## Some more toric Gorenstein varieties

As a follow-up to my previous post that encountered toric Gorenstein varieties on quite a general, abstract level I thought that I would write something much more computational. The result is my attempt to survey of some of the beautiful phenomenonology of Gorenstein toric varieties, starting with the classification of Gorenstein weighted projective spaces (equivalently, reflexive simplices). They turn out to biject with ‘unit partitions’

$1=\frac{1}{q_0}+\dots+\frac{1}{q_n}$

for $q_i\in\mathbb{N}$, with weighted projective spaces of dimension $n$ corresponding to partitions of length $n+1$. These are related to the Sylvester sequence $s_k$ defined by $s_1=2$ and

$s_{k+1}=1+\prod_{m\leq k}s_m$

There’s a famous result of Curtiss from 1921 stating that the largest possible value of $q_i$ in a partition of length $n$ is $s_n-1$ that, with a touch of toric geometry, yields bounds on the degree of weighted projective spaces (equivalently, volumes of reflexive simplices). Lastly, I describe a similarly famous result of Stanley stating that a reasonably general graded ring is Gorenstein iff the numerator of its Hilbert series is palindromic. As an immediate consequence, one sees that a Fano polytope is reflexive iff its Ehrhart series has a palindromic numerator. Wonderful.

## Toric Gorenstein varieties

It’s been sometime since I said that I was going to start a series on the ‘foundations’ of toric geometry. In many ways toric geometry is the antithesis of a ‘foundational’ subject: its primary benefits are ease of computation and experimentation, and the point is that it shifts technical difficulty into combinatorics and away from less accessible areas. Anyway, I wrote some notes adapted from the book of Cox-Little-Schenk.

The main focus of the notes is on toric Fano varieties and toric Gorenstein varieties. I’ve already expounded on the former quite regularly, so I will make some brief comments about the latter. From a high-level standpoint, Gorenstein varieties are the possibly singular varieties one would consider in order to try to build up a reasonable duality theory similar to the smooth case. They are the varieties for which the dualising sheaf $\omega_X$ is a line bundle, and hence for which $K_X$ is Cartier. There’s of course a purely algebraic incarnation of this as ‘Gorenstein rings’, which were studied by Grothendieck, Serre, famously by Bass, and by many others in the mid-20th century.

Stanley and others pioneered the extraordinarily ‘symmetric’ combinatorics of Gorenstein rings and varieties from the 1970s onwards; I make some remarks in my notes. One particular claim that I am quite fond of is the following. Toric Gorenstein Fano varieties correspond to ‘reflexive polytopes’. There are 16 reflexive polygons – it’s quite fun to find them all! – and they all have the property that the sum of the number of their boundary lattice points and the number of boundary lattice points on their dual is 12. There’s a geometrical proof of this via Noether’s formula (or RR for smooth surfaces):

$\chi(\mathcal{O}_X)=\frac{K_X^2+e(X)}{12}$

where $e(X)$ is the topological Euler characteristic. Since $\chi(\mathcal{O}_X)=1$ for a toric variety, there’s the 12. The other two terms turn out to be given by the two boundary lattice point counts I mentioned.

## Symplectic embeddings III

I’ve been working through Hutchings’ lovely survey paper on symplectic embedding problems and, as the next episode of the symplectic embeddings series, I enclose my notes. To be honest, Hutchings’ exposition is pretty accessible even for someone who doesn’t know a huge amount of symplectic geometry; I mostly just expand on some of the main arguments and computations. That said, I did intend for my notes to be compatible with some of my other posts on symplectic [toric] geometry, as well as carrying especial emphasis on the points that might be particularly enjoyable for an algebraic geometer or algebraic combinatorist. In particular, there are some interesting methods at play in some of the proofs alluding to the sort of ‘functoriality’ of the moment polytope of a toric symplectic manifold that I have been pondering of late.

