In 2023 I proved an improved duality theoremand vanishing theorem for Witt divisorial sheaves. This year I further improved the result and expandedit to include nef and big divisors in the case of surfaces. Since then I have been working on applyingthe result to obtain interesting results in the original setting of a smooth surfacein positive characteristic、as well as providing illuminating real-world examples.

Research Title: Vanishing and Duality of Witt-divisorial sheafcohomologyGoal: Improvement ofthe duality theorem for cohomology of Witt divisorial sheaves.Computation of an instructive example. Application to vanishingtheorems.Results: In 2022 I publisheda duality result for sheaf cohomology of Witt divisorial sheaves(Duality for Witt Divisorial Sheaves, Arkiv för Matematik 60 (2022),no. 1, 107-124.). However, for general Q-Cartier divisors, there wassome unknown torsion left that prevented a proof of proper vanishing.This torsion was caused by the first derived limit of a projectivesystem possibly being non-zero.This year I improvedupon the duality theorem, as initially intended. Using an inductionargument I was able to show vanishing of the above mentioned firstderived limit, and so to remove the torsion from the result. As acorollary I used the new and improved duality to prove an improvedvanishing theorem, which now completely lacks any torsion, even forgeneral Q-divisors. This expands Tanaka’s initial vanishing result,which was the inspiration for the 2022 paper as well as this work, togeneral Q-Cartier divisors. The result can be seen on the arXiv(arXiv: 2305.17893).Since then I havebeen working on applying the result to further generalize thevanishing in certain settings. In order to find the best way forwardI consulted with several international colleagues, and solicitedcomments and suggestions at several talks. Talks:March 2023,Utsunomiya University 研究集会“Duality andvanishing of Witt-divisorial sheaves in positive characteristic”.November 2023, NihonUniversity 特異点セミナー“Vanishing andduality of Witt-divisorial sheaves in char p”.