequality is by the fact that the \(\eta_1\) is primitive on \(\mathcal{X}_{t_0}\).
We claim that the middle term in (7 ) is negative. For the \((n,0)\)-form \(u'\), we can write locally \[u'=u_z dz+\sum_ {\lambda, j}u_{z_\lambda t_j}d\hat{z}_\lambda\wedge dt_j+(\text{terms with more than one t_j}),\] where \(dz=dz_1\wedge\dots \wedge dz_n\) and \(d\hat{z}_\lambda\) means \(dz_1\wedge \dots \wedge dz_n\) omitting \(dz_\lambda\); here \(u_z\) and \(u_{z_\lambda t_j}\) simply mean the coefficients, not differentiation. By a degree count, we see that \(-c_n p_* ( u'\wedge\overline{u'}\wedge \partial\bar{\partial} \phi e^{-\phi})\) is equal to \[\begin{align} -c_n\int_{\mathcal{X}_{t_0}} e^{-\phi} \big(&\sum_{j,k} u_z dz\wedge \overline{u_zdz} \wedge\phi_{j\bar{k}} dt_j\wedge d\bar{t}_k \\+&\sum_{\lambda,j,k} u_z dz\wedge \overline{u_{z_\lambda t_k}d\hat{z}_\lambda\wedge dt_k}\wedge \phi_{j\bar{\lambda}} dt_j\wedge d\bar{z}_\lambda \\+&\sum_{\lambda,j,k} u_{z_\lambda t_j}d\hat{z}_\lambda\wedge dt_j\wedge \overline{u_zdz}\wedge\phi_{\lambda \bar{k}} dz_\lambda \wedge d\bar{t}_k \\+ &\sum_{\lambda,\mu,j,k} u_{z_\lambda t_j}d\hat{z}_\lambda\wedge dt_j\wedge \overline{u_{z_\mu t_k}d\hat{z}_\mu\wedge dt_k}\wedge\phi_{\lambda \bar{\mu}} dz_\lambda\wedge d\bar{z}_\mu \big) \end{align}\] which can be organized as \[\label{4468} \begin{align} -c_n \sum_{j,k}\int_{\mathcal{X}_{t_0}} e^{-\phi}\big(|u_z|^2\phi_{j\bar{k}}+&\sum_{\lambda} (-1)^{n-\lambda+1}u_z \overline{u_{z_\lambda t_k}} \phi_{j\bar{\lambda}}+\sum_{\lambda} (-1)^{n-\lambda+1}\overline{u_z} u_{z_\lambda t_j}\phi_{\lambda\bar{k}}\\+&\sum_{\lambda,\mu} (-1)^{\lambda+\mu}u_{z_\lambda t_j} \overline{u_{z_\mu t_k}}\phi_{\lambda \bar{\mu}} \big) dz\wedge d\bar{z}\wedge dt_j\wedge d\bar{t}_k. \end{align}\tag{9}\] Applying the above \((1,1)\)-form to \((v_0,\bar{v}_0)\), we see that \(-c_n p_* ( u'\wedge\overline{u'}\wedge \partial\bar{\partial} \phi e^{-\phi})(v_0,\bar{v}_0)\) is equal to \[\begin{align}\label{4469} -c_n \int_{\mathcal{X}_{t_0}} e^{-\phi}\big(|u_z|^2\phi_{1\bar{1}}+&\sum_{\lambda} (-1)^{n-\lambda+1}u_z \overline{u_{z_\lambda t_1}} \phi_{1\bar{\lambda}}+\sum_{\lambda} (-1)^{n-\lambda+1}\overline{u_z} u_{z_\lambda t_1}\phi_{\lambda\bar{1}}\\+&\sum_{\lambda,\mu} (-1)^{\lambda+\mu}u_{z_\lambda t_1} \overline{u_{z_\mu t_1}}\phi_{\lambda \bar{\mu}} \big) dz\wedge d\bar{z}. \end{align}\tag{10}\] If we denote the matrix \((\phi_{\lambda\bar{\mu}})\) by \(A\), the column vector \((\phi_{\lambda\bar{1}})\) by \(\phi_1\), and the column vector \(((-1)^{n-\lambda+1}u_{z_\lambda t_1})\) by \(B_1\), then the expression inside the big parenthesis in (10 ) can be written as \[\label{44610} |u_z|^2\phi_{1\bar{1}}-\sum_{\lambda,\mu}\phi_{\lambda\bar{1}}\phi^{\lambda\bar{\mu}}\phi_{1\bar{\mu}}|u_z|^2+\|\sqrt{A^{-1}}\phi_1\overline{u_z}+\sqrt{A}\overline{B_1}\|^2.