Specifically, I was looking for analogues for the following fundamental properties:

- Generically, a perverse sheaf is just a locally constant sheaf of finite rank.
- The support of a perverse sheaf is a complex analytic space, and admits a Whitney stratification—a locally finite partition into complex analytic submanifolds of the ambient manifold along which the cohomology sheaves are local systems, and
- the normal data is a finite dimensional vector space concentrated in a single degree, independent of chosen generically (this last property is sometimes called the
**microloca****l**characterization of perversity, and can be found as definition 10.3.7 in Kashiwara-Schapira’s*Sheaves on Manifolds*).

There are *many* obstacles to reaching an understanding of the appropriate analogues for irregular perverse sheaves, and I do hope to eventually cover all of them. In this post, I hope to address parts of all these things. I’ll be broadly following Sabbah’s “Recent Advances in Holonomic D-modules” and adding in some extra explanation where I needed it.

If you can get your foot in the door and get around the use of ind-sheaves and their enhancement and the instrumental role of the convolution functor , then an irregular -constructible (ind-)sheaf is one which, locally in the subanalytic topology on , “comes from” an ordinary constructible sheaf with a “trivial filtration” (see section 6.6 of Kashiwara-Schapira’s “Regular and Irregular Holonomic D-modules”). If you look through section 5 of Kuwagaki’s “Irregular Perverse Sheaves”, then irregular constructibility is a property that is defined after finitely many complex blow-ups to a normal crossing divisor, on the real blow-up of that divisor, in terms of irregular constant sheaves that look nothing like what your average joe topologist thinks a constant sheaf should be.

The core of the issue, alluded to in the last post, is the distinction between formal solutions and convergent solutions, and asymptotic expansions of exact solutions in terms of exponential factors (like we talked about in our brief discussion on the Airy equation). These expansions are often the most important property of the solutions, as the arise all over the place in physics (basically every time you use special functions or Fourier analysis), even if they are not the exact analytic solution. The key to understanding all of this, I believe, is **Sabbah-Mochizuki-Kedlaya’s Hukuhara-Levelt-Turritin theorem**.

Yes, people really call it that.

A Meromorphic connection on a complex disk with a single regular singular point at has a basis of solutions given by a multivalued functions times entire functions (like times a convergent power series, with maybe some logarithmic terms thrown in if two of the exponents are repeated or differ by an integer). Under Riemann-Hilbert, these turn into perverse sheaves where the local system in degree -1 corresponds to the flat sections away from 0, and the data of is completely determined by the monodromy of these flat sections and any confluence in the eigenvalues of the exponents of the terms . The moral of this being, the topological properties of comes from understanding the associated ODEs, and every perverse sheaf is of this form for some . Analogously, we’ll need to first understand what a basis of solutions “looks like” for irregular singularities before we can talk about their topological properties.

Our main coefficient rings in the previous two posts were the local ring and its fraction field . We’ll now need to introduce the **formal completion** of ,

the ring of formal power series in at 0. By considering a Meromorphic connection as a -module, we get a natural inclusion functor into the category of -modules by extension of scalars. For connections with regular singularities, this map is always an isomorphism (moreover, it is an isomorphism if and only if the connection has regular singularities), proved by Malgrange in dimension 1, and Kashiwara-Kawai in general. Hence, in the irregular setting, we expect different behavior depending on whether we work over or over . The first of these differences, the place we’ll start, is the following:

**Theorem**(Levelt-Turrittin)

*Let , and let be its formal completion as a -module. Then, there is a finite subset $latex \Phi \subseteq \mathscr{O}_{X,0}(*0)*/\mathscr{O}_{X,0} \cong z^{-1}\mathbb{C}[z^{-1}]$ *and an isomorphism of -modules*

where . That is, up to formal isomorphism, a Meromorphic connection is a finite direct sum of connections with exponential factor times a connection with regular singularity. For , set

.

Then, the Levelt-Turrittin theorem says that is formally isomorphic to a direct sum of connections of the form (where is a connection with regular singularity).

We’re lying a bit here, though; the full statement of the theorem is that this formal decomposition is only true after possible finite ramification of the variable . That is, if sends for a positive integer , then the decomposition holds for .

So, in the case of the Airy equation with irregular singularity at infinity, the associated Meromorphic connection is formally isomorphic to

where is a disk around inside . This sum reflects the fact that the Airy functions can be asymptotically approximated by linear combinations

.

The issue we brought up with these approximations last post was that the exact solutions to the Airy equation are entire functions, but these exponential functions are multivalued. Hence, any such approximation *cannot hold* with the same values for all values of as it travels counterclockwise around infinity. This is generally known as the **Stokes Phenomenon**; the isomorphism of formal meromorphic connections guaranteed by the Levelt-Turrittin theorm does not in general lift to a global isomorphism of meromorphic connections. We can, however, locally lift the formal isomorphism on arc neighborhoods of the singular point. But, what do we mean by locally?

**The Real Blow-up and Arc Neighborhoods**

We want to work locally around each angle , and the arc neighborhoods introduced last post are rectangular if we work in polar coordinates. The space with these coordinates is called the **real oriented blowup of X at 0**, denoted with natural map

is not a complex manifold, so it doesn’t really make sense to talk about a sheaf of holomorphic functions on all of . But it is a real manifold with boundary, so there is a well-defined sheaf of smooth functions . On the complement of boundary , there is a natural subsheaf of holomorphic functions , so the only trouble is understanding the right analogue of “holomorphic function” on the boundary. We’d like these functions to satisfy (something like) the Cauchy-Riemann equation and have a power series representation everywhere, even the boundary.

Is even well-defined on ? Sort of. If we work in polar coordinates on , then a quick computation tells us that

as operators on . So, we can talk about the operator on that acts on smooth functions and produces a smooth function with maybe a logarithmic pole at 0 (by just diving both sides of the expression by ). So, in analogy with how on itself we can realize as the flat sections of , we define the sheaf of **holomorphic functions on **, denoted , to be the flat sections of the connection

.

That is, . If (with , $a < b$ with is a proper arc neighborhood of 0, then .

**What does this give us? Asymptotic Expansions**.

The sheaf is only really a new object for us on the boundary of , , so let’s focus on that. Denote ; this is a subsheaf of . What sort of functions does it contain?

