04.03.19 · algebraic-geometry / cohomology

Perverse sheaves Perv(X) — pointer + foundations

shipped3 tiersLean: none

Anchor (Master): Beilinson-Bernstein-Deligne *Faisceaux pervers* (*Astérisque* 100, 1982) §§2.1, 4.0, 5.0–6.2 (perverse t-structure, intermediate extensions, decomposition theorem with weights); Goresky-MacPherson *Intersection Homology I* (*Topology* 19, 1980) and *Intersection Homology II* (*Invent. Math.* 71/72, 1983); Kashiwara 1984 *Publ. RIMS* 20 + Mebkhout 1980 thesis (Riemann-Hilbert correspondence); Saito 1988 *Publ. RIMS* 24 (mixed Hodge modules); de Cataldo-Migliorini 2005 *Ann. Sci. ÉNS* 38 (Hodge-theoretic proof of decomposition theorem); Hotta-Takeuchi-Tanisaki *D-Modules, Perverse Sheaves, and Representation Theory* (Birkhäuser 2008); Ngô 2010 *Publ. Math. IHÉS* 111

Intuition Beginner

A constructible complex on an algebraic variety $X$ is a complex of sheaves whose cohomology sheaves are locally constant of finite rank along each stratum of a stratification — the basic objects of topological cohomology on a singular space. The constructible derived category $D_{c}^{b} (X)$ assembles all such complexes, with the usual derived-category morphisms. The standard cohomology functors $H^{i}$ extract the cohomology sheaves of a complex one degree at a time. Perverse sheaves are constructible complexes that satisfy a single condition tying together all the cohomology sheaves of a complex into one balanced object — the support dimensions of the cohomology sheaves shrink as the cohomological degree grows, in a way that is symmetric under Verdier duality.

The condition is the support-dimension inequality $dim supp H^{i} (F) \leq - i$ for all $i$ . Read it this way: the cohomology sheaf in degree $i$ lives on a closed subset of $X$ whose dimension is at most $- i$ . In degree $- d$ the support is allowed to be all of $X$ (an open stratum of dimension $d$ ); in degree $- d + 1$ the support must drop to dimension $\leq d - 1$ ; and so on.

The dual condition under Verdier duality $D$ gives the other half of the perverse t-structure, and the intersection of the two halves is the category $Perv (X)$ — the heart of the middle-perversity t-structure on $D_{c}^{b} (X)$ . Despite being a heart of a t-structure on a triangulated category, $Perv (X)$ is an honest abelian category, and every object has a finite composition series in the Jordan-Hölder sense.

The everyday analogy is a soil-profile diagram in stratigraphy. Each horizontal stratum of soil has its own composition and age, and the "perverse" condition says that the deeper strata extend further in horizontal area than the surface ones — the topsoil is the most spatially extensive, the bedrock the most localised. A perverse sheaf is the analogue of a balanced soil profile where the horizontal-extent constraint scales correctly with depth. The simple perverse sheaves are pure-stratum objects (intersection cohomology complexes), and every perverse sheaf is built from these simples by a finite sequence of extensions.

Visual Beginner

A diagram showing the bounded constructible derived category $D_{c}^{b} (X)$ of a complex algebraic variety $X$ , split by the middle-perversity t-structure into two halves $^{p} D^{\leq 0} (X)$ and $^{p} D^{\geq 0} (X)$ , with their intersection labelled $Perv (X)$ as the abelian heart. The diagram also indicates the BBD decomposition theorem: for a proper morphism $f : X \to Y$ between complex varieties with $X$ smooth, the derived direct image $R f_{*} IC_{X}$ decomposes as a direct sum of shifted intersection cohomology complexes $⨁_{i} IC_{\overline{Z_{i}}} (L_{i}) [d_{i}]$ in the constructible derived category $D_{c}^{b} (Y)$ .

The picture captures the structural shape: the perverse t-structure is determined by a single support-dimension inequality and its Verdier dual, the heart is the abelian category $Perv (X)$ , and the BBD decomposition theorem says that proper pushforwards of pure perverse sheaves split as direct sums of simple summands. A reader who internalises this picture will recognise the same template every time perverse sheaves appear — support-dimension inequality, Verdier-dual half, abelian heart, simple objects as intermediate extensions, decomposition theorem for proper pushforwards.

Worked example Beginner

Compute the two smallest interesting perverse sheaves on $X = C$ (the complex line), and verify the support-dimension condition by hand. This is the smallest example where the perverse t-structure has more than one isomorphism class of simple object.

Step 1. Identify the two basic complexes. Take the constant sheaf $Q_{C}$ on $C$ (the locally constant sheaf with stalk $Q$ ), shifted to put it in degree $- 1$ : $F_{1} = Q_{C} [1]$ has $H^{i} (F_{1}) = Q_{C}$ when $i = - 1$ and zero otherwise. Take the skyscraper sheaf $Q_{{0}}$ at the origin: $F_{2} = Q_{{0}}$ has $H^{i} (F_{2}) = Q_{{0}}$ when $i = 0$ and zero otherwise.

Step 2. Verify the support-dimension condition for $F_{1}$ . The condition is $dim supp H^{i} (F_{1}) \leq - i$ for all $i$ . The only nonzero cohomology is in degree $i = - 1$ , where $supp H^{- 1} (F_{1}) = C$ has dimension $1$ . The inequality reads $1 \leq - (- 1) = 1$ , which holds with equality. For all other $i$ , $H^{i} (F_{1}) = 0$ has empty support of dimension $- \infty$ , and the inequality holds. So $F_{1} \in^{p} D^{\leq 0}$ .

Step 3. Verify the support-dimension condition for $F_{2}$ . The only nonzero cohomology is in degree $i = 0$ , where $supp H^{0} (F_{2}) = {0}$ has dimension $0$ . The inequality reads $0 \leq - 0 = 0$ , which holds with equality. For all other $i$ , the inequality holds vacuously. So $F_{2} \in^{p} D^{\leq 0}$ .

Step 4. Verify the Verdier-dual condition. The Verdier dual on $C$ is computed using the dualising complex $ω_{C}^{∙} = Q_{C} [2]$ (the shifted constant sheaf, since $C$ is smooth of complex dimension $1$ ). The Verdier dual of $F_{1} = Q_{C} [1]$ is $D (Q_{C} [1]) = R H o m (Q_{C} [1], Q_{C} [2]) = Q_{C} [1] = F_{1}$ , so $F_{1}$ is self-Verdier-dual. The Verdier dual of $F_{2} = Q_{{0}}$ is $D (Q_{{0}}) = Q_{{0}}$ by the analogous calculation (using that the inclusion of the origin is a proper map and applying the duality $i_{*} D = D i_{!}$ ). So $F_{2}$ is also self-Verdier-dual.

Step 5. Conclude. Both $F_{1}$ and $F_{2}$ satisfy the support-dimension condition and are self-Verdier-dual, hence both lie in the heart $^{p} D^{\leq 0} \cap^{p} D^{\geq 0} = Perv (C)$ . These are two distinct simple perverse sheaves on $C$ : $F_{1} = IC_{C} = Q_{C} [1]$ is the intersection cohomology complex on the whole line (dimension $1$ , constant local system), and $F_{2} = IC_{{0}} (Q) = Q_{{0}}$ is the intersection cohomology complex on the origin (dimension $0$ , constant local system).

What this tells us. On the complex line $C$ , the two basic perverse sheaves are the shifted constant sheaf $Q_{C} [1]$ and the skyscraper $Q_{{0}}$ at any point. They satisfy the support-dimension condition with equality in their respective degrees, are stable under Verdier duality, and exemplify the two basic types of simple objects in $Perv (C)$ : intermediate extensions from open strata (here the open dense $C^{*}$ or all of $C$ ) and from zero-dimensional closed strata (here the origin).

Check your understanding Beginner

Exercise (easy, multiple choice).

The category $Perv (X)$ of perverse sheaves on a complex algebraic variety $X$ is:

A. The bounded constructible derived category $D_{c}^{b} (X)$ itself B. The heart of the middle-perversity t-structure on $D_{c}^{b} (X)$ C. The category of locally constant sheaves on $X$ D. The category of D-modules on $X$

Hint

A t-structure on a triangulated category has a heart, which is an abelian category sitting inside the triangulated category. Perverse sheaves arise from a specific t-structure determined by a support-dimension condition.

Answer

B. Perverse sheaves form the heart of the middle-perversity t-structure on the bounded constructible derived category $D_{c}^{b} (X)$ . Option A is the ambient triangulated category, not the heart. Option C describes a different category (locally constant sheaves are constructible but not generally perverse, and the heart of the standard t-structure on $D_{c}^{b} (X)$ is the category of all constructible sheaves $Cons (X)$ , distinct from $Perv (X)$ ). Option D is the source side of the Riemann-Hilbert correspondence (regular holonomic D-modules), which is equivalent to but not literally the same as perverse sheaves.

Exercise (easy, true-false).

True or false: every perverse sheaf on a complex algebraic variety has a finite Jordan-Hölder filtration whose simple subquotients are intersection cohomology complexes attached to (smooth locally closed subvariety, irreducible local system) pairs.

Hint

The category $Perv (X)$ is abelian, Artinian, and Noetherian — that is the structural fact one is invited to remember about it.

Answer

True. $Perv (X)$ is an abelian, Artinian, and Noetherian category (proved in BBD 1982 §4.0), so every object has a finite composition series whose subquotients are simple. The simple objects in $Perv (X)$ are classified (BBD §4.3) as the intermediate extensions $IC_{\overline{Z}} (L) = j_{! *} (L [d_{Z}])$ of irreducible local systems $L$ on smooth locally closed subvarieties $Z \subset X$ of dimension $d_{Z}$ , where $j : Z ↪ X$ is the inclusion. This classification is what makes perverse sheaves a tractable abelian category — every perverse sheaf is built from a finite multiset of $(Z, L)$ pairs by repeated extension.

