Advanced topics in Geometry

AY 2023/2024

1st semester, 48 hours.

Lectures and exercises (in variable ratio), for Mathematics.

Recommended textbooks

• J.-P. Demailly "Complex analytic and differential geometry"

• D. Huybrechts "Complex geometry. An introduction"

• F. Zheng "Complex differential geometry"

• C. Voisin "Hodge Theory and Complex Algebraic Geometry I"

• P. Griffiths, J. Harris "Principles of algebraic geometry"

Examination procedures

The exam will consist in an oral test.

Exercises sheets

• Exercises sheet n° 1 (basics on holomorphic functions, differential forms, complex manifolds, vector bundles).

• Exercises sheet n° 2 (holomorphic vector bundles & Co). Still in progress

Logbook (click here for the course notes; starting from 28/11 you can find notes in Italian clicking here and looking about the corresponding topics)

1. 03/10/2023 (Lecture) Aula C, 17:15 - 19:00.
   Introduction to the course, and logistic. Basics on complex linear algebra: (quasi-)complex structure on a real vector space, from real to complex and viceversa, realification of a complex basis, complex linear and real linear maps in terms of complex structures and matrices, standard orientation of a complex vector space. Complexification of a real vector space, complexification of the quasi-complex structure, eigenspaces of the complexified complex structure. Examples.
   You may want to also take a look here.

2. 05/10/2023 (Lecture) Aula G, 08:15 - 10:00.
   Recap. Various duals, and their complexification and consequent decomposition. Construction of natural bases on all these spaces, exterior algebra and its bigraduation, example: real (p,p)-forms in coordinates. Definition of holomorphic function of several complex variables, Cauchy-Riemann equations, polydiscs vs balls, distinguished boundary of a polydisc, Cauchy integral formula in several variables.

3. 10/10/2023 (Lecture) Aula C, 17:15 - 19:00.
   Polydiscs, distinguished boundary of a polydisc, Cauchy integral formula in several variables, power series expansion, consequences. Analytic continuation, Cauchy inequalities, Liouville's Theorem, Hartogs and Riemann Extension Theorems. The ring of germs of holomorphic function at the origin is the ring of convergent power series, it is a local noetherian ring which is factorial. Germ of a set, analytic germ, analytic subsets, correspondence between ideal of an analytic germ and zeros of an ideal of germs of holomorphic functions, irreducibility, analytic Nullstellensatz, local representation of analytic germs (analytic Noether's Normalization Lemma).

4. 12/10/2023 (Lecture) Aula G, 08:15 - 10:00.
   Holomorphic maps with values in C^m, biholomorphisms, real vs complex Jacobian of a holomorphic map, inverse and implicit function theorem, constant rank theorem, the Jacobian of a bijective holomorphic map does not vanish. Complex manifolds, examples, sheaf of holomorphic functions on a complex manifold, locally closed analytic subsets and complex submanifolds of a complex manifolds, complex projective varieties are closed analytic subsets. Sheaf of meromorphic functions on a complex manifold, zeros and poles of a meromorphic function.

5. 17/10/2023 (Lecture) Aula C, 17:15 - 19:00.
   Meromorphic functions field of a connected complex manifold, and algebraic dimension. Regular point of analytic subsets, structure of irreducible components of closed analytic subsets with respect to the connected components of the regular points. Group of analytic q-cycles and Weil divisors. Order of a meromorphic function along an irreducible divisor, divisor of a meromorphic function, order as a valuation on the meromorphic functions field. Quasi-complex structure on the real tangent space of a complex manifold.

6. 19/10/2023 (Lecture) Aula G, 08:15 - 10:00.
   Quasi-complex structures on a real differentiable manifold, quasi-complex manifolds, integrable complex structures, Nijenhuis tensor, statement of the Newlander-Nirenberg Theorem. Quasi-complex structure on S^2 and S^6, Hopf problem. Holomorphic and anti-holomorphic tangent bundle, decomposition of k-forms in (p,q)-forms, corresponding decomposition of the de Rham exterior differential. Dolbeault complex and Dolbeault cohomology on a complex manifold, Bott-Chern cohomology. The Bott-Chern cohomology is finer than both Dolbeault and de Rham cohomology, Dolbeault and de Rham cohomology are in general not comparable.

