Sabbah-Mochizuki-Kedlaya’s Hukuhara-Levelt-Turrittin Theorem

When I first started learning about irregular perverse sheaves, I struggled to find a clear answer to the question: “what does an irregular perverse sheaf look like, in the simplest cases? What should be my intuition?” If I don’t know about the motivation from differential equations, how can I picture them? This is not yet feasible given the current state of the literature; moreover, there is no general introduction to this motivation if you care about getting your hands on these new objects. Aside from the dumb name, perverse sheaves are some of the most natural objects in complex geometry, so I was hoping for a similar situation in the irregular world. This introduction doesn’t exist yet, but I hope to eventually write one.

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 \mathbf{P}^\bullet is a complex analytic space, and admits a Whitney stratification—a locally finite partition into complex analytic submanifolds S of the ambient manifold M along which the cohomology sheaves H^k(\mathbf{P}^\bullet)_{|_S} are local systems, and
  • the normal data \mu_\xi(\mathbf{P}^\bullet) is a finite dimensional vector space concentrated in a single degree, independent of \xi \in T_S^*M chosen generically (this last property is sometimes called the microlocal 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 \overset{+}{\otimes}, then an irregular \mathbb{R}-constructible (ind-)sheaf is one which, locally in the subanalytic topology on M, “comes from” an ordinary \mathbb{R}-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 (M,\nabla) on a complex disk with a single regular singular point at 0 has a basis of solutions given by a multivalued functions times entire functions (like z^\alpha times a convergent power series, with maybe some logarithmic terms thrown in if two of the exponents \alpha are repeated or differ by an integer). Under Riemann-Hilbert, these turn into perverse sheaves DR(M,\nabla) where the local system in degree -1 corresponds to the flat sections \ker \nabla away from 0, and the data of DR(M,\nabla) is completely determined by the monodromy of these flat sections and any confluence in the eigenvalues of the exponents of the terms z^\alpha. The moral of this being, the topological properties of DR(M,\nabla) comes from understanding the associated ODEs, and every perverse sheaf is of this form for some (M,\nabla). 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 \mathscr{O}_{X,0} \cong \mathbb{C}\{z\} and its fraction field \mathscr{O}_X(*0) \cong \mathbb{C}\{z\}[z^{-1}]. We’ll now need to introduce the formal completion of \mathscr{O}_{X,0},

\widehat{\mathscr{O}}_{X,0} := \varprojlim_{i \geq 1} \mathscr{O}_{X,0}/{\mathfrak{m}_{X,0}}^i \cong \mathbb{C}[[z]]

the ring of formal power series in z at 0. By considering a Meromorphic connection as a \mathbb{C}\{z\}[z^{-1}]-module, we get a natural inclusion functor into the category of \mathbb{C}[[z]][z^{-1}]-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 \mathbb{C}\{z\} or over \mathbb{C}[[z]]. The first of these differences, the place we’ll start, is the following:

Theorem(Levelt-Turrittin)

Let (M,\nabla) \in Conn(X;\{0\}), and let (\widehat{M},\widehat{\nabla}) be its formal completion as a \mathbb{C}[[z]][z^{-1}]-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 \mathbb{C}[[z]][z^{-1}]-modules

(\widehat{M},\widehat{\nabla}) \xrightarrow{\widehat{\lambda}} \bigoplus_{\varphi \in \Phi} \left (\mathscr{O}_X(*0)^{d_\varphi},d+d\varphi+C_\varphi \frac{dz}{z} \right)

where C_\varphi \in M_{d_\varphi}(\mathbb{C}). 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 \varphi \in z^{-1}\mathbb{C}[z^{-1}], set

\mathcal{E}^\varphi := (\mathscr{O}_X(*0),d+d\varphi).

Then, the Levelt-Turrittin theorem says that (M,\nabla) is formally isomorphic to a direct sum of connections of the form \mathcal{E}^\varphi \otimes \mathcal{R}_\varphi (where \mathcal{R}_\varphi 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 z. That is, if \rho: X \to X sends z \mapsto z^p for a positive integer p, then the decomposition holds for \widehat{\rho^*M}.

So, in the case of the Airy equation (\partial_z^2 -z)u = 0 with irregular singularity at infinity, the associated Meromorphic connection is formally isomorphic to

\bigoplus_{\pm} \left (\mathcal{O}_Y(*\infty),d+d(\pm \frac{2}{3}z^\frac{3}{2})+\frac{1}{4} \frac{dz}{z} \right )

where Y is a disk around \infty inside \mathbb{P}^1. This sum reflects the fact that the Airy functions can be asymptotically approximated by linear combinations

c_1 z^{-\frac{1}{4}}e^{\frac{2}{3}z^{\frac{3}{2}}}+c_2 z^{-\frac{1}{4}}e^{-\frac{2}{3}z^{\frac{3}{2}}}.

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 c_1,c_2 for all values of z 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 \theta \in S^1, and the arc neighborhoods S_r(a,b) = \{ z \in X  | 0 < |z| < r, a < \arg z < b\} 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 \overset{\thicksim}{X} := S^1 \times [0,1) with natural map

\varpi : \overset{\thicksim}{X} \to X

(e^{i\theta},r) \mapsto re^{i\theta}

\overset{\thicksim}{X} is not a complex manifold, so it doesn’t really make sense to talk about a sheaf of holomorphic functions on all of \overset{\thicksim}{X}. But it is a real manifold with boundary, so there is a well-defined sheaf of smooth functions \mathscr{C}_{\overset{\thicksim}{X}}^\infty. On the complement of boundary \varpi^{-1}(0) \cong S^1, there is a natural subsheaf of holomorphic functions {\mathscr{O}_{\overset{\thicksim}{X}}}_{|_{\overset{\thicksim}{X}-\varpi^{-1}(0)}} \cong \mathscr{O}_{X^*}, 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 \partial_{\overline{z}}f(z) = 0 and have a power series representation everywhere, even the boundary.

Is \overline{\partial} even well-defined on \overset{\thicksim}{X}? Sort of. If we work in polar coordinates (r,e^{i\theta}) on \overset{\thicksim}{X}, then a quick computation tells us that

\overline{z}\partial_{\overline{z}} = \frac{1}{2}\left (r\partial_r+i\partial_\theta \right )

as operators on C_{\overset{\thicksim}{X}}^\infty. So, we can talk about the operator \overline{\partial} := \partial_{\overline{z}} on \widetilde X 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 \overline{z}). So, in analogy with how on X itself we can realize \mathscr{O}_X as the flat sections of C_X^\infty \xrightarrow{\overline{\partial}} C_X^\infty \otimes_{\mathscr{O}_{\overline{X}}} \Omega_{\overline{X}}^1, we define the sheaf of holomorphic functions on \widetilde X, denoted \mathcal{A}_{\widetilde X}, to be the flat sections of the connection

C_{\widetilde X}^\infty \xrightarrow{\overline{\partial}} C_{\widetilde X}^\infty \otimes_{\varpi^{-1}\mathscr{O}_{\overline{X}}} \varpi^{-1}\Omega_{\overline{X}}^1(\log \overline{\{0\}}).

That is, \ker \overline{\partial} =: \mathcal{A}_{\widetilde X}. If S_\epsilon(a,b) (with 0 < \epsilon \ll 1, $a < b$ with 0 < b-a < 2\pi is a proper arc neighborhood of 0, then \mathcal{A}_{\widetilde X}(S_\epsilon(a,b)) \cong \mathscr{O}_X(S_\epsilon(a,b)).

What does this give us? Asymptotic Expansions.

The sheaf \mathcal{A}_{\widetilde X} is only really a new object for us on the boundary of \widetilde X, \varpi^{-1}(0) \cong S^1, so let’s focus on that. Denote {\mathcal{A}_{{\widetilde X}}}_{|_{S^1}} = \mathcal{A}; this is a subsheaf of i^{-1}j_*\mathscr{O}_{X^*}. What sort of functions does it contain?

Let e^{i\theta} \in S^1, and let’s look at the stalk \mathcal{A}_{e^{i\theta}}, say on some small proper arc neighborhood S_r(a,b) containing \theta, and take a function f \in \mathcal{A} defined on this neighborhood. Then, f admits an asymptotic expansion at 0; that is, there is a formal series c = \sum_{n \geq 0} c_n z^n \in \mathbb{C}[[z]] such that, for all N \in \mathbb{N}, and every closed subsector W \subset S_r(a,b), we can find a constant C_{N,W} > 0 such that the estimate

|f(z)-\sum_{n=0}^{N-1}c_n z^n| < C_{N,W}|z|^N

holds for all z\in W. 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 f is asymptotic to a series c \in \mathbb{C}[[z]] above, we write f \thicksim c. An example of something outside from ordinary stuff like Taylor series is

e^{-\frac{1}{z}}\int_{1}^{1/z} \frac{e^t}{t} \, dt \thicksim \sum_{n \geq 0} n! z^{n+1}

which clearly doesn’t converge on any neighborhood of the origin. The sheaf \mathcal{A} on S^1 consists of all those smooth functions on C_{\widetilde X}^\infty that locally admit an asymptotic expansion on the boundary of \widetilde X. It contains things like e^{-1/z}, but only for those \theta \in S^1 where \theta \in (3\pi/2,\pi/2) \mod 2\pi (where e^{-1/z} \thicksim 0). It also contains all of our exponential factors e^\phi whenever \textnormal{Re}\phi < 0. We’ll see more later about this sheaf \mathcal{A}_{\widetilde X} with moderate growth and rapid decay functions and the Borel-Ritt Lemma.

