Hodge Theory Fall 2026 – SIMIS

Course Information

• Lecturer: Chenglong Yu
• Office: SIMIS 1106
• E-mail: yuchenglong {at} simis.cn
• Location: SIMIS 1010
• Time: Fridays, 9:50–12:30 (Shanghai time), except the holidays listed below
• Duration: Fudan teaching weeks 1–16, including holiday breaks (14 regular Friday meetings)
• First class: September 11, 2026
• TA: 黎总标
• TA e-mail: zbli24@m.fudan.edu.cn

Syllabus

The course introduces complex manifolds, Kähler manifolds, Hodge structures, and variations of Hodge structures. Through examples of curves, K3 surfaces, Calabi–Yau manifolds, and families of hypersurfaces, we will study the corresponding period domains and period maps.

Course Schedule

The schedule follows weeks 1–16 of the Fudan University 2026–2027 academic calendar. The two holiday Fridays are marked below; there are 14 regular Friday meetings between September 11 and December 25, 2026.

Class meetings: SIMIS 1010, 9:50–12:30 (Shanghai time). Week numbers follow the Fudan academic calendar and include holiday weeks.

Fudan week	Date (Friday)	Topics / arrangement
1	September 11, 2026	Topic 1 (part I). Covered: the definition and basic properties of de Rham cohomology; the Hodge star and associated inner products on differential forms. Not yet covered: Hodge (harmonic-form) decomposition in the infinite-dimensional setting. Topic 1 notes (PDF)
2	September 18, 2026	Regular class
3	September 25, 2026	No class — Mid-Autumn Festival holiday
4	October 2, 2026	No class — National Day holiday
5	October 9, 2026	Regular class
6	October 16, 2026	Regular class
7	October 23, 2026	Regular class
8	October 30, 2026	Regular class
9	November 6, 2026	Regular class
10	November 13, 2026	Regular class
11	November 20, 2026	Regular class
12	November 27, 2026	Regular class
13	December 4, 2026	Regular class
14	December 11, 2026	Regular class
15	December 18, 2026	Regular class
16	December 25, 2026	Regular class

Fudan weeks 17–18 (December 28, 2026–January 8, 2027) are reserved for final examinations, with regular classes suspended. January 1 is the New Year’s Day holiday; it is not a regular meeting of this course.

Calendar information checked on September 8, 2026.

Lectures

Topic 1: From de Rham Cohomology to Harmonic Forms (PDF)

Complete Topic 1 notes, with proofs, examples, and exercises. These include material for later classes; the schedule records what has been covered so far.

Topic 2: Complex Geometry and the Kähler Hodge Decomposition (PDF)

Homework assignments

To be announced.

Projects

To be announced.

References

To be announced.