Kaehler-Einstein metrics on toric manifolds

I had a wonderfully unexpected intersection of interests with my officemate who works with PDEs a few weeks ago. He works on optimal transport problems; a motivation for which is finding Kaehler-Einstein metrics on Fano manifolds (or smooth Fano varieties). Tian, one of the pioneers of the theory, wrote that Kaehler-Einstein metrics are designed to be ‘canonical’ choices of metric on certain types of manifold and, to that end, are very well motivated. The issue, of course, is that they don’t always exist. The statement my officemate shared that intrigued me was the following…

If a toric Fano manifold admits a Kaehler-Einstein metric, then the barycentre of its moment polytope is the origin.

A good reason for this is that the barycentre turns out to be a geometric invariant of the manifold called the ‘Futaki character’ that measures the obstruction to the existence of a Kaehler-Einstein metric. There’s an interesting and, to me at least, not yet fully explored feature of the combinatorics at play in which both the algebra (from toric algebraic geometry) and the analysis (from the Einstein PDE governing the metric) fix a particular translate of the moment polytope, which is only defined a priori up to translation. Otherwise the ‘barycentre’ doesn’t even make sense!

Anyway, here are my notes elaborating on the claim above and the nearby theory based on a paper of Mabuchi.

There’s also an element of Lie theory brought in by the fact that manifolds with a Kaehler-Einstein metric should have that the identity component of their automorphism group is reductive, which is interestingly complementary to discussion I’ve posted in the past of root systems associated to the ‘discrete’ part of the automorphism groups del Pezzo surfaces. More on this soon.

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Clusters and the semicanonical basis

In the final instalment of this trilogy comparing Lusztig’s semicanonical basis on \mathbb{C}[N] with Fomin-Zelevinsky’s cluster basis, I will discuss Geiss-Leclerc-Schroer’s construction of the cluster structure on \mathbb{C}[N] in dual terms to Lusztig’s Lagrangian construction of U\mathfrak{n}, which then makes the comparison transparent.

Their construction also clarifies why the Dynkin types \Gamma of the preprojective algebras \text{PP}(\Gamma) with only finitely many indecomposable modules coincide with the types for which the cluster structure on \mathbb{C}[N] is finite-type, where N is a maximal unipotent subgroup inside a simple Lie group of type \Gamma.

As a bonus, I included some notes – adapted from a recent talk by Paul Hacking here – that act as a prologue for my Lie groups class paper; they elaborate on one of the entrances cluster algebras make into mirror symmetry, and how this fosters a geometrical expectation of some sort of ‘semicanonical’ basis in situations where they arise.

Clusters in Lie theory

To continue on from where my previous post left off, many objects in Lie theory naturally carry the structure of a cluster algebra. Indeed, for many people – including Fomin and Zelevinsky – this was a substantial part of the motivation for the arcane definitions of the theory. The prototypical collection of cluster variables is the ‘flag minors’ on the base affine space of \text{SL}_n(\mathbb{C}) that obey generalised Pluecker relations, which are the archetypal exchange relations.

This next part outlines the wonderfully aesthetic cluster structure on base affine spaces, as well as starting to tie the cluster variables to the indecomposable modules over the preprojective algebra of the same Dynkin type as the group, so that the coincidence of finitude between these two objects in types A_n for n\leq 4 that I mentioned last time can be explained geometrically.

To prepare this post, I read the notes from Fomin’s ICM address in 2010. It was probably one of the most pleasant articles that I’ve read for a while; I highly recommend it. it contains some interesting historical commentary too that I wasn’t very familiar with. I come in with largely a mirror symmetric interest in cluster algebras, but the original motivation – asides from the compelling examples – was to study ‘positive varieties’. The basic idea is that, given a complex variety X equipped with some ‘natural’ set of functions \mathbf{\Delta}, one considers the positive variety

X_+:=\{x\in X:\Delta(x)\geq0\text{ for all }\Delta\in\mathbf{\Delta}\}.

This is often some sort of cone, for example in the case that X=N is a maximal unipotent subgroup of a simple Lie group of type ADE and \mathbf{\Delta} is the set of flag minors. The positive variety can contain a bizarre amount of information about the original variety: in the previous example, the strata in the cone know about the corresponding Bruhat order on Schubert cells. In their study of these objects, Fomin and Zelevinsky realised that a good source of ‘natural’ functions was cluster variables, and that hence the varieties to consider are cluster varieties.

Anyway, in the final part of this trilogy on clusters and semicanonical bases in Lie theory the two sides will be reconciled, allowing cluster techniques to help compute semicanonical basis elements.

