The integrality conjecture and the cohomology of preprojective stacks
-
Ben Davison
Abstract
We study the Borel–Moore homology of stacks of representations of preprojective algebras
1 Introduction
1.1 Background
This paper concerns the Borel–Moore homology of stacks of representations of preprojective algebras
Via dimensional reduction, we study the Borel–Moore homology of the stack of
1.2 Purity
In DT theory, as well as many of the other subjects this paper touches on, we are typically interested in motivic invariants.
See e.g. [22, 27] and references therein for extensive background on motivic DT theory.
This means that we are interested in invariants
so that a motivic invariant in the first sense induces one in the second sense.
A very basic example of a motivic invariant is the Euler characteristic of a complex of vector spaces
Recall that a mixed Hodge structure on a rational vector space 𝑉 is the data of an ascending weight filtration
Since both the E series and weight series involve an alternating sum over cohomological degrees, they are motivic invariants.
We say that a cohomologically graded mixed Hodge structure ℒ is pure if its 𝑎th cohomologically graded piece is pure of weight 𝑎, i.e. if
1.3 The purity theorem
Let 𝑄 be a quiver with set of vertices
We define
This space is symplectic, via the natural isomorphism
We identify
Theorem A ([7, Conjecture 3.1])
Fix a quiver 𝑄 and a dimension vector
We prove a more general version of Theorem A, concerning Borel–Moore homology of stacks of semistable
In Theorem A, the symbol
Theorem A concerns compactly supported cohomology.
Since
1.3.1 Okounkov’s conjecture
Theorem A is a singular stack-theoretic cousin of the result that the cohomology of Nakajima quiver varieties is pure, with Hodge polynomial expressible as a polynomial in
In a little more detail, it is conjectured that the cohomological Hall algebra
1.4 From DT theory to symplectic geometry
Consider the following general setup, of which our situation with
This function is 𝐺-invariant and so defines a function
Note that
The link between DT theory and symplectic geometry is completed by the observation of [14, Section 4.2] (see also [32]) that, associated to any quiver 𝑄, there is a tripled quiver with potential
Putting all of this together, the cohomological DT theory of
We use cohomological DT theory to prove powerful theorems regarding the right-hand side, and deduce results regarding the left-hand side.
1.5 BPS sheaves and their supports
We prove Theorem A via an analysis of BPS sheaves.
These were introduced in [10], in the course of the proof of the relative cohomological integrality/PBW theorem for the critical cohomological Hall algebras introduced by Kontsevich and Soibelman [28].
This theorem states that, for a symmetric quiver
Although the direct image of the mixed Hodge module of vanishing cycles along
Unless the pair
Lemma 1.1 (Lemma 4.1)
Let 𝑥 be a point in
This is a crucial lemma on the way to proving purity of BPS cohomology for
1.6 2d BPS sheaves
Aside from purity of BPS cohomology, one of the main applications of the support lemma is that it enables us to define 2d BPS sheaves.
Let
The pure intersection complex
They categorify the Kac polynomials; we elaborate upon this in Section 8.
They are Verdier self-dual (see Section 4.2), which we expect to have a role in producing geometric doubles of BPS Lie algebras.
Their hypercohomology carries a Lie algebra structure, the BPS Lie algebra
They are pure as mixed Hodge modules, enabling us to relate generators of
1.7 Serre subcategories
A Serre subcategory
Because many of our results can be stated in the category of mixed Hodge modules[2] on
1.8 Structural results
We prove two general structural results (Theorems C and D) regarding the compactly supported cohomology of stacks
Let 𝑄 be a quiver, let
where
and
Taking the Hodge series of both sides of this isomorphism yields the equality
regardless of whether the compactly supported cohomology of
1.8.1 PBW/integrality isomorphism
Fix a quiver 𝑄, a stability condition
This graded mixed Hodge module carries a Hall algebra structure; see Section 9.1 for details.
