Hodge decomposition theorem on strongly Kähler Finsler manifolds

Abstract

Faran posed an open problem about analysis on complex Finsler spaces: Is there an analogue of the \(\bar \partial \)-Laplacian? Is there an analogue of Hodge theory? Under the assumption that (M, F) is a compact strongly Kahler Finsler manifold, we define a \(\bar \partial \)-Laplacian on the base manifold. Our result shows that the well-known Hodge decomposition theorem in Kahler manifolds is still true in the more general compact strongly Kahler Finsler manifolds.