Aim
Provide attendees enough information of Fukaya category and derived category of coherent sheaves to understand the main statement of homological mirror symmetry. \[D^{\pi}{\rm Fuk}(X) \cong D^{b}{\rm Coh}(X^{\vee})\] Also, I hope this learning seminar will give us a good chance to communicate with others in Tsinghua University, so feel free to invite others!
Time and Place
Start from 2022.02.27.
Every Sunday, 09:30-11:00
Jinchunyuan West Building, Conference Room 1. (近春园西楼，第一会议室)
- Introduction to mirror symmetry, 02.27, Weilin Su
- Derived category and triangulated category, 03.06, Nantao Zhang
- Derived functors and some example of derived category of coherent sheaves, 03.13, Nantao Zhang
- Fukaya category, 03.27 & 04.03, Zhuoming Lan
- HMS for elliptic curves, 04.10 & 04.17, Nantao Zhang
- Constructible sheaves and Fukaya category, 04.24, Jiawei Zhou
- Introduction to toric varieties, 05.08, Jinghao Yu
- A categorification of Morelli’s theorem, 5.22 & 6.5, Luyu Zheng
- T-duality and homological mirror symmetry for toric varieties, 6.12, Nantao Zhang
Prerequisite
Some basic knowledge about differential geometry, homological algebra and category theory is required. For algebraic geometry, we assume everybody knows at least the definition of scheme and coherent sheaf. Hartshorne chapter 2 and 3 is the standard reference. Professor Sun Shenghao(孙晟昊) will teach algebraic geometry next semester but mainly about algebraic varieties (For example, Hartshorne 1). So you have to learn the language of schemes by yourself either in winter vacation or during the algebraic geometry course. Knowing algebraic topology and complex geometry will be helpful.
Syllabus
Currently, there are three speakers, Su Weilin(苏蔚琳) will talk about part 1, Lan Zhuoming will talk about 3 and Zhang Nantao will talk about 2,4,5. Everyone is welcomed to give a talk on either the topics or other aspects of homological mirror symmetry.
- History of homological mirror symmetry
- Derived category
- Fukaya category
- Derived category of coherent sheaves
- Homological mirror symmetry for elliptic curve
We choose this order because the history serves as the introduction and backgroud of the topic. The derived category is a natural sucessor of the course “Commutative algebra and homological algebra”. In case somebody doesn’t know much about algebraic geometry, we tend to put the derived category of coherent sheaves later and deal with symplectic geometry side first. You can know more about algebraic geometry from the course taught by Professor Sun Shenghao. As an example, we will prove the homological mirror symmetry for elliptic curve at last.
Notes
Below is a working draft of the lecture notes which will be continually edited and expanded during the seminar. Any comments/suggestions are welcomed.
The notes of first part of the seminar (Until 04-24): Seminar Notes I. Due to Covid-19, the seminars were then moved online.
- Notes of Jinghao Yu: Toric Varieties.
- Notes of Luyu Zheng: FLTZ.
- Notes of Nantao Zhang: FLTZ2.
References
For derived category of coherent sheaves
- Algebraic Geometry, R. Hartshorne, 1977
- Fourier-Mukai Transformation in Algebraic Geometry, D. Huybrechts, 2006
For Fukaya category
- A beginner’s introduction to Fukaya categories, D. Auroux, arxiv:1301.7056
- Floer cohomology and Fukaya category, Z. Lan, Web
For HMS for elliptic curve
- An Introduction to Homological Mirror Symmetry and the Case of Elliptic Curves, A. Port, arxiv:1501.00730
- Categorical mirror symmetry: the elliptic curve, A. Polishchuk, E. Zaslow, arXiv:math/9801119
More will be added during the seminar.