\tag{11}\] Combining (10 ) and (11 ), we get \[\label{44611} \begin{align} &-c_n p_* ( u'\wedge\overline{u'}\wedge \partial\bar{\partial} \phi e^{-\phi})(v_0,\bar{v}_0)\\=&-c_n\int_{\mathcal{X}_{t_0}}e^{-\phi}\big(|u_z|^2(\phi_{1\bar{1}}-\sum_{\lambda,\mu}\phi_{\lambda\bar{1}}\phi^{\lambda\bar{\mu}}\phi_{1\bar{\mu}})+\|\sqrt{A^{-1}}\phi_1\overline{u_z}+\sqrt{A}\overline{B_1}\|^2\big)dz\wedge d\bar{z}. \end{align}\tag{12}\] Meanwhile, by Lemma 1, the assumption in Theorem 2 implies that \(\phi_{1\bar{1}}-\sum_{\lambda,\mu}\phi_{\lambda\bar{1}}\phi^{\lambda\bar{\mu}}\phi_{1\bar{\mu}}>0\) on the fiber \(\mathcal{X}_{t_0}\), so the integrand in (12 ) is semipositive on the fiber \(\mathcal{X}_{t_0}\). Actually, the integrand in (12 ) is positive somewhere on the fiber \(\mathcal{X}_{t_0}\). This is because \(u(t_0)\neq 0\), \(\mathbf{u}|_{\mathcal{X}_{t_0}}\) cannot be identically zero, hence \(u_z\) is nonzero somewhere.
As a consequence, from formula (12 ), we deduce that \(-c_n p_* ( u'\wedge\overline{u'}\wedge \partial\bar{\partial} \phi e^{-\phi})(v_0,\bar{v}_0)\) is negative, as claimed. All in all, the right hand side in (7 ) is negative, so \(\partial \bar{\partial} H(u,u)(v_0,\bar{v}_0)< 0\) at \(t_0\). By (5 ), we get \(H(\Theta^V u_0,u_0)(v_0,\bar{v}_0)> 0\).
We first give a characterization of weak RC-positivity. It is a variant of [29], [30], and [31]. Let \(p:\mathcal{X}^{m+n}\to Y^m\) be the fibration with the Hermitian line bundle \((L,h)\to \mathcal{X}\) in the introduction. We consider \(\beta\in C^{\infty}(\mathcal{X}, p^*(\wedge^{1,1}T^*Y))\) and call \(\beta\) positive if the matrix \((\beta_{i\bar{j}})\) in the local expression \(i\sum_{i,j}\beta_{i\bar{j}}dt_i\wedge d\bar{t}_j\) is positive.
Lemma 2. Assume the curvature \(\Theta\) of \(h\) is positive on every fiber \(\mathcal{X}_t\). For \(1\leq k\leq m\), the following are equivalent.
The curvature \(\Theta\) has at least \(n+k\) positive eigenvalues at every point in \(\mathcal{X}\).
The horizontal component \(\Theta_\mathcal{H}\) has at least \(k\) positive eigenvalues at every point in \(\mathcal{X}\).
There exists a positive \(\beta\in C^{\infty}(\mathcal{X}, p^*(\wedge^{1,1}T^*Y))\) such that the sum of any \(m-k+1\) eigenvalues of \(\Theta_\mathcal{H}\) with respect to \(\beta\) is positive.
Moreover, when \(k=1\), statement C can be rephrased as \(\Theta_\mathcal{H}\wedge \beta^{m-1}>0\) or equivalently \(\Theta^{n+1}\wedge \beta^{m-1}>0\).
Proof. The equivalence between A and B can be seen by the decomposition \(\Theta=\Theta_\mathcal{H}+\Theta_\mathcal{V}\) and formulas (2 ) and (3 ). To prove that C implies B, we denote the eigenvalues of \(\Theta_\mathcal{H}\) with respect to \(\beta\) by \(\gamma_1\leq\ldots \leq \gamma_m\). Since \(0<\gamma_1+\cdots+\gamma_{m-k+1}\), we see that \(\gamma_{m-k+1}\) must be positive, so \(\Theta_\mathcal{H}\) has at least \(k\) positive eigenvalues.
