| 研究実績の概要 |
I published paper (a) constructing perverse sheaves of categories on the complex plane, thought of as the complexification of a one-parameter GIT stability space. In a new preprint (b), I constructed and studied perverse sheaves of categories on Riemann surfaces. I also studied possible extensions of the results to multi-parameter variation of GIT.In a picture outlined by Kapranov-Schechtman, the complex plane above can be taken as a chart on some Riemann surface with non-trivial topology reflecting relations between derived symmetries. In preprint (b), I worked this out in detail for standard flops, including flops of orbifold projective spaces. I also constructed a schober on a (partial compactification) of a stringy Kaehler moduli space: this has interesting mirror symmetry applications, discussed in 8.I worked on extending preprint (a) to multi-parameter variations of GIT, to provide tools to construct perverse sheaves of 3-fold categories (research area 1). I was not able to establish a general theory, but I obtained useful results in examples.
Bondal-Kapranov-Schechtman published an important paper (c): I worked to combine my results with theirs, to make progress in research area 1.(a) Perverse schobers and wall crossing (FY2017, published IMRN),(b) Perverse schobers on Riemann surfaces: constructions and examples (FY2017, submitted),(c) Perverse schobers and birational geometry (Bondal, Kapranov, and chechtman, arXiv:1801.08286, published Selecta Math)
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
The results of preprint (b) are satisfying, unifying known features of the behaviour of derived categories for toric flops. I believe such results will hold in much greater generality, so that extending them to other flops is a fruitful new area of research. The results raise new questions in mirror symmetry, which were not anticipated at the time of writing the research proposal: these are discussed in 8.My work on multi-parameter variation of GIT has made progress towards the goals of research area 1. This work has been slowed by technical problems in defining perverse sheaf of categories on higher-dimensional bases, however this issue has now been addressed by Bondal-Kapranov-Schechtman in paper (c). Although I have not been able to establish a general theory, I have calculated examples and established technical tools which will be useful in continuing research.
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| 今後の研究の推進方策 |
In preprint (b), I constructed a schober on a stringy Kaehler moduli space for a standard 3-fold flop. Via mirror symmetry, I believe this can be related to a naturally-defined schober on the mirror complex structure moduli space: generic fibres of this schober will be given by Fukaya categories. This should allow the perverse sheaves of categories in paper (a) and preprint (b) to be reinterpreted in Fukaya-categorical terms, yielding new insights. To achieve this, I will study Fukaya categories appearing in toric mirror symmetry, for which much is known.These toric results should also be a testing ground for more general results.I will continue to work to extend the methods of paper (a) to multi-parameter variation of GIT, to make progress in research area 1.My position at IPMU ends in May 2018, so time is limited: I therefore plan to focus on specific examples related to the sl(3) case addressed in paper (c). In paper (a), I discussed a concept of `categorified intersection cohomology': I would like to test whether the examples fit this framework. As part of my work, I will travel to UK in Apr 2018 to consult with E. Segal and M. Wemyss, who are experts on derived categories and GIT.
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| 次年度使用額が生じた理由 |
The publication of paper (c) by Bondal-Kapranov-Schechtman in Jan 2018 has given new opportunities to make progress in research area 1. To take advantage of these, I plan to extend the working period by two months. I also plan to use funds to travel to UK in Apr 2018 for consultation with E. Segal and M. Wemyss, experts on derived categories and GIT, and for possible further collaboration and conference travel.
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