Exercise (easy, multiple choice).

On $X = C$ (the complex line), which of the following is a perverse sheaf?

A. The constant sheaf $Q_{C}$ in degree $0$ (unshifted) B. The shifted constant sheaf $Q_{C} [1]$ in degree $- 1$ C. The constant sheaf $Q_{C}$ in degree $2$ D. The cone of an arbitrary morphism $Q_{C} [1] \to Q_{{0}}$ in $D_{c}^{b} (C)$

Hint

The support-dimension condition $dim supp H^{i} (F) \leq - i$ has to hold for all $i$ . On $C$ , the constant sheaf has support all of $C$ (dimension $1$ ), so it must appear in degree $i = - 1$ to satisfy the inequality with equality.

Answer

B. The shifted constant sheaf $Q_{C} [1]$ is perverse on $C$ , because its only nonzero cohomology is in degree $- 1$ with support all of $C$ of dimension $1$ , satisfying $1 \leq - (- 1) = 1$ . Option A fails: the unshifted constant sheaf $Q_{C}$ has cohomology in degree $0$ with support of dimension $1$ , violating $1 \leq - 0 = 0$ . Option C fails for the same reason in a different degree. Option D is generally not perverse — the cone of a morphism between perverse sheaves need not be perverse, since the perverse t-structure is not closed under cones in general (it produces long exact sequences in $Perv (C)$ , not direct decompositions).

Formal definition Intermediate+

Let $X$ be a complex algebraic variety (or a complex analytic space, or a finite-type scheme over an algebraically closed field, with adapted notions of constructibility and stratification). Fix a coefficient field $k$ (typically $Q$ or $\overline{Q}_{ℓ}$ in the étale setting) and a stratification $X = ⨆_{α} S_{α}$ by smooth locally closed subvarieties such that closures of strata are unions of strata (a Whitney stratification suffices).

Definition (constructible complex). A complex $F^{∙}$ in the bounded derived category $D^{b} (Sh (X, k))$ of sheaves of $k$ -modules on $X$ is constructible with respect to the stratification if each cohomology sheaf $H^{i} (F^{∙})$ is a locally constant sheaf of finite-rank $k$ -modules on each stratum $S_{α}$ . The full subcategory of constructible complexes is the bounded constructible derived category $D_{c}^{b} (X) \subset D^{b} (Sh (X, k))$ . It is closed under the Verdier dual $D_{X} = R H o m (-, ω_{X}^{∙})$ , where $ω_{X}^{∙} = a_{X}^{!} k$ is the dualising complex associated to the structure map $a_{X} : X \to pt$ via the six-functor formalism 04.03.16; for $X$ smooth of complex dimension $d$ , $ω_{X}^{∙} = k_{X} [2 d]$ (with a Tate twist in the $ℓ$ -adic setting that we suppress).

Definition (middle-perversity t-structure on $D_{c}^{b} (X)$ , Beilinson-Bernstein-Deligne 1982 §2.1). Define two full subcategories of $D_{c}^{b} (X)$ : $$ {}^p!D^{\le 0}(X) := { \mathcal{F}^\bullet \in D^b_c(X) : \dim \mathrm{supp}, \mathcal{H}^i(\mathcal{F}^\bullet) \le -i \text{ for all } i }, $$ $$ {}^p!D^{\ge 0}(X) := { \mathcal{F}^\bullet \in D^b_c(X) : \mathbb{D}_X(\mathcal{F}^\bullet) \in {}^p!D^{\le 0}(X) }. $$ The pair $(^{p} D^{\leq 0},^{p} D^{\geq 0})$ is a t-structure on $D_{c}^{b} (X)$ (in the sense of 04.03.18), called the middle-perversity t-structure. Equivalently, $^{p} D^{\geq 0} (X)$ is the subcategory of $F^{∙}$ such that for every $i$ , the costalk $i_{x}^{!} F^{∙} \in D^{\geq - d i m \overline{{x}}}$ for every point $x \in X$ (Mac Pherson's reformulation).

Definition (perverse sheaves; BBD 1982 §2.2). The heart of the middle-perversity t-structure is the abelian category of perverse sheaves: $$ \mathrm{Perv}(X) := {}^p!D^{\le 0}(X) \cap {}^p!D^{\ge 0}(X). $$ By the general BBD theorem [04.03.18 §1.3.6], $Perv (X)$ is an abelian category, with short exact sequences corresponding to distinguished triangles in $D_{c}^{b} (X)$ all three of whose terms lie in $Perv (X)$ .

Definition (perverse cohomology). The cohomological functor of the middle-perversity t-structure is $$ {}^p!H^n : D^b_c(X) \to \mathrm{Perv}(X), \qquad {}^p!H^n := {}^p\tau_{\le 0} \circ {}^p\tau_{\ge 0} \circ [n]. $$ For every distinguished triangle $F^{∙} \to G^{∙} \to H^{∙} \to F^{∙} [1]$ in $D_{c}^{b} (X)$ , applying ${^{p} H^{n}}_{n \in Z}$ produces a long exact sequence in $Perv (X)$ .

Definition (intermediate extension and simple perverse sheaves). Let $Z \subset X$ be a smooth locally closed irreducible subvariety of complex dimension $d_{Z}$ , with locally closed embedding $j : Z ↪ X$ , and let $L$ be an irreducible local system of $k$ -vector spaces on $Z$ (i.e., an irreducible representation of $π_{1} (Z, z_{0})$ on a finite-dimensional $k$ -vector space). The intermediate extension is $$ j_{!}(\mathcal{L}[d_Z]) := \mathrm{im}\bigl( {}^p!H^0(j_! \mathcal{L}[d_Z]) \to {}^p!H^0(j_ \mathcal{L}[d_Z]) \bigr) \in \mathrm{Perv}(X), $$ where the morphism is the natural one induced by the adjunction $j_{!} \to j_{*}$ on $D_{c}^{b}$ , and $im$ is taken in the abelian category $Perv (X)$ . Write $IC_{\overline{Z}} (L) := j_{! *} (L [d_{Z}])$ when $Z ↪ X$ is understood; this is the intersection cohomology complex of $\overline{Z}$ with coefficients in $L$ .

Definition (simple objects of $Perv (X)$ , BBD 1982 §4.3). The simple objects of $Perv (X)$ are exactly the intersection cohomology complexes $IC_{\overline{Z}} (L)$ for pairs $(Z, L)$ with $Z \subset X$ smooth locally closed irreducible and $L$ an irreducible local system on $Z$ , up to the equivalence $(Z, L) \sim (Z^{'}, L^{'})$ iff $\overline{Z} = \overline{Z^{'}}$ and $L ≅ L^{'}$ on the common open dense subset of $Z \cap Z^{'}$ .

Definition (Verdier-self-duality of $IC$ ). For $Z$ open in its closure and $L$ equipped with a non-degenerate symmetric (or alternating) self-pairing $L \otimes L \to k_{Z}$ , the intersection cohomology complex satisfies $D_{X} (IC_{\overline{Z}} (L)) ≅ IC_{\overline{Z}} (L^{\lor}) ≅ IC_{\overline{Z}} (L)$ as perverse sheaves on $X$ .

Counterexamples to common slips

The condition $dim supp H^{i} (F) \leq - i$ controls cohomology sheaves, not the complex's own degrees. A complex concentrated in a single degree $n$ as a sheaf is perverse iff its support has dimension exactly $- n$ ; the constant sheaf $k_{X}$ on a smooth variety $X$ of dimension $d$ is not perverse, but $k_{X} [d]$ is. The shift is not optional cosmetic data.
The middle-perversity t-structure on $D_{c}^{b} (X)$ is not the standard t-structure. The standard t-structure has heart $Cons (X)$ (constructible sheaves placed in degree $0$ ); the perverse t-structure has heart $Perv (X)$ . These are different abelian categories and the two t-structures are exchanged by no functor — they are simply two different t-structures on the same triangulated category, each with its own cohomology, its own simples, and its own decomposition behaviour under proper pushforward.
Verdier duality on $IC_{\overline{Z}} (L)$ takes the local system $L$ to its dual $L^{\lor}$ . Verdier duality is not the identity on perverse sheaves in general; the simples are self-Verdier-dual only when the underlying local system is. For $L = k$ the constant local system (any rank), $L^{\lor} = L$ and $IC$ is self-dual; for higher-rank local systems without self-pairings, duality acts as a non-identity permutation of simples.
The simple objects of $Perv (X)$ are classified by pairs $(Z, L)$ — both the stratum and the local system — not by the stratum alone. Two distinct local systems $L_{1}, L_{2}$ on the same smooth locally closed $Z$ produce non-isomorphic simple perverse sheaves $IC_{\overline{Z}} (L_{1}) \neq ≅ IC_{\overline{Z}} (L_{2})$ .
The intermediate extension $j_{! *}$ is generally not equal to either $j_{!}$ or $j_{*}$ . It is the image of the morphism $^{p} H^{0} (j_{!} L [d_{Z}]) \to^{p} H^{0} (j_{*} L [d_{Z}])$ in $Perv (X)$ , and it sits strictly between $j_{!}$ and $j_{*}$ when $j$ is not closed. Confusing $j_{! *}$ with $j_{*}$ produces an object that is generally not perverse and has the wrong simple-summand structure under the decomposition theorem.

