Theme 1 was successfully executed. In joint work with R.P. Thomas, I found an explicit description of the “vertical component” of the moduli space of stable pairs on K_S in the case S has a holomorphic 2-form cutting out a smooth curve. “Vertical” refers to stable pairs with underlying support curve being a thickening “into the fibre direction” of a reduced curve in the zero section. As anticipated, there are interesting applications to enumerative questions; in this case spin Hurwitz numbers. We have written 1 paper on this topic (51 pages, submitted) and I have given 4 talks on this topic at leading conferences/workshops. Theme 1 also led to a side project with A. Gholampour (and parts with B. Young) on generating functions of higher rank sheaves on threefolds. This led to a publication in IMRN (89 pages) and Selecta Math. (19 pages) as well as a very recent further preprint (33 pages, to be submitted). I have given 2 talks on this topic in leading international conferences/workshops. Output:
M. Kool and R. P. Thomas, Stable pairs with descendents on local surfaces I: the vertical component, 51 pages, arXiv:1605.02576
M. Kool, A. Gholampour, and B. Young, Rank 2 sheaves on toric 3-folds: classical and virtual counts, 62 pages, to appear in IMRN, doi.org/10.1093/imrn/rnw302. arXiv:1509.03536
M. Kool and A. Gholampour, Rank 2 wall-crossing and the Serre correspondence, Selecta Math. 23 (2017) 1599-1617. arXiv:1602.03113
M. Kool and A. Gholampour, Higher rank sheaves on threefolds and functional equations, 33 pages, arXiv:1706.05246
Theme 2 was successfully executed. In joint work with L. Göttsche, I found virtual refinements of a formula from the physics literature due to Vafa-Witten. Specifically, we conjecture a formula for the virtual \chi_y genera of moduli spaces of rank 2 stable sheaves on general type surfaces. We have written 1 paper in this topic (36 pages, submitted) and we are currently writing a second paper. I have given 2 talks on this topic in two seminars conferences/workshop.
M. Kool and L. Göttsche, Virtual refinements of the Vafa-Witten formula, 36 pages, arXiv:1703.07196
The above work is openly accessible through arXiv.org. In addition, the above work has been disseminated at 18 invited talks at (mostly international) seminars/conferences/workshops.