Moduli spaces and their cohomologies

Most problems I am interested in are related to moduli spaces and their invariants. Here is a (very biased) introduction to this topic.

Moduli spaces

Let me first explain briefly what moduli spaces are. A moduli space is a space that parametrizes a class of objects. In algebraic geometry, some of the first examples one learns about are projective space, the moduli of lines through the origin in a vector space, and the Grassmannian, the moduli of linear subspaces of a fixed dimension in a vector space. Another early example is the moduli of line bundles on a smooth projective complex curve $C$. For example, the degree-zero line bundles form the Jacobian $\operatorname{Pic}^0(C)$. In algebraic topology, classifying spaces play a similar role: homotopy classes of maps $S\to BU(r)$ classify rank-$r$ complex vector bundles on a CW complex $S$. The idea behind their construction (behind the functor of points approach) is that families of objects are described by maps to a fixed space.

Such spaces are interesting for several reasons, see the answers collected by Rahul Pandharipande in Reflections on moduli space. As far as I see, there are two main initial reasons to study them: the hope that the moduli space reveals something about the general member of the class parametrized, and the fact that moduli spaces are new spaces, with interesting geometries and symmetries, worth studying in their own right. One also later learns that they are connected to many other areas of mathematics.

It is truly remarkable that moduli spaces of objects in algebraic geometry can themselves be studied inside algebraic geometry. A priori, there is no reason that objects defined by polynomial equations should themselves fit into parameter spaces defined by polynomial equations. To do this properly, however, one has to enlarge the class of spaces under consideration, first from varieties to schemes, and then to stacks.

There are two big directions in the study of moduli spaces in algebraic geometry. One is parametrizing varieties. The most famous example is the moduli of curves. This example goes back to Riemann’s 1857 paper Theorie der Abel’schen Functionen, where he counted $3g-3$ local parameters for a compact Riemann surface of genus $g\ge2$. The use of the word moduli in this context also goes back to this paper. The second direction is parametrizing objects on a fixed variety. The most famous examples are the moduli of vector bundles on a curve, with line bundles, mentioned above, as a special case. Of course, one can also combine the two directions. In this post, I will mostly discuss the second kind.

A first large class of examples is that of moduli of coherent sheaves on varieties. Coherent sheaves include vector bundles, and thus we recover the example of vector bundles mentioned above. But they also include ideal sheaves, where we obtain the Hilbert scheme of a variety, or sheaves supported on curves, where we obtain a moduli space of bundles on a moving curve inside our initial variety. One also studies problems originating in topology using constructions from algebraic geometry, for example, moduli of local systems on a real manifold, such as the surface underlying a Riemann surface. Many of these moduli problems have local descriptions in terms of representations of associative algebras constructed from quivers. A quiver is simply a directed finite graph. These are also objects of independent interest in algebra and Lie theory.

Compactifications and stability

With moduli spaces, a standard issue is constructing the “right” space. When one first learns the subject, “right” means “a proper variety”. For the moduli of genus $g$ curves, one has to add certain singular curves, namely stable nodal curves, to obtain

where $\overline{\mathcal M}_g$ is a smooth proper Deligne-Mumford stack with a projective coarse moduli space, see The irreducibility of the space of curves of given genus.

For moduli of objects, the collection one starts with may be too large, or the proposed moduli space may fail to be separated. To select a manageable part, one usually imposes a stability condition. For example, if $E$ is a vector bundle on a smooth projective curve, its slope is

The bundle is semistable if $\mu(F)\leq\mu(E)$ for every nonzero proper subbundle $F\subset E$, and stable if the inequality is always strict. So the “moduli space of vector bundles” one may first encounter is actually the moduli space of semistable vector bundles of fixed rank and degree. There are two moduli spaces here: the moduli stack of semistable vector bundles, which is smooth, but a stack, and the associated variety (the good moduli space), which may be singular and identifies a semistable bundle with its associated polystable bundle.