7. 24/10/2023 (Lecture) Aula C, 17:15 - 19:00.
   Holomorphic maps between complex manifolds, the pull-back of a (p,q)-form by a holomorphic map is still a (p,q)-form, induced morphism in Dolbeault cohomology by a holomorphic map. The Dolbeault-Grothendieck lemma. Definition of real or complex vector bundle, system of transition matrices, cocycle condition, functorial operations on vector bundles, especially in terms of the defining cocycles.

8. 26/10/2023 (Lecture) Aula F, 08:15 - 10:00.
   Examples of vector bundles defined via cocycles, holomorphic vector bundles, sheaves of sections of vector bundles as locally free sheaves of modules over smooth/holomorphic functions, the Dolbeault operator on (p,q)-forms with values ina holomorphic vector bundle, Dolbeault cohomology of holomorphic vector bundles, morphisms of vector bundles. Tautological line bundle on the projective space, and its transition functions.

9. 31/10/2023 (Lecture) Aula C, 17:15 - 19:00.
   Reprise on the tautological line bundle on the projective space, and its relatives O(k). Local holomorphic frames, trivializations, transition functions. Description of global holomorphic sections of O(k), structure of the graded algebra direct sum of the global holomorphic sections of O(k), relation with the homogeneous coordinate ring of P^n. The line bundle O(-1) is not topologically trivial. Example: the holomorphic tangent bundle to P^1 is isomorphic to O(2).

10. 02/11/2023 (Lecture) Aula F, 08:15 - 10:00.
    Reprise on the isomorphism between the holomorphic tangent bundle to P^1 and O(2). Canonical line bundle of a complex manifold, the canonical lline bundle of P^n is O(-n-1). Euler exact sequence, determinant of a short exact sequence of vector bundles. Structure of group on the set of isomorphism classes of holomorphic line bundles, the Picard group of a complex manifold. Hermitian metrics on complex vector bundles, special case of line bundles. Disquisition on the Chern connection, Chern curvature of a Hermitian holomorphic line bundle and its well-definiteness. The Chern curvature of a Hermitian holomorphic line bundle is a closed real (1,1)-form, which thus give a 2-cohomology class in the real de Rham cohomology of. First Chern class of a holomorphic line bundle.

11. 07/11/2023 (Lecture) Aula C, 17:15 - 19:00.
    The first Chern class of a holomorphic line bundle does not depend on the chosen metric, weights of the metric for duals and tensor products of line bundles, computation of the curvature of the tautological line bundle of the projective space of an Hermitian vector space, the Fubini-Study form, computation of the volume of the Fubini-Study form on P^1. Definition of (semi)positive (1,1)-form, local expression, one-to-one correspondence between positive (1,1)-forms, Hermitian metrics on the holomorphic tangent bundle, and J-invariant Riemannian metrics on the real tangent bundle. Kähler forms, Käshler manifolds, Riemannian volume form in terms of the top exterior power of the fundamental 2-form, exterior powers of a Kähler form are always d-closed but never d-exact on a compact manifold, a compact Kähler manifold has non trivial even de Rham cohomology.

12. 09/11/2023 (Lecture) Aula F, 08:15 - 10:00.
    A metric is Kähler if and only if it is tangent at order two to the flat metric in suitable holomorphic coordinates. Normal holomorphic coordinates vs local holomorphic normal frame. Examples of power series expansions to compute curvatures. The pull-back of a (semi)positive (1,1)-form by a holomorphic map is a semipositive (1,1)-form and it's positive if the starting form is positive and the map is an immersion. Examples of Kähler manifolds, and of non Kähler manifolds. Definition of modification, of simple normal crossing divisor, and statement of Hironaka's desingularization theorem for complex spaces. Complex manifolds with arbitrarily large fundamental group. A compact Kähler manifold such that the pull-back of one of its Kähler forms to the universal cover is d-exact has arbitrarily large fundamental group.