Let 𝜚 be the slope function defined with respect to a stability condition
Then there is an isomorphism
Moreover, there is a PBW isomorphism
Since
1.9 Positivity of restricted Kac polynomials
For an arbitrary quiver 𝑄, it was proven by Kac in [23] that, for each dimension vector
In the case of the degenerate stability condition, for which all modules are semistable of the same slope, and so the superscript 𝜁 and the subscript 𝜃 can be dropped, (1.5) gives
Taking weight series of both sides of (1.6) yields
where
where the + subscript means that
The mere existence of isomorphism (1.6) can tell us something highly nontrivial about
In [3, 43], Bozec, Schiffmann and Vasserot define the subcategory of nilpotent, *-semi-nilpotent and *-strongly semi-nilpotent
Theorem E (Theorem 8.2, Remark 8.4)
Let 𝑄 be an arbitrary finite quiver, and let
1.10 Conventions
For 𝐺 a complex algebraic group, we set
We continue to write
Wherever an object appears with a subscript that is a bold Roman letter, that letter refers to a dimension vector, and
For 𝒟 a triangulated category equipped with a t structure, we define the total cohomology functor
2 Quiver representations
2.1 Quivers and potentials
Throughout the paper, 𝑄 will be used to denote a finite quiver, i.e. a pair of finite sets
A potential on a quiver 𝑄 is an element
and we extend this definition linearly to general 𝑊. We define the Jacobi algebra
associated to the quiver with potential
Given a quiver 𝑄, we denote by
We denote by
where each
Define
to be the subspace of morphisms
Proof
From the relations
These are precisely the conditions for the elements
2.2 Moduli spaces
Given an algebra 𝐴, presented as a quotient
and the action is by simultaneous conjugation.
We define
As substacks of
and denote by
We define
A stability condition for 𝑄 is defined to be an element of
For a fixed stability condition
We define the slope of a dimension vector
If 𝜌 is a representation of 𝑄, we define
If 𝜁 is a King stability condition, then for each
We denote by
the morphism from the stack to the coarse moduli space.
At the level of points, this map takes a semistable representation 𝜌 to the direct sum of the subquotients appearing in the Jordan–Hölder filtration of 𝜌, considered as an object in the category of 𝜁-semistable representations of slope
We denote by
We define two pairings on
Again, we will drop the subscript 𝑄 when the choice of quiver is obvious from the context.
For
We denote by
If 𝒮 is a Serre subcategory of the category of
the inclusion of the polystable
and denote the inclusion
3 Cohomological DT theory
3.1 Vanishing cycles and mixed Hodge modules
Let 𝑋 be a smooth complex variety, and let 𝑓 be a regular function on it.
Set
and we define the functor
Alternatively, define
Then we can define
If 𝑋 is a quasiprojective complex variety, and so there is a closed embedding
If 𝑓 is a regular function on the smooth variety 𝑋, then
We define
Explicitly, an object of
Similarly, we define
For general 𝑋, we define
We define
This may be achieved either purely formally, or by embedding
Let 𝑋 be a complex variety such that each connected component contains a connected dense smooth locus.
In this paper, we will shift the definition of the intersection complex mixed Hodge module for 𝑋 so that it is pure of weight zero, while its underlying element in
If 𝑋 is a smooth connected variety, we set
3.2 Pushforwards from stacks
Assume that 𝑋 is a complex variety, acted on by an algebraic group 𝐺, following [10, Section 2] how to define
Let
where the limit is constructed, and exists, as in [10, §2.2]. Similarly, we define
This can be seen as a special case of the previous construction, with
Let
3.3 Dimensional reduction
Given a decomposition
to obtain a natural transformation
This is a cohomological analogue of the dimensional reduction theorem of [1].
It implies (see [5, Corollary A.7]) that if 𝑋 is the total space of a 𝐺-equivariant affine fibration
3.4 Integrality and PBW isomorphisms
Let 𝑄 be a finite quiver.
We consider
The GIT quotient
If 𝑋 is a commutative monoid in the category of locally finite type complex schemes, with finite type monoid morphism
The following theorem allows for the definition of BPS sheaves and BPS cohomology. It is a cohomological lift of the property known in DT theory as integrality.
Theorem 3.5 ([10, Theorem A][3])
Fix a QP
a slope
and define
Then there are isomorphisms of objects in
Since Verdier duality naturally commutes with the vanishing cycles functor, and since
3.4.1 (3d) BPS cohomology
Let 𝒮 be a Serre subcategory of the category of
where the isomorphism follows from Verdier self-duality of the BPS sheaf.
The cohomologically graded mixed Hodge structure
carries a Hall algebra multiplication, defined in [28, 5], via pullback and pushforward of vanishing cycle sheaves; see Section 9.1 for a generalisation of the construction.
Applying the natural transformation
Applying
Theorem 3.6 (PBW theorem [10, Theorem C])
The morphism
extending 𝑔 via the Hall algebra multiplication on the target is an isomorphism.