Key theorem with proof Intermediate+

Theorem (BBD decomposition theorem; Beilinson-Bernstein-Deligne 1982 Astérisque 100 §6.2.5; Gabber 1981). Let $f : X \to Y$ be a proper morphism of complex algebraic varieties with $X$ smooth, and let $k$ be a field of characteristic zero. Then in $D_{c}^{b} (Y, k)$ the derived direct image of the shifted constant sheaf decomposes as a direct sum of shifted intermediate extensions: $$ Rf_* k_X[\dim X] \cong \bigoplus_{i \in \mathbb{Z}} , {}^p!H^i\bigl(Rf_* k_X[\dim X]\bigr)[-i], $$ and each perverse cohomology summand ${}^p!H^i(Rf_ k_X[\dim X]) \in \mathrm{Perv}(Y)$ is a semisimple perverse sheaf, hence decomposes as a direct sum of simple intersection cohomology complexes* $$ {}^p!H^i\bigl(Rf_* k_X[\dim X]\bigr) \cong \bigoplus_{(\overline{Z_\alpha}, \mathcal{L}\alpha)} \mathrm{IC}{\overline{Z_\alpha}}(\mathcal{L}\alpha)^{\oplus m{i, \alpha}} $$ indexed by pairs $(\overline{Z_{α}}, L_{α})$ of irreducible closed subvarieties of $Y$ equipped with irreducible local systems on smooth open dense parts, with non-negative integer multiplicities $m_{i, α}$ .

Proof. The argument has four steps. First, reduce to the $ℓ$ -adic setting over a finite field via spreading out and base change. Second, apply Deligne's purity theorem to obtain that $R f_{*} k_{X} [dim X]$ is pure of weight $dim X$ in the sense of Frobenius weights. Third, use the BBD theory of weights on perverse sheaves to derive that every pure perverse sheaf is geometrically semisimple. Fourth, transfer back to the original characteristic-zero setting via specialisation.

Step 1: spread out. A smooth proper map $f : X \to Y$ of complex algebraic varieties is defined over a finitely generated $Z$ -algebra $A \subset C$ , by standard spreading-out arguments (EGA IV, §8): there exists a smooth proper morphism $f_{A} : X_{A} \to Y_{A}$ over $Spec A$ with generic fibre the original $f$ . Choosing a maximal ideal $m \subset A$ produces a residue field $A / m = F_{q}$ for some prime power $q$ , and the special fibre $f_{q} : X_{q} \to Y_{q}$ is a smooth proper morphism over $F_{q}$ . By the smooth proper base change theorem (SGA 4, Exposé XII), the étale cohomology of the special fibre agrees with the étale cohomology of the generic fibre, and the comparison between $ℓ$ -adic and Betti cohomology on the generic fibre (Artin's comparison, Deligne) reduces the decomposition statement in characteristic zero to its $ℓ$ -adic analogue over $F_{q}$ .

Step 2: purity. By Deligne's purity theorem (Weil II, Publ. Math. IHÉS 52, 1980), the derived direct image $R f_{q, *} \overline{Q}_{ℓ, X_{q}} [dim X_{q}]$ is pure of weight $dim X_{q}$ as a complex of $ℓ$ -adic sheaves on $Y_{q}$ : every Frobenius eigenvalue on each cohomology stalk has complex absolute value $q^{(w - i) /2}$ where $w = dim X$ and $i$ is the perverse-cohomology degree. The purity statement requires $f$ proper (for the proper-base-change comparison) and $X$ smooth (to ensure the constant sheaf $\overline{Q}_{ℓ, X} [dim X]$ is pure of weight $dim X$ on $X$ before pushforward; equivalently, that it is the intersection cohomology complex of $X$ , which holds for $X$ smooth).

Step 3: semisimplicity of pure perverse sheaves. BBD §5.3.8 proves that every pure perverse sheaf $P \in Perv (Y_{q}, \overline{Q}_{ℓ})$ on a variety over $F_{q}$ is semisimple after pullback to the algebraic closure $\overline{F}_{q}$ . The proof uses the weight filtration $W_{∙} P$ on a mixed perverse sheaf and the fact that the graded pieces of a pure perverse sheaf are forced to be direct sums of simples by the rigidity of weight filtrations under Frobenius. The simples of $Perv (Y_{q}, \overline{Q}_{ℓ})$ are the intersection cohomology complexes $IC_{\overline{Z_{α}}} (L_{α})$ for $(\overline{Z_{α}}, L_{α})$ pairs over $\overline{F}_{q}$ . Hence the perverse cohomology sheaves $^{p} H^{i} (R f_{q, *} \overline{Q}_{ℓ} [dim X])$ are direct sums of shifted IC complexes.

Step 4: degeneration of the perverse Leray spectral sequence. The perverse Leray spectral sequence $E_{2}^{p, q} =^{p} H^{p} (^{p} H^{q} (R f_{*} k_{X} [dim X])) \Rightarrow H^{p + q} (X, k)$ degenerates at $E_{2}$ for proper $f$ with $X$ smooth — equivalently, the direct sum decomposition $R f_{*} k_{X} [dim X] ≅ ⨁_{i}^{p} H^{i} (R f_{*} k_{X} [dim X]) [- i]$ in $D_{c}^{b} (Y)$ holds without higher correction. This degeneration is a consequence of the semisimplicity proved in Step 3 (semisimple pure perverse sheaves admit no nonzero extensions to higher perverse-cohomology degrees) and the standard machinery of t-structure spectral sequences applied to the middle-perversity t-structure. Transferring back to characteristic zero via Step 1's spreading-out completes the proof in the original complex setting. $□$

Bridge. The BBD decomposition theorem builds toward an abelian-category understanding of proper pushforwards in the derived category of constructible sheaves, and the foundational reason the construction works is that the weight filtration coming from $ℓ$ -adic Frobenius on a pure perverse sheaf forces semisimplicity, which is then transferred to characteristic zero via spreading-out and Artin comparison. The bridge is the perverse t-structure, which packages the support-dimension condition into an abelian category whose simples (the intersection cohomology complexes) are exactly what proper pushforwards of smooth varieties decompose into. Putting these together, every proper map of complex varieties with smooth source produces a direct-sum decomposition of its constant sheaf's pushforward into a multiset of $(\overline{Z_{α}}, L_{α})$ pairs, and the geometry of the map is read off from the multiplicities.

This pattern appears again in 04.03.18 (t-structures and hearts), where the perverse t-structure is the canonical non-standard example with deep geometric content, and in 04.03.16 (the six-functor formalism), where the interaction between Verdier duality, proper pushforward, and the perverse t-structure produces the decomposition statement. The central insight is that perverse sheaves are the abelian category that organises constructible cohomology of singular and stratified spaces in a single uniform setting, and the decomposition theorem is the statement that proper maps respect this organisation — the cohomology of $X$ decomposes along the strata of $Y$ in a way that is canonical and computable.

Exercises Intermediate+

Exercise 1 (easy, short-answer).

State the support-dimension condition defining $^{p} D^{\leq 0} (X)$ and verify it for the shifted constant sheaf $k_{X} [d]$ on a smooth complex variety $X$ of dimension $d$ .

Hint

The condition is $dim supp H^{i} (F) \leq - i$ for all $i$ . The only nonzero cohomology of $k_{X} [d]$ is in degree $- d$ .

Answer

The condition is $F \in^{p} D^{\leq 0} (X)$ iff $dim supp H^{i} (F) \leq - i$ for all $i \in Z$ .

For $F = k_{X} [d]$ with $X$ smooth of complex dimension $d$ : the only nonzero cohomology sheaf is $H^{- d} (k_{X} [d]) = k_{X}$ , which has support all of $X$ , of complex dimension $d$ . The condition reads $d \leq - (- d) = d$ , holding with equality. For all $i \neq = - d$ , $H^{i} (k_{X} [d]) = 0$ has empty support of dimension $- \infty$ , and the condition holds vacuously. Hence $k_{X} [d] \in^{p} D^{\leq 0} (X)$ . Combined with Verdier-self-duality ( $D_{X} (k_{X} [d]) = k_{X} [d]$ for $X$ smooth of dimension $d$ , since $ω_{X}^{∙} = k_{X} [2 d]$ ), one obtains $k_{X} [d] \in Perv (X)$ — in fact $k_{X} [d] = IC_{X}$ , the intersection cohomology complex on the whole smooth variety. Rubric: full credit for the condition statement and the verification.

Exercise 2 (easy, short-answer).

For $X = C^{2}$ and $Z = {0} \subset X$ , identify the intermediate extension $j_{! *} (k_{{0}} [0])$ of the constant rank-one local system on the origin to all of $C^{2}$ as an object of $Perv (C^{2})$ .

Hint

The point ${0}$ is a smooth zero-dimensional subvariety, and $j : {0} ↪ C^{2}$ is a closed embedding. For a closed embedding, $j_{!} = j_{*}$ in the constructible derived category.

Answer

For a closed embedding $j : Z ↪ X$ , the proper-pushforward $j_{!}$ and the ordinary pushforward $j_{*}$ agree, so $j_{!} (F) = j_{*} (F)$ for any $F \in D_{c}^{b} (Z)$ . Both are exact functors on $Perv$ for closed embeddings, and the morphism $^{p} H^{0} (j_{!}) \to^{p} H^{0} (j_{*})$ is the identity. Hence the intermediate extension is $$ j_{!}(k_{{0}}[0]) = j_(k_{{0}}[0]) = k_{{0}} = \mathbb{Q}{{0}}, $$ the skyscraper sheaf at the origin (with rational coefficients here for concreteness). As a perverse sheaf on $C^{2}$ , $\mathrm{IC}{{0}}(k) = k_{{0}} $ha ss u pp or t o f d im e n s i o n$ 0 $in co h o m o l o g i c a l d e g r ee$ 0 $, s a t i s f y in g$ 0 \le 0 $. I t i so n eo f t h es im pl e p er v er ses h e a v es in$ \mathrm{Perv}(\mathbb{C}^2) $, w i t h$ (Z, \mathcal{L}) = ({0}, k) $. R u b r i c : f u l l cr e d i t f or t h e$ j_! = j_*$ observation for closed embeddings and the identification with the skyscraper sheaf.

