# 2015 AG Institute in Utah: Day 3

Day four of the 2015 AG Institute has just ended!! As with the intro post for this conference I am a bit delayed getting this post up, and so this post will only cover things through Day 3 (Wed, July 16)  of the conference. Day 4 and beyond will be covered in a future post.

Day 3 was sort of a rest day and so there were no lectures in the afternoon. I spent much of this time starting to explore Salt Lake City. First I tried to hike up a small hill located behind the dorms. (It looks like it would give  a great view of the city.) However, the trail I wanted to hike was closed

Salt Lake from head of the trail

So instead of hiking I ended up exploring campus and a bit of the city. Didn’t end up seeing to much, but I did see Ute Stadium and what I think is the 2002 Olympic torch!

Where the reign of Harbaugh will begin!

The University of Utah campus is completely dry (see the post-script for more about this) and where we are staying is fairly far away so the first few days of the conference saw little of the usual night time conference socialization. However, someone discovered the Marriott near campus has a bar, and so there was an invasion of mathematicians.

Finally, a quite rundown of a few of the cooler things I have heard/learned so far — this is in no way an exhaustive list. Disclaimer: All of this is based on the talks I’ve seen this week and I do not understand much of it. Any errors are due to me and not the speakers :

• Green-Lazarsfeld Secant Conjecture (Gavril Farkas): The Green-Lazarsfeld Secant (GLS) conjecture is a generalization of Green’s conjecture concerning the syzygies of a curve. The set up for the conjecture is as follows: we have a smooth projective curve $C$ of genus $g$ and a very ample line bundle $L$ on $C$. Now $|L|$ determines an embedding of $C$ into $\mathbb{P}^n$ where $n= \text{deg}(L)-g$. We let $S_C$ be the homogenous coordinate ring of $C$ with respect to this embedding and $S$ the homogenous coordinate ring of the ambient space i.e. the polynomial ring. (Remember $S_C$ is really really dependent on $L$.) Now one way to study $S$, and so study $C$, is by looking at its minimal free resolution, and the associated Betti numbers. (If $F_{\cdot}\rightarrow S\rightarrow 0$ is a free resolution then each $F_i$ is free and so is of the form $S(-j)^{b_{ij}}$ were $S(-j)$ is $S$ with its grading twisted down by $j$. Recall this is a resolution as graded modules so we need our maps to respect the grading hence the twisting. These $b_{ij}$ are called the Betti numbers of $S_X$.) The general idea behind the (GLS) conjecture is that these the level of ampleness of $L$ restrain these $b_{ij}$. Explicitly (as presented here) the GLS conjecture says:

Conjecture (GLS): Let $C$ is a smooth projective curve of genus $g$ and $L$ be a globally generated line bundle of degree $d$ on $C$ such that
$d\geq 2g+p+1-2h^1(C, L)-\text{Cliff}(C)$
then $L$ fails $N_p$ if and only if $L$ is not $(p+1)$-very ample.

Certainly this conjecture is a mouthful, but my understanding is that it basically boils down to the idea that if $L$ has a very very large degree then ampleness of $L$ determines when the Betti numbers start to vanish. Note the conjecture gets its name because the condition that $L$ is not $(p+1)$-very ample is equivalent to saying the embedding of $C$ induced by $|L|$ has a $(p+2)$-secant $\latex p$-plane. (Not sure exactly why…) Regardless the big point of the talk was the following theorem:

Theorem (Farkas-Kemeny): The GLS conjecture holds for a general curve and general line bundle.

Note the GLS conjecture is a generalization of Green’s conjecture for canonically embedded curves, which Claire Voisin proved was true for general curves about 8 years ago. My understanding of the proof is that satisfying the GLS conjecture is an open condition (not sure why…) in a moduli space (not sure precisely which…) hence to prove it for a general curve it suffices to cook up examples of the theorem. Farkas and his co-author Michael Kemeny have recently managed to do this!! I did not follow there exact constructs, but somehow utilize K3 surfaces — I think Voisin’s work also goes via a similar route. This is a talk I really want to understand better, and would like to know more about the general area.

• Groups of Birational Transforms (Serge Cantat): Serge Cantat has given a series of great talks discussing the complexity of the group $\text{Bir}(V)$ of birational transformations of a variety $V$. The general theme of this has been that as $\text{dim} V$ gets larger the complexity of $\text{Bir}(V)$ increases. Cantat has presented many different incarnations of the this general idea, but one example I find particularly interesting is the following theorem:

Theorem (Cantat-Xie): If $\Gamma\subset \text{SL}_n(\mathbb{Z})$ is a finite index subgroup with $n\geq3$ and $V$ is a complex projective variety (irreducible) then if $\Gamma$ embeds into $\text{Bir}(V)$ then $\text{dim}(V)\geq n-1$. Moreover in the case of equality $V$ is rational and $\Gamma$ acts via conjugation by regular automorphisms.

