Mathematics / research
Complex manifolds and Kahler geometry
Connect holomorphic coordinates, Hermitian metrics, cohomology, and curvature through complex differential geometry.
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Module 01 research
Complex manifolds and Kahler geometry
Connect holomorphic coordinates, Hermitian metrics, cohomology, and curvature through complex differential geometry.
- 01
Complex manifolds and holomorphic coordinate changes
Build complex atlases through biholomorphic transitions and track derivatives, orientations, and differential forms across charts.
35 min - 02
Almost complex structures and integrability
Separate pointwise multiplication by i from the differential condition that produces genuine holomorphic coordinates.
38 min - 03
Hermitian metrics and complex differential forms
Fix metric and fundamental-form normalizations, distinguish bidegrees, and test positivity separately from closedness.
37 min - 04
Dolbeault operators and Dolbeault cohomology
Turn the bar-partial complex into global cohomology and state exactly which holomorphic sheaves its smooth resolutions compute.
39 min - 05
Holomorphic bundles and Chern connections
Construct the metric-compatible connection with prescribed holomorphic structure and keep its curvature sign fixed under frame changes.
40 min - 06
Kähler forms and Kähler identities
Relate closed positive forms, local potentials, adjoint commutators, and the factor of two between real and complex Laplacians.
40 min - 07
Hodge decomposition on compact Kähler manifolds
Use ellipticity and Kähler identities to split de Rham classes by type and derive checkable restrictions on Betti numbers.
40 min - 08
Chern curvature and positivity
Read positivity in complex tangent directions, distinguish line and vector-bundle notions, and integrate a correctly normalized Chern form.
41 min - 09
Ricci forms and the first Chern class
Take the trace of Chern curvature, verify determinant and canonical-bundle signs, and distinguish a zero class from a zero Ricci form.
40 min - 10
Complex Monge–Ampère and the Calabi–Yau theorem
Prescribe volume in a fixed compact Kähler class with explicit positivity, compatibility, and uniqueness conditions.
44 min