## Notes of Vincent Humiliere’s lecture

Singular support

Today, the goal is to define terms arising in the statement of the Morse Lemma for sheaf cohomology.

1. Definitions, examples

1.1. Propagation and micro-support

Given a sheaf ${\mathcal{F}}$, for every open set ${U}$, ${H^\cdot(U,\mathcal{F})}$ is defined via injective resolutions. The stalk ${\mathcal{F}_x}$ at point ${x}$ is the set of germs of sections of ${\mathcal{F}}$

Definition 1 Let ${X}$ be a manifold, ${x\in X}$, ${p\in T^*_x X}$. Say that ${\mathcal{F}}$ propagates at ${x}$ in the direction of ${p}$ if for all ${C^1}$ functions ${\phi}$ such that ${\phi(x)=0}$ and ${d\phi(x)=p}$, for all ${j}$, the natural map

$\displaystyle \begin{array}{rcl} \lim_{x\in U} H^j(U,\mathcal{F})\rightarrow \lim_{x\in U} H^j(U\cap\{\phi<0\},\mathcal{F}) \end{array}$

is an isomorphism.

Patrick Massot prefers an alternative definition. The above definition includes the case ${p=0}$, i.e. the support of ${\mathcal{F}}$. The following avoids this.

Say that ${\mathcal{F}}$ propagates at ${x}$ through ${H}$, a co-oriented hyperplane in ${T_xX}$, for all ${C^1}$ domains touching ${x}$ and contained on the preferred side of ${H}$, for all ${j}$, the map

$\displaystyle \begin{array}{rcl} \lim_{x\in U} H^j(U,\mathcal{F})\rightarrow \lim_{x\in U} H^j(U\cap D,\mathcal{F}). \end{array}$

is an isomorphism.

For ${j=0}$, it means that one-sided germs of sections uniquely extend to ordinary germs.

Definition 2 The micro-support (often called singular support) of ${\mathcal{F}}$ is the closure in ${T^*_x X}$ of the set of ${(x,p)}$ where ${\mathcal{F}}$ does not propagate. Notation: ${SS(\mathcal{F})}$.

• It is a cone.
• The projection of ${SS(\mathcal{F})}$ is contained in the support of ${\mathcal{F}}$.
• The intersection of ${SS(\mathcal{F})}$ with the zero section equals the support of ${\mathcal{F}}$ (indeed, then one can take ${\phi=0}$, then ${H^0(U,\mathcal{F})=0}$ for sufficiently small ${U}$; this implies that ${\mathcal{F}_x=0}$).

1.2. Examples

Example 1 Let ${k_X}$ be the constant sheaf. Then ${SS(k_X)=O_X}$ equals the zero section.

Indeed, one can choose a basis of neighborhoods ${U}$ such that ${U}$ and ${U\cap\{\phi<0\}}$ is contractible. These open sets have equal cohomologies, so the isomorphism takes place.

Example 2 Let ${k_V}$ be the constant sheaf on a closed, smooth domain ${V}$. Then ${SS(k_V)=\{(x,p)\,;\,x\in\partial X,\,p>0}$ on the conormal to ${\partial V}$ at ${x\}}$ equals the conormal of the boundary.

By definition, ${k_V(U)}$ consists of locally constant functions on ${U\cap V}$. Again, if ${p}$ is not conormal to ${\partial X}$ at ${x}$, one can choose a basis of neighborhoods ${U}$ such that ${U\cap V}$ and ${U\cap V\cap\{\phi<0\}}$ is contractible, the isomorphism takes place, propagation holds. Otherwise, take for ${\phi}$ a local correctly oriented equation of the boundary, and ${p=d\phi(x)}$. Then, for all neighborhoods ${U}$, ${U\cap V\cap\{\phi<0\}}$ empty, so cohomology vanishes, isomorphism fails for ${j=0}$, no propagation.

For the next example, the following lemma will be useful.

