On the holomorphic invariants for generalized Kähler-Einstein metrics (Q2518786)

The main result of the paper under review is the following: Let \((M,g)\) be a Fano manifold with Kähler metric \(g\) which represents the Chern class \(c_1(M)\). Let \(\nu_{\omega_g}\) the extremal Kähler vector field. For the multiplier Hermitian structure of type \(\sigma(s)=-\log(s+c)\) for some \(c>\alpha_M\) with respect to \(X=-\nu_{\omega_g}\), the holomorphic invariant \(F^{\sigma}_X\) vanishes if and only if \(\alpha_M<1\). Before explaining the result and its notations, let us state a corollary of the previous result, which was already found by different methods in \textit{T. Mabuchi} [Tohoku Math. J., II. Ser. 53, No. 2, 171-182 (2001; Zbl 1040.53084)]: \(\mathbb{CP}^2\#\overline{\mathbb{CP}^2}\) admits generalized Kähler-Einstein metrics in \(c_1(M)\). Notice that by \textit{Y. Matsushima} [Nagoya Math. J. 11, 145--150 (1957; Zbl 0091.34803)] the above example is a non Kähler-Einstein manifold, so that the above generalized metrics are not the standard ones. Now, we briefly explain the notations. Let \(M\) be a \(n\)-dimensional Fano manifold with Kähler form \(\omega\) in the canonical class and let \(h_\omega\) be the Einstein discrepancy function defined by \[ \text{Ric}(\omega)-\omega=\frac{i}{2\pi}\partial\overline\partial h_{\omega},\qquad \int_M e^{h_\omega}\,\omega^n=\int_M\omega^n. \] The Futaki invariant \(F:\mathfrak h(M)\to\mathbb C\), where \(\mathfrak h(M)\) is the Lie algebra of holomorphic vector fields, given by \[ F(X)=\int_M X(h_\omega)\,\omega^n, \] is well-known to be an obstruction for the existence of Kähler-Einstein metrics in Fano manifolds, that is Kähler metrics that are proportional to their Ricci form. Following \textit{T. Mabuchi} [Tohoku Math. J., II. Ser. 53, No.~2, 171--182 (2001; Zbl 1040.53084)], if the Futaki invariant is non vanishing the metric \(\omega\) is said to be generalized Kähler-Einstein if the complex gradient vector field \[ \text{grad}^{\mathbb C}_{\omega}(1-e^{h_\omega}):=\frac{1}{i}\sum_{j,k}\omega^{jk}\frac{\partial(1-e^{h_\omega})}{\partial\overline z_k}\,\frac{\partial}{\partial z_j} \] of \(1-e^{h_\omega}\) is holomorphic (here we have fixed local holomorphic coordinates \((z_j)\) and the \(\omega^{jk}\)'s are the components of the inverse of the metric \(\omega\) written in these coordinates). Let \[ \widetilde{\mathfrak k}_\omega:=\left\{\varphi\in C^{\infty}(M)\mid\text{grad}^{\mathbb C}_\omega\varphi\in\mathfrak h(M),\quad\int_M \varphi\,\omega^n=0\right\} \] and let \(\text{pr}: L^2(M,\omega)\to\widetilde{\mathfrak k}_\omega\) be the orthogonal projection, where \(L^2(M,\omega)\) is the Hilbert space of all real-valued \(L^2\) functions in \(M\) with respect to \(\omega\). Then, the extremal Kähler vector field is defined by \[ \nu_\omega:=\text{grad}^{\mathbb C}_\omega\text{pr}(s(\omega)-\widehat s)= \text{grad}^{\mathbb C}_{\omega}\text{pr}(1-e^{h_\omega}), \] where \(s(\omega)\) is the scalar curvature and \(\widehat s\) its average. Next, we let \(\alpha_M\) to the maximum over \(M\) of \(\text{pr}(1-e^{h_\omega})\), which can be seen to be independent of \(\omega\) varying in the canonical class. We are left to describe what a multiplier Hermitian structure of type \(\sigma\) is (here, \(\sigma\) is a real-valued smooth function with certain conditions on the first two derivatives). Suppose \(X\) is a non trivial Hamiltonian holomorphic vector field, that is to say, to each \(\omega\) in the canonical class such that the Lie derivative \(L_{X+\overline X}\omega=0\), we can associate a real-valued function \(\theta_{X,\omega}\in C^{\infty}(M)\) such that \[ X=\text{grad}^{\mathbb C}_{\omega}\theta_{X,\omega},\quad\int_M \theta_{X,\omega}\,\omega^n=0. \] Then, associated to each such \(\omega\) with zero Lie derivative with respect to \(X+\overline X\), there is a multiplier Hermitian metric of type \(\sigma\), defined as the conformal Kähler metric given by \[ \widetilde\omega:=\omega\exp(-\sigma(\theta_{X,\omega})/n). \] To finish with, the holomorphic invariant \(F^{\sigma}_X\) is an analogue of the Futaki invariant and it is an obstruction for the existence of Einstein multiplier Hermitian metrics. Its formal definition is: \[ F^{\sigma}_X(Y)=\int_M Y(h_\omega+\sigma(\theta_{X,\omega}))e^{-\sigma(\theta_{X,\omega})}\,\omega^n,\quad Y\in\mathfrak h(M). \]