Branched Covers and Fiberwise Degeneration along Weak Kähler Geodesics

Abstract

On the projective line, we construct a pair of smooth Kähler potentials whose weak geodesic has isolated quadratic zeros of the fiber form at two fixed points throughout an interior time interval. The geodesic is smooth near these zeros, and its space-time form has constant rank one there. For the same fixed endpoints, the minimum fiber eigenvalue of the nondegenerate Monge-Ampère approximations tends to zero, answering AIMPL Problem 2.5 in the negative. The construction works on every compact connected Riemann surface with a prescribed Kähler form and realizes any positive even order of vanishing. Taking products yields higher-dimensional examples in which the fiber rank drops by one along a smooth hypersurface while the space-time form retains maximal rank nearby. The proof starts from a semipositive model built from a branched cover. We make its endpoints strictly positive by local corrections and use comparison along holomorphic curves to prove that the weak geodesic still agrees with the model on an explicit interior region. There a ramification point of local degree $m$ gives a fiber zero of order exactly $2m-2$.