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

\mathscr{D}_{\widetilde X} = \mathcal{A}_{\widetilde X} \otimes_{\varpi^{-1}\mathscr{O}_X} \varpi^{-1}\mathscr{D}_X

And likewise define \mathcal{A}_{\widetilde X}(*0) = \mathcal{A}_{\widetilde X} \otimes_{\varpi^{-1}\mathscr{O}_X} \varpi^{-1}\mathscr{O}_X(*0). So, tensoring with \varpi^{-1}\mathscr{O}_X(*0) allows us turn meromorphic connections (which are \mathscr{O}_X(*0)-modules with connection) into \mathcal{A}_{\widetilde X}(*0)-modules with connection in a natural way. The point of all this is that, over \mathcal{A}_{\widetilde X}, 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 e^{i\theta} \in S^1, the formal isomorphism \widehat{\lambda} can be locally lifted as an isomorphism

\mathcal{A}_{\widetilde X} \otimes_{\varpi^{-1}\mathscr{O}_X}(M,\nabla) \xrightarrow{\lambda_\theta} \bigoplus_{\varphi \in \Phi} \left (\mathcal{A}_{\widetilde X}(*0)^{d_\varphi},d+d\varphi+C_\varphi \frac{dz}{z} \right)

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 S^1 = \varpi^{-1}(0), the exact description of the isomorphism may change as we travel around the circle (i.e., doesn’t necessarily give an isomorphism of \mathscr{O}_X(*0)-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 X with poles along a simple normal crossings divisor \Delta = \{z_1\cdots z_r = 0\}, then we have to assume that our connections have a good formal decomposition (which means that we can find a finite set \Phi \subset \in \mathscr{O}_X(*0)/\mathscr{O}_X that gives such a formal decomposition, and goodness means that pairwise choices of exponential factors \phi \neq \psi have divisor of zeroes \phi - \psi that is empty near 0). Assuming a good formal decomposition, we can then locally lift it to a real isomorphism over \mathcal{A}_{\widetilde X} (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 \leq 5). 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 (M,\nabla) on a space X with poles in a divisor D, and any point x \in D, there is an open neighborhood U of x finite sequence of blow-ups e: U^\prime \to U such that e^{-1}(D) = D^\prime has normal crossings and e^*(M,\nabla) has a good formal decomposition at each point of D^\prime

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 \varpi^{-1}(0) 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 z^\alpha 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 \{\lambda_\theta \}_{e^{i\theta} \in S^1} 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 \mathcal{A}_{\widetilde X} agrees with j_*\mathscr{O}_{X^*} away from \varpi^{-1}(0)), and so the difficultly will be in understanding both the formal structure at the origin, along with the family of isomorphisms \lambda_{\theta}, without knowing the exponential factors.

Deligne’s regular solution in dimension 1

In this post, I want to recall the elements of the regular Riemann-Hilbert Correspondence (but only in dimension 1, on our small disk around the origin). We’ll talk more about the category of Meromorphic Connections on X with singularities at 0, and how they’re just a different way of phrasing linear ODEs whose solutions have singularities at 0. From there, we examine the simplest class of solutions, those with regular singularities. These are, in general, multi-valued functions of z that satisfy an analytic condition called moderate growth. From there, we can state Deligne’s solution to the regular R-H correspondence, and start to understand more about the failure in the irregular setting. We end with some emergent phenomena that occur only for irregular singularities, which add to the many difficulties in proving the irregular R-H correspondence. Lots of things from this post are from chapter 5 of “D-modules, perverse sheaves, and representation theory“.

Meromorphic Connections = Fancy ODE’s with singularities

Let X be a small open ball around the origin in \mathbb{C}, \mathscr{O} := \mathscr{O}_{X,0}, and \mathscr{O}(*0) the field of fractions of \mathscr{O}, representing holomorphic functions with possible poles at 0. If z is a local coordinate on X, then \mathscr{O} \cong \mathbb{C}\{z\}, and \mathscr{O}(*0) \cong \mathbb{C}\{z\}[z^{-1}]. Then, recall that a meromorphic connection consists of the data of a free \mathscr{O}(*0)-module M and a \mathbb{C}-linear map \nabla : M \to M satisfying the Leibniz rule \nabla(fm) = \frac{df}{dz}u + f\nabla(u) for elements f \in \mathscr{O}(*0), u \in M. A morphism of meromorphic connections is just a map \phi : (M,\nabla) \to (N,\nabla) that is \mathscr{O}(*0)-linear from M to N and commutes with the action of \nabla. This is actually a bit of a simplification of connections–normally we’d specify \nabla as a map

M \xrightarrow{\nabla} M \otimes_\mathscr{O} \Omega_{X/\mathbb{C}}^1

Contracting this “universal” \nabla with elements of \mathscr{D}_X 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 0. Now, what does a general meromorphic connection look like?

On X with our local coordinate z, choosing a system of generators of M over \mathscr{O}_X(*0), say e_1(z),\cdots ,e_n(z) (these always exist globally on X because X is contractible). Then, looking at the action of \nabla on this basis gives us a connection matrix with values in \mathscr{O}(*0)

\nabla e_j =- \sum_{i=1}^n a_{ij}(z) e_i,

the negative sign is just a convention, to make the next expression nicer. Thus, if u(z) = \sum_i u_i(z)e_i(z) is a general element of M, then the above expression and the Leibniz rule gives us

\nabla (\sum_{i=1}^n u_i e_i) = \sum_{i=1}^n \left (\frac{du_i}{dz}-\sum_{j=1}^n a_{ij}u_j \right )e_i

Hence, the collection of flat sections of \nabla (those \vec{u} = \sum_i u_ie_i that satisfy \nabla u = 0) correspond to solutions of the system of linear ordinary differential equations

\frac{d \vec{u}}{dz} = A(z)\vec{u}

Likewise, any such differential equation gives rise to a connection–let \widetilde M be the space of solutions to the above differential equation. Then, this equation is a rule for how the symbol \frac{d}{dz} acts on the elements of \widetilde{M}, taken as the definition of \nabla.

The simplest case that arises from having meromorphic coefficients is when the connection matrix A(z) has at worst poles of degree one, i.e., if A(z) = \frac{A}{z} for some constant matrix A \in M_n(\mathbb{C}) (to compensate for maybe choosing a bad generating set for M over \mathcal{O}(*0), we only care if A(z) is “gauge equivalent” to a matrix of the form \frac{A}{z}). In the rank one case, this is a differential equation of the form

\frac{du}{dz} = \frac{\alpha}{z}u(z)

for some \alpha \in \mathbb{C}, and has fundamental solution u(z) = z^\alpha. What sort of function is this? When \alpha = 0,1,2,3,\cdots, z^\alpha is a globally defined holomorphic function (it’s just a monomial!), and when \alpha = -1,-2,-3,\cdots, z^\alpha is a globally defined meromorphic function on X that is holomorphic on X^*=X-\{0\}. When \alpha \notin \mathbb{Z}, we have to use the definition z^\alpha := \exp(\alpha \log(z)), where \log(z) 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 X^*, as we travel around the origin and analytically continue z^\alpha, the value of this function jumps due to monodromy. Precisely, if we let z \mapsto e^{2\pi \theta i}z as 0 \leq \theta \leq 1 varies, we find

z^\alpha \mapsto e^{2\pi i \alpha}z^\alpha := \exp\left \{\alpha(\log |z| +i(\arg(z)+2\pi)\right \}.

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

This is an example of a multi-valued function on X, and doesn’t lie in \mathcal{O} or \mathcal{O}(*0), and naturally form a ring (after fixing a universal cover of X^*). 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 \frac{A}{z} (we can even fudge a bit to allow A(z) \in M_n(\mathcal{O})) 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 z^\alpha given above; if M has rank n, we can find by the Frobenius method n linearly independent solutions of the form

z^\alpha_i \phi_i(z)

where \alpha_i \in \mathbb{C} and \phi_i(z) is a holomorphic function on X with f(0) \neq 0 (the expression is slightly more complicated if an exponent \alpha_i is repeated, or if some pair \alpha_i and \alpha_j 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

S_\epsilon(a,b) := \{z \in X^*  |  0<|z|<\epsilon, a < \arg(z) < b\}

for some \epsilon > 0 and 0 \leq a < b \leq 2\pi.

arc neighborhoods on which z^{\frac{1}{3}} is single-valued

Solutions to ODE’s with regular singularities on X, even multi-valued ones, always behave “like” meromorphic functions in an appropriate arc neighborhood of any particular angle \theta \in S^1. 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 u(z) grows only polynomially in |z|^{-1}, as if it were a Laurent series with only finitely many terms of negative degree.

More precisely, solutions u(z) to ODEs with regular singularities are said to have moderate growth at \theta \in S^1, in the sense that there are 0< \epsilon,\delta \ll 1, constant C_\theta > 0 and exponent N_\theta \in \mathbb{N} for which

|u(z)| \leq C_\theta|z|^{-N_\theta} (1)

on the arc neighborhood S_\epsilon(\theta-\delta,\theta+\delta). As \theta travels around S^1, the values of the constants C_\theta and N_\theta may change. We just say u(z) has moderate growth at 0 if it has moderate growth at every \theta \in S^1.

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

This analytic characterization is actually a necessary and sufficient condition; a meromorphic connection (M,\nabla) 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 Conn^{reg}(X;0) be the category of Meromorphic Connections with regular singularities at 0, and Conn(X^*) the category of flat connections on X^*.