Correspondences between moduli spaces

There are certain natural relations, or better, correspondences, between these moduli spaces. Some are fairly standard. For example, one can look at two vector bundles on a curve related by

where $\mathbf C_x$ is the length-one skyscraper sheaf at a point $x\in C$. Then $\deg E'=\deg E+1$, and the space of all such modifications maps to the stacks of vector bundles $\operatorname{Bun}_{r,d}(C)$ and $\operatorname{Bun}_{r,d+1}(C)$. These relations simply come from extensions of sheaves on a curve, here a vector bundle and a sheaf supported at a point. More generally, we will encounter such relations coming from abelian categories.

If one wants to study semistable vector bundles, one may restrict the correspondence to the locus on which both bundles are semistable. Getting ahead of myself, we will see in later posts why it can be useful to keep the full stacks: the correspondences there are often simpler and allow operations that do not preserve the semistable loci. These correspondences are natural, and of course one would like to make sense of them and use them. The machinery of Hall algebras does exactly this: a correspondence parametrizing short exact sequences turns “extensions” into a multiplication, or an action. I think Hall algebras capture most correspondences one may be interested in, but not all.

There are other types of relations between moduli spaces about which I hope to write later, but which are much more mysterious than the one above. Let me mention two. First, the same underlying topological space may have different descriptions as a moduli space. The classical example is the Narasimhan–Seshadri theorem. On a compact Riemann surface $C$ of genus at least two, stable vector bundles of rank $r$ and degree zero are classified by irreducible unitary representations of the fundamental group

up to conjugacy. Allowing direct sums of stable degree-zero bundles gives all unitary representations. The corresponding moduli spaces are naturally homeomorphic, and they are smooth manifolds when restricted to stable bundles and irreducible representations. So we have a description using holomorphic bundles and another using the fundamental group. See Narasimhan and Seshadri’s paper.

Yet other relations are dualities. One of the best-known examples involves Higgs bundles: vector bundles $E$ equipped with a map $\phi:E\to E\otimes K_C$, where $K_C$ is the cotangent line bundle of the curve. In the mirror-symmetry picture of Hausel and Thaddeus, suitable moduli spaces of $\mathrm{SL}_r$ and $\mathrm{PGL}_r$ Higgs bundles are mirror partners. The duality also predicts matching invariants.

The Euler characteristic and beyond

So, how does one study a moduli space? First, what types of questions does one ask about any variety? Some are qualitative, such as: is it smooth, does it have mild singularities, is it rational, is it of general type?

One may also ask quantitative questions, about numbers associated to a variety. The most obvious is the number of connected components. Yet another, less obvious one, is the Euler characteristic. It says something about the geometry of the space: on a compact smooth manifold without boundary, a nonzero Euler characteristic implies that every vector field has a zero.

The Euler characteristic is the alternating sum of the Betti numbers. For a complex algebraic variety $X$, these are

where $H^i(X,\mathbf Q)$ is rational singular cohomology. The zeroth Betti number is the number of connected components. Even if one is initially interested in numbers, it is a good idea to study these vector spaces: there are more structures (e.g. filtrations) and operations (e.g. pushforward, i.e. integration) on them, from which one can then deduce numerical consequences. For example, when $X$ is a smooth projective complex variety, the Hodge decomposition implies that every odd Betti number is even. Hard Lefschetz implies that the Betti numbers increase toward the middle when considered separately in even and odd degrees, and Poincaré duality gives the corresponding symmetry between Betti numbers whose degrees sum to the real dimension of the underlying manifold.

Singular cohomology is possibly the main linearization of a variety. Part of its centrality in the subject is that each $H^i(X,\mathbf Q)$ is a finite-dimensional $\mathbf Q$-vector space for a complex variety of finite type. One also has other “linearizations”, such as integral cohomology, (mixed) Hodge structures, topological $K$-theory, Chow groups, algebraic $K$-theory, and so on. Chow groups and algebraic $K$-groups can be much harder to compute, as they are not topological invariants, and can be infinite-dimensional even after tensoring with $\mathbf Q$.

So, we went from numbers to vector spaces. One may wonder whether there are objects which are to vector spaces as vector spaces are to numbers. These are categories, and then one can go on in a similar way to higher categories. To a space $X$, we may associate categories of constructible sheaves, $D$-modules, or coherent sheaves. Taking a Grothendieck group is one way of passing from a category back to an abelian group.