3.5 Framed moduli spaces
Let 𝑄 be a quiver.
For the moment, we do not assume that 𝑄 is symmetric.
Let
and
Given a King stability condition 𝜁 for 𝑄, and a slope
for sufficiently small positive 𝜖.
Let
We denote by
Let 𝜁 be a 𝜃-generic stability condition, and assume that
There is an isomorphism in the category
4 Purity and supports
In this section, we prove Theorem A. A crucial role in the proof is played by the support lemma (Lemma 4.1), which also enables us to prove Theorem B.
4.1 Proof of Theorem A
Fix a quiver 𝑄.
We define
the compactly supported cohomology and the restricted compactly supported cohomology, respectively, of the BPS sheaf from Theorem 3.5.
Recall that
Lemma 4.1 (Support lemma)
Let
There are isomorphisms of
Lemma 4.3 ([7, Theorem 3.4])
The
is pure, of Tate type.
Assuming Lemmas 4.1, 4.2 and 4.3, we argue as follows.
Proof of Theorem A
First, note that a graded mixed Hodge structure ℱ is pure, of Tate type, if and only if
are pure, of Tate type, and the theorem follows. ∎
Before coming to the proof of Lemmas 4.1, 4.2 and 4.3, we note the following.
For all
Proof
From Theorem (3.6), we deduce that there is an inclusion of mixed Hodge structures
The proof of both Lemmas 4.2 and 4.3 will use the dimensional reduction theorem, recalled as Theorem 3.4.
Let
If we let
We fix
Lemma 4.3 is proved in [7] by analysing the right-hand side of (4.4). Note that there is no overall Tate twist in (4.4) – the Tate twist in the definition of the left-hand side is cancelled by the Tate twist appearing in Theorem 3.4.
The first isomorphism in Lemma 4.2 is obtained in similar fashion.
Let
and let
Proof of Lemma 4.2
Since the map
the inclusion.
Taking the direct sum over all
To prove the existence of isomorphism (4.2), then it is sufficient to prove that
Fix a dimension vector 𝐝.
We let
Then
be the restriction of the action map.
This is an isomorphism since, for a module represented by a point in ℳ, there exists a
By the support lemma (Lemma 4.1), we have
as required. ∎
We complete the proof of Theorem A by proving the support lemma.
Proof of Lemma 4.1
To ease the notation, we prove the lemma under the assumption that 𝜁 is the degenerate stability condition: the proof for the general case is unchanged.
Since the support of
Let
and so, by (2.1) and Remark 3.1, there exists a
Under our assumptions, there are disjoint (analytic) open sets
By Lemma 4.5, proved below, there is an isomorphism
Applying (3.1) to the right-hand side of (4.7), we have isomorphisms
On the other hand, restricting the isomorphism of (3.1) to the left-hand side of (4.7) yields
Comparing (4.8) and (4.9), we deduce that
We deduce that
and so, since
For the final statement of the lemma, it suffices to prove that if 𝜌 is a simple
4.2 Proof of Theorem B
Firstly, by Lemma 4.1, the support of
We have seen in the proof of Lemma 4.2 that
Finally,
and Verdier self-duality of
We finish this section with the technical lemma appearing in the proof of Lemma 4.1.
Fix a decomposition
Let
This isomorphism does not follow directly from the Thom–Sebastiani isomorphism since we need to compare the vanishing cycle sheaf of the function
with the vanishing cycle sheaf for the function
Proof of Lemma 4.5
Again, it is sufficient to prove the lemma for the degenerate stability condition (the general case then follows by restriction to the 𝜁-semistable locus). Writing
where 𝐿 is the quiver with vertices
The space 𝑌 is the total space of the
Note that
Let
We claim that the natural morphism
is an isomorphism, where we denote by
Combining (4.11) and (4.12) yields the isomorphism
4.3 Calculating
H
c
(
M
(
Π
Q
)
d
,
Q
)
We use Theorem A and existing results on the 𝐸 series of
Recall the plethystic exponential defined in Section 1.9. We define the ring
of formal Laurent power series
such that, for each
is in
for each
for each
We define
where 𝔪 is the maximal ideal generated by
Here
There is an isomorphism of cohomologically graded,
5 Degree zero DT theory
5.1 Degree zero BPS sheaves
For
Let
For all
Proof
By the same argument as for Lemma 4.1, the support of
By the argument in the proof of Lemma 4.2,
On the other hand, by [1, Proposition 1.1], there is an equality
in the Grothendieck ring of mixed Hodge structures.