Theorem (Deligne): The restriction functor M \mapsto M_{|_{X^*}} induces an equivalence of categories

Conn^{reg}(X;0) \xrightarrow{\thicksim} Conn(X^*)

To show that we can always extend a given (N,\nabla) \in Conn(X^*) of rank n to an element of Conn^{reg}(X;0), we note that (N,\nabla) is uniquely determined by a monodromy representation \rho : \pi_1(X^*) \to Gl_{n}(\mathbb{C}) defined by the local system \ker \nabla. If we let \gamma correspond to the element 1 in the identification \pi_1(X^*) \cong \mathbb{Z}, then \rho is determined by the matrix C = \rho(\gamma) . We can then always find a matrix \Gamma \in M_{n}(\mathbb{C}) such that \exp(2 \pi i \Gamma)= C, which we then use to define a connection matrix on \widehat{N} := \mathscr{O}(*0)^{n} via

\nabla e_q = -\sum_{1 \leq p \leq n} \frac{\Gamma_{pq}}{z} \otimes e_p

where \{e_1,\cdots,e_n\} is the standard basis for \widehat{N}. The resulting meromorphic connection (\widehat{N},\nabla) clearly has a regular singularity at 0 and restricts to N on X^*.

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

Conn^{reg}(X;0) \xrightarrow{\thicksim} Loc(X^*)

(M,\nabla) \mapsto (M_{|_{X^*}},\nabla_{|_{X^*}}) \mapsto \ker \nabla_{|_{X^*}}

Where Loc(X^*) is the category of finite-rank \mathbb{C}-local systems on X^*.

Irregular Singularities and Stokes Phenomena

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

For example, take (\mathscr{O}(*0),d-\frac{1}{z^2}), whose associated linear ODE has fundamental solution u(z)=e^{\frac{1}{z}}. Then, u(z) has moderate growth at every \frac{\pi}{2} < \theta < \frac{3\pi}{2}. Why? On this region, \textnormal{Re} \frac{1}{z} < 0, and so |u(z)| = e^{\textnormal{Re} \frac{1}{z}} decays to 0 exponentially as z \to 0. u(z) does not have moderate growth on (3\pi/2,\pi/2) (\mod 2\pi). When \textnormal{Re} \frac{1}{z} > 0, say \theta = 0 for simplicity, then

e^{1/z} = \sum_{n =0}^\infty \frac{1}{z^n n!} > z^{-N}

for every exponent N > 0.

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 X by its restriction to X^*. We actually saw this in the last post–the Meromorphic Connections (\mathscr{O}(*0),d-df) are all non-isomorphic (for f \in \mathscr{O}(*0)/\mathscr{O}), but their local systems of flat sections are all isomorphic to the constant local system \mathbb{C}_{X^*}. 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

\frac{d^2 u}{dz^2} = zu

which has an irregular singularity at z= \infty, and its fundamental solutions Ai(z) and Bi(z) are approximated (in a rigorous sense, using asymptotic analysis) by linear combinations of the functions u_\pm(z) = z^{-1/4}e^{\pm \frac{2}{3}z^{3/2}} for large values of z. Now, here is the interesting thing: a solution Ai(z) of the Airy equation is an entire function of z, since the coefficient function (which is just z) is entire, but the functions u_\pm are multi-valued functions. Hence, as we let z \mapsto z e^{2\pi i}, around z=0, the “true solution” Ai(z) will return to its original value, but u_+ and u_- will not. Hence, Ai(z) and A(z e^{2\pi I}) cannot be represented by the same linear combination of u_+ and u_-.

This is known as the Stokes Phenomenon, where solutions to linear ODE’s can undergo drastic qualitative changes as \theta \in S^1 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.

Irregular Riemann-Hilbert Correspondence: introduction to the problem

One of the most successful bridges between analysis and algebraic geometry is the classical Riemann-Hilbert (R-H) correspondence between regular holonomic D-modules and perverse sheaves on complex manifolds, where \mathscr{D} is the sheaf of differential operators with holomorphic coefficients (proved independently by Kashiwara and Mebkhout in 1984). This correspondence is a far-reaching generalization of Hilbert’s 21st Problem asking about the existence of ordinary differential equations (ODE’s) with regular singularities on a Riemann surface with prescribed monodromy groups. A great introduction to this problem (prior to the work of Kashiwara and Mebkhout) is Katz’s 1976 paper “An overview of Deligne’s work on Hilbert’s 21st problem“. Loosely, it is the “correct” generalization of the correspondence between vector bundles with flat connection and local systems to the singular setting, which is itself a massive generalization of the local existence and uniqueness of ordinary differential equations.

The problem of extending the R-H correspondence to cover holonomic \mathscr{D}-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-\mathbb{C}-perverse sheaves, Kuwagaki’s irregular perverse sheaves, and Ito’s \mathbb{C}-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 \mathscr{D}-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 X is a open complex disk around the origin in \mathbb{C}. This is Deligne’s version–meromorphic connections (\mathscr{D}-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 \mathbb{C}, holonomicity of a \mathscr{D}_X-module M just translates into saying there is a finite set of points X_0 in X off of which M_{|_{X -X_0}} 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 M looks like at the points of X_0. Let’s just assume M has only one “singularity”, so X_0 =\{0\} and X is a connected open neighborhood of the origin. Then, holonomicity is equivalent to giving a finite dimensional \mathbb{C}-vector space M and a \mathbb{C}-linear map \nabla : M \to M that satisfies, for all m \in M and f \in \mathscr{O}_X(*0), the Leibniz rule:

\nabla(fm) = f\nabla(m) + \frac{df}{dz} m

(where z is a local coordinate on X with z(0) = 0).

The Problem

Let f \in \mathscr{O}_X(*0)/\mathscr{O}_X be a meromorphic function on X, and consider the meromorphic connection

(M,\nabla) = (\mathscr{O}_X(*0),d+df)

Then, the flat sections of M correspond to solutions of the differential equation \nabla u  =0, i.e., du = -udf. These flat sections form a rank one \mathbb{C}-local system \ker \nabla with stalk \mathbb{C}\cdot e^{-f} at every point of X^*:= X-\{0\}. The monodromy action is trivial, we find \textnormal{DR}_{X^*}(M) \cong \ker \nabla \cong \mathbb{C}_{X^*}, where \textnormal{DR}_{X^*} is the de Rham functor.

Now, here is the problem: f \in \mathscr{O}_X(*0)/\mathscr{O}_X 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 \mathscr{D}_X-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.

Some Microlocal Computations

A couple of posts ago, I mentioned the “microlocal” category D^b(X;\Omega), and I talked about a couple of neat things you could do with it.  Looking back, I’m finding myself unsatisfied with the level of detail in my examples (as well as the number of examples!), so I figured I’ll make a post solely dedicated to examples of D^b(X;\Omega)

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), \Omega a subset of the cotangent bundle T^*X.  Then, the category D^b(X;\Omega) is defined to be the localization of the bounded, derived category D^b(X) by the null system

Ob(D_{T^*X \backslash \Omega}^b(X)) = \{ F \in D^b(X) | SS(F) \cap \Omega = \emptyset \}.  

We then say that a morphism u: F \to G in D^b(X) is an isomorphism on  \Omega (or, an isomorphism in D^b(X;\Omega) ) if there exists a distinguished triangle F \overset{u}{\to} G \to H \overset{+1}{\to} in D^b(X) with SS(H) \cap \Omega = \emptyset.  If \Omega = \{p\} for some point p \in T^*X, we write D^b(X;p) instead of D^b(X;\{p\}) (because we’re lazy).  

I won’t be using this property, but it’s pretty neat, and is solely a consequence of D^b(X;\Omega) being a localization:

Hom_{D^b(X;\Omega)}(F,G) \cong \varinjlim_{F' \to F} Hom_{D^b(X)}(F',G) \cong \varinjlim_{G \to G'} Hom_{D^b(X)}(F,G')

where the limit is indexed over those morphisms F' \to F with target F (resp., G \to G' with source G) in D^b(X) that are isomorphisms on \Omega.   

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

0 \to A_{X \backslash Z} \to A_X \to A_Z \to 0

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

A_{X \backslash Z} \to A_X \to A_Z \overset{+1}{\to}

in the (bounded) derived category D^b(X) = D^b(A_X).  

Second, for E a finite dimensional real vector space, and \gamma \subseteq E a closed, convex cone, we set \gamma^\circ = \{ \xi \in E^* | \langle v, \xi \rangle \geq 0 \text{ for any} v \in \gamma\}.  In this case, we have the equality SS(A_\gamma) \cap \pi^{-1}(0) = \gamma^\circ (where \pi : T^*E \to E is the canonical projection). Generalizing this, if M is a closed submanifold of the manifold X, then SS(A_M) = T_M^*X is the conormal bundle to M 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 X= \mathbb{R}.  While we’re at it, let’s revisit the example I mentioned in the previous post, in which we cared about the closed subset Z = \{x \geq 0\} in \mathbb{R}, and the subset \Omega = \{(x;\xi) \in T^*\mathbb{R} | \xi > 0\} of the cotangent bundle of \mathbb{R}.  The relevant distinguished triangles to keep in our heads are then:

  1. A_{\{x < 0\}} \to A_{\mathbb{R}} \to A_{\{x \geq 0\}} \overset{+1}{\to}
  2. A_{\{x > 0\}} \to A_{\{x \geq 0\}} \to A_{\{0\}} \overset{+1}{\to}.

Using these, I want to show the isomorphisms A_{\{x \geq 0\}} \cong A_{\{x < 0\}}[1] \cong A_{\{0\}} in D^b(\mathbb{R};\Omega).  

We should start by calculating all those microsupports! For simplicity, we use the isomorphism T^*\mathbb{R} \cong \mathbb{R} \times \mathbb{R}^* \cong \mathbb{R}^2.  

