The first terms in the expansion of the Bergman kernel in higher degrees (Q2346680)

Let \((X,\omega,J)\) denote a compact Kähler manifold of complex dimension \(n\). Suppose further that we are given a holomorphic hermitian line bundle \((L,h^L) \) on \(X\), and a hermitian complex vector bundle \((E,h^E)\). The line bundle \(L\) is endowed with its Chern connection \(\nabla^L\) and \(E\) with a hermitian connection \(\nabla^E\). The curvatures are \(R^L=(\nabla^L)^2\) and \(R^E=(\nabla^E)^2\). Let \(g^{TX}\) be the Riemann metric on \(TX\) induced by \(\omega\) and \(J\). It induces a metric \(h^{\Lambda^{0, \bullet}}\) on \(\Lambda^{0, \bullet}(T^*X):=\Lambda^\bullet (T^{*\,(0,1)}X)\). By \(L^p\) we denote the \(p\)-th tensor power of \(L\) and define \[ \Omega^{0, \bullet}(X,L^p\otimes E)=C^\infty(X,\Lambda^{0, \bullet}(T^*X)\otimes L^p \otimes E) \] and the Dolbeault operator \(\bar \partial ^{ L^p \otimes E }: \Omega^{0, \bullet}(X,L^p\otimes E) \to \Omega^{0, \bullet +1}(X,L^p\otimes E)\), induced by the \((0,1)\)-part of \(\nabla^E\). Let \(\bar \partial ^{ L^p \otimes E.* }\) denote its dual with respect to the \(L^2\) product. Finally let \[ D_p=\sqrt{2} ( \,\bar \partial ^{ L^p \otimes E } + \bar \partial ^{ L^p \otimes E.* } \,)\,. \] This gives rise to a Bergman projector \(P_p\), which is defined as the orthogonal projection \(P_p: \Omega^{0, \bullet}(X,L^p\otimes E) \to \text{ker} (D_p)\). It has a smooth kernel \(P_p(x,y)\) with respect to the volume form \(dv_X(y)\), which is called the Bergman kernel. From now on we suppose that \(\omega = \frac{i}{2\pi} R^L\) (prequantization condition). The following asymptotic expansion is due to \textit{X. Dai} et al. [J. Differ. Geom. 72, No. 1, 1--41 (2006; Zbl 1099.32003)]: There exist sections \(b_r \in C^\infty (X, \text{End} (\Lambda^{0,\text{even}}(T^*X) \otimes E))\) such that for any \(k \in \mathbb N\) one has \[ p^{-n}P_p(x,x) =\sum_{r=0}^k b_r(x)p^{-r} + O(p^{-k-1}) , \] when \(p \to \infty\). By \(I_j\) we denote the orthogonal projection \(I_j:\Lambda^{0,\bullet}(T^*X) \otimes E\to \Lambda^{0,j}(T^*X)\otimes E\). Then one can state the first main result established by the authors as follows: For any \(k \in \mathbb N\) with \(k \geq 2j\) we have \[ p^{-n}I_{2j}P_p(x,x)I_{2j} = \sum_{r=2j}^k I_{2j}b_r(x)I_{2j} p^{-r} + O(p^{-k-1}) , \] when \( p \to \infty\). In a second main theorem the authors prove an explicit formula for the term \(I_{2j}b_{2j+1}I_{2j}\). Finally a third main result describes explicitly the term \(I_{2j}b_{2j+2}(x)I_{2j}\) under the condition that \[ I_{2j}b_{2j}(x)I_{2j}=I_{2j}b_{2j+1}(x)I_{2j}=0. \]