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Math 731: Topics in Algebraic Geometry I – Abelian Varieties

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Math 731: Topics in Algebraic Geometry I – Abelian Varieties MATH 731: TOPICS IN ALGEBRAIC GEOMETRY I { ABELIAN VARIETIES BHARGAV BHATT∗ Course Description. The goal of the first half of this class is to introduce and study the basic structure theory of abelian varieties, as covered in (say) Mumford's book. In the second half of the course, we shall discuss derived categories and the Fourier{Mukai transform, and give some geometric applications. Contents 1. September 5th..................................................................................... 3 1.1. Group Schemes................................................................................ 3 1.2. Abelian Schemes............................................................................... 5 2. September 7th..................................................................................... 6 2.1. Rigidity Results................................................................................ 6 2.2. Differential Properties.......................................................................... 10 3. September 12th.................................................................................... 10 3.1. Differential Properties (Continued)............................................................. 10 4. September 14th.................................................................................... 14 4.1. Cohomology and Base Change................................................................. 14 5. September 19th.................................................................................... 18 5.1. Cohomology and Base Change (Continued)..................................................... 18 6. September 21st.................................................................................... 21 6.1. The Seesaw Theorem.......................................................................... 21 6.2. The Theorem of the Cube...................................................................... 23 6.3. Applications to Abelian Varieties............................................................... 24 7. September 26th (Lecture by Zili Zhang)............................................................ 24 7.1. Introduction to Derived Categories............................................................. 25 7.2. The Derived Category of Coherent Sheaves..................................................... 26 7.3. Triangulated Categories........................................................................ 27 7.4. Derived Functors............................................................................... 28 8. September 28th (Lecture by Zili Zhang)............................................................ 29 8.1. Derived Functors (Continued).................................................................. 29 8.2. Duality........................................................................................ 31 9. October 3rd........................................................................................ 33 9.1. The Theorem of the Cube (Continued)......................................................... 33 9.2. The Mumford Bundle.......................................................................... 34 10. October 5th...................................................................................... 36 10.1. The Mumford Bundle (Continued)............................................................ 36 10.2. Projectivity of Abelian Varieties.............................................................. 36 ∗Notes were taken by Matt Stevenson, who is responsible for any and all errors. Special thanks to Devlin Mallory for taking notes on October 12th and November 7-16. Thanks also to Attilio Castano, Raymond Cheng, Carl Lian, Devlin Mallory, Shubhodip Mondal, & Emanuel Reinecke for sending corrections. Please email [email protected] with any corrections. 1 2 BHARGAV BHATT 11. October 10th..................................................................................... 39 11.1. Projectivity of Abelian Varieties (Continued).................................................. 39 11.2. Torsion subgroups............................................................................ 41 12. October 12th..................................................................................... 43 12.1. Torsion Subgroups (Continued)............................................................... 43 12.2. The Dual Abelian Variety..................................................................... 44 13. October 19th..................................................................................... 47 13.1. The Dual Abelian Variety (Continued)........................................................ 47 13.2. Group Schemes Are Smooth In Characteristic Zero............................................ 48 13.3. Quotients by Finite Group Schemes........................................................... 49 14. October 24th..................................................................................... 51 14.1. Construction of the Dual Abelian Variety..................................................... 51 15. October 26th..................................................................................... 53 15.1. Construction of the Dual Abelian Variety (Continued)........................................ 54 15.2. Duality....................................................................................... 56 16. October 31st...................................................................................... 57 16.1. Duality (Continued).......................................................................... 57 16.2. Fourier{Mukai Transforms.................................................................... 58 17. November 2nd.................................................................................... 59 17.1. Fourier{Mukai Transforms (Continued)....................................................... 60 17.2. The Fourier{Mukai Equivalence............................................................... 60 17.3. Properties of the Fourier{Mukai Equivalence.................................................. 62 18. November 7th.................................................................................... 63 18.1. Properties of the Fourier{Mukai Equivalence (Continued)..................................... 63 18.2. Homogeneous Vector Bundles................................................................. 64 18.3. Unipotent Vector Bundles..................................................................... 65 19. November 9th.................................................................................... 66 19.1. Cohomology of Ample Line Bundles........................................................... 66 20. November 14th................................................................................... 69 20.1. Cohomology of Ample Line Bundles (Continued).............................................. 69 20.2. Stable Vector Bundles on Curves.............................................................. 70 21. November 16th................................................................................... 71 21.1. Stable Vector Bundles on Curves (Continued)................................................. 71 21.2. Atiyah's Theorem on Vector Bundles on Elliptic Curves....................................... 73 22. November 21st.................................................................................... 74 22.1. Generic Vanishing Theorems.................................................................. 74 22.2. Picard & Albanese Schemes................................................................... 75 22.3. Hacon's Theorem............................................................................. 76 23. November 28th................................................................................... 78 23.1. Hacon Complexes & Hacon Sheaves........................................................... 78 24. November 30th................................................................................... 81 24.1. Hacon Complexes & Hacon Sheaves (Continued).............................................. 81 24.2. Proof of a Generic Vanishing Theorem........................................................ 82 24.3. Jacobians..................................................................................... 83 24.4. Symmetric Powers & Jacobians............................................................... 83 25. December 5th..................................................................................... 84 25.1. Symmetric Powers & Jacobians (Continued).................................................. 84 25.2. Jacobian & the Albanese...................................................................... 86 MATH 731: TOPICS IN ALGEBRAIC GEOMETRY I { ABELIAN VARIETIES 3 25.3. Determinants................................................................................. 88 26. December 7th..................................................................................... 88 26.1. Towards the Torelli Theorem.................................................................. 89 26.2. The Torelli Theorem.......................................................................... 92 References............................................................................................. 92 1. September 5th The main reference for the first part of the course is Mumford's Abelian varieties [Mum08]. For more modern treatments, see also [Con15, Mil08, EVdGM]. For some background material, see [BLR90,
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