Research Title: Vanishing and Duality of Witt-divisorial sheaf cohomology

Goal: Improvement of the duality theorem for cohomology of Witt divisorial sheaves. Computation of an instructive example. Application to vanishing theorems.

Results:

In 2022 I published a 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 was some unknown torsion left that prevented a proof of proper vanishing. This torsion was caused by the first derived limit of a projective system possibly being non-zero.

This year I improved upon the duality theorem, as initially intended. Using an induction argument I was able to show vanishing of the above mentioned first derived limit, and so to remove the torsion from the result. As a corollary I used the new and improved duality to prove an improved vanishing theorem, which now completely lacks any torsion, even for general Q-divisors. This expands Tanaka’s initial vanishing result, which was the inspiration for the 2022 paper as well as this work, to general Q-Cartier divisors. The result can be seen on the arXiv (arXiv: 2305.17893).

Since then I have been working on applying the result to further generalize the vanishing in certain settings. In order to find the best way forward I consulted with several international colleagues, and solicited comments and suggestions at several talks.

Talks:

March 2023, Utsunomiya University 研究集会

“Duality and vanishing of Witt-divisorial sheaves in positive characteristic”.

November 2023, Nihon University 特異点セミナー

“Vanishing and duality of Witt-divisorial sheaves in char p”.