Graduate Seminar
"Generalized Griffiths calculus"
Lorenz Wotzlaw
23 Jan 2006, 13:00 – 15:00
If 
is a smooth hypersurface in
, the Jacobian ring of 
calculates the graded parts of the Deligne-Hodge-structure on the primitive cohomology:
![$\displaystyle Gr^{p-1}_FH_0^{n-1}(X)\stackrel{\sim}{\longrightarrow} (\mathbb{C...
...s,x_n]/ \langle\partial_0F,\ldots,\partial_nF\rangle)_{(n-p)\text{deg}(X)-n-1} $](./teximages/4157/en/sfbltx4158img4.png){kind=small}
This is a well-known result of P. Griffiths, which extends to case of (quasi-) smooth complete intersections in toric varieties. It allows to represent Hodge-theoretical objects related to families of such varieties, like Higgs-bundles and Yukawa-couplings. Among the applications are global Torelli theorems and curve-counting in mirror symmetry.
The aim of the talk is to relax the smoothness condition and give a generalized ``Griffiths calculus'' for intersection cohomology of (families of) hypersurfaces in
with isolated singularities. The proof will use the theory of 
-modules, namely mixed Hodge modules of M. Saito. A special focus will be on nodal threefolds in
; here intersection cohomology is isomorphic to ordinary cohomology of a small resolution.
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