Let , and let’s look at the stalk , say on some small proper arc neighborhood containing , and take a function defined on this neighborhood. Then, admits an **asymptotic expansion **at 0; that is, there is a **formal** series such that, for all , and every closed subsector , we can find a constant such that the estimate

holds for all . We specify formal series because sometimes the functions we care about might be asymptotic to a divergent series; oftentimes power series solutions to differential equations are divergent, so we need this possibility. Poincaré developed the idea of asymptotic expansions in 1886, as a kind of analogy to Taylor series expansions of smooth functions, that still provide a useful approximation for divergent series. If is asymptotic to a series above, we write . An example of something outside from ordinary stuff like Taylor series is

which clearly doesn’t converge on any neighborhood of the origin. The sheaf on consists of all those smooth functions on that locally admit an asymptotic expansion on the boundary of . It contains things like , but only for those where (where ). It also contains all of our exponential factors whenever . We’ll see more later about this sheaf with moderate growth and rapid decay functions and the Borel-Ritt Lemma.

Okay, now we want to talk about differential operators up on , and the most natural way of doing this is to set

And likewise define . So, tensoring with allows us turn meromorphic connections (which are -modules with connection) into -modules with connection in a natural way. The point of all this is that, over , we can locally lift the formal isomorphism of the Levelt-Turrittin theorem to an isomorphism of honest-to-goodness connections.

**Theorem**(Hukuhara-Turrittin)

*For each , the formal isomorphism can be locally lifted as an isomorphism *

This is exactly an algebraification of the Stokes phenomenon. The Levelt-Turrittin theorem tells us the asymptotic behavior as we approach 0 (in the form of exponential factors tensored with regular connections), and the Hukuhara-Turrittin theorem tells us that, while this behavior is true locally around , the exact description of the isomorphism may change as we travel around the circle (i.e., doesn’t necessarily give an isomorphism of -modules). This if the reason why we can’t make a global choice of coefficients for the asymptotic expansions of exact solutions of the Airy equation.

Things are slightly more complicated in higher dimensions—these decompositions do not hold a priori hold in general without some assumptions. If we work consider meromorphic connections on a polydisk with poles along a simple normal crossings divisor , then we have to assume that our connections have a **good formal decomposition** (which means that we can find a finite set that gives such a formal decomposition, and goodness means that pairwise choices of exponential factors have divisor of zeroes that is empty near 0). Assuming a good formal decomposition, we can then locally lift it to a real isomorphism over (this is due to Hukuhara-Turrittin-Sibuya-Malgrange-Sabbah-Mochizuki). The goalposts are now moved to:

**“when does a meromophic connection have a good formal decomposition?”**

In dimension 2, Claude Sabbah conjectured that this held in general (e.g. for a general surface and divisor) after perhaps a finite sequence of point blow-ups (he proved that this was true for connections of rank ). Mochizuki was able to prove this in the projective algebraic setting in dimension 2 in 2008, and Kedlaya proved the general case in dimension 2 shortly after in 2009. Then again, Mochizuki prove the algebraic case in all dimensions, and Kedlaya proved the local analytic case in all dimensions.

**SO**

**Theorem (**Sabbah-Mochizuki-Kedlaya, Hukuhara-Levelt-Turrittin)

*Given a meromophic connection on a space with poles in a divisor , and any point , there is an open neighborhood of finite sequence of blow-ups such that has normal crossings and has a good formal decomposition at each point of *

Right now, in a small neighborhood of this post, I’ll only be focusing on the one dimensional case, so we won’t see all these details yet. But, this data is all absolutely essential in understanding the irregular Riemann-Hilbert correspondence in this case. It tells us what to expect of the structure of “solutions” as algebraic objects, and that all of the interesting behavior happens on inside the real blow-up. This is exactly the reason why irregular perverse sheaves are still generically local systems; the support condition for perverse sheaves carries through essentially unchanged to irregular perverse sheaves (modulo understanding how we embed perverse sheaves into this bigger category).

Monodromy of solutions is now a different beast; it is not determined by the multi-valuedness of the coefficients in the solutions, it is determined by the fact that we entire exact solutions with multi-valued approximations. “Irregular monodromy” then, is the data of the family of isomorphisms in the Hukuhara-Levelt theorem.

I can’t answer now all of the the broad questions I had at the beginning of this post, but this is the first step.

What comes next: the **Stokes filtration**. Away from the origin, the solutions of irregular holonomic D-modules are locally constant sheaves (since agrees with away from ), and so the difficultly will be in understanding both the formal structure at the origin, along with the family of isomorphisms , **without knowing the exponential factors . **

**Meromorphic Connections = Fancy ODE’**s** with singularities**

Let be a small open ball around the origin in , , and the field of fractions of , representing holomorphic functions with possible poles at . If is a local coordinate on , then , and . Then, recall that a **meromorphic connection** consists of the data of a free -module and a -linear map satisfying the Leibniz rule for elements . A morphism of meromorphic connections is just a map that is -linear from to and commutes with the action of . This is actually a bit of a simplification of connections–normally we’d specify as a map

Contracting this “universal” with elements of produces the original definition I gave.

These objects then form an Abelian category (since we’re really just restricting to a thick subcategory of Holonomic D-modules). Again, the point of all this is just to formalize both our fast-and-loose simplifications from last time, and to phrase the problem functorially. This category of meromorphic connections is just a simplified way of talking about (complex) linear ODE’s with (possible) singularities at . Now, what does a general meromorphic connection look like?

On with our local coordinate , choosing a system of generators of over , say (these always exist globally on because is contractible). Then, looking at the action of on this basis gives us a **connection matrix** with values in

,

the negative sign is just a convention, to make the next expression nicer. Thus, if is a general element of , then the above expression and the Leibniz rule gives us

Hence, the collection of **flat sections **of (those that satisfy ) correspond to solutions of the system of linear ordinary differential equations

Likewise, any such differential equation gives rise to a connection–let be the space of solutions to the above differential equation. Then, this equation is a rule for how the symbol acts on the elements of , taken as the *definition* of .