Exercise 3 (medium, short-answer).

Let $X = C$ and let $j : C^{*} ↪ C$ be the open inclusion of the punctured line. Let $L_{χ}$ be a nonconstant rank-one local system on $C^{*}$ corresponding to a character $χ : π_{1} (C^{*}) = Z \to k^{*}$ with $χ (1) \neq = 1$ . Compute $j_{!} (L_{χ} [1])$ , $j_{*} (L_{χ} [1])$ , and $j_{! *} (L_{χ} [1])$ as objects of $D_{c}^{b} (C)$ and identify which is perverse.

Hint

Use that for an open embedding $j$ , $j_{!}$ extends by zero and $j_{*}$ takes a sheaf to its "topologisation" at the boundary, with the stalk at the boundary point being the invariants of the local system under monodromy. The intermediate extension is the image inside the perverse t-structure.

Answer

For $j : C^{*} ↪ C$ with $L_{χ}$ nonconstant:

$j_{!} (L_{χ} [1])$ is the extension by zero: stalk at $0$ is zero. So $H^{- 1} (j_{!} (L_{χ} [1]))$ is $L_{χ}$ on $C^{*}$ and zero at $0$ , and all other cohomology vanishes. Support dimensions: degree $- 1$ has support $C^{*}$ (closure dimension $1$ , since the support condition uses the closure of the support of the cohomology sheaf), satisfying $1 \leq 1$ . So $j_{!} L_{χ} [1] \in^{p} D^{\leq 0}$ . The Verdier dual is $j_{*} (L_{χ}^{\lor} [1])$ , which is shown below not to be in $^{p} D^{\leq 0}$ unless we account for the monodromy at $0$ ; in fact $j_{!} L_{χ} [1]$ is perverse iff $χ$ has vanishing monodromy invariants at $0$ , which is the case for nonconstant $χ$ .
$j_{*} (L_{χ} [1])$ : stalk at $0$ is the monodromy-invariant part $L_{χ}^{π_{1} (C^{*})} = L_{χ}^{⟨ T ⟩}$ where $T = χ (1)$ is the monodromy. For $χ$ nonconstant, $T \neq = 1$ , so the invariants vanish. Hence $j_{*} (L_{χ} [1])$ has stalk zero at $0$ , and is identical to $j_{!} (L_{χ} [1])$ when $χ$ is nonconstant. So $j_{!} (L_{χ} [1]) = j_{*} (L_{χ} [1])$ in this case.
$j_{! *} (L_{χ} [1])$ : the intermediate extension is the image of $j_{!} \to j_{*}$ in $Perv$ , which equals both (they coincide for nonconstant $χ$ ). Hence $j_{! *} (L_{χ} [1]) = j_{!} (L_{χ} [1]) = j_{*} (L_{χ} [1])$ is the unique perverse extension.

This complex is the intersection cohomology complex $IC_{C} (L_{χ}) \in Perv (C)$ , a simple perverse sheaf attached to the pair $(C^{*}, L_{χ})$ — the underlying constructible complex has $L_{χ}$ on the open dense $C^{*}$ and nothing at the boundary point $0$ , because the nonconstant monodromy obstructs any extension. Rubric: full credit for the three computations and the identification of $j_{! *}$ with the perverse extension under nonconstant monodromy.

Exercise 4 (medium, short-answer).

State the BBD decomposition theorem for a proper morphism $f : X \to Y$ of complex algebraic varieties with $X$ smooth, and derive as a corollary the degeneration of the perverse Leray spectral sequence at $E_{2}$ .

Hint

The decomposition theorem says $R f_{*} k_{X} [dim X]$ decomposes as a direct sum of shifted intermediate extensions. The perverse Leray spectral sequence has $E_{2}^{p, q} =^{p} H^{p} (Y,^{p} H^{q} (R f_{*} k_{X} [dim X]))$ .

Answer

Theorem (BBD decomposition). For $f : X \to Y$ proper with $X$ smooth complex algebraic, $dim X = n$ : $$ Rf_* k_X[n] \cong \bigoplus_{i} {}^p!H^i(Rf_* k_X[n])[-i] \quad \text{in } D^b_c(Y), $$ and each summand $^{p} H^{i} (R f_{*} k_{X} [n])$ is semisimple in $Perv (Y)$ , decomposing as $$ {}^p!H^i(Rf_* k_X[n]) \cong \bigoplus_\alpha \mathrm{IC}{\overline{Z\alpha}}(\mathcal{L}\alpha)^{\oplus m{i, \alpha}}. $$

Corollary (degeneration of the perverse Leray spectral sequence). The perverse Leray spectral sequence is $$ E_2^{p,q} = {}^p!H^p(Y, {}^p!H^q(Rf_* k_X[n])) \Longrightarrow H^{p+q}(X, k)[n] = H^{p+q+n}(X, k). $$ The decomposition $R f_{*} k_{X} [n] ≅ ⨁_{i}^{p} H^{i} (R f_{*} k_{X} [n]) [- i]$ implies that the abutment $H^{p + q + n} (X, k)$ splits as $$ H^{p+q+n}(X, k) \cong \bigoplus_{i} {}^p!H^{p+q-i}(Y, {}^p!H^i(Rf_* k_X[n])), $$ i.e., the spectral-sequence terms in degrees $(p, q)$ with $p + q = p_{0} + q_{0}$ assemble directly without higher differentials. By a standard convergence argument, $d_{r} = 0$ for all $r \geq 2$ , so the spectral sequence degenerates at $E_{2}$ .

Application: for $f : X \to Y$ a proper smooth morphism (e.g., a smooth projective family), the Leray spectral sequence in classical cohomology degenerates at $E_{2}$ (Deligne 1968, with the algebraic-geometric proof via the Weil conjectures). The perverse decomposition theorem generalises this to any proper $f$ with $X$ smooth, including non-smooth $f$ with singular fibres — the singular-fibre contribution is captured by the IC complex multiplicities $m_{i, α}$ . Rubric: full credit for the theorem statement, the spectral-sequence statement, and the degeneration argument.

Exercise 5 (medium, short-answer).

State the Riemann-Hilbert correspondence of Kashiwara and Mebkhout, and identify what category sits opposite $Perv (X^{an}, C)$ under the equivalence for $X$ a smooth complex algebraic variety.

Hint

The opposite category is a category of differential modules with a regularity condition. The functor in one direction is the de Rham functor $DR (M) = Ω_{X}^{∙} \otimes_{D_{X}}^{L} M$ .

Answer

Theorem (Riemann-Hilbert correspondence; Kashiwara 1984 Publ. RIMS 20; Mebkhout 1980 thesis). Let $X$ be a smooth complex algebraic variety (or smooth complex analytic manifold). The de Rham functor $$ \mathrm{DR} : D^b_{rh}(\mathcal{D}X) \xrightarrow{\sim} D^b_c(X^{\mathrm{an}}, \mathbb{C}), \qquad \mathrm{DR}(\mathcal{M}) := \Omega^\bullet_X[\dim X] \otimes{\mathcal{D}_X}^L \mathcal{M} $$ is an equivalence of triangulated categories from the bounded derived category of regular holonomic $D_{X}$ -modules to the bounded constructible derived category of complexes of $C$ -sheaves on the underlying complex analytic space $X^{an}$ . The equivalence is t-exact for the natural t-structure on $D_{r h}^{b} (D_{X})$ (whose heart is the abelian category $Mod_{r h} (D_{X})$ of regular holonomic $D_{X}$ -modules) and the middle-perversity t-structure on $D_{c}^{b} (X^{an}, C)$ . Restricting to hearts gives an equivalence of abelian categories $$ \mathrm{Mod}_{rh}(\mathcal{D}_X) \xrightarrow{\sim} \mathrm{Perv}(X^{\mathrm{an}}, \mathbb{C}). $$

The category $Mod_{r h} (D_{X})$ consists of finitely generated $D_{X}$ -modules whose characteristic variety $Char (M) \subset T^{*} X$ is holonomic (Lagrangian in the symplectic geometry of $T^{*} X$ ) and whose connection at infinity is regular (no irregular singularities). The de Rham functor pairs a $D_{X}$ -module with its analytic solution complex. The Riemann-Hilbert correspondence is the modern incarnation of the original Hilbert 21st-problem question: which monodromy data come from differential systems with regular singularities, with the answer "all of them, via the perverse-sheaf encoding". Rubric: full credit for the theorem statement, the de Rham functor, the t-exactness statement, and the heart-level equivalence.

Exercise 6 (medium, short-answer).

Show that $Perv (X)$ is Artinian and Noetherian by reducing to the case of a single stratum and using that constructible sheaves on a smooth stratum (locally constant of finite rank) form an abelian category with finite Jordan-Hölder length.

Hint

Inductively cut $X$ along strata. On a single smooth stratum, perverse sheaves with constant cohomology dimension reduce to local systems, which are representations of the fundamental group on finite-dimensional vector spaces — these are Artinian and Noetherian by classical representation theory.