This progression appears naturally in topological quantum field theory (TQFT). A $d$-dimensional TQFT assigns a number to a closed $d$-manifold and a vector space, its state space, to a closed $(d-1)$-manifold. In extended versions, categories appear one dimension lower. These assignments are related by gluing. For example, closing up a cylinder takes the trace of the identity on its state space, producing its dimension, or its Euler characteristic in a graded version. See this paper of Atiyah.

In some theories, the state spaces and categories can come from moduli spaces. Compactifying a higher-dimensional theory on a curve can, for example, produce a two-dimensional theory involving a moduli space of Higgs bundles. A duality between theories is then expected to relate the whole collection: equivalences of categories, isomorphisms of state spaces, and equalities of numerical invariants.

All these may be thought of as linearizations of the space. I am using linearization informally here. Sometimes I use gadgets. I am not sure what the best term is. The point is that we replace a nonlinear space by an abelian group, a vector space, or a linear category, while keeping enough of its geometry for maps and correspondences to act on these objects.

I think these problems, of determining these linearizations, are natural if one is interested in a particular space. But they seem particularly well suited to geometric representation theory. Here one is interested in realizing algebras and representations of algebras using linearizations of spaces. Then one can use the machinery behind these linearizations, which is most powerful in the singular cohomology case, to answer questions in representation theory. But how does one construct these spaces, and then these actions? Well, one needs families of spaces related by correspondences. Moduli problems supply both: the objects give the spaces, and operations such as taking an extension or choosing a subobject give correspondences between them. Pulling back and pushing forward along these correspondences can then produce the desired operators, if the relevant maps allow it.

These spaces also underlie enumerative invariants. A counting problem becomes a question about the moduli space of the objects being counted. The answer may be the number of connected components, or an Euler characteristic, sometimes weighted, or an intersection number. For example, on a smooth proper moduli space $M$, conditions on the objects can give cohomology classes $\alpha_1,\ldots,\alpha_k$, and one considers

when the total degree is the top degree. Under suitable transversality assumptions, this counts objects satisfying all the conditions, with multiplicities. In general, these computations will not give the right enumerative invariant, but they are still interesting to study and are actually some of the numerics from field theories.

Tautological objects and the restriction map

What are some examples, and how does one study these invariants? Well, there are two structures that keep appearing on moduli spaces. The first, which we already mentioned, is correspondences between them. The second is tautological objects. A universal vector bundle, for example, lives naturally on

where $\operatorname{Bun}_{r,d}(C)$ is the moduli stack of vector bundles. The Künneth components of its Chern classes give classes on the moduli stack. Likewise, the universal curve gives the familiar tautological classes on moduli spaces of curves.

Actually, the only two obvious techniques I know to study the cohomology of a moduli space are these two: correspondences and tautological objects. I only know of other, magical, techniques, such as dualities.

It may happen that we restrict to semistable objects, which is the case if we want to eventually study an actual variety. The universal or tautological objects live naturally on the stack and are then restricted to the semistable part. The first examples where one sees this is practice are the moduli of vector bundles on a curve and for representations of a quiver. So the restriction map is useful to study.

Let’s discuss one example, that of the projective space $\mathbf P^n$, with $n\ge1$. Here and below, quotients such as $X/G$ mean quotient stacks. For the scalar $\mathbf C^*$-action on $\mathbf C^{n+1}$, we have

The complement is $B\mathbf C^*$. Since $\mathbf C^{n+1}$ contracts equivariantly to the origin, the cohomology upstairs is $H^\bullet(\mathfrak X,\mathbf Q)=\mathbf Q[h]$, with $h$ of degree two. Choose the sign so that $h$ restricts to the hyperplane class, which I will also call $h$. Then

Restriction is the quotient map, and the displayed basis gives an obvious section: lift each $h^i$ to the same monomial upstairs. Equivalently, we have the splitting

This is a splitting of graded vector spaces, not of rings: downstairs, one has $h^{n+1}=0$, while upstairs it is nonzero.

