Complements, index theorem, and minimal log discrepancies of foliated surface singularities

We present an extension of several results on pairs and varieties to foliated surface pairs. We prove the boundedness of local complements, the local index theorem, and the uniform boundedness of minimal log discrepancies (mlds), as well as establishing the existence of uniform rational lc polytopes. Furthermore, we address two questions posed by P. Cascini and C. Spicer on foliations, providing negative responses. We also demonstrate that the Grauert-Riemenschneider type vanishing theorem generally fails for lc foliations on surfaces. In addition, we determine the set of minimal log discrepancies for foliated surface pairs with specific coefficients, which leads to the recovery of Y.-A. Chen's proof on the ascending chain condition conjecture for mlds for foliated surfaces.