The simplest case that arises from having meromorphic coefficients is when the connection matrix has at worst poles of degree one, i.e., if for some constant matrix (to compensate for maybe choosing a bad generating set for over , we only care if is “gauge equivalent” to a matrix of the form ). In the rank one case, this is a differential equation of the form

for some , and has fundamental solution . What sort of function is this? When , is a globally defined holomorphic function (it’s just a monomial!), and when , is a globally defined meromorphic function on that is holomorphic on . When , we have to use the definition , where is **only a well-defined function on the complement of a choice of branch cut**. Moreover, while this is a perfectly valid solution on any open simply connected subset of , as we travel around the origin and analytically continue , **the value of this function jumps due to monodromy.** Precisely, if we let as varies, we find

By cutting up the domain of log, we can make it single-valued

This is an example of a **multi-valued function** on , and doesn’t lie in or , and naturally form a ring (after fixing a universal cover of ). We don’t need this entire ring quite yet (it contains other weird things like functions with essential singularities and Whitney functions), but we’ll come back to it.

**Regular Singularities and Moderate Growth**

Meromorphic connections which have connection matrices gauge equivalent to one of the form (we can even fudge a bit to allow ) are said to have a **regular singularity **at 0. This is a bad choice of terminology, since sometimes “regular” means “non-singular” in algebraic geometry, so perhaps “tame” would’ve been a better name since they are classified by the rank of the connection and the monodromy matrix. These are the mildest sorts of singularities that appear in the theory of meromorphic connections, and are the the subject of the classical Riemann-Hilbert correspondence (in D-module language) of Kashiwara and Mebkhout. The solutions to these sorts of differential equation are very similar to our example of given above; if has rank , we can find by the Frobenius method linearly independent solutions of the form

where and is a holomorphic function on with (the expression is slightly more complicated if an exponent is repeated, or if some pair and differ by an integer). These functions may be only well-defined single-valued functions on certain **arc neighborhoods** of 0, i.e., open sets of the form

for some and .

arc neighborhoods on which is single-valued

Solutions to ODE’s with regular singularities on , even multi-valued ones, always behave “like” meromorphic functions in an appropriate arc neighborhood of any particular angle . When we say “behaves like a meromorphic function”, we mean that as we approach 0 from inside that arc neighborhood, the norm of the solution grows only polynomially in , as if it were a Laurent series with only finitely many terms of negative degree.

More precisely, solutions to ODEs with regular singularities are said to have **moderate growth at **, in the sense that there are , constant and exponent for which

(1)

on the arc neighborhood . As travels around , the values of the constants and may change. We just say has moderate growth at 0 if it has moderate growth at every .

**For single-valued functions on , moderate growth at 0 is equivalent to being meromorphic along 0.**

This analytic characterization is actually a necessary and sufficient condition; **a meromorphic connection has a regular singularity at 0 if and only if all of its flat sections have moderate growth at 0. **Algebraically, this can be characterized by Fuch’s Criterion, but we will not focus on this perspective.

Aside from this interesting growth condition, or the simplicity of their solutions, why do we care about regular singularities? Let be the category of Meromorphic Connections with regular singularities at 0, and the category of flat connections on .

**Theorem (Deligne)**: *The restriction functor induces an equivalence of categories*

To show that we can always extend a given of rank to an element of , we note that is uniquely determined by a monodromy representation defined by the local system . If we let correspond to the element 1 in the identification , then is determined by the matrix . We can then always find a matrix such that , which we then use to define a connection matrix on via

where is the standard basis for . The resulting meromorphic connection clearly has a regular singularity at 0 and restricts to on .

Essentially, the singularities of connections with regular singularities are so mild that they are completely determined by their monodromy around the singular point. Recalling the previous post, we obtain the following:

**Corollary (Deligne): ***There is an equivalence of categories *

Where is the category of finite-rank -local systems on .

**Irregular Singularities and Stokes Phenomena**

We say a meromorphic connection has an irregular singularity at 0 if it doesn’t have a regular singularity at 0; hence, flat sections of **do not have moderate growth** at 0. What does this mean? First off, the absence of regularity doesn’t necessarily mean doesn’t satisfy (1) everywhere, just that there are some which have no arc neighborhood on which has polynomial growth in .

For example, take , whose associated linear ODE has fundamental solution . Then, has moderate growth at every . Why? On this region, , and so decays to 0 exponentially as . does not have moderate growth on . When , say for simplicity, then

for every exponent .

Aside from no longer having moderate growth, one of the other reasons Deligne’s theorem no longer holds in the irregular setting is that we can’t uniquely specify a general meromorphic connection on by its restriction to . We actually saw this in the last post–the Meromorphic Connections are all non-isomorphic (for ), but their local systems of flat sections are all isomorphic to the constant local system . In essence,** there are too many local systems with the same monodromy**, if we have no restrictions on the singularities.

So, why do we care about these ODEs if they’re so difficult to deal with? Many interesting special functions arising in engineering and physics arise as solutions to ODEs with irregular singularities. Take, for example, **the Airy equation **

which has an irregular singularity at , and its fundamental solutions and are approximated (in a rigorous sense, using asymptotic analysis) by linear combinations of the functions for large values of . Now, here is the interesting thing: a solution of the Airy equation is an entire function of , since the coefficient function (which is just ) is entire, but the functions are multi-valued functions. Hence, as we let , around , the “true solution” will return to its original value, but and will not. Hence, and **cannot be represented by the same linear combination** of and .

This is known as the **Stokes Phenomenon**, where solutions to linear ODE’s can undergo drastic qualitative changes as passes through certain angles. Specifically, these qualitative changes happen to the **asymptotic behavior** of the solutions in the different regions bounded by these special angles (sometimes called Stokes angles, or Stokes lines (or anti-Stokes lines, if you’re a physicist)). This is a phenomenon unique to the world of irregular singularities. We go through all of this extra work because sometimes it can be exceedingly difficult to express exact solutions in terms of elementary functions (or even special functions!), and the approximations in terms of exponential factors can often tell us much more about the “physical” properties the true solutions that are difficult to see from their formulas. We recommend Meyer’s “A Simple Explanation of the Stokes Phenomenon” for this perspective.

In the next post, I’ll talk some more about the formalism of approximation and asymptotic analysis, and how it naturally leads to a way of characterizing solutions of with irregular singularities via the Stokes filtration.

