# Student Differential Geometry Seminar — Spring 2020: The Inhomogeneous ˉ∂-Equation

## Organizers

- El Mehdi Ainasse and Jordan Rainone.

## Meeting times

- Fridays, 1:00 PM to 2:00 PM – starting Friday, January 31st 2020.

## Location

- Mathematics Building, Room 5-127 (5th floor).
- Zoom starting April 3rd, 2020.

## Description

Welcome to the webpage for the Spring 2020 iteration of the SDGS! This semester, we decided to go for the inhomogenous \(\bar{\partial}\)-equation and its (complex) geometric applications.

More precisely, the goal of this seminar is three-fold: (1) studying bits and pieces of several complex variables (SCV) with an analytic introduction to the \(\bar{\partial}\)-operator, (2) covering some basics in complex differential geometry with a geometric take on the \(\bar{\partial}\)-operator, and then (3) exploring applications of the (\(L^2\) theory) of the \(\bar{\partial}\)-operator to various geometric problems.

Any suggestions are welcome!

## References

Here are some links to relevant materials in different forms:

### Introductory articles:

### Books:

- Tasty Bits of Several Complex Variables, A whirlwind tour of the subject, By: Jiří Lebl. Version 3.2, October 1st, 2019, 182 pages.
- Complex Analytic and Differential Geometry. By: Jean-Pierre Demailly. Version of Thursday June 21, 2012.
- Holomorphic Function Theory in Several Variables - An Introduction. By: Christine Laurent-Thiébaut.
- Analytic and algebraic geometry : common problems, different methods, volume 17 of IAS/PARK CITY Mathematics Series. American Mathematical Soc., 2010. By: Jeffery D. McNeal and Mircea Mustata.

### Papers:

- \(L^2\) ESTIMATES FOR THE \(\bar{\partial}\) OPERATOR. By: Jeffery D. McNeal and Dror Varolin.
- A Survey on the \(L^2\) Extension Theorems. By: Takeo Ohsawa.

### Lecture notes:

- Bo Berndtsson’s notes.
- Notes for the course “MATH 710: TOPICS IN MODERN ANALYSIS II – \(L^2\)-METHODS” (taught by Mattias Jonsson), taken by Matt Stevenson.
- MSRI Summer School notes (cf. below.)
- SEVERAL COMPLEX VARIABLES. By: ZBIGNIEW BLOCKI.
- Complex analysis, the \(\bar{\partial}\)-Neumann problem, and Schrödinger operators. By: Friedrich Haslinger.
- Dror’s notes for the “CAUCHY-RIEMANN EQUATIONS IN HIGHER DIMENSIONS” program held at ICTS, Banglore.

### Videos:

- Dror Varolin’s SCGP talks.
- MSRI Summer School lectures. (cf. below.)

### MSRI Summer School — The ˉ∂-Problem in the Twenty-First Century:

## Schedule

Speaker | Date | Topic |
---|---|---|

Jae Ho Cho | 01/31/2020 | An introduction to the \(\bar{\partial}\)-method |

Pranav Upadrashta | 02/07/2020 | Basic properties of holomorphic functions of several variables |

Willie Rush Lim | 02/14/2020 | Plurisubharmonic functions |

Emily Schaal | 02/21/2020 | Pseudoconvexity |

Roberto Albesiano | 02/28/2020 | \(L^2\) estimates for the \(\bar{\partial}\)-operator |

El Mehdi Ainasse | 03/06/2020 | Hörmander’s theorem and its twisted versions |

Marlon de Oliveira Gomes | 03/13/2020 | The geometry of bundles and notions of positivity |

Jordan Rainone | 04/03/2020 | \(L^2\) estimates for \(\bar{\partial}\) on Riemann surfaces and applications |

Conghan Dong | 04/17/2020 | \(L^2\) estimates for the \(\bar{\partial}\)-equation on Kähler manifolds |

Conghan Dong | 04/24/2020 | Applications of \(L^2\) techniques on complex manifolds |

El Mehdi Ainasse | 04/24/2020 | Berndtsson’s Nakano-positivity theorem and optimal \(L^2\) extension theory (Slides) |

Roberto Albesiano | 05/1/2020 | The Kodaira embedding theorem |