Each semester we will cover some topic of relevance to some of the RTG projects or of general interest.

We meet every second week (during the lecture period), alternating between Berlin and Hannover. Each day there will be two talks by some of the Ph.D. students of the RTG. 

• • The talks in Hannover take place in room A036 (Mensa) from 12:00 to 13:30, and in room F342 (Main building) from 14:30 to 16:00.

  • The talks in Berlin take place in room 3.006 (Johann von Neumann-Haus) from 13:00 to 17:00.

In this seminar we will discuss étale cohomology, from its definition to its main properties, with the aim of understanding Deligne's proof of the Weil conjectures.

The main source will be Milnes's notes, from which we will select some sections. The remaining sources can be used to find alternative proofs or more streamlined presentations.

Preliminary schedule (more details here).

• October 20, Berlin:

  • Talk 1: short motivation (how to obtain a cohomology theory for algebraic varieties similar to singular cohomology for complex manifolds?), and basics on étale morphisms.

  • Talk 2: the étale fundamental group and local rings with respect to étale coverings.

• November 3, Hannover:

  • Talk 3: sites, specially the étale site of an algebraic variety, and sheaves on them.

  • Talk 4: sheaves on $X_{et}$, categorical properties and behaviour under morphisms.

• November 17, Berlin:

  • Talk 5: étale cohomology of sheaves, higher direct images and the Leray spectral sequence.

  • Talk 6: cohomology of curves.

• December 1, Hannover:

  • Talk 7: Proper and smooth base change.

  • Talk 8: Cohomology groups with compact support and the Gysin sequence.

• December 15, Berlin:

  • Talk 9: finiteness results, $\ell$-adic cohomology and comparison with singular cohomology in the compolex case.

  • Talk 10: Cycle maps and the Lefschetz fixed point Formula.

• January 12, Hannover:

  • Talk 11: The Weil conjectures and proof of part of them. 

  • Talk 12: The main bound on the size of the eigenvalues.

• January 26, Berlin:

  • Talk 13: Lefschetz pencils.

  • Talk 14: Proof of the Weil conjectures.

Summer Semester 2025: What determines an algebraic variety?

This semester we will read the recent book by Kollár, Lieblich, Olsson and Sawin "What determines an algebraic variety?". We will focus on its main result in characteristic 0, namely that homeomorphic normal projective geometrically irreducible algebraic varieties of dimension at least 4 are automatically isomorphic (i.e. the Zariski topology already determines the algebraic structure).

Temptative plan (more detailed version here):

• April 14th (Berlin)

  • Talk 1: Introduction and motivation of the main result.

  • Talk 2: The fundamental theorem of definable projective geometry.

• May 5th (Hannover)

  • Talk 3: Basics on divisors and divisorial structures. Which geometric properties are determined by them?

  • Talk 4: Definable subspaces in linear systems.

• May 12th (Hannover)

  • Talk 5: Reconstruction of varieties from divisorial structures (the case of infinite fields).

  • Talk 6: Algebraic and topological pencils.

• May 19th (Hannover)

  • Talk 7: Finite morphisms, degree functions and linear equivalence.

  • Talk 8: Linkage of divisors.

• May 26th (Hannover)

  • Talk 9: Linear similarity and Bertini-Hilbert dimension.

  • Talk 10: Linkage and residue fields.

• June 30th (Berlin)

  • Talk 11: Linkage and transversality.

  • Talk 12: Recovering linear equivalence from the Zariski topology.

Winter Semester 2024/2025: Hodge Theory

We discussed some features of Hodge-Theory, starting from the basic definitions of complex manifolds, the representation of cohomology classes by harmonic representatives leading to the Hodge decomposition in the case of compact Kähler manifolds. Depending on the interests of the participants, we will devote the final lectures to variations of Hodge structure, the Hodge conjecture or mixed Hodge structures.

Plan (more detailed version here):

• December 2nd (Berlin)

  • Generalities on complex manifolds: definition, decomposition of the complexified tangent and cotangent bundles, and of the exterior differential.Time permitting: Integrability of almost complex structures.

  • Kähler manifolds: Riemannian metrics and correspondence with (1,1)-forms, volume forms, Kähler metrics, characterizations and examples.

• December 16th (Hannover)

  • Cohomology and harmonic forms 1: Hodge-Star and adjoint operators, L²-metric, Laplacians.

  • Cohomology and harmonic forms 2: Symbols of differential operators, symbol of the Laplacians, decomposition theorem for elliptic operators. Time permitting: applications to vector bundles, Poincaré and/or Serre Duality.

• January 13th (Berlin)

  • Cohomology of compact Kähler manifolds 1: Kähler identities, comparison of the Laplacians and Hodge decomposition.

  • Cohomology of compact Kähler manifolds 2: Lefschetz operator in cohomology, primitive cohomology and Lefschetz decomposition, polarizations and Hodge index Theorem. Time permitting: Lefschetz Theorems.

• January 27th (Hannover)

  • Abstract Hodge Structures, Lefschetz' Theorem on (1,1)-classes: Definition of Hodge structures and associated filtrations, characterization for compact Kähler manifolds, polarizations of Hodge stuctures, Lefschetz' Theorem of (1,1)-classes and the Hodge conjecture.

  • Examples: Hodge structure of projective spaces. Hodge structures of weight 1: complex Tori and Abelian varieties, Picard and Albanese varieties and intermediate Jacobians. Hodge structures of weight 2: middle cohomology of a compact Kähler surface, Torelli theorem (Piatetski-Shapiro, Shafarevich) for K3 surfaces.