Pieter Belmans
http://pbelmans.ncag.info/
Tue, 09 Jul 2024 09:12:43 +0000Tue, 09 Jul 2024 09:12:43 +0000Jekyll v3.9.5QuiverTools<p style="background-color: rgb(255, 243, 205); border: 1px solid rgb(255, 238, 186); padding: 10px;">
<img style="float: left; margin-right: 10px" src="/assets/quivertools-logo.svg" width="70">
This post concerns <a href="https://quiver.tools">QuiverTools</a>.
If you use it for your research, please cite it using <a href="https://zenodo.org/doi/10.5281/zenodo.12680224" rel="nofollow"><img src="https://zenodo.org/badge/DOI/10.5281/zenodo.12680224.svg" alt="DOI" data-canonical-src="https://zenodo.org/badge/DOI/10.5281/zenodo.12680224.svg" style="max-width:100%;"></a>.
</p>
<h2>QuiverTools</h2>
<p>I am really excited to announce something that has been in the making for quite some time, and which is the result of the hard work with
<ul>
<li><a href="https://math.uni-paderborn.de/ag/arbeitsgruppe-algebra/team/hans-franzen">Hans Franzen</a>
<li><a href="https://giannipetrella.eu">Gianni Petrella</a>
</ul>
<p>Many questions in the representation theory and algebraic geometry of quivers and their moduli of representations
can be answered using very algorithmic methods.
This is what <a href="https://quiver.tools"><strong>QuiverTools</strong></a> is about.
We do not deal with actual representations (that is what <a href="https://folk.ntnu.no/oyvinso/QPA/">QPA</a> is for),
rather our focus is on the geometry of moduli spaces.
<h3>An example</h3>
Let us illustrate some things.
One of my favourite quiver moduli is the 6-dimensional Kronecker moduli space,
which Hans and I considered in <a href="https://doi.org/10.1093/imrn/rnad306">On Chow rings of quiver moduli</a>.
Let us define it in QuiverTools:
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="kn">from</span> <span class="nn">quiver</span> <span class="kn">import</span> <span class="o">*</span>
<span class="n">Q</span> <span class="o">=</span> <span class="n">KroneckerQuiver</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">QuiverModuliSpace</span><span class="p">(</span><span class="n">Q</span><span class="p">,</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span></code></pre></figure>
<p>Then we can compute some of its basic invariants as follows:
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">X</span><span class="p">.</span><span class="n">is_projective</span><span class="p">()</span> <span class="c1"># True
</span><span class="n">X</span><span class="p">.</span><span class="n">is_smooth</span><span class="p">()</span> <span class="c1"># True
</span><span class="n">X</span><span class="p">.</span><span class="n">dimension</span><span class="p">()</span> <span class="c1"># 6
</span><span class="n">X</span><span class="p">.</span><span class="n">picard_rank</span><span class="p">()</span> <span class="c1"># 1
</span><span class="n">X</span><span class="p">.</span><span class="n">index</span><span class="p">()</span> <span class="c1"># 3</span></code></pre></figure>
<p>Just like in the Hodge diamond cutter you can compute Betti numbers:
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">X</span><span class="p">.</span><span class="n">betti_numbers</span><span class="p">()</span> <span class="c1"># [1, 0, 1, 0, 3, 0, 3, 0, 3, 0, 1, 0, 1]</span></code></pre></figure>
<p>We also support the recent work involving Chow rings and rigidity questions for quiver moduli:
<figure class="highlight"><pre><code class="language-python" data-lang="python"><span class="n">eta</span> <span class="o">=</span> <span class="o">-</span><span class="n">Q</span><span class="p">.</span><span class="n">canonical_stability_parameter</span><span class="p">(</span><span class="n">d</span><span class="p">)</span> <span class="o">/</span> <span class="mi">3</span>
<span class="n">X</span><span class="p">.</span><span class="n">degree</span><span class="p">(</span><span class="n">eta</span><span class="p">)</span> <span class="c1"># 57
</span>
<span class="n">X</span><span class="p">.</span><span class="n">if_rigidity_inequality_holds</span><span class="p">()</span> <span class="c1"># True</span></code></pre></figure>
<p>This is only a very quick tour of what it can do, more explanations are to come!