This seemed quite surprising to me, and the proof uses p-adic dynamics! (In particular, that p-adic dynamics is much more well-behaved as compared to complex dynamics.) Notice from this we can also get the following corollary regarding the Cremona group. Recall that the nth Cremona group over $k$ is $\text{Bir}(\mathbb{P}^n_k)$. (Alternatively it is the group of automorphisms of $k(x_1,\ldots,x_n)$.

Corollary: $\text{Cr}_n(k)$ is isomorphic to $\text{Cr}_m(k)$ if and only if $n=m$. ($k=\mathbb{C}=\mathbb{Q}=\cdots$)

I’ve never really thought about the group of birational transformations, and I’ve found it really interesting — and to me surprising, although maybe I shouldn’t have — that such groups can be used to be prove facts about the geometry of a variety e.g. the moreover bit of the above theorem. (The slides for his talks are available here and videos should be posted shortly after the conference.)

• YTD Conjecture (Simon Donaldson): Finally Field’s medalist and Breakthrough prize winner Simon Donaldson has been giving a series of lectures on the YTD conjecture. These conjectures concern the existence of certain types of metrics on Kahler manifolds and connect algebraic geometry with symplectic and remanian geometry and analysis. Kahler geometry is not my cup of tea — I actually barely know the definition of a Kahler manifold, however, these lectures have served to show a surprising connection between metrics and geometric invariant theory (GIT).Let $G$ be a compact group and $G^c$ be the complexification of $G$ so that $G^c$ is a reductive complex Lie group. (Just think of $G^c$ as being a nice group i.e. $\text{GL}_n(\mathbb{C})$ or $\text{SL}_n(\mathbb{C})$). Suppose $V$ is a complex vector space on which $G^c$ acts linearly i.e. a complex representation of $G^x$. Hence $G^c$ acts on $\mathbb{P}(V)$. Now the motivating question of GIT is: How can we parameterize the orbits of $G^c$ on $\mathbb{P}V$? Put different how can we construct the moduli space of $G^c$ orbits?One approach to this is to let $R$ be the ring of invariants of polynomials on $\mathbb{P}V$. In nice cases $R$ will be finitely generated and so $\text{Proj}(R)$ is a somewhat reasonable thing to consider. Moreover $\text{Proj}(R)$ is close to being the moduli we want, but not quite :/. In order to understand what $\text{Proj}(R)$ actually parameterizes we introduce two different classes of points of $\mathbb{P}(V)$:
1. A point $v\in \mathbb{P}(V)$ is stable iff $\text{Orb}_{G^c}(v)$ is closed.
2. A point $v\in \mathbb{P}(V)$ is semi-stable iff $0\not\in \overline{\text{Orb}_{G^c}(v)}$.

Now $\text{Proj}(R)$ parameterizes semi-stable points. A theorem of Mumford called the Hilbert-Mumford criteria provides a way to distinguish between stable and semi-stable points in terms of certain numerical invariants related to one parameter subgroups. Not wanting to go into the details of this here — also because I am shaky on the details of this — the important thing to take away is there is an algebraic criteria for these notions.

That said it turns out these things can also be detected metrically!! In particular, if $N$ is a Hermitian metric on $\mathbb{P}(V)$ invariant under $G^c$ then we can re-phrase the definition of stability as follows:

1. A point $v\in \mathbb{P}(V)$ is stable iff $\text{Orb}_{G^c}(v)$ contains a point minimizing $N$.
2. A point $v\in \mathbb{P}(V)$ is semi-stable iff $N$ is bound from below away from zero on $\text{Orb}_{G^c}(v)$.

This itself seems really cool that somehow a metric can detect something that I thought about as being an algebraic property. That said maybe this is not to surprising since the actually definition of stability and semi-stability are fairly topological in nature. (Thoughts?) However, it gets even better than this. For each point $v\in V$ we may define a function $F_v(g):G^c\rightarrow\mathbb{R}$ by $F_v(g)=\text{log}(N(g\cdot V))$. This turns out to be invariant under $G$ and so descends to a function $F_v(g):G^c/G\rightarrow\mathbb{R}$. Finally we have that

1. A point $v\in \mathbb{P}(V)$ is stable iff $F_v(g)$ attains a minimum.
2. A point $v\in \mathbb{P}(V)$ is semi-stable iff $F_v(g)$ is bounded below.

This seems even more out there than the first re-definition of stability/semi-stablity. I tend to view the existence of minima as a fairly analytic property of a function so I found/find this really really surprising. (Of course I also have no idea how this is proved so that also adds to the intrigue.)

Finally I said that Donaldson was speaking about when Khaler manifolds had certain types of metrics, but so fair all I have talked about is the relationships between classical GIT and metrics. It turns out that this is sort of the motivation for how one goes about proving the existence of these metrics!!! Donaldson’s slides are available here, and video from the lectures should be posted at the end of the conference.

PS: Apparently the University of Utah heard about how crazy math conferences can be and wished to be proactive:

Included in the welcome packet.