The work of Atiyah–Bott and Kirwan gives a general framework for this picture. Atiyah and Bott studied bundles on a curve by stratifying the space of holomorphic structures according to how unstable they are. In finite-dimensional geometric invariant theory (GIT), the analogous decomposition is the Kempf–Ness stratification. For a smooth projective complex variety $X$ with an action of a reductive group $G$ and a chosen ample $G$-equivariant line bundle, write

where $X^{\mathrm{ss}}$ is the semistable locus and the $S_\beta$ are the unstable strata. A theorem of Kirwan says that the long exact sequences obtained by adding the strata break into short exact sequences. Choosing splittings gives

where $c_\beta$ is the complex codimension of $S_\beta$. Thus restriction to the semistable locus is surjective, which is referred to as Kirwan surjectivity, and the unstable strata account for its kernel. The splitting is generally noncanonical and is only a splitting of graded vector spaces. If semistability equals stability and the stabilizers are finite groups, the rational cohomology of the semistable stack is also the rational cohomology of the coarse GIT quotient.

One can then ask for a distinguished section of the restriction map. The projective-space example has one because the grading selects representatives. I will talk about this construction later, but let me just mention that, in the symplectic setting, there are such lifts, which go under the name of nonabelian stable envelopes or, more generally, BPS cohomology.

K-theory

Let’s talk about $K$-theory next, by which I actually mean only the Grothendieck group $K_0$. For a space, it is generated by coherent sheaves on the space with the relation $[E]=[E']+[E'']$ for each short exact sequence $0\to E'\to E\to E''\to 0$. Choose the character $q$ of $\mathbf C^*$ whose associated line bundle restricts to $\mathcal O(1)$ on $\mathbf P^n$. The analogous restriction is

Here there is no cohomological grading to select representatives. Instead, for any integer $w$, restriction identifies

Equivalently, we have the additive splitting

Thus each interval of $n+1$ consecutive weights gives a lift. There is no preferred interval without an additional choice. These choices are called windows or grade-restriction rules.

They have natural categorical lifts. Let $D^b\operatorname{Coh}(\mathfrak X)$ denote the bounded derived category of coherent sheaves on $\mathfrak X$, equivalently of $\mathbf C^*$-equivariant coherent sheaves on $\mathbf C^{n+1}$. The full triangulated subcategory generated by

and closed under direct summands restricts equivalently to $D^b\operatorname{Coh}(\mathbf P^n)$. Taking Grothendieck groups recovers the window $W_w$. For more general smooth GIT quotients, under suitable hypotheses, one imposes weight intervals along the fixed loci associated with the Kempf–Ness strata. These give window subcategories and hence analogous splittings on $K_0$.

For the development of this picture, see the work of Herbst–Hori–Page, Hori–Romo, Segal, Teleman, Ballard–Favero–Katzarkov, and Halpern-Leistner.

OK, so we learned that if one wants to study the cohomology of moduli spaces, one may need to look at correspondences between these spaces, tautological generation, and the problem of finding sections of the restriction map from the stack to the semistable locus. For the first one, we will use correspondences coming from Hall algebras. The third leads to BPS cohomology and quasi-BPS categories.

Lastly, we did not say much about the possible singularities of the moduli spaces of sheaves, which can be pretty terrible. The usual linearizations still make sense for singular moduli spaces, but may not be the most adequate ones to study them. For sheaves on a Calabi–Yau threefold, or representations of a quiver with potential, the moduli space is locally described as the critical locus of a function on a smooth space or stack. One therefore studies critical cohomology, built from vanishing cycles, which record how nearby fibers of that function degenerate. BPS cohomology captures basic building blocks of the critical cohomology.

This is part of cohomological, or refined, Donaldson–Thomas theory, see Szendrői’s survey and Davison’s ICM 2026 article. For a small sample of foundational papers, see Kontsevich–Soibelman and Brav–Bussi–Dupont–Joyce–Szendrői. I will come back to an overview of cohomological Donaldson-Thomasd theory in later posts.

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