## Talk Notes: Factoring Derived Categories (2)

< See Part 1

Last time, we talked about hearts of a triangulated category, and how they corresponded to subcategories that allowed us to uniquely factor objects in the derived category. Using this philosophy, we discovered that finding a way to factor the heart itself into two pieces (a ‘torsion pair’) gave us a way of making a new heart.

Today, I will talk about a specific example of an abelian category that doesn’t just factor into two pieces, but instead factors into $\mathbb{Q}$ pieces. This is the theory of semistable bundles on a projective curve. Motivated by this example, I will define a ‘slicing’ of a triangulated category, which is roughly when you have a heart that factors alot. Then, I will attempt to motivate why the ‘set of all slicings’ of a triangulated category is not the right thing to look at. The concept needs to be rigidified before the corresponding space is nice. Answering this is the notion of a ‘stability condition’, which I will define. Finally, I will talk about why this is a nice notion, and some current research into these spaces.

These notes are going up later than I intended; I gave the talk these cover earlier today. Grading reared its ugly head and I was only able to work on the talk for a few hours beforehand.

Stable Bundles

Let $X$ be a smooth, projective complex curve. Then for any vector bundle $B$ on $X$, I define the following quantity:

$slope(B)=\frac{deg(B)}{rank(B)}$

I also have the following pair of definitions:

Def. A vector bundle $B$ is semistable (resp. stable) if, for any sub-bundle $A\subset B$, $slope(A)\leq slope(B)$ (resp. $slope(A)).

The strength of these concepts comes from the following key fact: if $A$ and $B$ are both semistable bundles with $slope(A)>slope(B)$, then $Hom(A,B)=0$. This is because the image of $A$ in $B$ would have to be another vector bundle, but simple computations show that this bundle would have to have a higher slope than $A$ and a lower slope than $B$.

Since we would like to start talking about all coherent modules on $X$, and not just the locally free ones (vector bundles), we employ the following cheat. We simply declare any coherent module which admits no maps into a vector bundle is semisimple of slope $+\infty$. Then the above key fact still holds automatically.

The reason semistable bundles are important for factoring derived categories is the following theorem of Harder and Narasimhan.

Thm. Let $b$ be a coherent module on $X$. Then there is a series of inclusions:

$0 \rightarrow b_1 \rightarrow b_2 \rightarrow ... b_{n-1}\rightarrow b_n=b$

such that $b_{i-1}/b_i$ is semistable of slope $k_i$ and $k_i>k_{i+1}$.

Note: as always, this factorization is unique.

Notation: Let $\phi_-(b)=k_n$ and $\phi_+(b)=k_1$.

This is a suggestively similar to the property that hearts have, in that it tells us how to make anything as a sequence of extensions in a canonical way. One immediate advantage of this is that we can make a wide array of torsion pairs. For any real number $\lambda\in \mathbb{R}$, let

$T_\lambda=\{b\in Coh(X)\;s.t.\; \phi_-(b)\geq\lambda\}$, and

$F_\lambda=\{b\in Coh(X)\; s.t. \; \phi_+(b)<\lambda\}$.

These then clearly form a torsion pair; this means we get a whole family of new hearts of $D^b(Coh(X))$ parametrized by $\mathbb{R}$.

Having an $\mathbb{R}$ family of hearts has an unexpected consequence; it gives us a topology on the hearts that appear. By having an infinite number of linearly ordered pieces in which to factor the heart, we get a notion of ‘deforming’ a heart into a nearby heart. If possible, we would like to extend this topology to the set of all hearts.

So what did the Harder-Narasimhan filtration really mean for $D^b(Coh(X))$? Given any derived object, we could first use the defining property of the heart $Coh(X)$ to express it as a sequence of extensions by objects in shifts of the heart. Then, each object in the heart (or its shifts) could be expressed as a sequence of extensions by semistable objects of descending slope. Combining these two, any derived object can be uniquely expressed as a sequence of extensions by shifts of semistable objects of descending slope, with the assumption that a shift decreases the slope by $\omega$, some infinite quantity. Let’s turn this into a definition:

Def. A slicing of a triangulated category $D$ is a family of full subcategories $P(\phi),\; \phi\in\mathbb{R}$, such that:

1) $P(\phi)[1]=P(\phi+1)$.

2) If $\phi>\phi'$, and $a\in P(\phi),\;b\in P(\phi')$, then $Hom(a,b)=0$.