Answer

Argument by induction on the dimension of $X$ and on the number of strata. Stratify $X = U ⊔ F$ as an open dense $U$ and a closed complement $F$ of strictly smaller dimension, with $U$ smooth and $F$ a union of strata.

Base case. $X = U$ smooth and irreducible of dimension $d$ . Perverse sheaves on $U$ are concentrated in cohomological degree $- d$ (since the support-dimension condition forces all support to be in dimension $d$ , hence degree $- d$ ). So $Perv (U) ≃ Loc (U)$ , the category of local systems of finite-dimensional $k$ -vector spaces on $U$ (shifted by $d$ ), which is equivalent to finite-dimensional representations of $π_{1} (U)$ . This is an abelian category of finite length: every object has a finite composition series whose simple subquotients are irreducible representations of $π_{1} (U)$ , and chains of submodules terminate (descending chain condition = Artinian, ascending chain condition = Noetherian).

Inductive step. Given a perverse sheaf $P \in Perv (X)$ , write $j : U ↪ X$ for the open inclusion and $i : F ↪ X$ for the closed complementary inclusion. The recollement $$ {}^p!H^0(j_! j^* \mathcal{P}) \to \mathcal{P} \to {}^p!H^0(i_* i^* \mathcal{P}) \to {}^p!H^0(j_! j^* \mathcal{P})[1] $$ in $Perv (X)$ exhibits $P$ as a finite extension of (i) an object pushed forward from $Perv (U)$ and (ii) an object pushed forward from $Perv (F)$ . By the base case and the induction hypothesis (on dimension of $F < dim X$ ), both $Perv (U)$ and $Perv (F)$ have finite Jordan-Hölder length. Composition series of $P$ are built by interleaving composition series of $j^{*} P$ on $U$ and $i^{*} P$ on $F$ , producing a finite series. Hence every $P$ has finite Jordan-Hölder length in $Perv (X)$ , and $Perv (X)$ is Artinian and Noetherian.

The simple subquotients are simple perverse sheaves on $U$ pushed forward (intermediate extensions $j_{! *}$ of irreducible local systems on smooth locally closed strata) or simple perverse sheaves on $F$ (recursively decomposed). Combining: every simple of $Perv (X)$ is $IC_{\overline{Z}} (L)$ for some $(Z, L)$ pair, recovering the BBD §4.3 classification. Rubric: full credit for the induction setup, the base case, the recollement use, and the conclusion.

Exercise 7 (hard, short-answer).

State and explain the support theorem of Ngô 2010 Publ. Math. IHÉS 111 §7 for the Hitchin fibration: how the BBD decomposition theorem applied to a proper map of stacks (the Hitchin fibration) constrains which strata can carry simple summands of $R f_{*} IC$ .

Hint

The support theorem says that for a "good" proper morphism between certain stacks, the supports $\overline{Z_{α}}$ of the simple summands appearing in the decomposition theorem are forced to be of maximal dimension (the whole base, or specific large-codimension loci), eliminating many a priori possible smaller-dimensional supports.

Answer

Theorem (Ngô's support theorem; Publ. Math. IHÉS 111, 2010, §7.1.13). Let $f : M \to S$ be a proper morphism of finite-type Deligne-Mumford stacks over an algebraically closed field, with $M$ smooth of pure dimension and $f$ of relative dimension $d$ . Assume:

$f$ is abelian-fibred: a smooth open subset $S^{\circ} \subset S$ has fibres that are abelian varieties (or extensions thereof), and the abelian-variety structure varies algebraically.
The codimension condition: for every irreducible closed $Z ⊊ S$ with $Z \cap S^{\circ} = \emptyset$ on which the relevant cohomological local system has positive monodromy, $codim_{S} (Z) > δ$ where $δ$ is a specific invariant related to the dimension of the abelian fibres.

Then the simple summands $IC_{\overline{Z_{α}}} (L_{α})$ appearing in the BBD decomposition of $R f_{*} \overline{Q}_{ℓ} [dim M]$ have supports $\overline{Z_{α}}$ that are either (i) all of $S$ (so the simple is $IC_{S} (L_{α})$ for a local system $L_{α}$ on $S^{\circ}$ ) or (ii) of codimension exactly the predicted invariant; no intermediate-codimension supports appear.

Ngô applied this to the Hitchin fibration $h : M_{G} \to A_{G}$ for a reductive group $G$ , where $M_{G}$ is the moduli stack of $G$ -Higgs bundles on a smooth projective curve $C$ and $A_{G}$ is the Hitchin base. The Hitchin fibration is abelian-fibred (generic fibres are abelian varieties — Picard schemes of spectral curves), and the codimension condition is verified using the symmetry properties of $h$ and the structure of the elliptic part of $A_{G}$ . The support theorem then forces the simple summands of $R h_{*} \overline{Q}_{ℓ}$ over the elliptic regular open subset $A_{G}^{ell, reg}$ to be of maximal support (= $A_{G}^{ell, reg}$ itself), eliminating many possible smaller-codimension contributions. This is the geometric input that lets Ngô identify the cohomology of the Hitchin fibration with the right side of the Fundamental Lemma after Grothendieck-Lefschetz trace formula. Rubric: full credit for the theorem statement (abelian-fibred condition + codimension condition + maximal-support conclusion) and the application to the Hitchin fibration / Fundamental Lemma.

Exercise 8 (hard, short-answer).

For $X$ a smooth complex projective variety, identify the hypercohomology $H^{*} (X, IC_{X})$ with the classical singular cohomology $H^{*} (X, k)$ , and explain why this identification fails when $X$ is singular — what is computed instead, and what is its relation to ordinary singular cohomology?

Hint

For $X$ smooth, $IC_{X} = k_{X} [dim X]$ is the shifted constant sheaf, so the hypercohomology shifts by $dim X$ . For $X$ singular, $IC_{X}$ encodes intersection cohomology — a different invariant.

Answer

Smooth case. For $X$ a smooth complex projective variety of complex dimension $d$ , $IC_{X} = k_{X} [d]$ . By definition, $$ \mathbb{H}^n(X, \mathrm{IC}_X) = \mathbb{H}^n(X, k_X[d]) = H^{n+d}(X, k), $$ the shifted singular cohomology. So $H^{*} (X, IC_{X})$ recovers all of $H^{*} (X, k)$ , with a dimensional shift.

Singular case. For $X$ a complex projective variety that is not smooth, $IC_{X}$ is the intersection cohomology complex on the underlying singular space, which is not quasi-isomorphic to the shifted constant sheaf. The hypercohomology $$ IH^n(X, k) := \mathbb{H}^{n - d}(X, \mathrm{IC}_X) $$ is the intersection cohomology of $X$ (with middle perversity, due to Goresky-MacPherson 1980), an invariant distinct from the singular cohomology $H^{*} (X, k)$ . The relation:

For $X$ smooth, $I H^{*} (X) = H^{*} (X, k)$ — the two agree.
For $X$ singular but rational-homology-manifold (e.g., $X$ has only quotient singularities), $I H^{*} (X) = H^{*} (X, k)$ still.
For $X$ with worse singularities (e.g., a nodal projective variety), $I H^{*} (X)$ is a proper modification of $H^{*} (X, k)$ . The Poincaré-duality-respecting cohomology theory is $I H^{*}$ , not $H^{*}$ . Classical singular cohomology $H^{*} (X, k)$ fails Poincaré duality for singular $X$ ; intersection cohomology restores it via $I H^{n} (X) \otimes I H^{2 d - n} (X) \to k$ , a perfect pairing.
The BBD decomposition theorem applied to a resolution of singularities $π : X \to X$ (proper with $X$ smooth) gives $R π_{*} k_{X} [d] ≅ IC_{X} \oplus α ⨁ IC_{\overline{Z_{α}}} (L_{α}) [n_{α}],$ expressing $I H^{*} (X)$ as a canonical summand of $H^{*} (X)$ . The "extra" summands are supported on the singular locus of $X$ and capture the geometry of the resolution.

So $H^{*} (X, IC_{X})$ recovers $H^{*} (X, k)$ for smooth $X$ and intersection cohomology $I H^{*} (X)$ for singular $X$ ; the latter is the Poincaré-self-dual cohomology theory that singular cohomology fails to be on a singular space. Rubric: full credit for the smooth identification, the singular intersection-cohomology identification, the Poincaré duality property, and the resolution-of-singularities expression.

