Theory of schemes and stacks, Derived Categories and Intersection theory II
This course is targeting 3 major segments of algebraic geometry. The theory of schemes and stacks following Grothendieck’s EGA / SGA and Deligne’s work on stacks. The derived category of coherent sheaves, its properties and relevance to birational geometry and moduli theory, and intersection theory on schemes and stacks. Students who passed a basic course in algebraic geometry (e.g. Hartshorne’s first 3 chapters) must be able to follow the course with no further pre-requisite. The course will offer an extensive analysis of all three segments and aims at providing the detailed background for algebraic geometry students to perform advanced research on geometry of schemes and moduli, say in the context of birational geometry, enumerative algebraic geometry or derived algebraic geometry.
Lecturer
Date
21st February ~ 20th June, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Friday | 09:50 - 12:15 | Shuangqing-B725 | ZOOM 09 | 230 432 7880 | BIMSA |
Prerequisite
Basic course in Algebraic Geometry (first 3 chapters of Hartshorne)
Syllabus
1. Review of Scheme theory
• 1.1: Review of Schemes (Hartshorne or EGA I, II)
• 1.2. Review of Sheaf Cohomology (EGA 3)
• 1.3. Residue and Duality (EGA 10)
2. Theory of stacks
• 2.1 Champs Algebrique (G. Lamoun)
• 2.2. Sheaf Cohomology, Lefschetz Local to global theorems (SGA 2)
3. Intersection theory
• 3.1 Intersection theory and Grothendieck -Riemann-Roch (EGA 11, SGA 6)
• 3.2 Fulton’s deformation to normal cone (Fulton’s Intersection theory)
• 3.3. Computations (Fulton’s Intersection theory)
4. Derived categories of coherent sheaves:
4.1. Derived Category of coherent sheaves, Properties (FM transforms in AG, Daniel Huybrechts) 4.2 Derived Category and canonical bundle (FM transforms in AG, Daniel Huybrechts)
4.3. FM transforms (FM transforms in AG, Daniel Huybrechts)
4.4. Spherical and Exceptional objects (FM transforms in AG, Daniel Huybrechts)
5. Let us now compute! Intersection theory in derived category:
5.1. Counting Curves in derived category (Fulton’s intersect. theory and collective papers in DT/GW) 5.2. Counting surfaces in derived category (Fulton’s intersect. theory and collective papers in DT/GW)
Note: This is the collective syllabus for both Fall 2024 and Spring 2025 courses combined.
• 1.1: Review of Schemes (Hartshorne or EGA I, II)
• 1.2. Review of Sheaf Cohomology (EGA 3)
• 1.3. Residue and Duality (EGA 10)
2. Theory of stacks
• 2.1 Champs Algebrique (G. Lamoun)
• 2.2. Sheaf Cohomology, Lefschetz Local to global theorems (SGA 2)
3. Intersection theory
• 3.1 Intersection theory and Grothendieck -Riemann-Roch (EGA 11, SGA 6)
• 3.2 Fulton’s deformation to normal cone (Fulton’s Intersection theory)
• 3.3. Computations (Fulton’s Intersection theory)
4. Derived categories of coherent sheaves:
4.1. Derived Category of coherent sheaves, Properties (FM transforms in AG, Daniel Huybrechts) 4.2 Derived Category and canonical bundle (FM transforms in AG, Daniel Huybrechts)
4.3. FM transforms (FM transforms in AG, Daniel Huybrechts)
4.4. Spherical and Exceptional objects (FM transforms in AG, Daniel Huybrechts)
5. Let us now compute! Intersection theory in derived category:
5.1. Counting Curves in derived category (Fulton’s intersect. theory and collective papers in DT/GW) 5.2. Counting surfaces in derived category (Fulton’s intersect. theory and collective papers in DT/GW)
Note: This is the collective syllabus for both Fall 2024 and Spring 2025 courses combined.
Reference
Grothendieck EGA I,II, III, X, XI
Grothendieck SGA VI
Champs Algebrique (G. Lamoun)
Fourier Mukai Transform in Algebraic Geometry (Daniel Huybrechts) Intersection theory (W. Fulton)
Grothendieck SGA VI
Champs Algebrique (G. Lamoun)
Fourier Mukai Transform in Algebraic Geometry (Daniel Huybrechts) Intersection theory (W. Fulton)
Audience
Graduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Artan Sheshmani is a Professor of pure Mathematics, specialized in Algebraic geometry, Enumerative and Derived Geometry, and Mathematics of String Theory. He is a Professor at Beijing Institute of Mathematical Sciences and Applications in Beijing, and a senior personnel (Professor) at Simons Collaboration Program on Homological Mirror Symmetry ( Harvard University Center for Mathematical Sciences and Applications), and an Affiliate Faculty Member at Harvard University- MIT IAiFi (Institute for Artificial Intelligence and Fundamental Interactions). Between 2020 and 2023, he jointly held the visiting professor position at Institute for the Mathematical Sciences of the Americas at University of Miami, where he was part of the research collaboration program on "Hodge theory and its applications". During the past 5 years while at Harvard CMSA he was also a visiting professor at Harvard Physics department (2020-2022), and an Associate Professor of Mathematics at Institut for Mathematik (formerly the Center for Quantum Geometry of Moduli Spaces) at Aarhus University in Denmark (2016-2022). His work is mainly focused on Gromov Witten theory, Donaldson Thomas theory, Calabi-Yau geometries, and mathematical aspects of String theory. He studies geometry of moduli spaces of sheaves and curves on Calabi Yau spaces, some of which arise in the study of mathematics of string theory. In his research he has worked on understanding dualities between geometry of such moduli spaces over complex varieties of dimension 2,3,4 and currently he is working on extension of these projects from derived geometry and geometric representation theory point of view. In joint work with Shing-Tung Yau (BIMSA, YMSC, Tsinghua, Harvard Math, Harvard CMSA, and Harvard Physics departments), Cody Long (Harvard Physics), and Cumrun Vafa (Harvard Math and Physics departments) he worked on geometry moduli spaces of sheaves with non-homolomorphic support and their associated non-BPS (non-holomorphic) counting invariants. In 2019 he was one of the 30 winners of the IRFD "Research Leader" grant (approx 1M USD) on his project "Embedded surfaces, dualities and quantum number theory". The project was additionally co-financed by Harvard University CMSA and Aarhus University (Approx total. 400K USD). Detail of IRFD "Research Leader" grant: https://dff.dk/en/grants/research-leaders-2018.