]]> The problem of extending the R-H correspondence to cover holonomic -modules with irregular singularities has recently been settled by Kashiwara-D’Agnolo in general, and many others in special cases (e.g., Deligne, Malgrange, Mochizuki, Sabbah, and Kedlaya to name a few). These objects correspond topologically to enhanced perverse ind-sheaves (and several other equivalent Abelian categories, following Deligne‘s Stokes--perverse sheaves, Kuwagaki’s irregular perverse sheaves, and Ito’s -constructible enhanced ind-sheaves). I’ll refer to any of these equivalent Abelian categories as **irregular perverse sheaves.** A great short intro is Sabbah’s 2019 article “What are irregular perverse sheaves?“, and I’m basing much of this first post on his notes.

I plan to write several posts on this topic, things I wish I had when I first started trying to learn this subject. Since I’m coming from the purely topological/perverse sheaf side of the old (hah) R-H correspondence for regular holonomic -modules, I will assume you are also just as ignorant as I was coming in–it won’t be obvious to you why we suddenly need new topics from functional analysis or asymptotic analysis, like it wasn’t obvious to me. Neither will be the jump to ind-sheaves, instead of usual sheaves, or what exactly the extra variable is doing in enhanced ind-sheaves. These are all things I hope to talk about.

The easiest possible situation in which to understand the irregular R-H correspondence is when is a open complex disk around the origin in . This is **Deligne’s version–meromorphic connections (-module side) and Stokes-filtered local systems (topological side). **Before we get into details, you should know that, loosely, the difficulty will be in distinguishing exponential factors that pop up in solutions to these differential equations. Everything that follows will be in the hope that we can fix this problem.

**Holonomic D-modules and Meromorphic** **Connections**

In the local analytic setting in , holonomicity of a -module just translates into saying there is a finite set of points in off of which is an integrable connection of finite rank. The only thing that changes from the regular singularity case is that we are now making no assumptions about what looks like at the points of . Let’s just assume has only one “singularity”, so and is a connected open neighborhood of the origin. Then, holonomicity is equivalent to giving a finite dimensional -vector space and a -linear map that satisfies, for all and , the Leibniz rule:

(where is a local coordinate on with ).

**The Problem**

Let be a meromorphic function on , and consider the meromorphic connection

Then, the flat sections of correspond to solutions of the differential equation , i.e., . These flat sections form a rank one -local system with stalk at every point of . The monodromy action is trivial, we find , where is the de Rham functor.

Now, here is the problem: was arbitrary, but the local system we end up with doesn’t depend on the function we started with. That is, the **de Rham functor is no longer faithful** when we extend to holonomic -modules with possibly irregular singularities. The question, then, is what is a natural category of objects that correspond to solutions of differential equations with irregular singularities?

**Deligne’s answer** to this question is to keep the local systems away from the singularities, and add a filtration at the “boundary” that keeps track of how fast these solutions grow as they approach the singularity from different directions. These objects are called Stokes-filtered local systems. Easy enough, right? They are perhaps the simplest to understand solution of the problem, but they only really work well in dimension one (although there is some work in higher dimensions, especially Sabbah’s work Équations différentielles à points singuliers irréguliers et phénomène de Stokes en dimension 2), and we need to introduce some notions from asymptotic analysis to work with the filtration. These will be the focus of the next post.

So, let’s recap the construction. Let X be a smooth manifold (fine for our purposes here; usually we’d take X to be a real analytic manifold), a subset of the cotangent bundle . Then, the category is defined to be the localization of the bounded, derived category by the null system

.

We then say that a morphism in is an *isomorphism on * (or, an isomorphism in ) if there exists a distinguished triangle in with . If for some point , we write instead of (because we’re lazy).

I won’t be using this property, but it’s pretty neat, and is solely a consequence of being a localization:

where the limit is indexed over those morphisms with target (resp., with source ) in that are isomorphisms on .

Lastly, before I delve into the examples, we’ll need the following facts. First, for X a topological space, a closed subset, and A a commutative, unital ring with finite global dimension (say, ), there is a short exact sequence

in the category of sheaves of -modules. For us, this translates to: there is a distinguished triangle

in the (bounded) derived category .

Second, for a finite dimensional real vector space, and a closed, convex cone, we set . In this case, we have the equality (where is the canonical projection). Generalizing this, if is a closed submanifold of the manifold , then is the conormal bundle to in X.

**EXAMPLES!**

**Example 1.**

If we’re going to understand this at all, we should start in the easiest possible (not stupid) case: when . While we’re at it, let’s revisit the example I mentioned in the previous post, in which we cared about the closed subset in , and the subset of the cotangent bundle of . The relevant distinguished triangles to keep in our heads are then:

- .

Using these, I want to show the isomorphisms in .

We should start by calculating all those microsupports! For simplicity, we use the isomorphism .

- (the zero section of the cotangent bundle). If you think of as a (closed, convex) cone in itself, its polar set is just the zero vector. That’s the basic idea.
- (just think of as a closed submanifold of , and use fact 2 above).
- (you’ll have to think about this one, sorry. It’s not too bad…).
- (remember, is always a
*closed*subset!). - (see above, and use the triangle inequality for the microsupport).

Okay! Now we can show those isomorphisms. If we rotate the triangle in 1, (i.e., the distinguished triangle

)

we see that is an isomorphism on , since ! One down.

Similarly, if we rotate triangle 2, we see that is an isomorphism on , as . Easy!

**Example 2.**

Okay, now we’ll consider the subset and .

Right off the bat, we note that if is locally constant in an open neighborhood U of 0, (so ), implying in . I’ll leave you guys (okay, I know that nobody actually reads this) with a cliffhanger:

For all with , we have

in .

Why?

]]>

**Example 1. **So, for baby’s first example of a symplectic manifold, we look at , equipped with the 2-form (fyi, we write to denote the space of () -forms on ) which has constant value on , for all . If you understood proposition 4 from my post about symplectic linear algebra (here: https://brainhelper.wordpress.com/2014/05/04/symplectic-basics/), you shouldn’t have any trouble with this example. The biggest change here is needing to work with differential forms on , not just a single bilinear form.