The general approach taken is that, given this sequence of ECH capacities or necessary conditions for symplectic embeddings to exist, one wants to show that these conditions are also sufficient to guarantee that an embedding exists. The methodology passes through symplectic blowups to convert the problem into finding certain classes in (co)homology, which turns out to have the same numerics as the ECH capacities. Before that one also translates the problem into one of ‘ball packing’: embedding a disjoint union of balls into a single ball that, like many tales in symplectic geometry, has classical origins.

## Algebraic versus symplectic… toric manifolds

As the second and probably final (for the moment at least) comparative post between algebraic and symplectic geometry, I thought I’d compare their respective toric geometries and related combinatorics. ‘Symplectic’ as terminology was proposed by Weyl to reflect the similarities with complex geometry by replacing the Latin word ‘complex’ with its Greek equivalent. As this anecdote suggests, there is much shared between the two!

In the toric world, there are at least three ways to construct a toric manifold from some combinatorial data; for instance a polytope or a fan. One of them is by using lattice points (= sections for an algebraic geometer) to embed the relevant torus into some big projective space and take the closure. If you like, this is just applying Proj to the toric ample divisor corresponding to the polytope. Given a Delzant polytope $\Delta$ with lattice points as vertices – notice that this can sometimes (always?) be achieved by suitable scaling of the symplectic form – one can perform this process to obtain a toric variety, which will be a smooth subvariety of some projective space by the Delzant condition. In particular, it inherits a symplectic form from the restriction of the Fubini-Study form as a complex submanifold of a projective space.

The torus action it also carries turns out to be Hamiltonian with respect to this symplectic structure as witnessed by its moment map, which has moment polytope… $\Delta$. That is, by Delzant’s theorem (which says that there’s a unique way to reconstruct a symplectic toric manifold from a polytope) the algebraic and symplectic constructions of a toric manifold from a lattice Delzant polytope agree.

## Platonic solids and the McKay correspondence

I teach classes for a linear algebra course and my students had their midterm yesterday. As a break from course content I decided to take a few minutes during our lighthearted class today to tell them about some magic that happens in low dimensional matrix groups. I’ve posted enthusiastically about the McKay correspondence before and noted how it was my introduction to algebraic geometry. This particular part of it is a classical story that I’ve known for a while but never committed to LaTeX until now.

In brief, like so many classifications, the classification of finite subgroups of $\text{SL}_2(\mathbb{C})$ contains a few infinite families and then a small number of exceptional cases. In this case, the classification is parameterised by the simply-laced Dynkin diagrams, and the exceptionals correspond to the exceptional Dynkin diagrams $E_6,E_7,E_8$. Where they arise geometrically is a more interesting question, to which the answer is the Platonic solids. It was known to the Greeks that there are five ‘regular’ polyhedra that fall into three dual pairs, and hence produce 3 distinct symmetry groups. It follows by some quick quaternionic algebra that these groups – naturally embedded in $\text{SO}_3(\mathbb{R})$ – lift to finite subgroups in $\text{SU}_2(\mathbb{C})\subset\text{SL}_2(\mathbb{C})$, which account for the exceptional Dynkin labels.

Of course there is the undercurrent of algebraic geometry surging through this story, but it’s nice to relive a more elementary view of things on occasion.

## Symplectic embeddings II revisited

As the title suggests, this is not so much a new instalment of the symplectic embeddings saga; more a development of the last one. To set the scene again, I have been exploring the role lattice point counts play in symplectic embedding problems – specifically, those involving ellipsoids – and have some new comments to make, which I found interesting at the time.

While the coefficients of Ehrhart polynomials of lattice polygons have a clear intepretation, which is more or less Pick’s theorem, the coefficients appearing in Ehrhart quasipolynomials for rational polygons are less well understood. One can do some manipulations to make a bit of progress – for instance, the leading term of each polynomial constituting the quasipolynomial is always the volume of the $\ell\text{th}$ dilate of the polygon, where $\ell$ is its index – although the linear coefficients especially seem to be quite mysterious. The periodic nature of quasipolynomials mean that certain inequalities are satisfied between coefficients. These give rise to the claims made above and allow one to more efficiently test whether or not an ellipsoid can be symplectically embedded in another.