Clusters and the preprojective algebra

I was in the UK last week at a conference and a wedding, which were both very enjoyable but understandably led to no blogging. I hope to post some of my notes from the conference at some point; Gavril Farkas gave a lovely talk on Abelian varieties in particular.

Being thrust back into the semester on my return, it seems that I have a class paper to write for a Lie groups course. I’ve wanted to explore the appearance of cluster phenomena in Lie theory for a while and this seems like a good opportunity to do so. Given a simply connected simple complex Lie group G of Cartan-Killing type ADE with Lie algebra \mathfrak{g} and a maximal unipotent subgroup N with Lie algebra \mathfrak{n}, Lusztig constructed a ‘semicanonical basis’ of the enveloping algebra U\mathfrak{n}. This is ‘semicanonical’ because, using the duality between U\mathfrak{n} and the coordinate ring \mathbb{C}[N], one obtains a dual basis of \mathbb{C}[N] with the property that it contains a basis for all the irreducible representations of \mathfrak{g}, which are well-known to live inside \mathbb{C}[N].

These basis elements are generally hard to compute – involving Euler characteristics of some flag varieties – but many of them can be found using cluster mutation. \mathbb{C}[N] is a cluster algebra – not usually finite-type! – and the natural cluster structure found by Fomin-Zelevinsky coincides with the dual semicanonical basis of Lusztig in the sense that the cluster variables are part of the basis. This in particular establishes that they are linearly independent, which was only conjectured in general by Fomin-Zelevinsky!

Big picture aside, one of the objects needed to make this comparison is the preprojective algebra, which turns out to be an old friend from the McKay correspondence. I find it intriguing that one of the main techniques for extracting information about hard noncommutative algebras is by viewing them as path algebras of quivers with relations – I believe that the zenith of this point of view is Auslander-Reiten theory – and this approach works excellently here. I wrote some notes setting up the preprojective algebra and discussing some of its startling features and their interplay with cluster algebras, which should be a prelude to the detail of the semicanonical basis.

The toric ansatz

I gave a talk yesterday on mirror symmetry, in particular the toric ansatz. There are a few different versions of the actual ansatz but the common theme is that the mirror to a Fano variety should be a cluster variety with cluster charts indexed by the toric degenerations of the Fano. Ilten showed that polytope (or, at least polygon) mutations induce qG deformations between toric Fanos and so part of the mirror symmetry conjectures in this vein are that are enough mutations to connect all toric degenerations of a given Fano variety.

Of course there’s a problem if your Fano doesn’t admit any degenerations to a toric variety! This is the surjectivity component of said conjectures. I’ve discussed class TG – the class of Fano varieties that do degenerate to toric varieties – a few times in the past, and this talk was intended to be an exposition of the very limited current knowledge on the matter.

To complement the talk notes, I wrote out an example of moving between polytope mutations, Laurent polynomial (or ‘algebraic’) mutations, and cluster mutations (transition functions between the cluster charts on the mirror cluster variety). The cluster algebraic objects in play are really ‘cluster algebras with potential’. This is capturing the notion dating back to Przyjalkowski that a ‘weak mirror’ to a Fano is a Landau-Ginzburg model: a regular function on a torus, or a Laurent polynomial, with certain compatibilities regarding the enumerative geometry of the Fano. After all, a regular function on a cluster variety is locally just a Laurent polynomial on each cluster chart. The cluster framework incorporates all the (infinitely many) weak mirrors into a single place to create a canonical mirror object.

If there isn’t enough symplectic geometry for you here, note that there are emerging beautiful syntheses between toric degenerations and Lagrangian tori. Vianna recently constructed infinite collections of monotone Lagrangian tori inside smooth del Pezzo surfaces indexed by generalised Markov triples, exactly those that classify the degenerations of the same del Pezzo surfaces to (fake) weighted projective spaces from the work of Hacking-Prokhorov. Moreover, these share the same mutation properties as these degenerations from the work of Akhtar-Kasprzyk, who classified how fake weighted projective spaces (or Fano simplices) mutate. I’ll post again in more detail on this correspondence soon.

Symplectic embeddings IV

I’ve been struggling through Hutchings’ lecture notes on ECH capacities for bit of time now and so I thought that I may as well post my progress for some katharsis. Expect these notes to be a under development as a result. I’m principally interested in the case of ECH capacities of toric domains (as a starting point at least), and in the appearance of lattice point counts therein. There are several aspects of the calculations that intrigue me – quaint applications of Pick’s theorem, a different sort of duality on the level of norms instead of (well, maybe comparably to) polytopes, combinatorial descriptions of the ECH differential and relevant chain maps, a sumptuous formula for the action functional,… – that I attempted to capture and communicate. You can be the judge of that.

Thanks are due to Julian for spending an afternoon computing integrals with me.