  • SS(A_\mathbb{R}) = \mathbb{R} \times \{0\} (the zero section of the cotangent bundle).  If you think of \mathbb{R}* as a (closed, convex) cone in itself, its polar set is just the zero vector.  That’s the basic idea. 
  • SS(A_{\{0\}}) = T_0^*\mathbb{R} (just think of \{0\} as a closed submanifold of \mathbb{R}, and use fact 2 above).  
  • SS(A_{\{x \geq 0\}}) = \{(x,y) \in \mathbb{R}^2 | y x = 0; y , x \geq \} (you’ll have to think about this one, sorry.  It’s not too bad…). 
  • SS(A_{\{x > 0\}}) = \{(x,y) \in \mathbb{R}^2 | yx = 0; y \leq 0, x \geq 0\} (remember, SS(F) is always a closed subset!).  
  • SS(A_{\{x < 0\}}) = \{(x,y) \in \mathbb{R}^2 | yx = 0; y \geq 0, x \leq 0\} (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

A_{\{x \geq 0\}} \to A_{\{x < 0\}}[1] \to A_\mathbb{R}[1] \overset{+1}{\to})

we see that A_{\{x \geq 0\}} \to A_{\{x < 0\}}[1] is an isomorphism on \Omega, since SS(A_\mathbb{R}) \cap \Omega = \emptyset! One down.  

Similarly, if we rotate triangle 2, we see that A_{\{x \geq 0\}} \to A_{\{0\}} is an isomorphism on \Omega, as SS(A_{\{x > 0\}}) \cap \Omega = \emptyset.  Easy!

Example 2.

Okay, now we’ll consider the subset \Omega = \{(0,y) \in \mathbb{R}^2 | y > 0\} and D^b(\mathbb{R}; \Omega)

Right off the bat, we note that if F \in D^b(\mathbb{R}) is locally constant in an open neighborhood U of 0, SS(F|_U) \subseteq U \times \{0\} (so SS(F) \cap \Omega = \emptyset), implying F \cong 0 in D^b(\mathbb{R};\Omega).  I’ll leave you guys (okay, I know that nobody actually reads this) with a cliffhanger: 

For all a,b \in \mathbb{R} with -\infty \leq b < 0 \leq a \leq +\infty, we have

A_{[0,a)} \cong A_{[0,a]} \cong A_{[b,0)}[1] \cong A_{(b,0)}[1] in D^b(\mathbb{R};\Omega).  

Why?

Symplectic Basics III: The Cotangent Bundle

Now that we’ve gotten a little comfortable with the idea of a symplectic vector space, it shouldn’t take a huge leap of the imagination to say there’s a similar notion for (smooth) manifolds, too: they’re called symplectic manifolds (surprise).  A (real, smooth) symplectic manifold of dimension 2n consists of a pair (M,\omega), where M is the manifold, and \omega is a closed, non-degenerate 2-form on M (i.e., for all p \in M, \omega_p is an alternating bilinear form on the tangent space T_p M, d\omega = 0, and \frac{1}{n!} \omega^n is a volume form on M (equivalently, for all p \in M, \omega_p is a non-degenerate bilinear form on T_pM) ).  In analogy with the case with symplectic vector spaces, \omega picks out certain types of submanifolds of M. That is, for a submanifold N \subseteq (M,\omega), we say N is isotropic (resp., Lagrangian; resp., involutive) if, for all p \in N, T_p N \subseteq (T_pM, \omega_p) is an isotropic (resp., Lagrangian; resp., involutive) subspace.  So, at face value, there doesn’t seem to be much of a change from the ordinary case of symplectic vector spaces, at least regarding these special types of submanifolds.  Wrong, I was.  I hope to talk about some of these things today.

 

Example 1. So, for baby’s first example of a symplectic manifold, we look at M = \mathbb{R}^{2n}, equipped with the 2-form \sigma_n \in \Omega^2(\mathbb{R}^{2n}) (fyi, we write \Omega^k(M) to denote the space of (C^\infty) k-forms on M) which has constant value (\sigma_n)_p = \sigma_n on T_p \mathbb{R}^{2n} \cong \mathbb{R}^{2n}, for all p \in \mathbb{R}^{2n}. 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 \mathbb{R}^{2n}, not just a single bilinear form.

Let (x;y) = (x_1,\cdots,x_n;y_1,\cdots,y_n) \in \mathbb{R}^{2n} be a (global) choice of linear coordinates on \mathbb{R}^{2n}, and let p = (x_0;y_0) \in \mathbb{R}^{2n}.  Then,

(\sigma_n)_p(v_1;v_2) := d_py \wedge d_px (v_1;v_2) = \sum_{i=1}^n (d_p y_i \wedge d_p x_i)(v_1;v_2)

for v_1,v_2 \in T_p \mathbb{R}^{2n} \cong \mathbb{R}^{2n}.

Example 2. As far as I’m (currently) concerned, these are the most important examples of symplectic manifolds: given any (real, smooth) manifold M of dimension n, the cotangent bundle T^*M 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 \omega \in \Omega^2(T^*M).  There are essentially two ways to do this: first, a coordinate-free definition of \omega; after that, we’ll see what \omega 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…), \alpha \in \Omega^1(T^*M).

Let \pi : T^*M \to M be the projection map, (x;\xi) \in T^*M (i.e., x \in M is a point, and \xi \in T_x^* M is a covector at x).  Then, using the pullback  \pi^* : T^*M \to T^*(T^*M), we define the value of \alpha at (x;\xi) to be

\alpha_{(x;\xi)} := \pi_{(x;\xi)}^* (\xi) .

This seems stranger/more abstruse than it actually is.  From the definition of \pi_{(x;\xi)}^*, \pi_{(x;\xi)}^*(\xi) = \xi \circ d_{(x;\xi)} \pi (this checks out: \xi is, by definition, a linear map T_xM \overset{\xi}{\to} \mathbb{R}, and d_{(x;\xi)} \pi : T_{(x;\xi)}(T^*M) \to T_x M, so the composition at least makes sense).  Finally, we define the “canonical” symplectic form \omega on T^*M to be \omega := d\alpha (NB: some authors take \omega = -d\alpha.  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 (x) = (x_1,\cdots,x_n) \in M near x_0, with associated linear coordinates (x; \sum_{i=1}^n \xi_i dx_i) \in T^*M.  What does \alpha look like in these coordinates?  Well, we know \alpha_{(x;\xi)} := \pi_{(x;\xi)}^*( \xi), so we just plug the coordinate expressions in and wind up with:

\alpha= \pi^* (\sum_{i=1}^n \xi_i d x_i ) = \sum_{i=1}^n \xi_i d ( x_i \circ \pi) = \sum_{i=1}^n \xi_i dx_i

(since \pi(x;\xi) = x).  Hence, the canonical 1-form looks like \alpha = \sum_{i=1}^n \xi_i dx_i in the local coordinates (x;\xi); consequently, the symplectic form \omega has the coordinate expression

\omega = d\alpha = \sum_{i=1}^n d(\xi_i dx_i) = \sum_{i=1}^n d\xi_i \wedge dx_i

near x_0.  In particular, since \omega is exact, it is automatically a closed 2-form, and in these coordinates, it is easy to show that \omega = d\alpha 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 \alpha important/useful? Well, for one, it satisfies the following universal property:

proposition 1: The canonical 1-form \alphais uniquely characterized by the property that, for every 1-form \eta : M \to T^*M (i.e., \eta(x) = (x;\eta_x) \in T^*M is the graph of \eta), one has \eta^* \alpha = \eta.

proof.

First, we note that \eta^* : T^*(T^*M) \to T^*M, and

\eta^* \alpha = \sum_{i=1}^n \eta^*(\xi_i dx_i) = \sum_{i=1}^n (\xi \circ \eta) d (x_i \circ \eta)

= \sum_{i=1}^n \eta_i dx_i = \eta

is just the expression of \eta in the local coordinates (x; \sum_{i=1}^n \xi_i dx_i) on T^*M.  I’ll leave the proof of uniqueness of \alpha to the reader 🙂

end proof.

Secondly, the canonical 1-form gives a really easy way to produce Lagrangian submanifolds of (T^*M,\omega).  Let \eta be a smooth 1-form on M, and denote by \Lambda_\eta := \{ (x;\eta_x) \in T^*M | x \in M\} the graph of \eta.  Then,

proposition 2: \Lambda_\eta is a Lagrangian submanifold of (T^*M,\omega) if and only if \eta is closed.

proof:

Clearly, \Lambda_\eta is a smooth submanifold of T^*M of dimension n (it’s diffeomorphic to M itself).  Then,

\omega|_{\Lambda_\eta} = \eta^* \omega = \eta^* d\alpha = d(\eta^* \alpha) = d\eta,

So \omega vanishes on \Lambda_\eta if and only if d\eta = 0, i.e., if \eta is closed.

end proof.

(Aside: for a symplectic vector space (V,\sigma) (of dimension 2n), a basis \{e_1,\cdots,e_n;f_1,\cdots, f_n\} \subset V is called a symplectic basis, provided \sigma = \sum_{i=1}^n f_i^* \wedge e_i^*.  In such case, we obtain the relations

  • \sigma(e_i,e_j) = \sigma(f_i,f_j) = 0,
  • \sigma(e_i,f_j) = -\sigma(f_j,e_i) = -\delta_{ij}.

