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Toric varieties and convex polyhedra are intimately connected. Some of this can be found in standard text books (the connection between divisors and mixed volumes seems to be a popular example).

One of the most important objects that are associated to an algebraic variety is its derived category. So I'm wondering: are there any constructions or properties in convex geometry that are reflected in the derived category of associated toric varieties?

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Derived category of a toric variety has a full exceptional collection which was constructed by Kawamata using toric Minimal Model Program. As far as I know no good direct relation between the collection and the fan (polyhedron) of the variety is known.

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That's a great question!

In the last few years there is plenty of research on the subject - and a few (interesting) open questions.

Before Kawamta's result - note that toric manifolds satisfy $rk(K(X))= \chi(X)$ (this is a result on the fan), so conceptually they are a good candidates to have a full exceptional collection (as Kawamata's result shows), see Remark 2.3 in Uhera's work on derived categories of toric threefolds.

Kawamata's result (mentioned in the answer above) is very deep but the exceptional collections he constructs can be quite hard to track (the construction relies on various ideas from toric Mori theory). Pay attention that Kawamata's construction of e.c is not necessarily strong - which is a condition one typically wants an exceptional collection to satisfy.

If you will look at the first examples of (full strongly) exceptional collections discovered by Beilinson for $X=\mathbb{P}^n$ (which is, of course, toric) you will see that they are given by rather natural elements $\mathcal{E}_1 = \left \{ \mathcal{O},\mathcal{O}(1),...,\mathcal{O}(n) \right \} $ and $\mathcal{E}_2 = \left \{ \mathcal{O},\Omega^1(1),... , \Omega^n(n) \right \}$. In fact $\mathcal{E}_1$ is given by line bundles, that is elements of $Pic(X)$.

This led A. King to ask - which toric manifolds $X$ admit exceptional collections of line bundles in $Pic(X)$. For a while it was strongly believed that any toric manifold admits such a collection - but this was proved to be wrong by L. Hille and M. Perling.

In particular, this question (which toric admit e.c of line bundles) turned out to be quite an elusive question regarding toric manifolds. For instance - there is no clear algorithm that determines on the basis of the toric fan $\Sigma$ whether the corresponding manifold has such a collection.

On the other hand - there's plenty of research and results which shows that exceptional collections of line bundles for toric manifolds $X$ are a rather interesting class of exceptional collections - for instance just to mention a few - A. Bondal, Costa and Miro-Roig, H. Uehara, Borisov and Hua, Dey, Lason and Michalek, Bernardi and Tirabassi...

Personally, I am interested in connections between such collections - and ideas from mirror symmetry - they turn to be related to solutions of toric Landau-Ginzburg system (which are given in terms of the fan) - if your'e interested - there's some details here.

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  • $\begingroup$ Something that bothered me some five years ago: has Bondal's result been written down? (Costa and Miro-Roig seem to be using it without proof.) $\endgroup$ – Piotr Achinger Aug 6 '17 at 12:01
  • $\begingroup$ Yes, it was written down - you can find a very short summary of a talk he gave about the subject in oberwolfach reports, 3, 2006 - it's very short but is a very worthwhile read - it contains some fundamental ideas. $\endgroup$ – Yochay Jerby Aug 6 '17 at 12:08
  • $\begingroup$ Thank you. I know that short report quite well – it's incomplete, which is why I'm asking if anybody has seen a full proof 10 years later... $\endgroup$ – Piotr Achinger Aug 6 '17 at 13:38
  • $\begingroup$ To the best of my knowledge this is the only document written by Bondal on the matter - and that's typically the reference appearing on the matter. I agree, I also found it quite elusive how he constructs the Hom - between the elements. $\endgroup$ – Yochay Jerby Aug 6 '17 at 13:44

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