Renormalized Chern-Gauss-Bonnet formula for complete Kähler-Einstein metrics (Q2816473)

Let \(X\) be an \((n+1)\)-dimensional compact complex manifold with strictly pseudoconvex boundary \(M\) and with a complete Hermitian metric \(g\) of the form \(g=\partial\overline\partial\log(-1/\rho)\) near the boundary for a defining function \(\rho\) of \(M\). Let \(\Theta\) denote the Bochner curvature tensor of \(g\). The author considers the renormalized Gauss-Bonnet formula \(\int_X c_{n+1}(\Theta)=\chi(X)+\mu(M,\rho)\) and presents an integral representation for the correction term \(\mu(M,\rho)=\int_M F(R,A,r)\theta\wedge (d\theta)^n\), with \(\theta=(\sqrt{-1}/2)(\overline\partial\rho-\partial\rho)|_{TM}\), \(F(R,A,r)\) an invariant polynomial in the components of the Tanaka-Webster (TW) curvature \(R\), \(A\) the torsion of the TW connection and \(r\) the transverse curvature of \(\rho\). If \(X\) admits a pseudo-Einstein contact form, then \(\mu(M):=\mu(M,\rho)\) is independent of \( \rho\) and a CR-invariant of \(M\).NEWLINENEWLINEThe author gives two examples. Let \(L\) be a negative line bundle over a two-dimensional compact complex manifold \(Y\). Assume that \(h\) is an Hermitian metric on \(L\) such that \(g=\partial\overline\partial\log h>0\) is a Kähler-Einstein metric. Consider the tube domain \(X=\{v\in L\mid (v,v)<1\}\). The explicit integral formula for \(\mu(\partial X)\) shows that this invariant depends only on \(\chi(Y)\) and the scalar and Weyl curvature of \(g\) as a Riemannian metric. The second example deals with a family of Reinhardt domains \(\Omega_r=\{(z_1,z_2,z_3)\in\mathbb C^3\,|\,\sum_{i=1}(\log|z_i|)^2<r^2\}\), \(r>0\), and leads to the result \(\mu(\partial\Omega_r)=-\frac{20\pi}{27}\frac{1}{r^3}\). In particular, \(\partial\Omega_r\) and \(\partial\Omega_{r'}\) are CR-diffeomorphic only if \(r=r'\). It follows by a theorem of \textit{C. Fefferman} [Bull. Amer. Math. Soc. 80, 667--669 (1974; Zbl 0294.32018)] that \(\Omega_r\) and \(\Omega_{r'}\) are not biholomorphic equivalent if \(r\not=r'\). For related results concerning the Gauss-Bonnet formula for strictly pseudoconvex domains in \(\mathbb C^n\), see also [\textit{D. M. Burns} and \textit{C. L. Epstein}, Acta Math. 164, No. 1--2, 29--71 (1990; Zbl 0704.32005)].