3) For all $e\in D$, there exists a diagram:

$\begin{array}{ccccccccc} 0 & \rightarrow & e_1 & \rightarrow & e_2 & ... & e_{n-1} & \rightarrow & e_n=e\\ & \nwarrow & \downarrow & \nwarrow & \downarrow & & & \nwarrow & \downarrow \\ & & a_1 & & a_2 & & & & a_n\end{array}$

such that $a_i\in P(\phi_i)$ and $\phi_i>\phi_{i+1}$.

Define $\phi_-(e)=\phi_n$ and $\phi_+(e)=\phi_1$.

Note: as always, the definition automatically will imply the factorization is unique.

It should be clear from the above discussion that $D^b(Coh(X))$ has a slicing given by $P(\phi)=$ semistable bundles of slope $\tan(\pi(\phi-1/2))$ (for $\phi$ between 0 and 1, extending by shifts).

Given a slicing of $D$ and a real number $\lambda$, the full subcategory consisting of objects $e\in D$ such that $\phi_-(e)>\lambda$ and $\phi_+(e)\leq \lambda +1$ is a heart. Thus, any slicing determines an $\mathbb{R}/\mathbb{Z}=S^1$ family of hearts.

It is tempting at this point to consider the set of slicings on a given triangulated category and ask if it has a natural topology. One can define an (almost) metric on the set of all slicings by taking the max of the differences $\phi_-(e)-\phi'_-(e)$ and $\phi_+(e)-\phi'_+(e)$ as $e$ runs over all objects in the category. The problem one runs into here is that this new space is infinite-dimensional; the notion of slicing is too floppy. For instance, notice that for any order-preserving map from $\mathbb{R}$ into itself which fixes the integers, one can re-index the slicing to get a new slicing. If we want a space that is nice, we need a more rigid notion.

Interlude: K-theory of an Abelian Category

First, however, I would like to spend a small amount of time building up the tool of K-theory. Depending on your background, this might be a different kind of thing than you usually think about as K-theory, so be forewarned.

The motivating question here: Is there someway of talking about all the ‘pieces’ an object in an abelian category is made of, without writing down exactly how to make it? I am thinking of an extension

$0\rightarrow a\rightarrow b\rightarrow c\rightarrow 0$

as telling me that $b$ is ‘made out of’ $a$ and $c$. So, for instance, I consider the Harder-Narasimhan filtration as telling me how to ‘make’ any bundle from semi-stable bundles.

The problem is that there are often lots of interesting extensions between any two objects in an abelian category, and so an object is often very far away, conceptually, from a set of objects that it is made from. However, there do exist categories where there are no non-trivial extensions. Take, for example, the category of finite dimensional vector spaces. Knowing that a given vector space is made from a set of other vector spaces completely determines it: it is isomorphic to their direct sum. This immediately implies that every vector space is isomorphic to a direct sum of simple vector spaces. Furthermore, we recall that it is useful to talk about the ‘dimension’ of a vector space, which we can do without having a specific basis in mind. This is an example of a way in which we already prefer to think about what pieces an object is made from (how many copies of $\mathbb{C}$, without worrying about the details of how to make it from those pieces (choosing a basis).

This leads to the following construction. For any abelian category $A$, consider the (quite large) free abelian group $\mathbb{Z}A$ generated by the objects. Then, impose the relations that, given any short exact sequence

$0\rightarrow a\rightarrow b\rightarrow c\rightarrow 0$,

then $(a)+(c)=(b)$, where parenthesis indicate their image in $\mathbb{Z}A$. Call this quotient group $K(A)$ the K-group of $A$.

Given any object $a$ in $A$, its image $(a)$ in $K(A)$ only depends on the simple objects that $a$ can be made from. Therefore, in the case of f.d. vector spaces, we recover dimension, and in the case of coherent sheaves on a projective curve, we get something that only depends on the pieces of the Harder-Narasimham filtration.

Useful fact: Given a quiver $Q$, the K-group of its category of representations is just the free abelian group generated by the vertices of the quiver. The crude reason this is true is that an arrow in a quiver is equivalent to defining a non-trivial extensions of its source by its target. The K-group crushes out all this information, so it effectively throws away the arrows.

Using the prevailing mantra that the exact triangles in a triangulated category are the right analogs of short exact sequences, the K-group of a triangulated category can be defined completely the same. However, we now have the following observation. Given an exact triangle $a\rightarrow b\rightarrow c\rightarrow$, then $(a)+(c)=(b)$. But, $b\rightarrow c\rightarrow a[1]\rightarrow$ is also exact, so $(b)+(a[1])=(c)$. Therefore, the shift map induces multiplication by -1 on the K-group.