<p>If you have problems installing it, you can also run it inside your browser using <a href="https://mybinder.org/v2/gh/QuiverTools/mybinder-sage/master">this interactive binder notebook</a>.
<h3>Some comments</h3>
<ul>
<li>This release is really only a <code>v1.0</code>.
Let us know if you find issues, or have feature requests!
<li>We are also working on a Julia version, focusing more on performance.
Stay tuned for more information.
<li>We are also working on more detailed instructions:
for now there is the documentation taken from the docstrings,
which should be good enough if you have some experience with Sage and quiver moduli,
but a more user-friendly guide is in the works.
</ul>
Sun, 07 Jul 2024 00:00:00 +0000
http://pbelmans.ncag.info/blog/2024/07/07/quivertools/
http://pbelmans.ncag.info/blog/2024/07/07/quivertools/algebraic geometrymathematicsUpper-Rhine and Tributaries Algebraic Geometry Seminar<p>I am really happy to announce <strong><a href="https://urtags.info">URTAGS: the Upper-Rhine and Tributaries Algebraic Geometry Seminar</a></strong>,
a joint seminar with the universities of of Basel, Freiburg, Lorraine, Luxembourg, Saarbrücken, Strasbourg, Stuttgart, and Tübingen.
<p>The first edition is scheduled for <strong>June 25</strong>,
and 2 or 3 editions per year (twice in the centrally located Strasbourg, once on location) are planned.
<p>I hope to see some of you there!
Wed, 29 May 2024 00:00:00 +0000
http://pbelmans.ncag.info/blog/2024/05/29/urtags/
http://pbelmans.ncag.info/blog/2024/05/29/urtags/algebraic geometryNew paper: On decompositions for Fano schemes of intersections of two quadrics<p>I'm really happy to announce that we have a new paper on the arXiv: <a href="https://arxiv.org/abs/2403.12517"><strong>On decompositions for Fano schemes of intersections of two quadrics</strong></a>.
The <em>we</em> here refers to my coauthors Jishnu Bose, Sarah Frei, Ben Gould, James Hotchkiss, Alicia Lamarche, Jack Petok, Cristian Andres Rodriguez Avila, and Saket Shah.
Here is all of us:
<center>
<img src="/assets/bundles-group.jpg" width="600">
</center>
<p>This picture was taken June 2023 in upstate New York,
because the paper is a collaboration which started at the
<a href="https://www.ams.org/programs/research-communities/2023MRC-DerivedCategories">Mathematics Research Communities: Derived Categories, Arithmetic and Geometry</a>,
expertly organized by Matthew Ballard, Katrina Honigs, Daniel Krashen, Alicia Lamarche, and Emanuele Macrì,
where I led a group of PhD students and postdocs on a week-long adventure ride
on Fano schemes of intersections of two quadrics.
One of the projects we were working on has now materialized in the form of this preprint.
<h3>Decompositions for varieties</h3>
<p>In an earlier joint work with Sergey Galkin and Swarnava Mukhopadhyay we looked at
<a href="https://mathscinet.ams.org/mathscinet/article?mr=4557892">how the moduli space of rank-2 bundles on a curve "decomposes" in several ways</a>,
the following 2 being relevant here:
<ul>
<li>a semiorthogonal decomposition of its derived category
<li>an identity in the Grothendieck ring of varieties
</ul>
<p>This has led to what is called the BMGN conjecture (with Narasimhan independently conjecturing the semiorthogonal decomposition),
which is now settled by <a href="https://arxiv.org/abs/2108.11951">Tevelev–Torres</a> and <a href="https://arxiv.org/abs/2304.01825">Tevelev</a>.
<p>On the other hand,
the <em>very first example</em> of a semiorthogonal decomposition
was that of $Q_1\cap Q_2\subset\mathbb{P}^{2n+1}$, due to Bondal and Orlov,
where the derived category of a naturally associated hyperelliptic curve appears.
