# Torelli type theorem for sextic threefolds

I have a question to you:

In the work of Rapoport and Deligne they classify smooth complete intersections with Hodge level 1, which are the following:

• Complete intersection of two quadrics in $\mathbb{P}^{2n+3}$.
• Complete intersection of three quadrics in $\mathbb{P}^{2n+4}$.
• Cubic hypersurface in $\mathbb{P}^4$.
• Complete intersection of a cubic and a quadric in $\mathbb{P}^5$.
• Cubic hypersurface in $\mathbb{P}^6$.
• Quartic hypersurface in $\mathbb{P}^4$

For the first three cases there are Torelli type theorems for their Intermediate Jacobians.

My question is: Do you know any reference or idea for a proof in the case of the complete intersection of a cubic and a quadric in $\mathbb{P}^5$ (which is a Fano solid of index 1, whose Intermediate Jacobian is a p.p.a.v. of dimension 20), or something similar?

Thanks a lot!

• Some typos in your attributions : the case of an intersection of two quadrics is due to M. Reid (in his thesis), and the cubic threefold to Clemens-Griffiths.
– abx
Dec 12, 2013 at 12:47
• You say that Deligne-Rapoport classify the varieties of Hodge level 1. You probably meant to say the (smooth) complete intersections of Hodge level 1. Mar 14, 2014 at 12:54

## 1 Answer

For the cases which you are interested in there appears to be only partial results available in the literature. Namely, as special cases of more general results.

For quartic threefolds and cubic fivefolds one knows that a generic Torelli theorem holds. This is a special case of a generic Torelli theorem for certain hypersurfaces, due to Donagi [1].

I don't know of an analogue of Donagi's work for complete intersections, but here Flenner [2] has shown an infinitesimal Torelli theorem in certain cases. This in particular applies to the case of an intersection of a quadric and cubic.

Also, I believe that the Torelli theorem for intersections of two quadrics of odd dimension is actually due to Donagi [3], not Reid (at least I can't find this statement in Miles Reid's PhD thesis).

Good luck with this problem if you are working on it! If you make any progress, or know more than I do, please do contact me as I would be most interested.

References:

[1] - Donagi. Generic Torelli for projective hypersurfaces. Compositio Math. 50 (1983), no. 2-3, 325–353

[2] - Flenner. The infinitesimal Torelli problem for zero sets of sections of vector bundles. Math. Z. 193 (1986), no. 2, 307–322.

[3] - Donagi. Group law on the intersection of two quadrics. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 2, 217–239.