The mixed Hodge structure
and so, from (5.1), there is an isomorphism
For any constructible inclusion
We denote by
We set
Combining Theorems 3.5, 3.6 and 5.1 gives the following.
There is an isomorphism in
and a PBW isomorphism of ℕ-graded mixed Hodge structures
Proof
We construct the first isomorphism as a special case of (3.2); via the same argument, we then realise (5.2) as a special case of Theorem 3.6.
In fact, it is sufficient to construct the isomorphism in the case
5.2 Applications to surfaces and character stacks
Let
where the horizontal morphisms are the forgetful morphisms.
We denote by
as in the previous section.
We denote by
We denote by
from the stack of coherent sheaves of length 𝑑. We define
Let
and a PBW isomorphism of ℕ-graded mixed Hodge structures
Proof
We denote by
For an application to nonabelian Hodge theory, we set
of substacks of
There is a PBW isomorphism of ℕ-graded mixed Hodge structures
The CoHA structure on the left-hand side of this isomorphism is introduced and studied in [4].
Given
where the action is the simultaneous conjugation action.
The action of
is a smooth quasiprojective variety. It follows that there is an isomorphism
For
Conjecture 5.5 ([4, Conjecture 1.1])
There is an isomorphism of ℕ-graded cohomologically graded mixed Hodge structures
From Corollary 5.4, we deduce the
Conjecture 5.5 is true for
6 Generalisations of the purity theorem
6.1 The wall-crossing isomorphism in DT theory
The wall-crossing isomorphism in cohomological DT theory (e.g. [10, Theorem B]) provides a powerful way to deduce purity of Borel–Moore homology of moduli spaces of semistable quiver representations, for some stability condition 𝜁, from purity of Borel–Moore homology for some other stability condition
Fix a quiver 𝑄 and a stability condition
the set of Harder–Narasimhan types for
we denote
For 𝑄 a quiver,
where
in
If 𝑄 is symmetric, the function 𝑓 in the above proposition is identically zero.
For any stability condition
is pure, of Tate type.
Proof
For each
to the right-hand side of (6.2), and so we deduce that, as a sub-mixed Hodge module of a mixed Hodge module that is both an ordinary mixed Hodge module and pure, of Tate type by Lemma 4.2 and Theorem A, the mixed Hodge module (6.3) is a pure element of
6.2 Purity for stacks of semistable
Π
Q
-modules
Fix a quiver 𝑄 and a dimension vector 𝐝. There is a natural projection
induced by forgetting
For 𝑄 an arbitrary finite quiver,
is the identity.
Proof
Let 𝜌 be a
where
Let 𝑄 be a finite quiver, let
and so, by Theorem 6.5, taking duals, the mixed Hodge structure
Proof
Write
There are equalities
Thus the natural map
6.3 Framed quivers
For
to be the map forgetting the framing and remembering the associated graded object of the Jordan–Hölder filtration (in the category of 𝜁-semistable
Fix a finite quiver 𝑄, dimension vectors
Proof
Applying
On the other hand, from Corollary 4.4, each of the complexes
6.4 Critical cohomology of
Hilb
(
A
3
)
We consider again the special case in which
where the right-hand side of (6.7) is the usual Hilbert scheme parameterising codimension 𝑛 ideals
The mixed Hodge structure
It follows from our purity result that the Hodge polynomial
is equal to the weight polynomial
after the substitution
Indeed, we can determine the critical cohomology of
There is an isomorphism of ℕ-graded, cohomologically graded mixed Hodge structures,
Proof
By Corollary 6.6, the left-hand side of the expression in the corollary is pure, of Tate type, as is the right-hand side (by definition).
A cohomologically graded mixed Hodge structure that is pure, of Tate type, is determined by its weight polynomial.
The required equality of weight polynomials follows from the main result of [1], following on from the earlier paper [12], where an in-depth analysis of the case
6.5 Nakajima quiver varieties
Let 𝑄 be an arbitrary quiver, and let
Fix a slope
For each of the vertices
which are the usual Nakajima quiver variety relations [33, 34], while at the vertex ∞, the relation imposed is
By cyclic invariance of the trace,
is the usual Nakajima quiver variety, which we will denote
Each
and we recover the following corollary.