<p>It is possible to interpolate between this ur-example
and moduli spaces of rank-2 bundles,
by considering Fano schemes of linear subspaces on intersections of 2 quadrics.
For $Q_1\cap Q_2\subset\mathbb{P}^{2n+1}$,
we will denote these as $\mathrm{F}_k(Q_1\cap Q_2)$,
for $k=0,\ldots,g-1$.
For more on the geometry of these Fano schemes, I refer to our paper.
<p>But what should the decomposition be?!
<h3>Hodge numbers and conjectures</h3>
<p>To find out what the decomposition ought to be
we can use the <a href="https://github.com/pbelmans/hodge-diamond-cutter">Hodge diamond cutter</a>.
With a bit of optimism,
<ol>
<li>a geometrically meaningful identity in $\mathbb{Z}[x,y]$ on the level of Hodge diamonds
<li><em>might</em> be arising from a decomposition in the Grothendieck ring of varieties,
<li>which in turn gives an identity in the Grothendieck ring of categories,
<li>which in turn <em>might</em> be arising from an actual semiorthogonal decomposition.
</ol>
<p>This type of optimism is of course not always warranted,
but it is exactly what led the BGMN conjecture.
So this brings us to what we did:
<ul>
<li>we explained how to compute the Hodge numbers of these Fano schemes,
by bootstrapping from earlier works of Chen–Vilonen–Xue:
this is where we worked some magic with mixed Hodge modules;
<li>we found a mysterious (to me) formula for the multiplicity of each component,
which can be checked numerically to be the right thing
</ul>
<p>Throughout I've focused on the "hyperelliptic" case, where the intersection of two quadrics is odd-dimensional.
There is a parallel story for the even-dimensional case, which can be called the "stacky" case.
Here, <a href="https://arxiv.org/abs/2305.18058">Fonarev has obtained a conjectural semiorthogonal decomposition in the "maximal" case</a>,
similar to the BGMN conjecture being the maximal case, and $Q_1\cap Q_2$ being the minimal case.
<p>Our work thus describes aspects of the geometry of $\mathrm{F}_k(Q_1\cap Q_2)$ is
<ul>
<li>for all dimensions of $Q_1\cap Q_2$, independent of the parity,
<li>for all $k=0,\ldots,g-1$.
</ul>
and uses it to make some conjectures about these really interesting varieties.
If you're not convinced why these are interesting, just check out the previous installment of fortnightly links.
<p>Let me know if you have any comments or questions!
Wed, 20 Mar 2024 00:00:00 +0000
http://pbelmans.ncag.info/blog/2024/03/20/new-paper-mrc/
http://pbelmans.ncag.info/blog/2024/03/20/new-paper-mrc/algebraic geometrymathematicsPostdoc position in derived algebraic geometry with Sarah Scherotzke<p>My colleague <a href="https://math.uni.lu/scherotzke/">Sarah Scherotzke</a> is advertising a postdoc position in derived algebraic geometry
on <a href="https://www.mathjobs.org/jobs/list/24449">MathJobs</a>.
The focus is on derived algebraic geometry.
It is not inconceivable that you get to interact with me too,
given that Sarah and I lead
<a href="https://pbelmans.ncag.info/uni.lu/group">a joint research group on algebraic geometry and representation theory</a>.
<p>If you have any questions, you can email her,
or both of us (or just me, if you think I'm better placed to answer your question).
Fri, 15 Mar 2024 00:00:00 +0000
http://pbelmans.ncag.info/blog/2024/03/15/postdoc-position-sarah/
http://pbelmans.ncag.info/blog/2024/03/15/postdoc-position-sarah/mathematicsFortnightly links (174)<ul>
<li><p><a href="https://arxiv.org/abs/2402.18857v1">Lena Ji, Fumiaki Suzuki: Arithmetic and birational properties of linear spaces on intersections of two quadrics</a>
is a really cool preprint,
which studies $\mathrm{F}_r(Q_1\cap Q_2)$ for $Q_1\cap Q_2\subset\mathbb{P}^{2g+1}$ and $\mathbb{P}^{2g}$.