Let be a (global) choice of linear coordinates on , and let . Then,

for .

**Example 2.** As far as I’m (currently) concerned, these are the most important examples of symplectic manifolds: given any (real, smooth) manifold of dimension , the cotangent bundle has a canonical symplectic structure. This is important, so I’ll go through most of the details on this one.

We’re going to need to conjure up some “canonical” symplectic form . There are essentially two ways to do this: first, a coordinate-free definition of ; after that, we’ll see what looks like in a system of coordinates to get a better feel for what we’re doing. The gist is that, on any cotangent bundle, we get a really useful 1-form for free, called the *canonical 1-form *(or *tautological 1-form*, or *Liouville 1-form*, or the *Poincar\'{e}* 1-form…), .

Let be the projection map, (i.e., is a point, and is a covector at ). Then, using the pullback , we define the value of at to be

.

This seems stranger/more abstruse than it actually is. From the definition of , (this checks out: is, by definition, a linear map , and , so the composition at least makes sense). Finally, we define the “canonical” symplectic form on to be (NB: some authors take . It doesn’t really matter, since both choices give you a coordinate-independent construction of a symplectic form).

Now, suppose we have a local system of coordinates near , with associated linear coordinates . What does look like in these coordinates? Well, we know , so we just plug the coordinate expressions in and wind up with:

(since ). Hence, the canonical 1-form looks like in the local coordinates ; consequently, the symplectic form has the coordinate expression

near . In particular, since is exact, it is automatically a closed 2-form, and in these coordinates, it is easy to show that is non-degenerate, and that (had we started out with using coordinates, the above expression is independent of the coordinates chosen). The rest of the details are yours to check.

What’s the big deal? Why is important/useful? Well, for one, it satisfies the following universal property:

**proposition 1: **The canonical 1-form** **is uniquely characterized by the property that, for every 1-form (i.e., is the graph of ), one has .

**proof.**

First, we note that , and

is just the expression of in the local coordinates on . I’ll leave the proof of uniqueness of to the reader

**end proof.**

Secondly, the canonical 1-form gives a really easy way to produce Lagrangian submanifolds of . Let be a smooth 1-form on , and denote by the graph of . Then,

**proposition 2: ** is a Lagrangian submanifold of if and only if is closed.

**proof:**

Clearly, is a smooth submanifold of of dimension (it’s diffeomorphic to itself). Then,

,

So vanishes on if and only if , i.e., if is closed.

**end proof.**

(**Aside: **for a symplectic vector space (of dimension 2n), a basis is called *a symplectic basis*, provided . In such case, we obtain the relations

- ,
- .

(). Additionally, in a symplectic basis, the *Hamiltonian isomorphism* is given by

- .
- .

for (in the general case, is defined by the formula , where is the canonical pairing, and ). I forgot to talk about this in previous posts, and the Hamiltionian isomorphism has a really neat analogue for symplectic manifolds.

**end aside.**)

So, I’m giving you fair warning now: for the majority of examples/ situations I’ll talk about, the symplectic manifolds will always be of the form for some smooth manifold . Okay. I was talking (secretively) about symplectic bases of symplectic vector spaces, and what the Hamiltonian isomorphism looks like when expressed in such a basis. For vector spaces, all we needed was for the (duals of) the basis elements to fit together in a regular way to form the original symplectic form (i.e., ). But, for symplectic *manifolds*, the symplectic linear algebra takes place on the tangent space to every point; we don’t have the same freedom to choose “global” symplectic bases. The next best thing would be, of course, if we can at least always *locally * construct some analogue of symplectic bases, spanned by some frame of vector fields which would be a symplectic basis on each tangent space in a neighborhood of a point. Praise be upon us, for this is in fact the case: **Darboux’s theorem** guarantees that, for a smooth, 2n-dimensional symplectic manifold , for all , there exists an open neighborhood of with smooth coordinate chart (actually, this is a symplectomorphism as well!) such that for all (one sometimes calls a system of Darboux coordinates, instead of/in addition to their symplectic properties). Luckily for us and our cotangent bundles, on , any local system of cotangent coordinates on serves as a system of Darboux coordinates, by our construction of . The most important thing to take away from Darboux’s theorem, however, is that all symplectic manifolds of a given dimension are (locally) the same! No distinguishing local invariants to be found here.

On , the *Hamiltonian isomorphism* is the (fiber-wise) linear isomorphism , and we’ll use it to construct some of the most important objects in symplectic geometry: Hamiltonian vector fields (check out Noether’s theorem, Hamiltonian dynamics, etc.). Precisely, given a smooth (real-valued) function on an open subset , we define the *Hamiltonian vector field of f, *denoted by , to be the image of the differential under the Hamiltonian isomorphism .

If we’ve chosen some local coordinates on , our above discussion implies the vector field is given by

Indeed, , so , and (for ). Therefore,

from which result is immediate.

I really want to tie all this material together to talk about isotropic,Lagrangian, and involutive submanifolds of , but I’ve already rambled on for quite a while in this post. If you’ve been following along these past few posts, you’ll remember that my original motivation for talking about these symplectic objects in was for their relationship with the microsupport of sheaves in . More precisely, for all , is an involutive subset; if is also -constructible, then is a Lagrangian subset of . A little alarm should be going off in your head right now; there’s no reason at all for to be a smooth submanifold of (take, for example, to be the constant sheaf supported on a singular subset of ), so how do these definitions apply? What details of the definition would we need to modify, and what changes? Let’s find out, next time.

References:

- M. Kashiwara and P. Schapira,
*Sheaves on Manifolds*(Appendix A). - A. Cannas de Silva,
*Lectures on Symplectic Geometry*(http://www.math.ist.utl.pt/~acannas/Books/symplectic.pdf) - Alex Zamorzaev, notes from a course on introductory symplectic topology (http://math.stanford.edu/~alzaor/257a.html).

**proposition 1: **They’re always even dimensional!

**proof:**

Let be a symplectic vector space of dimension *m*. I claim then that *m *is even. Choose some basis of , so that we get a matrix representative of , i.e., for all . Since is alternating, is skew-symmetric, giving the relation . Hence,

.

So, if is odd, we must have . But, since is non-degenerate, , which forces to be even in the above equation. Done!

