Review of classical Hodge theory, Ze
Variation of Hodge structures, period maps, Max
Quasi-unipotency of monodromy of Z-polarized VHS on a punctured disk, Dingchang
Reference: Topics in transcendental algebraic geometry chapter 2, 3
VHS on a punctured disk, Ze
This talk will be on some analytic aspects of variation of Hodge structures on a punctured disk. The results have strong implications about VHS on quasi-projective manifolds. I'll first state the nilpotent orbit theorem which says Hodge bundles extend to a Deligne canonical extension over the origin, and introduce the limiting Hodge filtration. As a consequence of the theorem, any VHS whose monodromy representation is trivial along a subvariety will extend as a VHS, not just as a flat bundle. Time permitting, I'll talk about the theorem of fixed parts and rigidity of local systems underlying a VHS.
Reference: Schmid 1973; Sabbah-Schnell
LMHS and Clemens-Schmid exact sequence, Max
In this talk, I will construct a mixed Hodge structure on the cohomology of SNC varieties, which appear as the central fibers of semistable degenerations. Building on last week's talk, we will see that the cohomology of a nearby fiber also naturally carries a so-called limiting
mixed Hodge structure. The Clemens-Schmid exact sequence gives a convenient tool to compare these two MHS often yielding geometrically useful information. Throughout the talk, we will try to illuminate these concepts with some concrete examples and calculations.
Steenbrink's geometric construction of LMHS, Yipeng
We have previously discussed Schmid’s result showing that a limit mixed Hodge structure arises from a variation of Hodge structure. Steenbrink, on the other hand, provided a geometric approach to obtain the limit mixed Hodge structure from the geometry of the central fiber. Today, I will briefly introduce Steenbrink’s work on the geometric construction of the limit Hodge structure. If time permits, I will also explain how this result leads to the invariant cycle theorem, which plays a crucial role in the proof of the Clemens–Schmid exact sequence. The main references are Steenbrink’s “Limits of Hodge Structure” and Chapter 7 of Topics in Transcendental Algebraic Geometry.
Reference: Steenbrink Inventiones paper
Decomposition theorem and Clemens-Schmid exact sequence, Ze
In this talk I'll first briefly discuss a proof of the Clemens-Schmid exact sequence when the total space is smooth and without the map to be semi-stable. The tool is the decomposition theorem. Then I'll illustrate through some examples about how the information of the limiting mixed structure/monodromy can be related to the singularity/configuration of the central fiber.
Reference: Kerr-Laza Hodge theory of degenerations I
Torelli theorem for curves and abelian varieties, Conner