For an arbitrary quiver 𝑄, nonzero dimension vectors
7 The PBW and wall-crossing isomorphisms
7.1 Serre subcategories
Let
The standard construction for 𝑃 is as follows.
For a quiver 𝑄, let
Setting all
Setting
we obtain the condition for the Lusztig nilpotent variety if 𝑄 has no loops.
In general, the Serre subcategory
Example 7.3 ([2])
Set
7.2 Proof of Theorems C and D
Applying the functor
Pick a stability condition
in
Proof of Theorem 7.4
We consider the commutative diagram
with 𝑉 defined as in the proof of Theorem 6.4.
By Theorem 3.4, there are isomorphisms
Thus, applying
Proof of Theorem D
Let
where
Arguing as in the proof of Theorem 7.4, there are isomorphisms
giving isomorphism (1.4).
The construction of the PBW isomorphism is similar; via dimensional reduction and Lemma 6.3, there is an isomorphism
where
following from isomorphism (7.2). ∎
7.3 Applications for Nakajima quiver varieties
We explain the special case of Theorem C which gives rise to Hausel’s original formula for the Poincaré polynomials of Nakajima quiver varieties.
In brief, we choose
Let 𝑄 be a quiver, and let
where
by taking Poincaré series of the two sides of the isomorphism.
8 Restricted Kac polynomials
8.1 Definition
Next we explain how Theorem D enables one to define and categorify the Kac polynomial
Defining
Isomorphism (8.1) can be restated as saying that
Isomorphism (8.1) and the definition of
This is indeed a polynomial: despite its high-tech definition,
8.2 Positivity of Kac polynomials
A corollary of the existence of isomorphism (8.1) is that if
while Theorem 3.4 states that there is an isomorphism
On the other hand, by [32, Theorem 5.1], there is an equality
where
From Corollary 4.4, we deduce that each
is pure, and so
Theorem 8.1 ([19])
For a finite quiver 𝑄 and a dimension vector
8.3 Positivity of restricted Kac polynomials
For new positivity results, we turn to the examples of Serre subcategories appearing in the work of Bozec, Schiffmann and Vasserot – see Examples 7.1, 7.2 and 7.3 for the definitions.
Setting
for
For a finite quiver 𝑄, the Kac polynomials
Proof
The proof proceeds exactly as in the above reproof of Theorem 8.1, using the results and proofs of [3, 43] to deduce purity of
converging to
8.4 Verdier duality and nilpotent Kac polynomials
We finish our discussion of restricted Kac polynomials with a result relating
For a quiver 𝑄 and a dimension vector
Proof
The torus
This action preserves
Since 𝑇 contracts
Combining these isomorphisms gives the isomorphism in the proposition. The identity in the proposition then follows from the definitions. ∎
9 Deformations of Hall algebras
9.1 Kontsevich–Soibelman CoHAs
In [28], a method was given for associating a cohomological Hall algebra (CoHA for short) to the data of an arbitrary QP
If
For
For a quiver 𝑄 and
is a
From now on, we will only consider the case in which our quiver with potential is
For fixed
For the remaining part of the section, we will only consider the degenerate stability condition
Assume that the grading
We define
the underlying cohomologically graded mixed Hodge module or, equivalently,
We define
We let
such that each
We break the multiplication into two parts.
Fix a pair of dimension vectors
For 𝐺 an algebraic group with a fixed embedding
is defined as
for
where
Consider the composition of proper maps
where
Applying
Defining
where
We write
9.2 Degeneration
The extra equivariant parameters arising from the torus action on
Let
where
We define
defined by the morphism
The function
The space
Proof
By choosing a splitting
where
Each of the maps
as
The Leray spectral sequence
therefore satisfies
Set
In similar fashion, we obtain spectral sequences
converging to
respectively.
As in the construction of
Each of the spectral sequences
exists as in Section 3.2. We claim the following commutativity of limits:
using the shorthand
The argument for all three statements is the same: fixing 𝑝 and 𝑞, the limit
We may define the cohomological Hall algebra multiplication on
providing a lift of the natural morphism
in cohomology to a morphism in the category of algebra objects in the category of complexes of mixed Hodge structures.
Let
be as above a specialisation of a
Proof
First we consider the special case
Let the torus
The domain of
is an isomorphism of algebras.
Since
Let 𝔪 be the maximal homogeneous ideal in
is an isomorphism.