Possible Topics
HMS beyond the CY case:
- Matrix factorization category. [Dyckerhoff2011]
- Categories of singularities. [Orlov2004]
- HMS for manifolds of general type. [KKO+2009]
Examples of HMS
- HMS for projective plane. [Seidel2001]
- MHS for del Pezzo surface. [AKO2006]
- HMS for weighted projective space. [AKO2008]
- HMS for genus 2 curve. [Seidel2011]
- HMS for pairs of pants. [Sheridan2011]
- HMS for toric varieties. [FLT+2012]
- HMS for punctured spheres. [AAE+2013]
- HMS for quartic surface. [Seidel2015]
- HMS for Calabi-Yau hypersurface in projective space. [Sheridan2015]
- HMS for Fano hypersurface in projective space. [Sheridan2016]
- HMS for higher dimensional pairs of pants. [LP2020]
- HMS for hypersurface in \((\CC^{*})^{n}\). [AA2021]
[Dyckerhoff2011] Dyckerhoff, Compact Generators in Categories of Matrix Factorizations, Duke Mathematical Journal, 159(2), (2011). doi. ↩
[Orlov2004] Orlov, Triangulated Categories of Singularities and D-branes in Landau-Ginzburg Models, arXiv:math/0302304, (2004). ↩
[KKO+2009] Kapustin, Katzarkov, Orlov & Yotov, Homological Mirror Symmetry for Manifolds of General Type, Open Mathematics, 7(4), (2009). doi. ↩
[Seidel2001] Seidel, More about Vanshing Cycles and Mutation, 429-465, in in: Symplectic Geometry and Mirror Symmetry, edited by WORLD SCIENTIFIC (2001) ↩
[AKO2006] Auroux, Katzarkov & Orlov, Mirror Symmetry for Del Pezzo Surfaces: Vanishing Cycles and Coherent Sheaves, Inventiones Mathematicae, 166(3), 537-582 (2006). doi. ↩
[AKO2008] Auroux, Katzarkov & Orlov, Mirror Symmetry for Weighted Projective Planes and Their Noncommutative Deformations, Annals of Mathematics, 167(3), 867-943 (2008). ↩
[Seidel2011] Seidel, Homological Mirror Symmetry for the Genus Two Curve, arXiv:0812.1171 [math], (2011). ↩
[Sheridan2011] Sheridan, On the Homological Mirror Symmetry Conjecture for Pairs of Pants, Journal of Differential Geometry, 89(2), 271-367 (2011). doi. ↩
[FLT+2012] Fang, Liu, Treumann & Zaslow, T-Duality and Homological Mirror Symmetry for Toric Varieties, Advances in Mathematics, 229(3), 1873-1911 (2012). doi. ↩
[AAE+2013] Abouzaid, Auroux, Efimov, Katzarkov & Orlov, Homological Mirror Symmetry for Punctured Spheres, Journal of the American Mathematical Society, 26(4), 1051-1083 (2013). doi. ↩
[Seidel2015] Seidel, Homological Mirror Symmetry for the Quartic Surface, Memoirs of the American Mathematical Society, 236(1116), (2015). doi. ↩
[Sheridan2015] Sheridan, Homological Mirror Symmetry for Calabi\textendashYau Hypersurfaces in Projective Space, Inventiones mathematicae, 199(1), 1-186 (2015). doi. ↩
[Sheridan2016] Sheridan, On the Fukaya Category of a Fano Hypersurface in Projective Space, Publications math'ematiques de l’IH'ES, 124(1), 165-317 (2016). doi. ↩
[LP2020] Lekili & Polishchuk, Homological Mirror Symmetry for Higher-Dimensional Pairs of Pants, Compositio Mathematica, 156(7), 1310-1347 (2020). doi. ↩
[AA2021] Abouzaid & Auroux, Homological Mirror Symmetry for Hypersurfaces in $((\textbackslash mathbb\vphantom\C\vphantom\^*)^n\textbackslash$, arXiv:2111.06543 [math], (2021). ↩
[NZ2009] Nadler & Zaslow, Constructible Sheaves and the Fukaya Category, Journal of the American Mathematical Society, 22(1), 233-286 (2009). doi. ↩
[FLT+2011] Fang, Liu, Treumann & Zaslow, A Categorification of Morelli’s Theorem, Inventiones mathematicae, 186(1), 79-114 (2011). doi. ↩
Organizer
Zhang Nantao (张南涛), Lan Zhuoming (兰倬铭)