As discussed previously, McDuff’s result states that, to test whether or not one can embed $E(a,b)$ into $E(c,d)$, one must establish that the sequences

$\mathcal{N}(a,b)\leq\mathcal{N}(c,d).$

Ideally one would have some result saying that this holds iff the first blah entries of each are bounded in this way, but such a claim seems hard to come by, if possible with a uniform bound on blah. Anyway, at least one can test it via a finite check, which I improved to $2\ell-1$ over $3\ell$

## Algebraic versus symplectic… blowups

It’s hard to miss the many parallels between symplectic toric geometry and algebraic toric geometry, which makes learning one having already experienced much of the other quite peculiar. I’m going to write a few posts outlining some of the similarities (and differences) between toric geometry in the algebraic and symplectic contexts, starting with blowups. Blowups provide, for me at least, a good insight into the level of functoriality present between the combinatorics and geometry at hand. For instance, on the algebraic side they arise from the simplest of maps – the identity – on the level of lattices, but are sufficiently rich to provide such resolutions as the Jung-Hirzebruch resolution discussed in previous posts (which is the minimal resolution of isolated surface quotient singularities) and describe $G\text{-Hilb}$. It has been remarked to me that the combinatorial presentation of blowups on the symplectic side is more appealing, which I agree with to a point: one is more overtly replacing a torus fixed point with a projective space. Though of course there one can’t so easily (or at all?) treat the nonsmooth case. Swings and roundabouts.

## Moment maps

Motivated by these tales of symplectic embeddings and my desire to get acquainted with both sides of the mirror in mirror symmetry, I have been writing some [developing] notes on symplectic geometry. The extract featured here focuses on moment maps and Delzant’s theorem; an introduction to symplectic toric geometry, if you will. These are intended to be a reference tool and springboard for future posts – some of which will compare the algebraic and symplectic flavours of toric geometry – and are heavily based on the notes of De Silva and of Jeffrey.

## Studio G-Hilb

The McKay correspondence is one of my favourite sagas in algebraic geometry, partly because of the role it played in my mathematical upbringing. I went through the Dynkin classification of finite subgroups of $\text{SL}_2(\mathbb{C})$ via the Du Val singularities with my DRP student last semester, which is one angle of approach. Another is via equivariant Hilbert schemes. The classical Hilbert schemes parameterise certain types of subschemes of some fixed space (for instance, subschemes of $\mathbb{P}^n$ with fixed Hilbert polynomial). For $G$ a finite group, the equivariant version utilised here parameterises $G$-clusters, which are the schematic version of group orbits. This moduli space is called the $G$-Hilbert scheme.

I computed the minimal or crepant resolution of the Du Val singularities of type $A_n$ (the cyclic quotient singularities of the form $\frac{1}{r}(1,-1)$) in a previous post using toric methods. In this note I present two more guises of this resolution, both as moduli spaces. One is the $G$-Hilbert scheme, and the other is as a moduli space of quiver representations of the McKay quiver of $G$. Both are very natural; especially the former which comes with a map to the quotient $\mathbb{A}^2/G$. The extraordinary thing is that these often poorly behaved moduli spaces are smooth varieties and hence supply a resolution of the quotient. They also led the way for $3$-dimensional versions of the McKay correspondence, where crepant resolutions are no longer unique and where there is only a far less elegant and concise classification of finite subgroups of $\text{SL}_3(\mathbb{C})$, but where the $G$-Hilbert scheme is still a computable crepant resolution!

## Symplectic embeddings II

I’ve been thinking a lot more about symplectic embeddings and the associated combinatorics recently after my first post on the topic, aided by some very useful conversations with my friend Julian and with Hutchings himself! I thought that I would post an extract from my developing TeX document on the subject, which focuses on the role lattice point counts play in the iconic example of ellipsoids. In particular, in viewing a modified version of the ECH capacities associated to an ellipsoid – counting how many capacities stay below each integer value – one gets a direct connection to Ehrhart theory. During another conversation about this today, I realised that one can use the rational version of Ehrhart’s theorem to reduce comparing two sequences of capacities to a finite check, albeit an awkward one best left to computers.