It's been quiet over here, because I was focusing on writing all kinds of things. Two of these are now on the arXiv, and the title of this blogpost is the chimera you get by combining the titles of

  1. Rigidity and Schofield's partial tilting conjecture for quiver moduli, joint with Ana-Maria Brecan, Hans Franzen, Gianni Petrella, and Markus Reineke
  2. Vector fields and admissible embeddings for quiver moduli, joint with Ana-Maria Brecan, Hans Franzen, Markus Reineke

I'm really proud of these works, for several reasons:

  • It's the first paper with my PhD student Gianni Petrella. Congratulations Gianni!
  • It answers questions Hans and I dreamt of solving back in 2017, using tools we had no idea existed back in the day.
  • It simultaneously resolves an old conjecture by Aidan Schofield, whose work I greatly admire.
  • It is another example of the philosophy that moduli spaces of vector bundles on curves and moduli spaces of quiver representations behave in very similar ways.
  • They have consecutive arXiv id's.

The main results

The main technical results of the paper can be combined into a single theorem statement, at the cost of losing some generality. Recall that there exists a universal representation $\mathcal{U}=\bigoplus_{i\in Q_0}\mathcal{U}_i$ on the moduli space $\mathrm{M}^{\theta\text{-st}}(Q,\mathbf{d})$ of $\theta$-stable representations of dimension vector $\mathbf{d}$, and our main technical result tells you how to compute the cohomology of the summands of the endomorphism bundle. Because of a (usually hidden) choice of normalization in the definition of $\mathcal{U}$, the cohomology of $\mathcal{U}_i$ depends on this choice, whereas that of $\mathcal{U}_i^\vee\otimes\mathcal{U}_j$ doesn't.

The condition of being strongly amply stable is a strengthening of the ample stability introduced by Reineke--Schröer (which says that the unstable locus has codimension at least 2).

Theorem Let $Q$ be an acyclic quiver, $\mathbf{d}$ an indivisible dimension vector, $\theta$ a stability parameter for which every semistable representation is stable, and assume that $\mathbf{d}$ is strongly $\theta$-amply stable. Then \[ \mathrm{H}^k(\mathrm{M}^{\theta\text{-st}}(Q,\mathbf{d}),\mathcal{U}_i^\vee\otimes\mathcal{U}_j) \cong \begin{cases} e_j \mathbf{k}Q e_i & k=0 \\ 0 & k\geq 1 \end{cases} \] where $e_j \mathbf{k}Q e_i$ is the vector space spanned by paths from $i$ to $j$. The non-trivial isomorphism sends a path $a_\ell\cdots a_1$ from $i$ to $j$ to the composition of the morphisms $\mathcal{U}_{a_m}\colon\mathcal{U}_{\mathrm{s}(a_m)}\to\mathcal{U}_{\mathrm{t}(a_m)}$ for $m=1,\ldots,\ell$.

On the methods

The statement of the technical result contains two parts: an explicit description of the global sections, and a vanishing of the higher cohomology. The two papers use wildly different methods to establish these, which is why they are indeed two separate papers.

For the vanishing, we use Teleman quantization (the Wikipedia link is for the symplectic version). It allows one to compute the cohomology on a GIT quotient by computing it on the quotient stack, before throwing out the unstable locus. The punchline is that the quotient stack for moduli of quiver representations is the quotient of an affine space by a reductive group, thus there is no higher cohomology on the quotient stack! The hard work is in checking that one can apply Teleman quantization, for which lots of weight calculations are necessary.

For the global sections, we use geometric invariant theory, by reducing the calculation to that of a line bundle on a different moduli space, and appealing to the celebrated Le Bruyn–Procesi result to compute this.

There is no dependency between the two results.

The applications

Sure, this all sounds fun, but why is this interesting? Very briefly, the 4 terms in the titles of the papers refer to the following:

  1. using the 4-term sequence for quiver moduli one can prove that moduli spaces of quiver representations are rigid;
  2. the cohomology vanishing is in fact inequivalent to $\mathcal{U}$ being a partial tilting object, as conjectured by Schofield;
  3. the global sections allow one to compute that the first Hochschild cohomology of the path algebra $\mathbf{k}Q$ (for which Happel gave a very explicit presentation) is isomorphic to the vector fields of the moduli space (at least as a vector space, conjecturally also as Lie algebras);
  4. the global sections also allow one to compute the endomorphism algebra of the partial tilting object $\mathcal{U}$, which turns out to be the path algebra $\mathbf{k}Q$, thus setting up an admissible embedding.
For the latter two applications, it is really the combination of the vanishing and the global sections which is needed to prove them. For the first two applications, only the vanishing is needed.

Similarities between moduli spaces of vector bundles and representations

These applications are inspired by what we already knew for moduli spaces of vector bundles on curves:

  • back in 1975, Narasimhan–Ramanan showed that the automorphism group of the moduli space is finite, and that the first-order deformations of the moduli space are identified with those of the curve;
  • in the last decade, there has been a flurry of activity by several authors (starting with Narasimhan and Fonarev–Kuznetsov) to show that the universal vector bundle sets up a fully faithful Fourier–Mukai functor.

We manage to find the quiver analogues for these two results:

  • the vector fields of the moduli space are now isomorphic to the first Hochschild cohomology of the path algebra, which are the appropriate symmetries to consider, and depending on the algebra this can be a very rich space, or zero;
  • the appropriate analogue of a Fourier–Mukai functor in this noncommutative setting is indeed fully faithful.

Using an idea Theo, Lie and I applied earlier for Hilbert schemes of points on surfaces we even relate the two, explaining how the fully faithful functor allows one to obtain the results on the vector fields and rigidity.

This was a long post, but it goes to show how proud I am of these results, and how pleasant it was to collaborate with my excellent coauthors on this. I am more than happy to answer questions you might have about this work!