10 Shuffle algebras, torsion-freeness and noncommutativity
10.1 Definition
Fix a weight function
Set
Here
Let
Then, since
The morphisms
The morphism
is an algebra morphism, where the domain and target are given the KS cohomological Hall algebra structure.
Proof
For
The shuffle algebra
be the morphism defined by taking the sum of
We define
10.2 Torsion-freeness
The algebra
Let 𝑄 be a finite quiver, and let
Proof
The passage from torsion-freeness to all of the other statements of the theorem is as explained in [42], so we focus on torsion-freeness. The proof for this is a modification of [43, Proposition 4.6]; the original statement of this result in [43], and its proof, require modification, which we indicate.
Firstly, by assumption, 𝑇 contains two one-dimensional tori
is an isomorphism by [15, Theorem 6.2].
It is thus enough to prove that
is an embedding.
Therefore, it is sufficient to show that the right-hand side of (10.1) has no
where
is an isomorphism.
So it is sufficient to show that
Set
The same proof works with
It is possible for the strata
10.3 Noncommutativity
Theorem 10.2 enables explicit calculations inside
We use explicit calculations in the algebra
By Theorem 5.1, there is an isomorphism of cohomologically graded vector spaces
so that, for each
The commutator
Proof
Set
and define
First we calculate the commutator in
This element has cohomological degree −2.
We claim that the unique nonzero element of cohomological degree less than −2 in
Thus
It follows that
There exists a nonzero scalar
Proof
In [10], it is shown that, for a general (symmetric) QP
and the associated graded algebra is supercommutative.
In particular,
Via (3.1), for general (symmetric) quiver
So
A version of the following result is to be found in [24]. Using ideas from the proof of Corollary 10.5, we give an alternative proof.
The ℚ-vector space
Proof
Set
So 𝛽 can be written as a linear combination of elements
But then
The same proof(s) demonstrate that
10.4 Torsion
We show that torsion-freeness of
Let
is not torsion-free.
The natural map
For
Proof of Proposition 10.7
Set
is injective.
We start with the
is nonzero since, by Lemma 10.4, it is nonzero after tensoring with
we have seen that
Now let
restricted to
since
Funding source: European Research Council
Award Identifier / Grant number: 320593
Award Identifier / Grant number: 759967
Funding statement: During the writing of this paper, I was a postdoctoral researcher at EPFL, supported by the Advanced Grant “Arithmetic and physics of Higgs moduli spaces” No. 320593 of the European Research Council. During redrafting of the paper, I was supported by the starter grant “Categorified Donaldson–Thomas theory” No. 759967 of the European Research Council. I was also supported by a Royal Society university research fellowship.
Acknowledgements
I would like to thank Sasha Minets, Tristan Bozec, Olivier Schiffmann, Eric Vasserot, Davesh Maulik and Victor Ginzburg for illuminating conversations that contributed greatly to the paper. In particular, the idea for the proof of the crucial “support lemma” (Lemma 4.1) came from seeing Victor Ginzburg talk about the results of [13] at the Warwick EPSRC symposium “Derived Algebraic Geometry, with a focus on derived symplectic techniques”, and the final section of the paper benefitted greatly from Sasha Minets’ careful reading of an earlier draft.
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Articles in the same Issue
- Frontmatter
- Proof of the Michael–Simon–Sobolev inequality using optimal transport
- Existence and symmetry of periodic nonlocal-CMC surfaces via variational methods
- Semi-continuity of conductors, and ramification bound of nearby cycles
- The integrality conjecture and the cohomology of preprojective stacks
- Graphical solutions to one-phase free boundary problems
- Quasi-projectivity of images of mixed period maps
- Higher order Kirillov--Reshetikhin modules for 𝐔 q (A n (1)), imaginary modules and monoidal categorification
- The set of local 𝐴-packets containing a given representation
Articles in the same Issue
- Frontmatter
- Proof of the Michael–Simon–Sobolev inequality using optimal transport
- Existence and symmetry of periodic nonlocal-CMC surfaces via variational methods
- Semi-continuity of conductors, and ramification bound of nearby cycles
- The integrality conjecture and the cohomology of preprojective stacks
- Graphical solutions to one-phase free boundary problems
- Quasi-projectivity of images of mixed period maps
- Higher order Kirillov--Reshetikhin modules for 𝐔 q (A n (1)), imaginary modules and monoidal categorification
- The set of local 𝐴-packets containing a given representation