Lean formalization Intermediate+

lean_status: none — Mathlib does not support perverse sheaves at any level of the stack. The intended formalisation reads schematically:

import Mathlib.CategoryTheory.Triangulated.Basic
import Mathlib.AlgebraicGeometry.Sheaf
import Mathlib.Topology.Sheaves.Constructible  -- aspirational

namespace Codex.AlgGeom.PerverseSheaves

variable {X : Type*} [TopologicalSpace X] [AlgebraicVariety X]
  (k : Type*) [Field k] [CharZero k]

/-- A complex of k-sheaves on X is constructible with respect to a fixed
    stratification if each cohomology sheaf is locally constant of finite
    rank on each stratum. -/
def IsConstructible (F : Cochain (Sheaf k X)) : Prop := sorry

/-- The bounded constructible derived category of X. -/
def DbConstructible (X : Type*) [TopologicalSpace X] [AlgebraicVariety X]
    (k : Type*) [Field k] : Type* := sorry

/-- The Verdier dualising complex of X via the structure map a_X : X → pt. -/
noncomputable def dualisingComplex : DbConstructible X k := sorry

/-- The Verdier dual functor. -/
noncomputable def verdierDual :
    DbConstructible X k ⥤ (DbConstructible X k)ᵒᵖ := sorry

/-- The half ^pD^{≤ 0}(X) of the middle-perversity t-structure. -/
def pNegHalf : Set (DbConstructible X k) :=
  { F | ∀ i, supportDim (cohomologySheaf F i) ≤ -i }

/-- The half ^pD^{≥ 0}(X) of the middle-perversity t-structure. -/
def pPosHalf : Set (DbConstructible X k) :=
  { F | verdierDual.obj F ∈ pNegHalf }

/-- The middle-perversity t-structure on D^b_c(X). -/
noncomputable def perverseTStructure :
    Codex.Triangulated.tStructure (DbConstructible X k) where
  D_le := pNegHalf
  D_ge := pPosHalf
  shift_le := sorry
  shift_ge := sorry
  orthogonality := sorry
  truncation_triangle := sorry

/-- The category of perverse sheaves on X = the heart of the perverse
    t-structure. -/
def PerverseSheaves : Type* :=
  FullSubcategory (· ∈ Codex.Triangulated.tStructure.heart perverseTStructure)

/-- The category of perverse sheaves is abelian. -/
instance : Abelian (PerverseSheaves X k) := sorry

/-- The category of perverse sheaves is Artinian and Noetherian. -/
instance : IsArtinian (PerverseSheaves X k) := sorry
instance : IsNoetherian (PerverseSheaves X k) := sorry

/-- The intermediate extension of a local system on a smooth locally closed
    subvariety. -/
noncomputable def intermediateExtension
    {Z : Set X} (hZ : IsSmoothLocallyClosed Z)
    (L : LocalSystem k Z) (hL : Irreducible L) :
    PerverseSheaves X k := sorry

/-- The intersection cohomology complex of (Z, L). -/
noncomputable def IC {Z : Set X} (hZ : IsSmoothLocallyClosed Z)
    (L : LocalSystem k Z) (hL : Irreducible L) :
    PerverseSheaves X k := intermediateExtension hZ L hL

/-- Simple objects of Perv(X) are classified by (Z, L) pairs (BBD §4.3). -/
theorem simples_classified (P : PerverseSheaves X k) (hP : Simple P) :
    ∃ (Z : Set X) (hZ : IsSmoothLocallyClosed Z)
      (L : LocalSystem k Z) (hL : Irreducible L),
      P ≅ IC hZ L hL := sorry

/-- BBD decomposition theorem: for proper f : X → Y with X smooth, the
    derived direct image of the shifted constant sheaf decomposes as a
    direct sum of shifted IC complexes. -/
theorem BBD_decomposition
    {X Y : Type*} [AlgebraicVariety X] [AlgebraicVariety Y]
    [IsSmooth X] (f : X → Y) (hf : IsProper f) (n := dim X) :
    ∃ (decomp : ∀ (i : ℤ), PerverseSheaves Y k),
      Rf_star (constSheaf X k)[n] ≅
        ⨁ (i : ℤ), (decomp i)[-i] ∧
      ∀ i, Semisimple (decomp i) := sorry

/-- Riemann-Hilbert correspondence: regular holonomic D-modules on a smooth
    complex variety are equivalent to perverse sheaves on the analytic
    space. -/
theorem riemann_hilbert {X : Type*} [SmoothComplexVariety X] :
    RegularHolonomicDModules X ≌ PerverseSheaves Xan ℂ := sorry

end Codex.AlgGeom.PerverseSheaves

The proof gap is enormous. The construction of $D_{c}^{b} (X)$ requires the constructibility predicate on cohomology sheaves of complexes of sheaves, which in turn requires a stratification API that Mathlib does not have. The Verdier dualising complex requires the six-functor formalism (also a Mathlib gap, see 04.03.16) including the existence of $a_{X}^{!}$ for a structure map $X \to pt$ . The middle-perversity t-structure depends on the t-structure package (also a gap, see 04.03.18) plus the support-dimension condition formulated correctly. The intermediate extension $j_{! *}$ requires the perverse-cohomology functors and the image-in-an-abelian-category construction in $Perv$ , all dependent on the previous gaps. The BBD decomposition theorem requires Deligne's purity and weight filtration in the $ℓ$ -adic setting plus spreading-out and Artin comparison, none of which are formalised. The Riemann-Hilbert correspondence requires the de Rham functor on regular holonomic D-modules plus the equivalence statement, again far beyond Mathlib's current scope.

Each component is formalisable in principle. The Mathlib roadmap to perverse sheaves is the long-form contribution agenda: stratifications, constructible sheaves, six functors with the dualising complex, the t-structure package, the perverse t-structure with the support-dimension condition, intermediate extensions, and the decomposition theorem — in that order. A natural intermediate target is the smooth case (where $IC_{X} = k_{X} [dim X]$ is just a shifted constant sheaf and many statements simplify) before tackling the full singular theory.

Advanced results Master

Theorem (BBD decomposition theorem; Beilinson-Bernstein-Deligne 1982 §6.2.5; Gabber 1981; Saito 1988 in Hodge-theoretic form; de Cataldo-Migliorini 2005 transcendental proof). Let $f : X \to Y$ be a proper morphism of complex algebraic varieties with $X$ smooth, and let $k$ be a field of characteristic zero. Then in $D_{c}^{b} (Y, k)$ : $$ Rf_* k_X[\dim X] \cong \bigoplus_{i \in \mathbb{Z}} {}^p!H^i(Rf_* k_X[\dim X])[-i], $$ and each perverse cohomology summand is a semisimple perverse sheaf on $Y$ : $$ {}^p!H^i(Rf_* k_X[\dim X]) \cong \bigoplus_\alpha \mathrm{IC}{\overline{Z\alpha}}(\mathcal{L}\alpha)^{\oplus m{i, \alpha}}. $$

The decomposition theorem is the deepest single statement about proper maps in algebraic geometry. It implies a vast generalisation of the hard Lefschetz theorem (the simple summands of $R f_{*}$ admit hard-Lefschetz isomorphisms on each intersection cohomology piece), degeneration of every spectral sequence attached to $f$ (Leray, perverse Leray) at $E_{2}$ , and the topological invariance of intersection cohomology of singular projective varieties under resolution of singularities. The proof has three independent paths: Deligne's Frobenius-weight proof in Weil II (1980) lifted to perverse sheaves by BBD §5–§6; Saito's mixed-Hodge-module proof using polarised pure Hodge modules; and de Cataldo-Migliorini's 2005 transcendental Hodge-theoretic proof using the relative hard Lefschetz on cohomology of fibres. The applications are pervasive: geometric Langlands (Beilinson-Bernstein, Mirković-Vilonen), Ngô's Fundamental Lemma, Kazhdan-Lusztig theory, and the topology of Hilbert schemes and moduli spaces.

Theorem (Riemann-Hilbert correspondence; Kashiwara 1984 Publ. RIMS 20; Mebkhout 1980 thesis). Let $X$ be a smooth complex algebraic variety or smooth complex analytic manifold. The de Rham functor $$ \mathrm{DR} : D^b_{rh}(\mathcal{D}X) \xrightarrow{\sim} D^b_c(X^{\mathrm{an}}, \mathbb{C}), \quad \mathrm{DR}(\mathcal{M}) := \Omega^\bullet_X[\dim X] \otimes{\mathcal{D}X}^L \mathcal{M} $$ *is a t-exact equivalence of triangulated categories, restricting to an equivalence of abelian categories on hearts: $\mathrm{Mod}{rh}(\mathcal{D}_X) \xrightarrow{\sim} \mathrm{Perv}(X^{\mathrm{an}}, \mathbb{C})$.*

The Riemann-Hilbert correspondence solves the inverse problem to differential systems with regular singularities: given a constructible system of finite-dimensional $C$ -vector-space data on the analytic topology of $X$ (encoded as a perverse sheaf), it is the de Rham realisation of a unique regular holonomic D-module. The equivalence is functorial under pullback, proper pushforward, tensor product, and Verdier/duality on both sides, compatible with the six-functor formalism. Applications: Beilinson-Bernstein localisation transfers between $g$ -modules and D-modules on the flag variety, and through Riemann-Hilbert, to perverse sheaves on the flag variety — the bridge from representation theory to geometry that underlies the proof of the Kazhdan-Lusztig conjectures (Beilinson-Bernstein 1981 C.R. Acad. Sci. Paris 292; Brylinski-Kashiwara 1981 Invent. Math. 64).

Theorem (classification of simple perverse sheaves; BBD 1982 §4.3). Let $X$ be a complex algebraic variety. The set of isomorphism classes of simple objects of $Perv (X)$ is in bijection with the set of pairs $(\overline{Z}, L)$ where $\overline{Z} \subset X$ is an irreducible closed subvariety and $L$ is an irreducible local system on the smooth locus $Z = \overline{Z}^{reg}$ (up to isomorphism), via the intermediate-extension construction $(\overline{Z}, L) \mapsto IC_{\overline{Z}} (L)$ .

The classification theorem is the structural backbone of perverse-sheaf theory. Combined with the Artinian + Noetherian property, every perverse sheaf has a finite multiset of $(\overline{Z}_{α}, L_{α})$ "pieces" appearing in its Jordan-Hölder filtration, and the Grothendieck group $K_{0} (Perv (X))$ is generated by these classes. The classification is exploited in geometric representation theory (the simple perverse sheaves on the affine Grassmannian classify the irreducible representations of the Langlands dual group via the geometric Satake equivalence) and in the theory of mixed Hodge modules (each simple perverse sheaf with an appropriate Hodge structure becomes a simple polarised pure Hodge module).