**end proof.**

A quick application of the “rank-nullity” theorem for a subspace applied to implies ; hence, all Lagrangian subspaces of are of dimension . Similarly, if and , then is Lagrangian.

Of course, from plain old linear algebra, we know that, given any Lagrangian subspace of , there exists an dim subspace such that . The question now is:

**proposition 2: **Can the complementary subspace be chosen such that is also Lagrangian?

**proof:**

Given Lagrangian, choose an isotropic space such that ( is a proper subspace, so we can always at least choose a line that satisfies this property, and lines are always isotropic). If , is not a subset of . Indeed, if it were, then we’d have (by “symplectic duality” given by regarding as the annihilator of w.r.t. ) . Since (so ), we contradict the assumption that is isotropic: i.e., we assumed . Thus, does not contain . Choose then some . Then, is an isotropic subspace (as is bilinear and alternating) satisfying . Arguing by induction on , we get our Lagrangian space . Perfect.

**end proof.**

Why is this relevant?

Recall the space of -dim subspaces of , called the *Grassmannian* and denoted . It is a smooth manifold of dimension . My main goal is to introduce the *Lagrangian Grassmannian * of Lagrangian planes in , a closed (smooth) submanifold of of dimension . Also, it’s compact! (implicitly, this is **proposition 3**)

**proof:**

Let . I claim that there is a bijection between the spaces (which is open in the subspace topology of in ) and the space of (real) quadratic forms on (a real vector space of dimension ).

Set (where is the algebraic dual of ), equipped with the standard symplectic form . With respect to , the subspace is Lagrangian (follows immediately from the definition of and the assumption ). By proposition 4 from the last post, we know there exists a symplectic map with . So, we might as well work inside . Let . Then, (it’s an easy exercise to show) that is the “graph” of a (unique!) linear map , that is, we can write (this is actually very similar to how you construct the smooth atlas on the ordinary Grassmannian manifold; check it out!).

Since , . Using the fact that is the graph of the matrix (suppose we’ve chosen a basis), this tells us

i.e., is a symmetric matrix. With a basis fixed for (giving a basis of by taking the dual basis for ), we know that the collection of real symmetric matrices is linearly isomorphic to the space of real quadratic forms on (cf. the wiki page on quadratic forms, or whatever linear algebra reference you hold dear), which has the desired dimension. Since this map is linear, it’s (a fortiori) smooth. Compactness follows trivially from the fact that is compact and is closed in . The only thing left to check is that the transition maps between different charts are smooth. Screw that; left to the reader.

**end proof.**

In the up and coming posts, we generalize our “symplectic vector spaces” to get *symplectic manifold: *these are smooth (real, at least for us) manifolds equipped with a closed, non-degenerate 2-form that gives each tangent space the structure of a symplectic vector space. Neat, right? Will there be some sort of “standard” symplectic manifold, like we saw last post (cf. example 1)?

(The answer is yes).

References:

- M. Kashiwara and P. Schapira,
*Sheaves on Manifolds*(Appendix A). -
Alex Zamorzaev, notes from a course on introductory symplectic topology (http://math.stanford.edu/~alzaor/257a.html).

A *symplectic vector space* is a pair , where is a finite dimensional real vector space (henceforth, all vector spaces for us will be finite dimensional over ), and is a *symplectic form* on ; that is, is a non-degenerate, alternating, bilinear form on . Let’s play with an example to get acquainted.

**Example 1. **Let be a vector space, , and let be the canonical pairing of and . Define a bilinear form on by

for . Naturally, I claim that is a symplectic form on . By construction, is alternating and bilinear, so we only need to check non-degeneracy. Let be such that, for all , . That is, for all ,

.

By non-degeneracy of , setting yields , and setting yields (remember, that equality was assumed to hold for all elements of !). Hence, , implying is non-degenerate. Note that, for , the form looks a lot like the determinant map! ().

Now, for a subspace of a symplectic vector space , we associate its *symplectic complement*, or *symplectic orthogonal.*

.

This is where we get the notions of isotropic, involutive, and Lagrangian subspaces: a subspace of is

*isotropic*if ,*involutive*if , and*Lagrangian*if .

Let’s end with some easy examples:

**Example 2.** A line is always an isotropic subspace.

Let be a non-zero vector, so that every element of is of the form for some . Then, the fact that is bilinear and alternating implies that .

**Example 3. **A hyperplane is always an involutive subspace.

Let be non-zero. If , then since is a hyperplane, we must have (by , I mean the line spanned by the non-zero vector ), so that every element of is of the form , for some and . But, since , we must have

by bilinearity. Since was arbitrary, the non-degeneracy of yields , a contradiction. Thus, , so is an involutive subset.

Naturally, when we introduce new structures on spaces, we want to identify those morphisms that “preserve” that structure. In this case, it’s the symplectic form. A linear map is called *symplectic* provided . That is, for all , we have (by definition of the pullback)

.

A symplectic map that is also invertible is called *a symplectomorphism.*

Just like every vector space is modeled on for some (upon choosing a basis), all symplectic vector spaces of dimension are symplectomorphic to , where

() for each (cf: example 1). This isn’t TOO hard to show, but it takes a little bit to work through all the necessary details. I don’t feel like writing this one out; you’ll just have to take my word for it (or, you know, work it out yourself).

That being said, there is a similar result that I do want to show you. It’s pretty clear that, for each , the subspace is a Lagrangian subspace of (i.e., ). As it turns out, is the prototype for all Lagrangian subspaces:

**Proposition 4: **Given any symplectic vector space of dimension , and Lagrangian , there exists a symplectic map sending to .

**proof: **Assume that we’ve proved the result for all dimensions (for , the Lagrangian subspaces are all just lines through the origin in , and the desired symplectic map is just a rotation about the origin). We want to then show the result for dimension . Okay. Let be a Lagrangian subspace, . Pick some non-zero. Since is non-degenerate, there exists some such that . As is Lagrangian, this gives . Set

;

with the restriction , is a symplectic space. Of course, from , the only thing to check is that is non-degenerate (if is non-zero, there exists some such that . By the definition of , we must have . It follows that ).