13. 14/11/2023 (Lecture) Aula C, 17:15 - 19:00.
    End of the proof of the fact that a compact Kähler manifold such that the pull-back of one of its Kähler forms to the universal cover is d-exact has arbitrarily large fundamental group. Positive line bundles, examples, the restriction of a positive line bundle to a submanifold is positive, Kähler forms do not necessarily come from a positive line bundle, statement of the Kodaira Embedding Theorem. Group of Cartier divisors and its isomorphism with the group of Weil divisors on locally factorial varieties, natural homomorphism from the group of Cartier (and thus Weil, on a locally factorial variety) divisors and the Picard group, description of holomorphic sections of a line bundle in terms of meromorphic functions with controlled zeros and poles, the Poincaré dual of the class of an irreducible hypersurface.

14. 16/11/2023 (Lecture) Aula F, 08:15 - 10:00.
    The Poincaré dual of the class of an irreducible hypersurface equals the first Chern class of the associated holomorphic line bundle. Explicit computation for a point in a compact Riemann surface. Basics about linear differential operators action on sections of vector bundles: definition, principal symbol, ellipticity, formal adjoint, structure of the kernel and cokernel of an elliptic operator on a compact manifold. Connection on a vector bundle, Laplace operator associated to a connection on a Euclidean/Hermitian vector bundle on an oriented Riemannian manifold, and its ellipticity when compatible with the metric structure.
    Some notes (way more than what we have done during the lecture) about differential linear operators and ellipticity can be found (in Italian) here.

15. 22/11/2023 (Lecture) Aula C, 17:15 - 19:00.
    Space of harmonic p-forms with value in a Hermitian vector bundle, harmonic if and only if D and D* closed, Hodge isomorphism between harmonic forms and de Rham cohomology of a unitary flat vector bundle, Poincaré duality, various Laplace operators of the Chern connection of a holomorphic Hermitian vector bundle, isomorphism between harmonic (p,q)-forms with value in a holomorphic Hermitian vector bundle and Dolbeault cohomology, Serre's duality. Lefschetz operator and its adjoint, their (graded) commutator. Akizuki-Nakano identity.

16. 23/11/2023 (Lecture) Aula F, 08:15 - 10:00.
    Recap of the previous episode, hints around the Levi problem and Hormander's solution of the dbar-equation in pseudoconvex domains. Particular case of the Akizuki-Nakano identity for the trivial flat line bundle with connection given by the de Rham differential: Hodge Decomposition Theorem and Hodge symmetry. The odd Betti numbers of a compact Kähler manifold are even. Hodge numbers, Hodge diamond and its symmetries, hints about the deformation of complex structures (Ehresmann's lemma and the Fischer-Grauert theorem) and the variation of Hodge structures. Statement of the ddbar-Lemma and the following application: how to prescribe the curvature of a holomorphic line bundle on a compact Kähler manifold.

17. 28/11/2023 (Lecture) Aula C, 17:15 - 19:00.
    Recap on Hodge decomposition for compact Kähler manifolds, the Dolbeault cohomology of the projective space. Proof of the ddbar-Lemma, and consequences: the Hodge decomposition is independent of the metric, how to prescribe the curvature of a holomorphic line bundle, a line bundle is positive if and only if its first Chern class can be represented by a Kähler form. Bochner-Kodaira-Nakano inequality, Akizuki-Nakano Vanishing Theorem for positive or negative line bundles, the example of O(k) on the projective space. 

18. 30/11/2023 (Lecture) Aula F, 08:15 - 10:00.
    Effective line bundles, linear systems, base locus of a linear system. Description of the one-to-one correspondence between {base point free linear systems on a compact complex manifold} and {non degenerate holomorphic maps to complex projective spaces modulo projective transformations}.

19. 05/12/2023 (Lecture) Aula C, 17:15 - 19:00.
    Sufficient criteria ofr the Kodaira map in order to be injective and immersive in terms of properties of sections. A line bundle on a compact Riemann surface is positive/ample if and only if its first Chern class is positive (hints about Nakai-Moishezon ampleness criterion), compact Riemann surfaces are projective. Natural map for the global sections of a holomorphic vector bundles twisted by O_X(-p) and global holomorphic sections of the vector bundle vanishing at p.

