LMS-CMI School
“Derived categories, Weyl algebras and Hodge theory”

Lecture 1: Homology, cohomology, intersection pairings. Examples from algebraic geometry: weak Lefschetz theorem, Grassmanians.

Lecture 2: Local systems, stratifications, constructible sheaves. Examples on curves.

Lecture 3: The derived category of constructible sheaves. Intersection cohomology complexes.

Lecture 4: T-structures, glueing, perverse sheaves, examples.

Lecture 5: Decomposition theorem. Examples.

Lecture 1: Differential operators, D-modules, pullback and pushforward on affine varieties.

Lecture 2: D-modules on general varieties. Kashiwara theorem.

Lecture 3: Coherent D-modules. Characteristic variety. Duality.

Lecture 4: Holonomic D-modules. Relation with local systems. Riemann-Hilbert correspondence on curves.

Lecture 5: Regular connections. Riemann-Hilbert correspondence.

Lecture 1: Differental forms. Dolbeault cohomology. Hodge decomposition. Example: projective nonsingular curves.

Lecture 2: The “Kähler package”: Hard Lefschetz theorem, Riemann bilinear relations. Application: degeneration of the Leray spectral sequence of a smooth projective map.

Lecture 3: Mixed Hodge structures. The yoga of weights. The global invariant cycle theorem. The theorem on semisimplicity of monodromy. Example: a quasiprojective, possibly singular curve.

Lecture 4: The ample cone of a projective variety and its boundary points. The Hard Lefschetz theorem for “lef” line bundles. The decomposition theorem for semi-small maps.

Lecture 5: A sketch of the proof of the decomposition theorem.