Now, set . We need to show is Lagrangian in , and . Since , and , it follows that is an isotropic subspace. Is it maximally isotropic in (i.e., Lagrangian?). If not, there would exist an isotropic subspace with . But then, would be isotropic in , and . But this is a contradiction, since an isotropic subspace of must have dimension ! (exclamation, not factorial. whoops). Thus, is Lagrangian. For the second part of the claim, we note that , and the above shows , so .

Okay, here’s where we invoke the inductive hypothesis: there exists a symplectic map sending to . Then, the map

via

is symplectic, and sends to . Oh, by the way: we define the form . By assumption, we know . Since I’m lazy, and this calculation is pretty messy, let’s write , . Then, by algebra:

as , and hence (() and by expanding the term in terms of ). So, is symplectic. , because , and . Done!

**end proof.**

Next time, I’ll talk some more about Lagrangian subspaces and some facts about the Lagrangian Grassmanian of a symplectic vector space.

Reference: M. Kashiwara and P. Schapira, *Sheaves on Manifolds* (Appendix A).

.

Specifically, in addition to being -conic and closed, is also a real subanalytic, isotropic subset of (with respect to the usual symplectic structure on the cotangent bundle).

**Example:**

One of the easiest examples of this is when we consider the constant sheaf on X. Suppose we’re using as our base ring for , and let be a closed submanifold. To , we can associate the sheaf (considered as a complex of sheaves concentrated in degree zero), whose stalks are if , and if . Then, it’s not too hard to show the equality , the full conormal bundle to in .

Another good toy example is when , and we’re considering the constant sheaf on a closed interval , extended by zero to all of . What would this sheaf’s microsupport be?

Since almost all the objects we’ll be interested in are in , we should explore the microsupport of these complexes as deeply as we can, from as many perspectives as we can find (morally, the “Yoneda lemma” approach (http://mathoverflow.net/a/3223)).

The perspective I want to talk about today is from the (triangulated) derived categories , from Kashiwara and Schapira’s *Sheaves on Manifolds *([K-S]), that are associated to subsets . Briefly, is the localization of by the full subcategory of objects whose microsupport is disjoint from . Then, we want to say that working “microlocally” with some complex of sheaves on means to consider as an object of the category for some subset . This should seem pretty reasonable; If , then has no interesting microlocal behavior on and can’t really interact in a meaningful way with other, non-zero, objects of .

So, now it’s time for the actual details. Let be a subset of . We define to be the full subcategory of consisting of those objects such that . Additionally, this is a triangulated category, with distinguished triangles (naturally) those of whose objects belong to . Then, is a null-system in , and we may define the relevant localization. That is, we set (where ). In the case we’ll usually use, for some covector , we’ll be lazy and write for the localized category.

One of the first things we can say about is: a morphism in becomes an *isomorphism *in (or, we say, “on “) if we can embed in a distinguished triangle $latex \mathcal{A}^\bullet \overset{u}{\to} \mathcal{B}^\bullet \to \mathcal{C}^\bullet \overset{+1}{\to}$ with .

Let’s play with some examples, and see how working in differs from working in .

**Example:**

Let . To the closed subset , there is a distinguished triangle in :

from this triangle, it follows that the complexes and are isomorphic in (i.e., there is a morphism between the complexes that induces an isomorphism on cohomology).

Let’s now consider the same distinguished triangle, but this time in the category , where . After a quick calculation, we obtain:

- ,
- , and
- .

whence it follows that in , as well as in .

**Why should you care? **

Okay, that’s cute. But why are these categories useful? Why should you bother to play with at all?

Earlier in this post, I mentioned that, for constructible, is an -conic, subanalytic, isotropic subset of . Similarly, using , I want to give a “microlocal” characterization of those whose microsupport is contained in an involutive submanifold of . Particularly, when the microsupport is contained in the conormal bundle to some submanifold of (as it turns out, the microsupport is ALWAYS an involutive subset of the cotangent bundle). Hopefully, this pursuit will tell us something new and interesting about the objects of our favorite category, . Eventually (not this time!) we’ll put these two things together, and see what happens along a Lagrangian submanifold; this leads to Kashiwara and Schapira’s notion of *pure sheaves*. These are pretty neat.

**Proposition 1: **([6.6.1, K-S])

Suppose is a closed submanifold of , and let denote the associated smooth, closed embedding. Let (non-zero, or else this is a stupid exercise), and let . Let denote the projection of the cotangent bundle. Assume further that, in a neighborhood of , . Then, there exists a such that in .

(Quick sketch: Basically, we want to successively, microlocally, cut up by restricting to various subsets of , and throw away those pieces whose microsupport doesn’t meet .)

**Detailed Sketch:**

By induction on the codimension of in , we may assume is a hypersurface, say given by , with . Set , so we have the open embeddings . By [6.3.1, K-S], there is an inclusion

,

where (in our case here) consists of the zero section of , together with those covectors where . Vaguely, the operation characterizes those covectors arising as limits of sequences of covectors in (see [6.2.3, K-S]).

Since is not in the right hand side, we must have . Hence, from the adjunction distinguished triangle

in , we see that in . So, we might as well have assumed from the start that . Similarly, we find (by [6.3.1, K-S]) that , and see that in , using the distinguished triangle

(where is the closed complement of in ), and using the fact that . All together, this gives us the desired isomorphism in .

Q.E.D.

**Proposition 2:** ** **[6.6.1, K-S]

Same as the set up of proposition 1, but now we assume that in a neighborhood of . Then, there exists such that in .

Since , the previous proposition applies, and we may assume in for some . In particular, this tells us in a neighborhood of . So, the assumption that , together with [5.4.4, K-S] (this tells us that consists of covectors such that , and ) give us that is contained in the zero section, in a neighborhood of : Since , the condition implies ) that in a neighborhood of .

Next, let be the canonical map to a point. Since , there exists a such that . Hence, in , and we’re done.

Q.E.D.

Now, what can we say our constructible case ? Keywords: non-degenerate covectors, and Morse modules. But that’s for another time.

]]>Let’s start with a simple example. Let V be a (finite dimensional) real vector space, and let be two convex (non-empty) open subsets of V.