Now what if the category at hand is $D^b(A)$? Every object must factor into a series of extensions by shifts of objects in the heart. Therefore, in the K-group, every element must be the sum of elements generated by the heart; ie, $K(D^b(A))=K(A)$. So as long as we only think about derived categories, K-groups aren’t anything new.

It is also an interesting consequence of this that if two abelian categories are (bounded) derived equivalent, they must have isomorphic K-groups. For those who know the Beilinson equivalence, this shows that $K(\mathbb{P}^1)=\mathbb{Z}\oplus\mathbb{Z}$.

Stability Conditions

We now have the tools we need to define a stability conditions, but lets try a small amount of motivation first. Given a short exact sequence of vector bundles on a smooth projective curve

$0\rightarrow a\rightarrow b\rightarrow c\rightarrow 0$,

one cannot compute $slope(b)$ in terms of $slope(a)$ and $slope(c)$. This is frustating, because one can compute both the degree and the rank of $b$ from the degree and rank of the consituent elements. Therefore, I can try to beef up the concept of slope by using the complex numbers:

$Z(a)=-deg(a)+i\;rank(a)$

This number $Z(a)$ is called the central charge. It is clear that I can recover the slope from the central charge by just taking the complex phase. But now, given a short exact sequence as above, $Z(b)=Z(a)+Z(c)$. To put this in fancier language, $Z$ defines a homomorphism from $K(Coh(X))$ to $\mathbb{C}$.

This leads to the following big defintion.

Def. A stability condition on a triangulated category $D$ is a slicing $P$, together with a homomorphism $Z$ from $K(D)$ to $\mathbb{C}$ such that for every non-zero element $e$ in any $P(\phi)$,

$Z(e)=m(e)\exp(i\pi\phi)$

for $m(e)$ a positive real number called the mass of $e$.

The existance of a central charge limits how much the slicing can be flopped around, which would make one hope that the space of stability conditions is nice. This is largely true, modulo one technical detail. For this, we need to delve into a technical defintion that the uninterested reader is invited to skip.

Def. Given a slicing on $D$, let $P_\epsilon(\phi)$ be the extension-closed full subcategory generated by $P(\lambda)$, for $\phi-\epsilon<\lambda<\phi+\epsilon$.

Then a slicing is locally-finite if, for all $\phi$, there is an $\epsilon$ such that $P_\epsilon(\phi)$ has only finite length chains of non-zero monomorphisms and epimorphisms. Here, a monomorphism is any arrow that is the first arrow of an exact triangle contained entirely in $P_\epsilon(\phi)$, and a similar definition for epimorphisms.

This definition is unfortunately unintuitive, but its purpose is reasonable. We want to talk about the space of all stability conditions, but we want to rule out ‘degenerate’ stability conditions, like those with all the non-zero objects sitting in $P(\mathbb{Z})$. Locally-finiteness will assure us that too much information is never crammed into one piece of the slicing.

This leads us to a big theorem, which is the main reason for these two talks.

Thm. (Bridgeland, 2002) Let $Stab(D)$ be the space of locally-finite stability conditions on a triangulated category $D$. Then $Stab(D)$ is a manifold with dimension bounded above by the number of generators of $K(D)$.

Additionally, if $D=D^b(Coh(X))$ for $X$ a smooth projective manifold, then this manifold is finite-dimensional.

It is important to point out that at no point does the theorem assert that the space is non-empty. Arguably the hardest part of working with these spaces for all but the most basic categories is showing the existance of even one stability condition.

For triangulated categories that do not correspond to smooth projective manifolds, it is an area of active research to show that this manifold is still finite-dimensional. For example, I believe this is known for the derived category of Kleinian singularies.

This space also has actions of both $Aut(D)$ and $\mathbb{C}^\times$ on it, which has proved useful for understanding $Aut(D)$. This is one of the main areas in which stability conditions are expected to be productive. It is also not known whether all spaces of stability conditions are contractible, which would mean that there was effectively only one choice of a stability condition (provided there was any).

Alright, I think I’ll shut up, since this is a pretty long post as it is. Anyone still curious about stability conditions is encouraged to read some of Bridgeland’s pre-prints on the arxiv, which are very readable and informative.