Theorem (Saito's mixed Hodge modules; Saito 1988 Publ. RIMS 24). Let $X$ be a complex algebraic variety. There exists an abelian category $MHM (X)$ of mixed Hodge modules on $X$ , fibred over $Perv (X, Q)$ via a faithful functor $rat : MHM (X) \to Perv (X, Q)$ that forgets the Hodge data. Each object of $MHM (X)$ carries a polarised mixed Hodge structure on its underlying perverse sheaf, and the six functors $f^, f_*, f_!, f^!, \otimes, R\mathcal{H}om $l i f t f r o m$ D^b\mathrm{Perv}(X) \simeq D^b_c(X, \mathbb{Q}) $t o$ D^b\mathrm{MHM}(X)$ in a compatible way.*

Saito's theory is the Hodge-theoretic refinement of perverse sheaves: every perverse sheaf $F \in Perv (X, Q)$ "comes from" a mixed Hodge module $M \in MHM (X)$ with $rat (M) = F$ , and the Hodge structure on the cohomology $H^{*} (X, M)$ refines the rational cohomology. The decomposition theorem in the Hodge-theoretic setting (Saito's version) is sharper: the simple summands of a proper pushforward are not only IC complexes but pure Hodge modules of a specific weight, and the weight filtration is canonical. Applications: Saito's mixed Hodge modules are the input to the recent topological-mirror-symmetry program (Konstevich-Soibelman; Donaldson-Thomas invariants via vanishing cycles of mixed Hodge modules on moduli spaces of sheaves) and to the proof of the BBD decomposition theorem via Hodge theory (de Cataldo-Migliorini 2005, 2009).

Theorem (de Cataldo-Migliorini Hodge-theoretic decomposition; Ann. Sci. ÉNS 38, 2005). Let $f : X \to Y$ be a proper morphism of complex algebraic varieties with $X$ smooth. The BBD decomposition theorem holds with the simple summands $IC_{\overline{Z_{α}}} (L_{α})$ underlying pure Hodge modules of explicit weight, with the decomposition compatible with the polarised Hodge structures on intersection cohomology.

The de Cataldo-Migliorini proof of the decomposition theorem is the modern transcendental approach: it avoids the spreading-out + characteristic- $p$ + Frobenius machinery of BBD §6 and instead works entirely in the complex Hodge-theoretic setting using the relative hard Lefschetz theorem on cohomology of fibres of $f$ and the polarisation of intersection cohomology. The argument is conceptually cleaner and exposes the geometric mechanism (the relative-hard-Lefschetz semisimplicity input) rather than relying on Deligne's purity. The proof has been generalised to algebraic stacks, to the relative setting, and to the equivariant setting.

Theorem (geometric Satake equivalence; Lusztig 1983 Astérisque 101–102; Ginzburg 1995; Mirković-Vilonen 2007 Ann. Math. 166). Let $G$ be a connected reductive complex algebraic group and let $Gr_{G} = G ((t)) / G [[t]]$ be the affine Grassmannian of $G$ . The category $Perv_{G [[t]]} (Gr_{G}, Q)$ of $G [[t]]$ -equivariant perverse sheaves on $Gr_{G}$ is a Tannakian category equivalent to the category $Rep (G_{Q}^{\lor})$ of finite-dimensional rational representations of the Langlands dual group $G^{\lor}$ : $$ \mathrm{Perv}_{G[[t]]}(\mathrm{Gr}G, \mathbb{Q}) \simeq \mathrm{Rep}(G^\vee\mathbb{Q}). $$ The tensor structure on the left comes from the fusion product (convolution of perverse sheaves under the Beilinson-Drinfeld Grassmannian), and the tensor structure on the right is the standard tensor product of representations.

The geometric Satake equivalence is the foundational input to the geometric Langlands program: it identifies the unramified local Langlands correspondence on the spherical side with a geometric correspondence between perverse sheaves on the affine Grassmannian and representations of the dual group. The proof uses the BBD decomposition theorem applied to the convolution morphism $Gr_{G} \times_{G [[t]]} Gr_{G} \to Gr_{G}$ (a proper map of stratified ind-schemes) to define the fusion product, and the semisimple structure on $Perv_{G [[t]]} (Gr_{G}, Q)$ implied by BBD. The Mirković-Vilonen 2007 paper provides the modern characteristic-free formulation of the equivalence and establishes its functoriality.

Theorem (Ngô's support theorem and the Fundamental Lemma; Ngô 2010 Publ. Math. IHÉS 111). The geometric proof of the Fundamental Lemma for Lie algebras uses the BBD decomposition theorem applied to the Hitchin fibration $h : M_{G} \to A_{G}$ for a reductive group $G$ over a function field. The support theorem (§7.1.13) constrains the simple perverse-sheaf summands of $Rh_ \overline{\mathbb{Q}}_\ell$ over the elliptic regular open of the Hitchin base to be of maximal support, which together with the abelian-variety symmetry of the Hitchin fibres produces the trace-formula identities equivalent to the Fundamental Lemma.*

Ngô's 2010 proof is the deepest single application of the BBD decomposition theorem to date. The Fundamental Lemma, conjectured by Langlands-Shelstad in 1979 and a longstanding obstacle to the trace-formula-based attack on the Langlands program, is proved via the geometric mechanism of the Hitchin fibration's perverse-sheaf decomposition. The proof uses the support theorem (a refinement of BBD applicable in the abelian-fibred setting), the Goresky-Kottwitz-MacPherson endoscopic stratification of the Hitchin base, and the relative Lefschetz hard-Lefschetz theorem for Hitchin fibres. Ngô received the Fields Medal in 2010 for this work.

Synthesis. The perverse-sheaf framework builds toward an abelian-category understanding of constructible cohomology on singular and stratified spaces, and the foundational reason it works is that the middle-perversity t-structure combines the support-dimension condition with Verdier duality to produce an abelian heart whose simple objects are exactly the intersection cohomology complexes attached to (stratum, local system) pairs. The package is dual to itself under Verdier duality, with proper pushforward respecting the t-structure (the BBD decomposition theorem) and the open-closed recollement providing the inductive structure. The central insight is that perverse sheaves are the canonical abelian category for studying singular varieties — they restore Poincaré duality through intersection cohomology, support the decomposition theorem for proper maps, classify their simples through stratified local systems, and bridge to D-modules via Riemann-Hilbert and to Hodge theory via Saito's mixed Hodge modules.

The formalism appears again in 04.03.18 (t-structures and hearts), where the middle-perversity t-structure is the canonical non-standard example; in 04.03.16 (six-functor formalism), where Verdier duality and proper pushforward interact with the perverse t-structure to produce the decomposition theorem; in 03.04.14 (hypercohomology of complexes of sheaves), where $H^{*} (X, IC_{X})$ computes intersection cohomology of singular projective $X$ ; and in 04.03.02 (local systems), where the data of an irreducible local system on a smooth locally closed subvariety is exactly half of the input to a simple perverse sheaf. The recursion stabilises: every perverse sheaf decomposes into intermediate extensions of local systems on strata, every proper morphism's derived pushforward decomposes into shifted intermediate extensions on the base, and the resulting framework is the abelian-category-level shadow of derived algebraic geometry on singular spaces.

Full proof set Master

Proposition (abelianness of $Perv (X)$ ). $Perv (X)$ is an abelian category.

Proof. By the BBD §1.3.6 theorem [04.03.18 Key Theorem], the heart of any t-structure on a triangulated category is abelian. The middle-perversity pair $(^{p} D^{\leq 0},^{p} D^{\geq 0})$ is a t-structure on $D_{c}^{b} (X)$ (the BBD axioms TR1–TR3 of 04.03.18 are verified directly: shift stability follows from the dimension-bound being shifted by $1$ , orthogonality follows from Verdier duality and the dimension inequality, and the truncation triangle is constructed via the standard truncation-and-glue argument along strata using the open-closed recollement). Hence $Perv (X) =^{p} D^{\leq 0} \cap^{p} D^{\geq 0}$ is the heart of a t-structure on the triangulated category $D_{c}^{b} (X)$ , and therefore abelian. The short exact sequences in $Perv (X)$ correspond bijectively to distinguished triangles in $D_{c}^{b} (X)$ all three of whose terms lie in $Perv (X)$ . $□$

Proposition (Artinian + Noetherian property). Every object of $Perv (X)$ has a finite Jordan-Hölder composition series whose simple subquotients are intersection cohomology complexes.

Proof. Argument by induction on the dimension of $X$ and on the number of strata, using the recollement triangle for an open-closed decomposition $X = U ⊔ F$ with $U$ smooth open dense and $F$ closed of smaller dimension (Exercise 6 above). The base case $X$ smooth and irreducible reduces $Perv (X) ≃ Loc (X)$ to the category of finite-dimensional $π_{1} (X)$ -representations, which is Artinian and Noetherian by classical representation theory. The inductive step pulls in the recollement to extend composition series across the open-closed cut. Combining all simple subquotients across the stratification produces the classification: every simple of $Perv (X)$ is $IC_{\overline{Z}} (L)$ for $(Z, L)$ a smooth locally closed irreducible subvariety with an irreducible local system. $□$

Proposition (Verdier duality on $IC$ ). For $Z$ smooth locally closed irreducible in $X$ of dimension $d_{Z}$ with $j : Z ↪ X$ and $L$ an irreducible local system on $Z$ , $$ \mathbb{D}X(\mathrm{IC}{\overline{Z}}(\mathcal{L})) \cong \mathrm{IC}_{\overline{Z}}(\mathcal{L}^\vee), $$ where $L^{\lor} = H o m (L, k_{Z})$ is the dual local system.

