Program of the course

0. Basic concepts. Ringed spaces. Compact manifolds. Structure sheafs on manifolds of various types. Forgetting functors.


1. Elements of manifolds topology. Orientation. Connected sums, their relation with blow-ups in the case of algebraic surfaces. Overview of the cohomological theories. One-dimensional manifolds.


2. Compact topological surfaces. The smooth structure and its uniqueness. Coincidence of the almost complex and complex structures. Complete topological classification of surfaces. Moduli spaces of algebraic curves.


3. Smooth structures. Smoothing of the $3$-manifolds; the uniqueness of the smooth structure on them. Examples of non-smoothable manifolds. The sphere $\mathbb S^7$ and the group of smooth structures on it.


4. Almost complex structures. Definitions. Examples of the even-dimensional manifolds admitting no almost complex structures; the sphere $\mathbb S^4$. The geography of almost complex $4$-dimensional manifolds.


5. Complex structures. Integrability of almost complex structures. The examples of non-integrable almost complex structures. The sphere $\mathbb S^6$, almost complex structure on it and the problem of existence of a complex structure.


6. Kahler metrics on complex varieties. The relation between symplectic, kahler and almost complex structures. The products of odd-dimensional spheres as complex manifolds and the possibility of introduction the kahler structures on them; the Hopf surface $\mathbb S^1 \times \mathbb S^3$


7. Complex and algebraic varieties. The complex varieties, admitting and not-admitting an algebraic structure. Complex tori and abelian varieties. K3-surfaces.


8. Various problems.

Program

Lecturer
Shabat Georgii Borisovich

Financial support
The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no.  075-15-2022-265).

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center