(1 \leq i,j \leq n).  Additionally, in a symplectic basis, the Hamiltonian isomorphism H : V^* \to V is given by

  • H(e_j^* ) = -f_j.
  • H(f_j^*) = e_j.

for 1\leq i \leq n (in the general case, H : V^* \to V is defined by the formula \langle \theta , v \rangle = \sigma(v,H(\theta)), where \langle \bullet, \bullet \rangle : V^* \times V \to \mathbb{R} is the canonical pairing, and \theta \in V^*, v \in V).  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 (T^*M,\omega) for some smooth manifold M.  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., \sigma = \sum_{i=1}^n f_i^* \wedge e_i^*).  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 (M,\omega), for all p \in M, there exists an open neighborhood U of p with smooth coordinate chart (x_1,\cdots,x_n;y_1,\cdots,y_n) : U \overset{\thicksim}{\to} \mathbb{R}^{2n} (actually, this is a symplectomorphism as well!) such that \omega_q = \sum_{j=1}^n d_q y_j \wedge d_q x_j for all q \in U (one sometimes calls (x;y) a system of Darboux coordinates, instead of/in addition to their symplectic properties).  Luckily for us and our cotangent bundles, on (T^*M,\omega), any local system of cotangent coordinates (x; \xi) on T^*M serves as a system of Darboux coordinates, by our construction of \omega.  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 (T^*M, \omega), the Hamiltonian isomorphism is the (fiber-wise) linear isomorphism H : T^*(T^*M) \overset{\thicksim}{\to} T(T^*M), 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 f on an open subset U \subseteq T^*M, we define the Hamiltonian vector field of f, denoted by H_f, to be the image of the differential df under the Hamiltonian isomorphism H : T^*(T^*M) \overset{\thicksim}{\to} T(T^*M).

If we’ve chosen some local coordinates (x;\xi) on T^*M, our above discussion implies the vector field H_f is given by

H_f = \sum_{j=1}^n \frac{\partial f}{\partial \xi_j} \frac{\partial}{\partial x_j} - \frac{\partial f}{\partial x_j} \frac{\partial }{\partial \xi_j}

Indeed, \omega = \sum_{j=1}^n d \xi_j \wedge dx_j, so H(dx_j ) = -\frac{\partial }{\partial \xi_j}, and H(d\xi_j) = \frac{\partial}{\partial x_j}  (for 1 \leq j \leq n).  Therefore,

H_f := H(df) = \sum_{j=1}^n (\frac{\partial f}{\partial x_j}H(dx_j) + \frac{\partial f}{\partial \xi_j} H(d \xi_j) )

 from which result is immediate.

I really want to tie all this material together to talk about isotropic,Lagrangian, and involutive submanifolds of (T^*M,\omega), 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 T^*M was for their relationship with the microsupport SS(F) of sheaves in D^b(M).  More precisely, for all F \in D^b(M), SS(F) \subseteq T^*M is an involutive subset; if F is also \mathbb{R}-constructible, then SS(F) is a Lagrangian subset of T^*M.  A little alarm should be going off in your head right now; there’s no reason at all for SS(F) to be a smooth submanifold of T^*M (take, for example, F to be the constant sheaf supported on a singular subset of M), 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:

Symplectic Basics II : The Lagrangian Grassmanian

Before we proceed with Lagrangian stuff, I should really talk about a fundamental property of (finite dimensional, real!) symplectic vector spaces:

proposition 1: They’re always even dimensional!

proof:

Let (E,\sigma) be a symplectic vector space of dimension m.  I claim then that is even.  Choose some basis of E, so that we get a matrix representative A of \sigma, i.e., \sigma(x,y) = x^T A y for all x, y \in E.  Since \sigma is alternating, A is skew-symmetric, giving the relation A^T = -A.  Hence,

det(A) = det(A^T) = det(-A) = (-1)^m det(A).

So, if m is odd, we must have det(A) = 0.  But, since \sigma is non-degenerate, det(A) \neq 0, which forces m to be even in the above equation.  Done!

end proof.

A quick application of the “rank-nullity” theorem for a subspace W \subseteq E applied to \sigma implies \dim W + \dim W^\perp = m = 2n; hence, all Lagrangian subspaces of E are of dimension n = \frac{1}{2} \dim E.  Similarly, if \dim W = n and W \subseteq W^\perp, then W is Lagrangian.

Of course, from plain old linear algebra, we know that, given any Lagrangian subspace \lambda_0 of E, there exists an n-dim subspace \lambda_1 \subseteq E such that E = \lambda_0 \oplus \lambda_1.  The question now is:

proposition 2: Can the complementary subspace \lambda_1 be chosen such that \lambda_1 is also Lagrangian?

proof:

Given \lambda_0 Lagrangian, choose an isotropic space \rho \subseteq E such that \lambda_0 \cap \rho = \{0\} (\lambda_0 is a proper subspace, so we can always at least choose a line that satisfies this property, and lines are always isotropic). If \rho^\perp \neq \rho,  \rho^\perp is not a subset of \lambda_0 + \rho.  Indeed, if it were, then we’d have (by “symplectic duality” given by regarding \rho^\perp as the annihilator of \rho w.r.t. \sigma) \lambda_0 \cap \rho^\perp \subseteq \rho.  Since \rho^\perp \cap \lambda_0 \subseteq \rho \cap \lambda_0 = \{0\} (so \rho^\perp \cap \lambda_0 = \{0\}), we contradict the assumption that \rho is isotropic: i.e., we assumed \dim \rho^\perp > 0.  Thus, \rho + \lambda_0 does not contain \rho^\perp.  Choose then some x \in \rho^\perp \backslash (\lambda_0 + \rho).  Then, \rho + \langle x \rangle is an isotropic subspace (as \sigma is bilinear and alternating) satisfying (\rho + \langle x \rangle) \cap \lambda_0 = \{0\}.  Arguing by induction on \dim \rho, we get our Lagrangian space \lambda_1.  Perfect.

end proof.

 Why is this relevant?

Recall the space of n-dim subspaces of E, called the Grassmannian and denoted G(E,n).  It is a smooth manifold of dimension n^2.  My main goal is to introduce the Lagrangian Grassmannian \Lambda (E) of Lagrangian planes in E, a closed (smooth) submanifold of G(E,n) of dimension \frac{n(n+1)}{2}. Also, it’s compact! (implicitly, this is proposition 3)

proof:

Let \lambda \in \Lambda(E).  I claim that there is a bijection between the spaces \Lambda_\lambda(E) := \{ \mu \in \Lambda(E) | \lambda \cap \mu = \{0\} \} (which is open in the subspace topology of \Lambda(E) in G(E,n)) and the space of (real) quadratic forms on \mathbb{R}^n (a real vector space of dimension \frac{n(n+1)}{2}).

Set W = \lambda \oplus \lambda^* (where \lambda^* := \text{Hom}_\mathbb{R}(\lambda,\mathbb{R}) is the algebraic dual of \lambda), equipped with the standard symplectic form \omega((x,\xi);(x',\xi')) := \langle x', \xi \rangle - \langle x,\xi' \rangle.  With respect to \omega, the subspace \lambda \cong \lambda \oplus \{0 \} is Lagrangian (follows immediately from the definition of \omega and the assumption \lambda \in \Lambda(E)).  By proposition 4 from the last post, we know there exists a symplectic map \psi : (E,\sigma) \to (W,\omega) with \psi(\lambda) = \lambda \oplus \{0\}.  So, we might as well work inside W.  Let \mu \in \Lambda_\lambda(W).  Then, (it’s an easy exercise to show) that \mu is the “graph” of a (unique!) linear map A : \lambda^* \to \lambda, that is, we can write \mu = \{ (Ay^*,y^*) | y^* \in \lambda^*\} (this is actually very similar to how you construct the smooth atlas on the ordinary Grassmannian manifold; check it out!).

Since \mu \in \Lambda_\lambda(W) \subseteq \Lambda(W), \omega|_{\mu} = 0.  Using the fact that \mu is the graph of the matrix A (suppose we’ve chosen a basis), this tells us

\langle Ay_2^*,y_1^* \rangle = \langle y_2^*, Ay_1^* \rangle

i.e., A is a symmetric matrix.  With a basis fixed for E (giving a basis of W by taking the dual basis for \lambda^*), we know that the collection of n \times n real symmetric matrices is linearly isomorphic to the space of real quadratic forms on \mathbb{R}^n (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 G(E,n) is compact and \Lambda(E) is closed in G(E,n).  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:

Symplectic Basics I: Symplectic Linear Algebra

Last post I mentioned some types of subsets of the cotangent bundle, associated to the bundle’s natural symplectic structure (i.e., the isotropic, involutive, and Lagrangian subsets). What was I talking about? Back to basics! Today, I want to talk about some “symplectic linear algebra.”

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

Example 1. Let V be a vector space, E := V \oplus V^*, and let \langle \cdot, \cdot \rangle : E \to \mathbb{R} be the canonical pairing of V and V^*. Define a bilinear form \sigma on E by

\sigma((x_1,\xi_1);(x_2,\xi_2)) := \langle x_2, \xi_1 \rangle - \langle x_1 , \xi_2 \rangle

for (x_i,\xi_i) \in E.  Naturally, I claim that \sigma is a symplectic form on E. By construction, \sigma is alternating and bilinear, so we only need to check non-degeneracy.  Let (x_1,\xi_1) \in E be such that, for all (x_2,\xi_2) \in E, \sigma((x_1,\xi_1);(x_2,\xi_2)) = 0.  That is, for all (x_2,\xi_2) \in E,

\langle x_2, \xi_1 \rangle = \langle x_1, \xi_2 \rangle.

By non-degeneracy of \langle \cdot, \cdot \rangle, setting x_2 = 0 yields \xi_1 = 0, and setting \xi_2 = 0 yields x_1 = 0 (remember, that equality was assumed to hold for all elements of E!).  Hence, (x_1,\xi_1) = (0,0), implying \sigma is non-degenerate.  Note that, for V = \mathbb{R}, the form \sigma looks a lot like the determinant map! (\sigma((x_1,y_1);(x_2,y_2)) = x_2 y_1 - x_1 y_2).