To prove that B implies C, we first fix a positive \(\beta_0\) in \(C^{\infty}(\mathcal{X}, p^*(\wedge^{1,1}T^*Y))\) and denote the eigenvalues of \(\Theta_\mathcal{H}\) with respect to \(\beta_0\) by \(\gamma_1\leq\ldots \leq \gamma_m\). We know that \(\gamma_{m-k+1}>0\) since there are at least \(k\) positive eigenvalues for \(\Theta_\mathcal{H}\). We are going to construct \(\beta\) using the arguments in [31] (see also [29] and [30]).
Consider the positive numbers \(A:=\inf_{\mathcal{X}}\gamma_{m-k+1}>0\), \(B:=\sup_{\mathcal{X}}\max_j |\gamma_j|>0\), and \(\varepsilon:=1/(m-k+1)\). Let \(\psi_\varepsilon\) be in \(C^{\infty}(\mathbb{R},\mathbb{R})\) such that \[\psi_\varepsilon(x)=x \text{ for } x\geq A,\,\,\, \psi_\varepsilon(x)\geq x \text{ for } 0\leq x \leq A,\,\,\, \psi_\varepsilon(t)=B/\varepsilon \text{ for } x\leq 0.\] By [31], \(\psi_\varepsilon(\Theta_\mathcal{H})\) is in \(C^{\infty}(\mathcal{X}, p^*(\wedge^{1,1}T^*Y))\) and positive. For a point \((t,z)\in \mathcal{X}\), if \(\{\xi_1,\ldots, \xi_m\}\) is a basis of \(T^*_t Y\) such that \(\beta_0(t,z)=\sum \xi_j\wedge \bar{\xi}_j\), then \[\Theta_\mathcal{H}(t,z)=\sum \gamma_j\xi_j\wedge \bar{\xi}_j \text{ and } \psi_\varepsilon(\Theta_\mathcal{H})(t,z)=\sum \psi_\varepsilon(\gamma_j) \xi_j\wedge \bar{\xi}_j.\] Therefore, the eigenvalues of \(\Theta_\mathcal{H}\) with respect to \(\psi_\varepsilon(\Theta_\mathcal{H})\) are \(\gamma_j/\psi_{\varepsilon}(\gamma_j)\). For \(1\leq j\leq m\),
\[\left\{ \begin{array}{l} \text{if } 0\leq \gamma_j, \text{ then } \gamma_j\leq \psi_\varepsilon(\gamma_j) \text{ and } 0\leq \gamma_j/\psi_\varepsilon(\gamma_j)\leq 1. \\ \text{if } \gamma_j<0, \text{ then } \psi_\varepsilon(\gamma_j)=B/\varepsilon \text{ and } -\varepsilon\leq \gamma_j/(B/\varepsilon)\leq 0. \end{array} \right.\] All in all, \(-\varepsilon\leq \gamma_j/\psi_\varepsilon(\gamma_j)\leq 1\) for \(1\leq j \leq m\). On the other hand, for \(m-k+1\leq j\leq m\), we have \(A\leq \gamma_j\), hence \(\psi_\varepsilon(\gamma_j)=\gamma_j\) and \(\gamma_j/\psi_\varepsilon(\gamma_j)=1\).
As a result, the sum of any \(m-k+1\) eigenvalues of \(\Theta_\mathcal{H}\) with resect to \(\psi_\varepsilon(\Theta_\mathcal{H})\) is at least \(1-\varepsilon(m-k)=\varepsilon>0\) as we chose \(\varepsilon=1/(m-k+1)\). By setting \(\beta=\psi_\varepsilon(\Theta_\mathcal{H})\), the proof is complete.