Lemma 3 For all short exact sequences of sheaves

$\displaystyle \begin{array}{rcl} 0\rightarrow\mathcal{F}_{1}\rightarrow\mathcal{F}_{2}\rightarrow\mathcal{F}_{3}\rightarrow 0, \end{array}$

for all permutations of indices ${\{i,j,k\}=\{1,2,3\}}$,

$\displaystyle \begin{array}{rcl} SS(\mathcal{F}_{i})\subset SS(\mathcal{F}_{j})\cup SS(\mathcal{F}_{k}),\quad SS(\mathcal{F}_{i})\Delta SS(\mathcal{F}_{j})\subset SS(\mathcal{F}_{k}). \end{array}$

Proof uses the long exact sequence in cohomology and Five Lemma.

Example 3 Let ${k_\Omega}$ be the constant sheaf on an open domain ${\Omega}$ with smooth boundary. Then ${SS(k_\Omega)=O_{\bar{\Omega}}\cup\nu^*\partial \Omega}$.

Remember that sections of ${k_\Omega}$ on ${U}$ are locally constant functions on ${U\cap \Omega}$ that vanish near ${U\cap\partial \Omega}$. By construction, there is a short exact sequence

$\displaystyle \begin{array}{rcl} 0\rightarrow k_{\Omega}\rightarrow k_X \rightarrow k_{X\setminus\Omega}\rightarrow 0. \end{array}$

The Lemma implies that ${SS(k_\Omega)}$ is contained in and contains the union of ${O_X=SS(k_X)}$ and ${\nu^*\partial \Omega=SS(k_{X\setminus\Omega})}$.

Example 4 Let ${k_W}$ be the constant sheaf on a closed submanifold. Then ${SS(k_W)=\{(x,p)\,;\,x\in W,\,ker(p)\supset T_xW\}=\nu^*W}$ equals the conormal space to ${W}$.

Locally on can pick a submanifold ${Z}$ such that ${\partial Z=W}$. Use the exact sequence

$\displaystyle \begin{array}{rcl} 0\rightarrow k_{Z}\rightarrow k_{\bar{Z}} \rightarrow k_{\partial Z}\rightarrow 0. \end{array}$

Question: are there easy examples where non propagation is seen in higher cohomology ? Consider non smooth closed sets.

The idea of conormal sets extends to mildly singular sets, like squares.

1.3. Micro-support for objects of the derived category

Notation. For ${Z}$ a closed set, define a sheaf ${\Gamma_Z \mathcal{F}}$ by

$\displaystyle \begin{array}{rcl} \Gamma_Z \mathcal{F}(U)=ker(\mathcal{F}(U)\rightarrow\mathcal{F}(U\setminus Z)). \end{array}$

For ${\Omega}$ open,

$\displaystyle \begin{array}{rcl} \Gamma_\Omega \mathcal{F}(U)=\mathcal{F}(U\cap \Omega). \end{array}$

Lemma 4 With this notation, the isomorphism

$\displaystyle \begin{array}{rcl} \lim_{x\in U} H^j(U,\mathcal{F})\rightarrow \lim_{x\in U} H^j(U\cap\{\phi<0\},\mathcal{F}) \end{array}$

holds if and only if for all ${j}$,

$\displaystyle \begin{array}{rcl} H^j (\Gamma_{\{\phi\geq 0\}}(\mathcal{F}))= 0. \end{array}$

Indeed, there is a short exact sequence

$\displaystyle \begin{array}{rcl} 0\rightarrow \Gamma_{\{\phi\geq 0\}}\mathcal{F}\rightarrow \mathcal{F}\rightarrow \Gamma_{\{\phi> 0\}}\mathcal{F}\rightarrow 0. \end{array}$

Note that ${H^j(U, \Gamma_{\phi\geq 0})}$ can be denoted by ${(R^j \Gamma_{\{\phi\geq 0\}})(\mathcal{F})(U)}$. This variant of the definition of propagation opens the way to generalization to the derived category of sheaves.

Definition 5 Let ${\mathcal{F}^{\cdot}}$ be an object of the bounded derived category of sheaves on ${X}$, ${D^b(Sh(X))}$. The micro-support ${SS(\mathcal{F}^{\cdot})}$ is the closure of all ${(x,p)}$ such that there exists a smooth function ${\phi}$ with ${\phi(x)=0}$, ${d\phi(x)=p}$ such that

$\displaystyle \begin{array}{rcl} (R\Gamma_{\{\phi\geq 0\}}(\mathcal{F}^{\cdot}))_x \not=0. \end{array}$

Alexandru Oancea: what does ${(R^j \Gamma_{\{\phi\geq 0\}})(\mathcal{F})}$ exactly mean ?

