Restricted Bergman kernel asymptotics (Q2839945)

Let \(L\) be a big line bundle over a smooth complex projective variety \(X\) and \(Z \subseteq X\) a subvariety. By \(\iota: Z \hookrightarrow X\) we denote the inclusion map. For an integer \(m\) let NEWLINENEWLINE\[NEWLINEH^0\big(X|Z,\,{\mathcal O}(mL)\big):=\text{Im\,}\Big[\iota^*:H^0\big(X,{\mathcal O}(mL)\big) \longrightarrow H^0\big(Z,{\mathcal O}(mL)\big)\Big]NEWLINE\]NEWLINE NEWLINENEWLINEIf \(p\) is the dimension of \(Z\), the restricted volume of \(L\) is defined by NEWLINENEWLINE\[NEWLINE\text{Vol\,}_{X|Z}(L):=p!\limsup_{m\to \infty} \,\frac{\text{dim\,} H^0\big(X|Z,\,{\mathcal O}(mL)\big)}{m^p} \,. NEWLINE\]NEWLINE Next assume that \(h_L\) is a smooth hermitian metric on \(L\) and \(\varphi\) a smooth weight function. Then, given a volume form \(d \mu\) on \(Z\), the restricted Bergman kernel of \((Z,mL,h_L^me^{-m\varphi},d\mu)\) is defined as follows: NEWLINE\[NEWLINEB_{X|Z} (m \varphi):= |s_{m,1}|_{m\varphi}^2 +...+ |s_{m,N(m)}|_{m\varphi}^2\,,NEWLINE\]NEWLINE where \(\{s_{m,1},...,s_{m,N(m)}\}\) denotes a complete orthonormal basis of \(H^0\big(X|Z,{\mathcal O}(mL)\big)\) with respect to the norm \(\| s \|_{m\varphi}^2:=\int_Z \iota^* h_L^m (s,s) e^{-m\iota^*\varphi} d\mu\).NEWLINENEWLINEThe author considers the case that \(Z\) is not contained in the augmented base locus of \(L\). As a main result he establishes an asymptotic formula for \(B_{X|Z} (m \varphi)\), namely NEWLINE\[NEWLINE p!\frac{B_{X|Z} (m\varphi)}{m^p} d\mu \longrightarrow \langle (\theta +dd^c P_{X|Z}\varphi)^p\rangleNEWLINE\]NEWLINE in the sense of currents, if \(m \to \infty\). Here, \(\theta\) dentotes the Chern curvature form of \(h_L\) and \(P_{X|Z}\varphi\) the equilibrium weight associated to \(\varphi\).NEWLINENEWLINEThen, as an application, he proves integral representations for restricted volumes.