For the moreover part, it is clear that, when \(k=1\), statement C can be rephrased as \(\Theta_\mathcal{H}\wedge \beta^{m-1}>0\). For the equivalence to \(\Theta^{n+1}\wedge \beta^{m-1}>0\), we will use the following formula: \[\label{wzw32formula} \Theta^{n+1}\wedge\beta^{m-1} =(n+1)! (m-1)! \sum_{i,j} \beta^{i\bar{j}}\det M(i\bar{j}) \det(\beta) \big(\bigwedge^m_{k=1} i dt_k\wedge d\Bar{t}_k\wedge \bigwedge^n_{\lambda=1} i dz_\lambda\wedge d\Bar{z}_\lambda\big).\tag{13}\] Here \(M(i\bar{j})\) for fixed \(i\) and \(j\) is a matrix defined by \[\label{matrix} M(i\bar{j}):=\left ( \begin{array}{cccc} \phi_{t_i\bar{t}_j} & \phi_{t_i\bar{z}_1} & \cdots & \phi_{t_i\Bar{z}_n}\\ \phi_{z_1\bar{t}_j} & \phi_{z_1\bar{z}_1} & \cdots & \phi_{z_1\bar{z}_n}\\ \vdots & \vdots & \ddots & \vdots \\ \phi_{z_n\bar{t}_j}& \phi_{z_n\bar{z}_1} &\cdots & \phi_{z_n\bar{z}_n} \end{array} \right )\tag{14}\] with respect to local coordinates \((t_1,\dots,t_m)\) in \(Y\) and local coordinates \((t_1,\dots,t_m,z_1,\dots, z_n)\) in \(\mathcal{X}\), and \(\phi\) is a local weight for the metric \(h\), namely \(h=e^{-\phi}\). Formula (13 ) is a variant of formula (3) in [25]. To prove formula (13 ), it suffices to consider a fixed point \((t,z)\) in \(\mathcal{X}\). Let \(\alpha\) be a Hermitian metric on \(Y\) such that \(p^*\alpha=\beta\) at \((t,z)\). We then apply formula (3) in [25] to deduce formula (13 ). (Formula (13 ) is related to the Wess–Zumino–Witten equation, see [32]–[35]).
According to Schur’s formula, we have \[\label{Schur} \phi_{i\bar{j}}-\sum_{\lambda,\mu} \phi_{i\bar{\mu}}\phi^{\lambda \bar{\mu}}\phi_{\lambda \bar{j}}=\frac{\det M(i\bar{j})}{\det (\phi_{\lambda\bar{\mu}})}.\tag{15}\] A reminder on the notation: \(M(i\bar{j})\) for fixed \(i\) and \(j\) is itself a matrix given in (14 ), but \((\phi_{\lambda\bar{\mu}})\) is a matrix with \((\lambda,\mu)\) entry equal to \(\phi_{\lambda\bar{\mu}}\). By formula (13 ), the sign of \(\Theta^{n+1}\wedge \beta^{m-1}\) is decided by \(\sum_{i,j}\beta^{i\bar{j}}\det M(i\bar{j})\) which has the same sign with \[\label{sign} \sum_{i,j}\beta^{i\bar{j}} (\phi_{i\bar{j}}-\sum_{\lambda,\mu} \phi_{i\bar{\mu}}\phi^{\lambda \bar{\mu}}\phi_{\lambda \bar{j}})\tag{16}\] by formula (15 ) and the fact \((\phi_{\lambda\bar{\mu}})\) is positive definite. But the sign of \(\Theta_{\mathcal{H}}\wedge \beta^{m-1}\) is also decided by (16 ) according to (2 ). All together, \(\Theta^{n+1}\wedge \beta^{m-1}>0\) if and only if \(\Theta_{\mathcal{H}}\wedge \beta^{m-1}>0\). ◻
The case we care about most in this paper is when \(k=1\) in Lemma 2 because it corresponds to weak RC-positivity. The difficulty in proving a theorem like Theorem 1 or Theorem 2 for weak RC-positivity is that the \(\beta\) in Lemma 2 is on \(\mathcal{X}\), so it does not quite fit into Berndtsson’s computation, especially formula (12 ). So, we raise the question:
Is it possible to choose \(\beta\) in Lemma 2 so that \(\beta=p^*\alpha\) for some Hermitian metric \(\alpha\) on \(Y\)?
This question is somewhat bold because if it is possible to choose \(\beta=p^*\alpha\), then we can use [25] to deduce that if \(E\) is weakly RC-positive, then \(S^k\otimes \det E\) has positive mean curvature for \(k\geq 0\). Moreover, it is even possible to use [25] to deduce that if \(E\) is weakly RC-positive, then \(E\) has positive mean curvature. Since positive mean curvature implies RC-positivity ([1]), this would mean that RC-positivity, weak RC-positivity, and mean curvature positivity are all equivalent. Such an equivalence is conjectured for tangent bundle \(TX\) in [2].
Erdős Center, HUN-REN Rényi Institute, Reáltanoda utca 14, H-1053, Budapest, Hungary
wuuuruuu@gmail.com
Mathematics Subject Classification 32U05, 32L05, 32J25.
Keywords: RC-positivity, uniform RC-positivity, direct images.↩︎