Now, for a subspace W of a symplectic vector space (V,\sigma), we associate its symplectic complement, or symplectic orthogonal.

W^\perp := \{ x \in V | \sigma(x,y) = 0 \text{ for all }y \in W\}.

This is where we get the notions of isotropic, involutive, and Lagrangian subspaces: a subspace W of (V,\sigma) is

  • isotropic if W \subseteq W^\perp,
  • involutive if W^\perp \subseteq W, and
  • Lagrangian if W = W^\perp.

Let’s end with some easy examples:

Example 2.  A line \ell is always an isotropic subspace.

Let x \in \ell be a non-zero vector, so that every element of \ell is of the form tx for some t \in \mathbb{R}.  Then, the fact that \sigma is bilinear and alternating implies that \ell \subseteq \ell^\perp.

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

Let x \in H^\perp be non-zero.  If x \notin H, then since H is a hyperplane, we must have \langle x \rangle + H = V (by \langle x \rangle, I mean the line spanned by the non-zero vector x), so that every element y of V is of the form y = tx + z, for some t \in \mathbb{R} and z \in H.  But, since x \in H^\perp, we must have

\sigma(x,y) = \sigma(x,tx + z) = t\sigma(x,x) + \sigma(x,z) = 0

by bilinearity.  Since y \in V was arbitrary, the non-degeneracy of \sigma yields x = 0, a contradiction.  Thus, x \in H, so H 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 \varphi : (V_1,\sigma_1) \to (V_2,\sigma_2) is called symplectic provided \sigma_1 = \varphi^* \sigma_2.  That is, for all x,y \in V_1, we have (by definition of the pullback)

\sigma_1(x,y) = \sigma_2(\varphi(x),\varphi(y)).

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

Just like every vector space is modeled on \mathbb{R}^n for some n (upon choosing a basis), all symplectic vector spaces of dimension 2n are symplectomorphic to (\mathbb{R}^{2n},\sigma_n), where

\sigma_n((x,y);(x',y')) := \sum_{j=1}^n (x_j'y_j - x_j y_j')

(x = (x_1,\cdots,x_n), y= (y_1,\cdots,y_n)) for each n \geq 1 (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 n \geq 1, the subspace Z_n := \mathbb{R}^n \oplus \{0 \} is a Lagrangian subspace of (\mathbb{R}^{2n},\sigma_n) (i.e., Z_n = \{(x_1,\cdots,x_n,y_1,\cdots,y_n) | y_i = 0, 1 \leq i \leq n \}).  As it turns out, Z_n is the prototype for all Lagrangian subspaces:

Proposition 4: Given any symplectic vector space (V,\omega) of dimension 2n, and Lagrangian \lambda \subseteq (V,\omega), there exists a symplectic map \psi : (\mathbb{R}^{2n},\sigma_n) \to (V,\omega) sending Z_n to \lambda.

proof: Assume that we’ve proved the result for all dimensions \leq n-1 (for n=1, the Lagrangian subspaces are all just lines through the origin in \mathbb{R}^2, and the desired symplectic map is just a rotation about the origin).  We want to then show the result for dimension n.  Okay.  Let \lambda \subseteq (V,\omega) be a Lagrangian subspace, \dim V = 2n.  Pick some e_1 \in \lambda non-zero.  Since \omega is non-degenerate, there exists some f_1 \in V such that \omega(e_1,f_1)=1.  As \lambda is Lagrangian, this gives f_1 \notin \lambda.  Set

\overset{\thicksim}{V} := \{x \in V | \omega(x,e_1)=\omega(x,f_1) = 0\};

with the restriction \overset{\thicksim}{\omega} := \omega|_{\overset{\thicksim}{V}}, (\overset{\thicksim}{V},\overset{\thicksim}{\omega}) is a symplectic space.  Of course, from \omega, the only thing to check is that \overset{\thicksim}{\omega} is non-degenerate (if x \in \overset{\thicksim}{V} \cap (\overset{\thicksim}{V})^\perp is non-zero, there exists some y \in V such that \omega(x,y) \neq 0.  By the definition of \overset{\thicksim}{V}, we must have y \notin \overset{\thicksim}{V}.  It follows that x = 0).

Now, set \overset{\thicksim}{\lambda} := \lambda \cap \overset{\thicksim}{V}.  We need to show \overset{\thicksim}{\lambda} is Lagrangian in \overset{\thicksim}{V}, and \lambda = \overset{\thicksim}{\lambda} + \langle e_1 \rangle.    Since \overset{\thicksim}{\omega}|_{\overset{\thicksim}{\lambda}} = \omega|_{\overset{\thicksim}{\lambda}}, and \lambda = \lambda^\perp, it follows that \overset{\thicksim}{\lambda} is an isotropic subspace.  Is it maximally isotropic in \overset{\thicksim}{V} (i.e., Lagrangian?).  If not, there would exist an isotropic subspace \mu with \overset{\thicksim}{\lambda} \subset \mu \subseteq \overset{\thicksim}{V}.  But then, \mu + \langle e_1 \rangle would be isotropic in (V,\omega), and n = \dim \lambda < \dim (\mu + \langle e_1 \rangle.  But this is a contradiction, since an isotropic subspace of V must have dimension \leq n! (exclamation, not factorial. whoops).  Thus, \overset{\thicksim}{\lambda} is Lagrangian.  For the second part of the claim, we note that \lambda \subseteq \overset{\thicksim}{\lambda} + \langle e_1 \rangle, and the above shows \dim \lambda = \dim (\overset{\thicksim}{\lambda} + \langle e_1 \rangle ) = n, so \lambda = \overset{\thicksim}{\lambda} + \langle e_1 \rangle.

Okay, here’s where we invoke the inductive hypothesis: there exists a symplectic map \varphi_{n-1} : (\mathbb{R}^{2n-2},\sigma_{n-1}) \to (\overset{\thicksim}{V},\overset{\thicksim}{\omega}) sending Z_{n-1} to \overset{\thicksim}{\lambda}.  Then, the map

\varphi_n : (\mathbb{R}^2,\sigma_1) \oplus (\mathbb{R}^{2n-2},\sigma_{n-1}) \to (V,\omega) via

(x,y;z) \mapsto x e_1 + y f_1 + \varphi_{n-1}(z)

is symplectic, and sends Z_n to \lambda.  Oh, by the way: we define the form \sigma_1 \oplus \sigma_{n-1}((x_1,y_1;z_1);(x_2,y_2;z_2)) := \sigma_1((x_1,y_1);(x_2,y_2)) + \sigma_{n-1}(z_1,z_2).  By assumption, we know \sigma_{n-1} = \varphi_{n-1}^* \overset{\thicksim}{\omega}. Since I’m lazy, and this calculation is pretty messy, let’s write X_1 = x_1 e_1 + y_1 f_1, X_2 = x_2 e_1 + y_2 f_1.  Then, by algebra:

\omega(X_1+ \varphi_{n-1}(z_1),X_2 + \varphi_{n-1}(z_2)) = \omega(X_1,X_2) + \omega(X_1,\varphi_{n-1}(z_2) + \omega(\varphi_{n-1}(z_1),X_2) + \omega(\varphi_{n-1}(z_1),\varphi_{n-1}(z_2))

= \omega(X_1,X_2) + \omega(\varphi_{n-1}(z_1),\varphi_{n-1}(z_2) = \sigma_1((x_1,y_1);(x_2,y_2)) + \sigma_{n-1}(z_1,z_2)

as \varphi_{n-1}(z_i) \in \overset{\thicksim}{V}, and hence \omega(\varphi_{n-1}(z_i),e_1) \omega(\varphi_{n-1}(z_i),f_1) = 0 ((i= 1,2) and by expanding the term \omega(X_1,X_2) in terms of e_1,f_1).  So, \varphi_n is symplectic.  \varphi_n(Z_n) = \lambda, because \varphi_n(x_1,0;z) = x_1 e_1 + \varphi_{n-1}(z) \in \lambda = \overset{\thicksim}{\lambda} + \langle e_1 \rangle, and \varphi_{n-1}(Z_{n-1}) = \overset{\thicksim}{\lambda}.  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).

Working Microlocally

So, by now, we have some ideas about what these object SS(\mathcal{F}^\bullet) \subset T^*X are, as (co)directions of non-propagation of sections of the various cohomology sheaves of the complex \mathcal{F}^\bullet.  But that’s really only scratching the surface, as all we’ve done so far is fiddle around with the definition of microsupport.  For a general complex of sheaves \mathcal{F}^\bullet \in D^b(X) (X a real, smooth manifold), there isn’t much hope for mere mortals like ourselves in calculating SS(\mathcal{F}^\bullet).  But, things get a little brighter when we restrict to those objects whose cohomology sheaves are all constructible; for \mathcal{F}^\bullet \in D_c^b(X), there exists a stratification \mathfrak{S} of X by smooth submanifolds such that

SS(\mathcal{F}^\bullet) \subseteq \bigcup_{S \in \mathfrak{S}} \overline{T_S^* X}.

Specifically, in addition to being \mathbb{R}^+-conic and closed, SS(\mathcal{F}^\bullet) is also a real subanalytic, isotropic subset of T^*X (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 \mathbb{Z} as our base ring for D^b(X), and let N \subseteq X be a closed submanifold.  To N, we can associate the sheaf \mathbb{Z}_N^\bullet \in D_c^b(X) (considered as a complex of sheaves concentrated in degree zero), whose stalks are (\mathbb{Z}_N^\bullet)_x \cong \mathbb{Z} if x \in N, and \cong 0 if x \notin N.  Then, it’s not too hard to show the equality SS(\mathbb{Z}_N^\bullet) = T_N^*X, the full conormal bundle to N in X.