This is a very well-behaved moduli space attached to the intersection of 2 quadrics,
whose geometry is a source of great fun,
as it is closely related to the geometry of a hyperelliptic (resp. stacky) curve.
Ji and Suzuki obtain some <em>strong</em> results on the rationality of these moduli spaces,
putting some of the earlier results into a single pretty and coherent picture.
<p>It does not appear in their introduction,
but I in particular like Theorem 4.13,
which relates the existence of a maximal linear subspace on $Q_1\cap Q_2\subset\mathbb{P}^{2g+1}$
to the vanishing of a natural Brauer class on this associated hyperelliptic curve.
<li><p><a href="https://people.math.ethz.ch/~rahul/ModuliReflections.pdf">Reflections on moduli space</a>
is a collection of comments, musings, answers to the question "Why are you interested in moduli spaces"
asked by Rahul Pandharipande to several of his colleagues.
There are some really interesting comments in there!
One which I particularly like
as it directly applies to me,
but I hadn't realized it before,
is
<blockquote>
When I learned about moduli spaces, it was at a time when I generally liked all sorts of stuff (literature/cinema) that was self-referential, so a statement like <em>the set of all geometric objects of a certain type is itself a geometric object</em> was really music to my ears.
</blockquote>
<li><p><a href="https://arxiv.org/abs/2402.15685v1">Yujiro Kawamata: On formal non-commutative deformations of smooth varieties</a>
is a really cool preprint which explains how to do deformations of $\operatorname{coh}X$
without having to deal with those pesky gerby deformations.
In other works, one is only concerned with the piece $\mathrm{H}^0(X,\bigwedge^2\mathrm{T}_X)\oplus\mathrm{H}^1(X,\mathrm{T}_X)\subset\mathrm{HH}^2(X)$.
The corresponding deformation-obstruction calculus is worked out
in a way that is immediately applicable by an algebraic geometer.
</ul>
Tue, 05 Mar 2024 00:00:00 +0000
http://pbelmans.ncag.info/blog/2024/03/05/fortnightly-links-174/
http://pbelmans.ncag.info/blog/2024/03/05/fortnightly-links-174/fortnightly linksmathematicsScholarships for studying at the University of Luxembourg<p>I doubt this post is relevant to the readers of this blog,
but maybe you are supervising or advising someone for whom it is useful.
I am not an expert on these things,
but it seems that there are some upcoming deadlines
for <a href="https://www.uni.lu/life-en/financial-support/scholarships/">scholarships at the University of Luxembourg</a>,
which thus could be of some interest to people who want to come and study at our university.
<p>Please spread the word!
Mon, 04 Mar 2024 00:00:00 +0000
http://pbelmans.ncag.info/blog/2024/03/04/scholarships-at-unilu/
http://pbelmans.ncag.info/blog/2024/03/04/scholarships-at-unilu/mathematicsI've joined the editorial board of Experimental Mathematics<p>This has been on the website for a few months now,
but I have joined the editorial board for <a href="https://www.tandfonline.com/journals/uexm20">Experimental Mathematics</a>,
following the "reboot" led by <a href="https://kasprzyk.work">Al Kasprzyk</a>.
<p>Please submit your interesting and high-quality experimental papers,
so that the journal can continue its role as a leading journal
in this particular niche!
Tue, 20 Feb 2024 00:00:00 +0000
http://pbelmans.ncag.info/blog/2024/02/20/editor-for-experimental-mathematics/
http://pbelmans.ncag.info/blog/2024/02/20/editor-for-experimental-mathematics/metaAn update for Grassmannian.info: 2 big exceptional quantum spectra<p>A little while ago I let my computer waste some electricity on computing the quantum spectra
of two exceptional Grassmannians.
For more on the quantum spectrum,
see <a href="/blog/2020/11/29/spectra-on-grassmannian/">this blogpost for more information</a>.