Example 5 Let ${X}$ be a complex manifold. Let ${\mathcal{O}_X}$ denote the sheaf of holomorphic functions on ${X}$. Let ${P}$ be a differential operator with holomorphic coefficients and let ${\mathcal{P}}$ be its principal symbol (a function on ${T^*X}$). Consider the following complex of sheaves

$\displaystyle \begin{array}{rcl} \mathcal{F}^{\cdot} : \quad 0\rightarrow\mathcal{O}_X\stackrel{\mathcal{P}}{\rightarrow} \mathcal{O}_X\rightarrow 0. \end{array}$

Then

$\displaystyle \begin{array}{rcl} SS(\mathcal{F}^{\cdot})=P^{-1}(0). \end{array}$

This is not a easy fact. The Cauchy-Kowalewskaya theorem is used. It motivated initially the concept of micro-support.

1.4. Direct image

Let ${M}$, ${N}$ be symplectic manifolds. Let ${\Lambda\subset \bar{M}\times N}$ be a Lagrangian submanifold, then, for every subset ${B\subset M}$, ${\Lambda(B)}$ is the projection on ${N}$ of the intersection of ${B\times\Lambda}$ with diagonal${\times N}$ in ${M\times\bar{M}\times N}$.

Proposition 6 Let ${f:X\rightarrow Y}$ be a ${C^1}$ map which is proper on the support of ${\mathcal{F}^{\cdot}}$. Then

$\displaystyle \begin{array}{rcl} SS(Rf_* \mathcal{F}^{\cdot})\subset \Lambda_f(SS(\mathcal{F}^{\cdot})), \end{array}$

this is associated with the graph of ${f}$ which is a Lagrangian submanifold of ${\overline{T^*X}\times T^*Y}$.

Note that

$\displaystyle \begin{array}{rcl} \Lambda_f=\{(x,\xi,y,p)\in \overline{T^*X})\times T^*Y\,;\,f(x)=y,\,\xi=p\circ df(x)\}. \end{array}$

Proof relies on ${f_* \circ \Gamma_{\{\phi\geq 0\}}=\Gamma_{\{\phi\circ f\geq 0\}}}$, and a passage to the derived category which uses spectral sequences. Assume that ${\mathcal{F}}$ propagates at ${(x,p\circ df(x))}$, ${p\in T^*Y}$, for all ${x\in f^{-1}(y)}$. Let ${\phi:Y\rightarrow{\mathbb R}}$ satisfy ${\phi(y)=0}$, ${d\phi(y)=p}$. One needs to show that

$\displaystyle \begin{array}{rcl} R\Gamma_{\{\phi\geq 0\}}(Rf_*\mathcal{F})_y=0. \end{array}$

Take ${\psi=\phi\circ f}$. By assumption, ${R\Gamma_{\{\psi\geq 0\}}(\mathcal{F})_x =0}$ for all ${x\in f^{-1}(y)}$. One first shows that

$\displaystyle \begin{array}{rcl} R\Gamma_{\{\phi\geq 0\}}(Rf_*\mathcal{F})_y=(Rf_*(R\Gamma_{\{\psi\geq 0\}}(\mathcal{F})))_y, \end{array}$

and, second, that

$\displaystyle \begin{array}{rcl} (Rf_*(R\Gamma_{\{\psi\geq 0\}}(\mathcal{F})))_y=\lim_{f^{-1}(y)\subset U}R\Gamma_{\{\psi\geq 0\}}(U,\mathcal{F}). \end{array}$

One uses the following fact.

Theorem 7 Let ${F}$, ${G}$ be left-exact functors between categories having enough injectives. Assume that ${G}$ send injectives to ${F}$-acyclic objects. Then

$\displaystyle \begin{array}{rcl} R(F\circ G)=R(F)\circ R(G). \end{array}$

C’est la preuve de ce théorème qui utilise des suites spectrales. Par exemple, le théorème implique la suite spectrale de Leray pour les fibrations.