20. 07/12/2023 (Lecture) Aula F, 08:15 - 10:00.
    Cohomological sufficient condition for injectivity and base-point-freenes for a complete linear system on a compact Riemann surface, basic trick in complex geometry solving a dbar equation in order to construct holomorphic sections of a line bundle. If L is a positive line bundle on a compact Riemann surface X then the cohomological sufficient condition is satisfied provided we take a sufficiently (explicit) high tensor power of L. Hurwitz formula and Gauss-Bonnet.

21. 12/12/2023 (Lecture) Aula C, 17:15 - 19:00.
    Cohomological sufficient condition for the Kodaira map in order to be an immersion, if L is a positive line bundle on a compact Riemann surface X then the cohomological sufficient condition is satisfied provided we take a sufficiently (explicit) high tensor power of L. Every sufficiently high power of a positive line bundle on a compact Riemann surface is very ample. Every sufficiently high power of an ample line bundle on a compact Riemann surface is very ample. Digression on the Kodaira-Iitaka dimension of a line bundle, the case of line bundles on a compact Riemann surface.

22. 14/12/2023 (Lecture) Aula F, 08:15 - 10:00.
    Kodaira dimension, manifolds of general type, digression on the birational classification of algebraic varieties, MMP, Iitaka fibration. Other consequences of the Kodaira embedding theorem for compact Riemann surfaces: a line bundle such that c_1(L)>2g(X) is very ample, the degree of the embedded Riemann surface via the Kodaira map of a very ample line bundle equals the first Chern class of the line bundle, every compact Riemann surface can be realized as a projective curve of degree 2g(X)+1. Every line bundle on a compact Riemann surface has a non zero meromorphic section, the first Chern class of a line bunbdle on a compact Riemann surface is integral.

23. 19/12/2023 (Lecture) Aula C, 17:15 - 19:00.
    Čech cohomology of sheaves on a topological space: Čech cochains, Čech differential, Čech cohomology with respect to an open covering,  morphisms induced by a refinement of open coverings, Čech cohomology as a direct limit. Leading example: the group of isomorphism classes of differentiable (resp. holomorphic) complex line bundles as the first Čech cohomology group of the sheaf of nowhere zero differentiable complex-valued functions (resp. holomorphic functions). Acyclicity of sheaves of modules over a fine sheaf of rings. Morphisms of sheaves induce morphisms in cohomology. When the space is paracompact a short exact sequence of sheaves induces a long exact sequence in cohomology, explicit description of the connecting homomorphism.
    Click here for some notes on sheaf cohomology by J.-P. Demailly.

24. 21/12/2023 (Lecture) Aula F, 08:15 - 10:00.
    Leray's Theorem on acyclic coverings, abstract de Rham-Weil isomorphism Theorem. Examples: de Rham isomorphism between Čech cohomology of the constant sheaf with values in C and de Rham cohomology of complex-valued forms, Dolbeault isomorphism between Čech cohomology of the sheaf of holomorphic p-forms with values in a holomorphic vector bundle and Dolbeault cohomology of the vector bundle. Holomorphic Euler characteristic, sketch of the proof of Kobayashi's theorem about simple connectedness of Fano manifolds. Exponential short exact sequence (both versions: complex-valued differentiable functions and holomorphic functions), integral first Chern class of a line bundle. The group of isomorphisms classes of differentiable complex line bundles is isomorphic to H^2(X,Z) via the integral first Chern class morphism. Irregularity of a compact Kähler manifold,  Jacobian variety and Néron-Severi group of a compact Kähler manifold. The Picard group of a compact Kähler manifold is an extension of the Néron-Severi group by the Jacobian variety. Lefschetz Theorem on (1,1)-classes: description of the Néron-Severi group as the space of integral (1,1)-classes, relation with the Hodge conjecture. The image of the intergal first Chern class in de Rham cohomology coincides with the real fisrt Chern class.