<p>The exceptional Grassmannians are:
<ul>
<li><a href="https://www.grassmannian.info/E7/3">E7/P3</a>, with Euler characteristic 2016
<li><a href="https://www.grassmannian.info/E7/5">E7/P5</a>, with Euler characteristic 4032
</ul>
<p>I believe slightly larger Grassmannians should also within reach, but the calculation,
which by no means was optimized,
did not finish on time.
Below you can see the largest quantum spectrum currently available on Grassmannian.info:
<img src="/assets/quantum-spectrum-E7-P5.png">
Thu, 15 Feb 2024 00:00:00 +0000
http://pbelmans.ncag.info/blog/2024/02/15/two-more-quantum-spectra/
http://pbelmans.ncag.info/blog/2024/02/15/two-more-quantum-spectra/algebraic geometryprogrammingmathematicsFortnightly links (173)<p>I didn't realize the last fortnightly links were posted in early September!
I can explain a few weeks of silence on the blog in December because of a planned surgery,
but the rest is solely to blame on some kind of inertia.
Anyway, time to get back into semi-regular blogging!
<ul>
<li><p><a href="https://arxiv.org/abs/2310.04029v2">Daniel Huybrechts, Dominique Mattei: Splitting unramified Brauer classes by abelian torsors and the period-index problem</a>
does something absolutely amazing.
The <em>period-index conjecture</em> is an important open problem,
trivial in dimension 1,
solved through hard work of de Jong and Lieblich in dimension 2,
and wide open in higher dimension
(except for abelian 3-folds, by amazing work of Hotchkiss and Perry).
In this paper,
it is shown that for every unramified Brauer class in the function field
(which means it comes from an Azumaya algebra on some smooth and proper model)
the index divides some power of the period.
This doesn't settle the period-index conjecture though,
because the power is not given in terms of the dimension,
but rather in terms of the geometry of curves on the variety.
<li><p><a href="https://arxiv.org/abs/2310.01090v1">Vladimiro Benedetti, Daniele Faenzi, Maxim Smirnov: Derived category of the spinor 15-fold</a>
checks the Kuznetsov–Smirnov conjecture
which relates the finer structure of small quantum cohomology
to the existence of full exceptional Lefschetz collections
for <a href="https://www.grassmannian.info/D5/5">the spinor 15-fold</a>.
Cool!
Next up, constructing its homological projective dual?
There are various open questions on the <a href="https://www.grassmannian.info/E7/7">Freudenthal variety</a> at the end,
in case you are looking for some hard problems to work on!
<li><p><a href="https://arxiv.org/abs/2312.06930">Nicolas Addington, Elden Elmanto: The Quillen-Lichtenbaum dimension of complex varieties</a>
introduces new derived invariants,
and it is extremely high on my must-read list,
but I haven't managed to do so yet.
Its appearance in these fortnightly links is thus also a reminder to myself.
<li><p><a href="https://chessapig.github.io/blog">Elliot Kienzle's blog</a>
is something I found through his <a href="https://twitter.com/chessapigbay">amazing Twitter feed</a>
which features the best drawings in mathematics I have ever seen.
Highly recommend to follow them if you are still on Twitter.
<li><p><a href="https://arxiv.org/abs/2309.05473">Tom Coates, Alexander M. Kasprzyk, Sara Veneziale: Machine learning the dimension of a Fano variety</a>
is about determining the dimension of a Fano variety
from its quantum period,
a "fingerprint" of a deformation class of a Fano variety
which is defined in terms of its Gromov–Witten theory (thus its symplectic geometry).
If it is to be a good fingerprint
(in the sense that it determines the deformation class)
it should certainly contain enough information to determine the dimension of the Fano variety.
The paper does not give a proof for this,
but using machine learning methods,
they realize that computers can be trained to predict the dimension pretty well.
Super-duper cool!