Another good toy example is when X = \mathbb{R}, and we’re considering the constant sheaf on a closed interval [a,b] \subset \mathbb{R}, extended by zero to all of \mathbb{R}.  What would this sheaf’s microsupport be?

Since almost all the objects we’ll be interested in are in D_c^b(X), 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 D^b(X;\Omega), from Kashiwara and Schapira’s Sheaves on Manifolds ([K-S]), that are associated to subsets \Omega \subseteq T^*X.  Briefly, D^b(X;\Omega) is the localization of D^b(X) by the full subcategory of objects whose microsupport is disjoint from \Omega.  Then, we want to say that working “microlocally” with some complex of sheaves \mathcal{A}^\bullet on X means to consider \mathcal{A}^\bullet as an object of the category D^b(X;\Omega) for some subset \Omega.  This should seem pretty reasonable; If SS(\mathcal{A}^\bullet) \cap \Omega = \emptyset, then \mathcal{A}^\bullet has no interesting microlocal behavior on \Omega and can’t really interact in a meaningful way with other, non-zero, objects of D^b(X;\Omega).

So, now it’s time for the actual details.  Let V be a subset of T^*X.  We define D_V^b(X) to be the full subcategory of D^b(X) consisting of those objects \mathcal{B}^\bullet such that SS(\mathcal{B}^\bullet) \subseteq V.  Additionally, this is a triangulated category, with distinguished triangles (naturally) those of D^b(X) whose objects belong to D_V^b(X).  Then, Ob(D_V^b(X)) is a null-system in D^b(X), and we may define the relevant localization.  That is, we set D^b(X;\Omega) := D^b(X)/Ob(D_V^b(X)) (where \Omega := T^*X \backslash V).  In the case we’ll usually use, \Omega = \{\omega\} for some covector \omega \in T^*X, we’ll be lazy and write D^b(X;\omega) for the localized category.

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

Let’s play with some examples, and see how working in D^b(X;\Omega) differs from working in D^b(X).

Example:

Let X = \mathbb{R}.  To the closed subset \{x | x \geq 0\}, there is a distinguished triangle in D^b(\mathbb{R}):

\mathbb{Z}_{\{x | x < 0\}}^\bullet \to \mathbb{Z}_\mathbb{R}^\bullet \to \mathbb{Z}_{\{x | x \geq 0\}}^\bullet \overset{+1}{\to}

from this triangle, it follows that the complexes \cdots 0 \to \mathbb{Z}_{\{x | x < 0\}} \to \mathbb{Z}_\mathbb{R} \to 0 \cdots and \mathbb{Z}_{\{x | x \geq 0\}}^\bullet [-1] are isomorphic in D^b(\mathbb{R}) (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 D^b(\mathbb{R};\Omega), where \Omega = \{ (x,\xi) \in T^*\mathbb{R} | \xi > 0 \}.  After a quick calculation, we obtain:

  • SS(\mathbb{Z}_\mathbb{R}^\bullet) = \mathbb{R} \times \{0\},
  • SS(\mathbb{Z}_{\{x | x \geq 0\}}^\bullet) = \{(0,\xi) | \xi \geq 0\} \cup \{(x,0)| x > 0 \}, and
  • SS(\mathbb{Z}_{\{x | x < 0 \}}^\bullet) = \{(0,\lambda) | \lambda \leq 0 \} \cup \{(x,0) | x < 0\}.

whence it follows that \mathbb{Z}_{\{x | x \geq 0 \}}^\bullet \cong \mathbb{Z}_{\{x | x < 0\}}^\bullet [1] in D^b(\mathbb{R}; \Omega), as well as \mathbb{Z}_{\{0\}}^\bullet \cong \mathbb{Z}_{\{x | x \geq 0\}}^\bullet in D^b(\mathbb{R}; \Omega).

Why should you care? 

Okay, that’s cute.  But why are these categories useful?  Why should you bother to play with D^b(X;\Omega) at all?

Earlier in this post, I mentioned that, for \mathcal{F}^\bullet constructible, SS(\mathcal{F}^\bullet) is an \mathbb{R}^+-conic, subanalytic, isotropic subset of T^*X.  Similarly, using D^b(X;\Omega), I want to give a “microlocal” characterization of those \mathcal{F}^\bullet \in D^b(X) whose microsupport is contained in an involutive submanifold of T^*X.  Particularly, when the microsupport is contained in the conormal bundle T_Y^*X to some submanifold Y of X (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, D_c^b(X).   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 Y is a closed submanifold of X, and let f: Y \hookrightarrow X denote the associated smooth, closed embedding.  Let p \in T_Y^*X (non-zero, or else this is a stupid exercise), and let \mathcal{F}^\bullet \in D^b(X).  Let \pi : T^*X \to X denote the projection of the cotangent bundle.   Assume further that, in a neighborhood of p,  SS(\mathcal{F}^\bullet) \subseteq \pi^{-1}(Y).  Then, there exists a \mathcal{G}^\bullet \in D^b(Y) such that \mathcal{F}^\bullet \cong f_* \mathcal{G}^\bullet in D^b(X;p).

(Quick sketch: Basically, we want to successively, microlocally, cut up SS(\mathcal{F}^\bullet) by restricting to various subsets of X, and throw away those pieces whose microsupport doesn’t meet p.)

Detailed Sketch:

By induction on the codimension of Y in X, we may assume Y is a hypersurface, say given by Y = \{\varphi = \}, with p = (x_0, d_{x_0} \varphi).  Set \Omega^{\pm} = \{x \in X | \pm \varphi(x) > 0 \}, so we have the open embeddings j_{\pm} : \Omega^\pm \hookrightarrow X. By [6.3.1, K-S], there is an inclusion

SS(Rj_{-*}j_{-}^*\mathcal{F}^\bullet) \subseteq SS(j_{-}^*\mathcal{F}^\bullet ) \hat{+} N^*(\Omega^-),

where (in our case here) N^*(\Omega^-) \subseteq T^*X consists of the zero section of T^*X, together with those covectors (y,\lambda d_y \varphi) \in T_Y^*X where \lambda < 0.  Vaguely, the \hat{+} operation characterizes those covectors arising as limits of sequences of covectors in SS(j_{-}^*\mathcal{F}^\bullet) + N^*(\Omega^-) (see [6.2.3, K-S]).

Since p is not in the right hand side, we must have p \notin SS(Rj_{-*}j_{-}^*\mathcal{F}^\bullet).  Hence, from the adjunction distinguished triangle

R\Gamma_{\{\varphi \geq 0\}} \mathcal{F}^\bullet \to \mathcal{F}^\bullet \to Rj_{-*}j_{-}^*\mathcal{F}^\bullet \overset{+1}{\to}

in D^b(X), we see that R\Gamma_{\{\varphi \geq 0\}}\mathcal{F}^\bullet \cong \mathcal{F}^\bullet in D^b(X;p).  So, we might as well have assumed from the start that supp(\mathcal{F}^\bullet) \subseteq \{\varphi \geq 0\}.  Similarly, we find (by [6.3.1, K-S]) that p \notin SS(Rj_{+!}j_{+}^*\mathcal{F}^\bullet), and see that \mathcal{F}^\bullet \cong \mathcal{F}_Y^\bullet = f_*f^*\mathcal{F}^\bullet in D^b(X;p), using the distinguished triangle

Rj_{+!}j_+^*\mathcal{F}^\bullet \to \mathcal{F}^\bullet \to i_{-*}i_-^*\mathcal{F}^\bullet \overset{+1}{\to}

(where i_- : \{ \varphi \leq 0 \} \hookrightarrow X is the closed complement of \Omega^+ in X), and using the fact that supp(\mathcal{F}^\bullet \subseteq \{\varphi \geq 0\}.  All together, this gives us the desired isomorphism \mathcal{F}^\bullet \cong f_*f^*\mathcal{F}^\bullet in D^b(X;p).

Q.E.D.

Proposition 2:  [6.6.1, K-S]

Same as the set up of proposition 1, but now we assume that SS(\mathcal{F}^\bullet) \subseteq T_Y^*X in a neighborhood of p.  Then, there exists M^\bullet \in D^b(\mathbb{Z}) such that \mathcal{F}^\bullet \cong M_Y^\bullet in D^b(X;p).

Since T_Y^*X \subseteq \pi^{-1}(Y), the previous proposition applies, and we may assume \mathcal{F}^\bullet \cong f_* \mathcal{G}^\bullet in D^b(X;p) for some \mathcal{G}^\bullet \in D^b(Y).  In particular, this tells us SS(\mathcal{F}^\bullet) = SS(f_*\mathcal{G}^\bullet) in a neighborhood of p.  So, the assumption that SS(\mathcal{F}^\bullet) \subseteq T_Y^*X, together with [5.4.4, K-S] (this tells us that SS(f_*\mathcal{G}^\bullet) consists of covectors (y,\xi) \in T^*X such that y \in Y, and \xi \circ d_y f = \xi \in SS(\mathcal{G}^\bullet)) give us that SS(\mathcal{G}^\bullet) is contained in the zero section, T_Y^*Y in a neighborhood of p:  Since SS(\mathcal{G}^\bullet) \subseteq T^*Y, the condition SS(f_*\mathcal{G}^\bullet) \subseteq T_Y^*X implies \xi = 0) that SS(\mathcal{G}^\bullet) \subseteq T_Y^*Y \cong Y \times \{0\} in a neighborhood of \pi(p).

