Derived categories related to the McKay correcepondence and Homological Mirror Symmetry

2015 Fiscal Year Final Research Report

• Project/Area Number: 23340011
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo Metropolitan University
• Principal Investigator: Uehara Hokuto 首都大学東京, 理工学研究科, 准教授 (80378546)
• Co-Investigator(Kenkyū-buntansha): TODA Yukinobu 東京大学, 国際高等研究所Kavli数物連携宇宙機構, 准教授 (20503882)
• Project Period (FY): 2011-04-01 – 2016-03-31
• Keywords: 導来圏 / トーリック多様体 / 楕円曲面

Outline of Final Research Achievements

We studied the derived categories of coherent sheaves on algebraic varieties from many aspects. We obtained (1) a counterexample to birational Torelli problem, (2) full exceptional collections on toric varieties by Frobenius morphisms, (3) a classification of exceptional sheaves on a Hirzebruch surface and (4) a description of the autoequivalences groups on minimal elliptic surfaces. We applied our study on Fourier--Mukai partners on elliptic surfaces to obtain (1). The result (2) is a generalization of the McKay correspondence. The study in (4) should be generalized in the cases of any algebraic surfaces.

Free Research Field

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