</ul>
</ul>
Wed, 14 Feb 2024 00:00:00 +0000
http://pbelmans.ncag.info/blog/2024/02/14/fortnightly-links-173/
http://pbelmans.ncag.info/blog/2024/02/14/fortnightly-links-173/fortnightly linksmathematicsNew papers: two papers on quiver moduli<p>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
<ol>
<li><a href="https://arxiv.org/abs/2311.17003"><strong>Rigidity and Schofield's partial tilting conjecture for quiver moduli</strong></a>,
joint with Ana-Maria Brecan, Hans Franzen, Gianni Petrella, and Markus Reineke
<li><a href="https://arxiv.org/abs/2311.17004"><strong>Vector fields and admissible embeddings for quiver moduli</strong></a>,
joint with Ana-Maria Brecan, Hans Franzen, Markus Reineke
</ol>
<p>I'm really proud of these works, for several reasons:
<ul>
<li>It's the first paper with my PhD student <a href="https://www.giannipetrella.eu">Gianni Petrella</a>.
Congratulations Gianni!
<li>It answers questions Hans and I dreamt of solving back in 2017,
using tools we had no idea existed back in the day.
<li>It simultaneously resolves an old conjecture by Aidan Schofield,
whose work I greatly admire.
<li>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.
<li><small>They have consecutive arXiv id's.</small>
</ul>
<h3>The main results</h3>
<p>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.
<p>The condition of being <em>strongly amply stable</em>
is a strengthening of the ample stability introduced by <a href="https://mathscinet.ams.org/mathscinet/article?mr=3683503"</a>Reineke--Schröer</a>
(which says that the unstable locus has codimension at least 2).
<p><strong>Theorem</strong>
<em>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 $m=1,\ldots,\ell$.</em>
<h3>On the methods</h3>
<p>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.
<p>For the <strong>vanishing</strong>,
we use <a href="https://en.wikipedia.org/wiki/Quantization_commutes_with_reduction">Teleman quantization</a> (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 <em>affine space</em> 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.
<p>For the <strong>global sections</strong>,
we use <a href="https://en.wikipedia.org/wiki/Geometric_invariant_theory">geometric invariant theory</a>,
by reducing the calculation to that of a line bundle on a different moduli space,
and appealing to the celebrated <a href="https://mathscinet.ams.org/mathscinet/article?mr=0958897">Le Bruyn–Procesi</a> result
to compute this.
<p>There is no dependency between the two results.
<h3>The applications</h3>
<p>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:
<ol>
<li>using the 4-term sequence for quiver moduli one can prove that moduli spaces of quiver representations are <em>rigid</em>;
<li>the cohomology vanishing is in fact inequivalent to $\mathcal{U}$ being a <em>partial tilting object</em>, as conjectured by Schofield;
<li>the global sections allow one to compute that the <em>first Hochschild cohomology of the path algebra</em> $\mathbf{k}Q$
(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);
<li>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 <em>admissible embedding</em>.
</ol>
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.
<h3>Similarities between moduli spaces of vector bundles and representations</h3>
<p>These applications are inspired by what we already knew for moduli spaces of vector bundles on curves:
<ul>
<li>back in 1975, <a href="https://mathscinet.ams.org/mathscinet/article?mr=0384797">Narasimhan–Ramanan</a>
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;
<li>in the last decade,
there has been a flurry of activity by several authors
(starting with <a href="https://mathscinet.ams.org/mathscinet/article?mr=3713871">Narasimhan</a> and <a href="https://mathscinet.ams.org/mathscinet/article?mr=3764066">Fonarev–Kuznetsov</a>)
to show that the universal vector bundle sets up a fully faithful Fourier–Mukai functor.
</ul>
<p>We manage to find the quiver analogues for these two results:
<ul>
<li>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;
<li>the appropriate analogue of a Fourier–Mukai functor in this noncommutative setting
is indeed fully faithful.
</ul>
<p>Using an idea <a href="https://mathscinet.ams.org/mathscinet/article?mr=3950704">Theo, Lie and I applied earlier for Hilbert schemes of points on surfaces</a>
we even relate the two,
explaining how the fully faithful functor allows one to obtain the results on the vector fields and rigidity.
<hr>
<p>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!
Wed, 29 Nov 2023 00:00:00 +0000
http://pbelmans.ncag.info/blog/2023/11/29/new-papers-quiver-moduli/
http://pbelmans.ncag.info/blog/2023/11/29/new-papers-quiver-moduli/algebraic geometrymathematics