\(L^2\) harmonic forms for a class of complete Kähler metrics (Q1396317)

One would like to establish versions of the Hodge theorem for Kähler manifolds in various complete, noncompact settings. Such results have been obtained on manifolds with cylindrical ends by Atiyah, Patodi, and Singer, and for Poincaré metrics by Zucker. Here the author examines this question for a larger class of metrics. They are complete on the complement of a divisor in a projective variety. Let \(\overline{M} \) be a complete smooth algebraic variety containing \(D,\) an ample divisor with normal crossings. The metrics considered are of the form \[ \omega _{f} = \frac{i}{2\pi } (\partial \overline{\partial } f (-\log \parallel s\parallel ^{2})) + K \omega _{0} , \] where \(s\) is a defining section of \([D] \) and \( \omega _{0} \) is a Kähler form on \( \overline{M} .\) The various objects appearing here must satisfy certain conditions, especially \(f.\) In fact, specific choices of \(f\) yield distinguished metrics appearing elsewhere in the literature. The first result is a vanishing theorem, stating that for the choices \(f(x) = x^{a} \) with \( a>1, \) or \( f(x) = e^{bx} \) with \( b>0,\) one has \( \mathcal{H} ^{i}_{(2)} (M, \omega _{f}) = 0, \) for \( i\not= n.\) In the latter case, that of \( e^{bx}, \) the middle-degree cohomology is isomorphic to the middle-degree primitive cohomology of \( \overline{M} .\) Further results describe the \(L^{2} \) holomorphic and anti-holomorphic forms.