Next, let g: Y \to \{pt\} be the canonical map to a point.  Since SS(\mathcal{G}^\bullet) \subseteq Y \times \{0\}, there exists a M^\bullet \in D^b(\mathbb{Z}) such that \mathcal{G}^\bullet \cong g^* M^\bullet.  Hence, \mathcal{F}^\bullet \cong f_* g^* M^\bullet \cong M_Y^\bullet in D^b(X;p), and we’re done.

Q.E.D.

Now, what can we say our constructible case SS(\mathcal{F}^\bullet ) \subseteq \bigcup_S T_S^*X?  Keywords: non-degenerate covectors, and Morse modules.  But that’s for another time.

Propagation of Sections

I mentioned last post that one should think of the microsupport of a complex of sheaves in terms of some loose idea of “propagation.”  I want to talk about that a little more now.  The gist I want you to walk away with is “the microsupport characterizes (co)directions of non-propagation.”  Whatever that means.

Let’s start with a simple example.  Let V be a (finite dimensional) real vector space, and let \Omega_0 \subseteq \Omega_1 be two convex (non-empty) open subsets of V.

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Let \mathcal{L}^\bullet 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 \Omega_0 \hookrightarrow \Omega_1 induces an isomorphism in (hyper)cohomology: R\Gamma(\Omega_1;\mathcal{L}^\bullet) \overset{\thicksim}{\to} R\Gamma(\Omega_0; \mathcal{L}^\bullet).  Ok, so what?

So, instead of just say “they have isomorphic cohomology,” you should be thinking “every section of \mathcal{L}^\bullet over \Omega_0 can be extended to all of \Omega_1,” or “sections of \mathcal{L}^\bullet can propagate from \Omega_0 to \Omega_1, in all ‘directions.'”

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

Pick some (non-zero, duh) covector \eta \in T_p^*U, the cotangent space of U to p.  By shrinking U if necessary, we can then assume that \eta is the differential of some smooth function L: U \to \mathbb{R}, and by translation, we can further assume that L(p) = 0.  Although lots of functions might have differential equal to \eta at p, for simplicity, we can just assume that L is some globally defined \mathbb{R}-linear form.  Since, locally, we can safely just picture covectors as ordinary vectors based at p, I like to think of \eta as the “normal vector” to the hyperplane L^{-1}(0) passing through p.  Which makes sense, of course, since we can think of the differential d_p L = \eta here as “the gradient vector \nabla L(p)“, which is always normal to level sets of L. 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).

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The whole point of this is that we want to use this function L to investigate possible changes in the cohomology of \mathcal{F}^\bullet as we move away from p, in the codirection \eta = d_p L.  How would we go about doing that?

Local Morse data!

Think about it.  First, let’s work inside some small open ball W = B_\epsilon^\circ(p) about p.  If every section of \mathcal{F}^\bullet over L^{-1}(-\infty,0) \cap W can be extended passed the “boundary” L^{-1}(0) \cap W, at least a little bit (say to L^{-1}(-\infty,\delta) \cap W for some small \delta > 0), then the cohomology of \mathcal{F}^\bullet over W should be isomorphic to the cohomology over L^{-1}(-\infty,0) \cap W, and this isomorphism should be induced by restriction.

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So, in fancy language, we’d say the relative cohomology complex of (W,W \cap L^{-1}(-\infty,0))  is zero.  Of course, it might happen to be the case that some sections can only propagate across L^{-1}(0) on small neighborhoods of p, so in order to pick up all such sections, we look at the limit \varinjlim_{p \in W \text{open}} \mathbb{H}^k(W , W \cap L^{-1}(-\infty,0); \mathcal{F}^\bullet) for k \in \mathbb{Z}.  A little bit of abstract nonsense then reveals this object is none other than LMD(L,p;\mathcal{F}^\bullet) = R\Gamma_{\{L \geq 0\}}(\mathcal{F}^\bullet)_p, the local Morse data of L at p, with respect to \mathcal{F}^\bullet.

Recalling my last post, this sort of vanishing tells us that the pair (p,\eta) \in T^*U is not in the microsupport of \mathcal{F}^\bullet! Before, I denoted this by \mu supp(\mathcal{F}^\bullet), but from now on, I think I’ll stick with the more popular notation: SS(\mathcal{F}^\bullet) (“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 “\gamma-topology” associated to a closed, convex cone \gamma 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 🙂

Microsupport and Propagation

So, last time, I briefly mentioned a sheaf-theoretic “Local Morse Datum” for a smooth (Morse) function f : M \to \mathbb{R} at a (non-degenerate) critical point p \in M

LMD(f,p) := R\Gamma_{\{f \geq 0\}}(\mathbb{Z}_M^\bullet)_p

which gives the integral cohomology of the “local Morse datum” of f at p, 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 f at p, with respect to the complex of sheaves \mathcal{A}^\bullet \in D^b(M), denoted

LMD(f,p; \mathcal{A}^\bullet ) := R \Gamma_{\{ f \geq 0 \}}(\mathcal{A}^\bullet )_p

 

Basically, you have the complex of sheaves, \mathcal{A}^\bullet, on M, and you consider sections of this “sheaf” whose support is contained in the subset \{f \geq 0\} := f^{-1}[0,\infty), and take the stalk cohomology at the point p \in f^{-1}(0).  Sort of like: the sections of the sheaf that propagate in the “positive direction” (where “positive” is taken to be with respect to f (really, the covector d_pf, 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 p, at least in the “positive” direction.  That’s supposed to be what I mean by “propagate” here.  

 

Why is this idea useful?  

 

Let X \subseteq M 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”), \mathfrak{S}.  To each stratum S \in \mathfrak{S}, we associate the following subset of the cotangent bundle of M, called the conormal space to S in M:

T_S^* M := \{ (p,\eta) \in T^*M | p \in S, \eta ( T_p S) = 0 \}

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

Let \overset{\thicksim}{f} : M \to \mathbb{R} be a smooth function, f := \overset{\thicksim}{f}|_X its restriction, \Sigma_\mathfrak{S} f the associated stratified critical locus of f.  It is any easy exercise to show that, when \mathfrak{S} is a Whitney (a) stratification, \Sigma_\mathfrak{S} f is a closed subset of X.  Similarly, the Whitney (a) condition is equivalent to requiring the equality:

\bigcup_{S \in \mathfrak{S}} T_S^*M = \bigcup_{S \in \mathfrak{S}} \overline{T_S* M}

Hence, when I say p \in \Sigma_\mathfrak{S} f, there is a unique stratum, say S, for which p \in \Sigma (f|_S) (since the strata are disjoint).  And, if you think about it, if p \in \Sigma (f|_S), we must have 

(p,d_p f) \in T_S^*M

as d_p (f|_S) = d_p f(T_p S) = 0.  Remember this.  

 

Local Stratified Acyclicity (LSA)

Say we have our Whitney stratification, \mathfrak{S}, of X \subseteq M.  Then, \mathfrak{S} satisfies something called LSA

for all \mathcal{F}^\bullet \in D_\mathfrak{S}^b(X) (this means the cohomology sheaves of \mathcal{F}^\bullet are all locally constant with respect to the strata of \mathfrak{S}), for all (germs of) stratified submersions f: (M,p) \to (\mathbb{R},0)

R \Gamma_{\{ f|_X \geq 0 \} }(\mathcal{F}^\bullet )_p = 0.  

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, \mathcal{F}^\bullet \in D^b(M), which encodes the “directions” at given points of M where sections of \mathcal{F}^\bullet “do not propagate.”  That is, the directions in which we should expect to detect changes in the cohomology (sheaves) of \mathcal{F}^\bullet.  Precisely, the Microsupport of \mathcal{F}^\bullet is the subset \mu supp(\mathcal{F}^\bullet ) \subseteq T^*M, characterized (in the negative…) by: 

(p,\eta) \notin T^*M if and only if there exists an open subset U of (p,\eta) in T^*M such that, for all smooth function germs f : (M,x) \to (\mathbb{R},0) with (x,d_x f) \in U, one has R \Gamma_{\{ f \geq 0 \}}(\mathcal{F}^\bullet )_x = 0.  

 

So, if (p,\eta) \in \mu supp(\mathcal{F}^\bullet), locally, we can say \eta = d_p f for some smooth function germ at p, and if you move in the direction of the “gradient flow” of f in the “positive direction,”  \mathcal{F}^\bullet will detect a change in the cohomology of M.  I like to picture f to be the germ of some linear form (say in a local coordinate system about p), and the gradient flow is like looking a family of cross sections of M near p.  Moreover, p 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: LMD(f,p; \mathcal{F}^\bullet ) \neq 0 if and only if (p,d_p f) \in \mu supp(\mathcal{F}^\bullet).  

 

Now, say we’ve got our sufficiently nice closed subset X \subseteq M, with Whitney stratification \mathfrak{S}, equipped with inclusion map i: X \hookrightarrow M.  Say we’ve got some smooth function f: M \to \mathbb{R}.  We know that, for p \in S \in \mathfrak{S}, p \in \Sigma (f|_S) if and only if (p, d_p f) \in T_S^*M.  Then, via the isomorphism

R \Gamma_{\{ f|_X \geq 0 \}}(\mathcal{F}^\bullet )_p \cong R \Gamma_{\{ f \geq 0 \}} (Ri_* \mathcal{F}^\bullet )_p

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

\mu supp(Ri_* \mathcal{F}^\bullet) \subset \bigcup_{S \in \mathfrak{S}} T_S^*M

 

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.