Let be a local system on V (i.e., a locally constant sheaf of real vector spaces, considered as a complex concentrated in degree zero). It’s well known that the inclusion of open subsets induces an isomorphism in (hyper)cohomology: . Ok, so what?

So, instead of just say “they have isomorphic cohomology,” you should be thinking “every section of over can be extended to all of ,” or “sections of can **propagate** from to , in all ‘directions.'”

The idea of localizing a sheaf at a point (say for the moment, is some open subset) is one of the most basic tools in an algebraic geometer’s tool box. You forget everything else about how might behave on , and just focus on its behavior very close to . That is, you’re examining its **local** behavior at . If we want to understand how the cohomology of might change as we move away from , we need to examine the **microlocal** behavior of at , in the space of directions from (really, however, we want ‘codirections’ associated to covectors based at . That part is coming).

Pick some (non-zero, duh) covector , the cotangent space of to . By shrinking if necessary, we can then assume that is the differential of some smooth function , and by translation, we can further assume that . Although lots of functions might have differential equal to at , for simplicity, we can just assume that is some globally defined -linear form. Since, locally, we can safely just picture covectors as ordinary vectors based at , I like to think of as the “normal vector” to the hyperplane passing through . Which makes sense, of course, since we can think of the differential here as “the gradient vector “, which is always normal to level sets of . In the general case, you can still make this association, but you’ll need some added structure on your space (e.g., a Riemannian metric).

The whole point of this is that we want to use this function to investigate possible changes in the cohomology of as we move away from , in the codirection . How would we go about doing that?

Local Morse data!

Think about it. First, let’s work inside some small open ball about . If every section of over can be extended passed the “boundary” , at least a little bit (say to for some small ), then the cohomology of over W should be isomorphic to the cohomology over , and this isomorphism should be induced by restriction.

So, in fancy language, we’d say the relative cohomology complex of is zero. Of course, it might happen to be the case that some sections can only propagate across on small neighborhoods of , so in order to pick up all such sections, we look at the limit for . A little bit of abstract nonsense then reveals this object is none other than , the local Morse data of at , with respect to .

Recalling my last post, this sort of vanishing tells us that the pair is not in the microsupport of ! Before, I denoted this by , but from now on, I think I’ll stick with the more popular notation: (“SS” stands for “singular support,” in case you were wondering).

In order to explore further properties of the microsupport of a complex of sheaves, we’ll need a little more machinery at our disposable. In particular, something called the “-topology” associated to a closed, convex cone inside a vector space, and the “non-characteristic deformation lemma,” a technical result at the core of the majority of the ensuing propagation theorems. At the risk of ranting even more, I’ll stop here. I know I haven’t done much new stuff in this post, but I hope I’ve helped elucidate the basic ideas/motivations behind microsupports.

Until next time

]]>which gives the integral cohomology of the “local Morse datum” of at , considered as a pair of spaces. Confusing, I know. It seems like complete overkill at this point, and it is.

So, let’s go deeper.

First, we’ll consider more general objects: the local Morse data of at , with respect to the complex of sheaves , denoted

Basically, you have the complex of sheaves, , on , and you consider sections of this “sheaf” whose support is contained in the subset , and take the stalk cohomology at the point . Sort of like: the sections of the sheaf that propagate in the “positive direction” (where “positive” is taken to be with respect to (really, the covector , but we’re not quite there yet ) ). If this stalk cohomology vanishes, then the local sections of the sheaf can be “extended” a little bit further away from , at least in the “positive” direction. That’s supposed to be what I mean by “propagate” here.

Why is this idea useful?

Let be a sufficiently nice closed subset, so we can give it a Whitney stratification (by which I mean, “satisfies Whitney’s condition (b) at all appropriate times”), . To each stratum , we associate the following subset of the cotangent bundle of , called the **conormal space to S in M****:**

which consists of covectors in the cotangent bundle which annihilate the various tangent spaces to , considered as subspaces of the tangent spaces to . Equivalently (after perhaps endowing with a Riemannian metric), you can think of these elements as hyperplanes in that contain the subspace .

Let be a smooth function, its restriction, the associated stratified critical locus of . It is any easy exercise to show that, when is a Whitney (a) stratification, is a closed subset of . Similarly, the Whitney (a) condition is equivalent to requiring the equality:

Hence, when I say , there is a unique stratum, say , for which (since the strata are *disjoint*). And, if you think about it, if , we must have

as . Remember this.

**Local Stratified Acyclicity (LSA)**

Say we have our Whitney stratification, , of . Then, satisfies something called **LSA**:

for all (this means the cohomology sheaves of are all locally constant with respect to the strata of ), for all (germs of) stratified submersions ,

.

I’m not going to prove this, but it’s a consequence of a result called the **non-characteristic deformation lemma** of Kashiwara and Schapira in *Sheaves on Manifolds*. This tells us that the cohomology sheaves of complexes of sheaves are locally constant if a certain (similar) vanishing condition occurs with germs of submersions.

And now, finally, I can talk about the microsupport of a complex of sheaves, , which encodes the “directions” at given points of where sections of “do not propagate.” That is, the directions in which we should expect to detect changes in the cohomology (sheaves) of . Precisely, the **Microsupport of ** is the subset , characterized (in the negative…) by:

if and only if there exists an open subset of in such that, for all smooth function germs with , one has .

So, if , locally, we can say for some smooth function germ at , and if you move in the direction of the “gradient flow” of in the “positive direction,” will detect a change in the cohomology of . I like to picture to be the germ of some linear form (say in a local coordinate system about ), and the gradient flow is like looking a family of cross sections of near . Moreover, will be a critical point whatever function we pick, by trivial application of LSA.

I also think this emphasizes the importance of the connection with Morse data: if and only if .

Now, say we’ve got our sufficiently nice closed subset , with Whitney stratification , equipped with inclusion map . Say we’ve got some smooth function . We know that, for , if and only if . Then, via the isomorphism

and an application of LSA, this quantity vanishes if and only if is a stratified submersion. Consequently, we see

Now…what does the microsupport reveal about the local Morse data of functions with stratified critical points? That’s where things will start to get interesting. Until next time.

References:

M. Kashiwara and P. Schapira; *Sheaves on Manifolds.*

J. Schurmann; *Topology of Singular Spaces and Constructible Sheaves.*