1 Prequisites
For this advanced course, previous exposure to a wide range of mathematics will be useful, but not fully required. We do not expect you to know all the prerequisites, but will recall everything as much as necessary.
• definition of differentiable manifolds and C∞-forms; integration on such a manifold. • From algebraic topology:
– definition of singular homology and cohomology via chains and cochains, and a few properties – definition of sheaf cohomology
• Some algebraic geometry:
– exposure to sheaf cohomology in the algebraic setting
– familiarity with varieties (over fields of characteristic 0)
– definition of a scheme, affine schemes, and interpreting varieties as schemes
• Some homological algebra (but most of this will be recalled and/or extended where necessary):
– definition of complexes, (co)homology of complexes
– long exact sequence of (co)homology associated to a short exact sequence of complexes
2 Aim of the course
The aim of the course is to give a thorough introduction to the (mixed) Hodge theory of smooth, quasi projective varieties over the complex numbers, and discuss various applications. Time permitting, we shall also discuss modestly singular varieties and, in particular, show how to compute the mixed Hodge structure on some examples.
Hodge theory originated with a result of W.V.D. Hodge, who in 1937 showed that the n-th cohomology group with complex coefficients of a smooth projective variety over the complex numbers (or more generally, a compact K¨ahler manifold) admits a more refined structure reflecting the complex structure of the manifold. This foundational result was later vastly generalized by Deligne (1971, 1974), who introduced the theory of mixed Hodge structures to encompass the cohomology of quasi-projective, and even singular, complex algebraic varieties. Over the decades, the resulting theory has become a cornerstone of modern algebraic geometry.
Mixed Hodge theory is deeply intertwined with transcendental number theory, particularly through the study of periods — integrals arising naturally in geometry — such as π, ζ(3), and many others. Several famous open conjectures suggest that all algebraic relations among such integrals must come from the underlying geometry. These conjectures offer a glimpse into what a future theory of transcendental numbers might look like. We will use this perspective as a rich source of motivation and examples.
We will begin the course with a motivational overview of these periods and conjectures. This will establish the need to handle arbitrary complex algebraic varieties (open or singular) and a cohomology theory of differentials sufficiently delicate to preserve number theoretic considerations (for example, when the variety is defined over a number field).
We shall then sketch classical Hodge theory, in the case of smooth projective (or K¨ahler) varieties, which is based on analysis. Next, we shall turn to the homological consequences, unearthed by Deligne, in the general case of smooth quasi-projective varieties. Here the nth cohomology group of the variety in terms of Hodge theory can also have ‘weights’ bigger than n, whereas in the smooth projective or K¨ahler case this weight is always n. Towards the end, we aim to discuss modern points of view on periods, and motives. Throughout the course, we will consider examples.
3 Rules about homework and exam
There will be four assignments throughout the course that will constitute in total 40% of the grade. The remaining 60% of the grade is determined by an oral exam. This oral exam will be based on the content of the lectures and a final non-marked assignment. The resit will have the same format, with a new non-marked assignment.
If your homework grade is lower than your exam grade, then your grade will be based entirely on your exam grade.
As a general requirement, you can only pass if you score at least 5.0 out of 10 on the oral exam (or resit for the oral exam).
4 Literature
The main references for the course are as follows, but we may use other sources for particular topics. We do not expect you to buy those, but they will be useful as references. Most of those are freely accessible through your institution.
• Bott–Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics 82, Springer Verlag, 1982.
• Burgos Gil and Fr´esan, Multiple Zeta Values; available on http://javier.fresan.perso.math.cnrs.fr/mzv.pdf. • Deligne, Th´eorie de Hodge II, Inst. Hautes Etudes Sci. Publ. Math., 1971 (4), p.5-57. ´
• Huber and M¨uller-Stach, Periods and Nori motives, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 65, Springer Verlag, 2017.
• Peters and Steenbrink, Mixed Hodge structures, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 52, Springer Verlag, 2008.
• Voisin, Hodge theory and complex algebraic geometry, I and II, Cambridge Studies in Advanced Mathe matics, 76 and 77, Cambridge University Press, 2007.
• Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cam bridge University Press, 1994.
- Docent: Rob de Jeu