Proof. The dualising complex on $Z$ smooth of complex dimension $d_{Z}$ is $ω_{Z}^{∙} = k_{Z} [2 d_{Z}]$ (with the same convention on $X$ via the structure map, since $X$ is irreducible). The Verdier dual of the shifted local system $L [d_{Z}]$ on $Z$ is $$ \mathbb{D}Z(\mathcal{L}[d_Z]) = R\mathcal{H}om(\mathcal{L}[d_Z], \omega^\bullet_Z) = R\mathcal{H}om(\mathcal{L}[d_Z], k_Z[2 d_Z]) = \mathcal{H}om(\mathcal{L}, k_Z)[d_Z] = \mathcal{L}^\vee[d_Z]. $$ By the duality interchange $\mathbb{D}X \circ j{!*} = j{!} \circ \mathbb{D}Z $o n t h e h e a r t o f t h e p er v er se t - s t r u c t u r e (a co n se q u e n ceo f$ \mathbb{D} $b e in g t - e x a c t f or t h e p er v er se t - s t r u c t u r e an d t h e d u a l i t y b e tw ee n$ j! $an d$ j_$ via Verdier), we have $$ \mathbb{D}X(\mathrm{IC}{\overline{Z}}(\mathcal{L})) = \mathbb{D}X(j{!}(\mathcal{L}[d_Z])) = j_{!}(\mathbb{D}Z(\mathcal{L}[d_Z])) = j{!*}(\mathcal{L}^\vee[d_Z]) = \mathrm{IC}{\overline{Z}}(\mathcal{L}^\vee). $$ For $L$ self-dual (e.g., $L = k_{Z}$ the constant local system, or $L$ equipped with a non-degenerate self-pairing), this gives $\mathbb{D}X(\mathrm{IC}{\overline{Z}}(\mathcal{L})) = \mathrm{IC}{\overline{Z}}(\mathcal{L}) $— t h e I C co m pl e x i sse l f - V er d i er - d u a l .$ \square$

Proposition (intermediate extension as image in $Perv$ ). Let $j : Z ↪ X$ be a locally closed embedding with $Z$ smooth irreducible of dimension $d_{Z}$ , and let $L$ be an irreducible local system on $Z$ . The intermediate extension $j_{!}(\mathcal{L}[d_Z]) $i sc ha r a c t er i se d a s t h e u ni q u e p er v er see x t e n s i o n$ \mathcal{P} \in \mathrm{Perv}(X) $w i t h$ j^* \mathcal{P} = \mathcal{L}[d_Z] $an d n o n o n - z er o p er v er ses u b - o bj ec t or p er v er se q u o t i e n t s u pp or t e d o n$ \overline{Z} \setminus Z$.*

Proof. Consider the natural morphism $α :^{p} H^{0} (j_{!} L [d_{Z}]) \to^{p} H^{0} (j_{*} L [d_{Z}])$ induced by the adjunction $j_{!} \to j_{*}$ on $D_{c}^{b} (X)$ and the perverse-cohomology functor. The image of $α$ in the abelian category $Perv (X)$ is by definition $j_{! *} (L [d_{Z}])$ .

Uniqueness. Suppose $P \in Perv (X)$ satisfies $j^{*} P = L [d_{Z}]$ and has no non-zero perverse sub-object or quotient supported on $\overline{Z} ∖ Z$ . Restricting the recollement triangle $i_{!} i^{!} P \to P \to R j_{*} j^{*} P$ to the perverse t-structure (where $i : \overline{Z} ∖ Z ↪ X$ is the closed complement), the perverse cohomology of $i_{!} i^{!} P$ is supported on $\overline{Z} ∖ Z$ and must vanish by hypothesis. So $P \sim^{p} H^{0} (R j_{*} j^{*} P) =^{p} H^{0} (j_{*} L [d_{Z}])$ at the perverse-cohomology level. Dually, $P$ is a quotient of $^{p} H^{0} (j_{!} j^{*} P) =^{p} H^{0} (j_{!} L [d_{Z}])$ . Together, $P$ is canonically the image of $α$ , which is $j_{! *} (L [d_{Z}])$ . Hence the characterisation is unique.

Existence. $j_{! *} (L [d_{Z}]) = im (α)$ has no non-zero perverse sub-object supported on $\overline{Z} ∖ Z$ because such a sub-object would have to come from the kernel of $α$ , which is forced to be zero by the construction (kernel of the natural map $^{p} H^{0} (j_{!}) \to^{p} H^{0} (j_{*})$ is supported on the boundary by adjunction, and the image construction quotients by it). Dually for non-zero perverse quotients. Hence $j_{! *} (L [d_{Z}])$ satisfies the universal property. $□$

Proposition (Beilinson reconstruction). Let $X$ be a complex algebraic variety. The natural functor $$ \mathrm{real} : D^b(\mathrm{Perv}(X)) \to D^b_c(X) $$ is a triangulated equivalence (Beilinson 1987 Funct. Anal. Appl. 21; BBD 1982 §3.1).

Proof sketch. The functor sends a complex of perverse sheaves to its associated constructible complex via the standard t-structure on $D^{b} (Perv (X))$ matched with the middle-perversity t-structure on $D_{c}^{b} (X)$ . Fully faithfulness is the equality $Hom_{D^{b} Perv} (P, Q [n]) = Hom_{D_{c}^{b}} (P, Q [n])$ for $P, Q \in Perv (X)$ , which holds because both Hom-groups are computed by the same perverse-cohomology long-exact sequence. Essential surjectivity uses the boundedness of $D_{c}^{b} (X)$ in the perverse t-structure (every constructible complex has finitely many non-zero perverse-cohomology sheaves) combined with the construction of any $F \in D_{c}^{b} (X)$ from its perverse-cohomology spectral sequence $^{p} H^{p} (^{p} τ_{\geq q} F) \Rightarrow^{p} H^{p + q} (F)$ , which converges by Artin vanishing on algebraic varieties. The key Artin vanishing input: for any constructible $F$ on $X$ algebraic of dimension $d$ , $^{p} H^{i} (F) = 0$ for $∣ i ∣ > d$ , bounding the perverse cohomology. $□$

Connections Master

t-Structure on a triangulated category 04.03.18. The perverse t-structure is the canonical non-standard example of a t-structure, defined by a support-dimension inequality and its Verdier dual. Its heart is the abelian category $Perv (X)$ , providing the abelian-category-level extraction of constructible cohomological information. The BBD framework studied here originated as the application that drove the development of the t-structure axiomatisation.
Six-functor formalism 04.03.16. The perverse t-structure interacts with the six operations in a controlled way: $j^{*}$ is t-exact for the perverse t-structure (open inclusion), $i_{!}$ and $i^{*}$ are right t-exact (closed inclusion), $i_{*}$ and $i^{!}$ are left t-exact, $R f_{!}$ is right t-exact for arbitrary $f$ , and $R f_{*}$ is left t-exact. Verdier duality $D$ is t-exact and interchanges $f^{*}$ with $f^{!}$ and $f_{!}$ with $f_{*}$ at the level of perverse t-structures.
Local systems 04.03.02. Local systems are the building blocks of perverse sheaves through the simple classification: every simple perverse sheaf is the intermediate extension of an irreducible local system on a smooth locally closed stratum. The data $(Z, L)$ is the "perverse signature" of a simple, and the assembly of simples by intermediate extension produces the full abelian category $Perv (X)$ .
Hypercohomology of a complex of sheaves 03.04.14. The intersection cohomology $I H^{*} (X)$ of a singular projective variety $X$ is the hypercohomology $H^{*- d} (X, IC_{X})$ . The hypercohomology spectral sequence on $IC_{X}$ computes $I H^{*}$ from local data, and the BBD decomposition theorem for a resolution of singularities $X \to X$ expresses $I H^{*} (X)$ as a canonical direct summand of $H^{*} (X)$ .
Sheaf cohomology — Leray spectral sequence general form 04.03.15. The perverse Leray spectral sequence $^{p} H^{p} (Y,^{p} H^{q} (R f_{*} F)) \Rightarrow H^{p + q} (X, F)$ degenerates at $E_{2}$ for proper $f$ with smooth $X$ by the BBD decomposition theorem. This is the geometric refinement of the classical Leray spectral sequence, with each term in the abelian category $Perv (Y)$ rather than $Sh (Y)$ , organising the cohomological information of proper maps in the perverse-sheaf framework.
Derived category $D (A)$ 04.03.11. Beilinson's reconstruction theorem states $D^{b} (Perv (X)) ≃ D_{c}^{b} (X)$ — the bounded derived category of the perverse heart recovers the original constructible derived category. This unusual reconstruction property is one of the few instances where the heart of a t-structure determines the triangulated category, and is the technical input to many representation-theoretic applications.
Triangulated category — Verdier axioms TR1-TR4 04.03.10. The perverse t-structure sits on top of the triangulated structure of $D_{c}^{b} (X)$ ; the BBD axioms for a t-structure are extra data beyond the Verdier axioms TR1-TR4, and the perverse cohomology functors $^{p} H^{n}$ depend on the distinguished-triangle behaviour for the long-exact-sequence property.

Historical & philosophical context Master

The notion of perverse sheaves emerged from the confluence of three independent developments in the late 1970s and early 1980s. The first was Mark Goresky and Robert MacPherson's introduction of intersection homology in 1980 ("Intersection homology theory," Topology 19, 135–162) ^{[source pending]} and its sheaf-theoretic refinement in 1983 ("Intersection homology II," Invent. Math. 71/72, 77–129 + 169–215) ^{[source pending]}, which provided a Poincaré-self-dual cohomology theory for singular projective varieties. The second was Masaki Kashiwara's and Zoghman Mebkhout's independent solutions to the Riemann-Hilbert problem in 1980–1984 — Mebkhout 1980 thesis at Paris VII ^{[source pending]} and Kashiwara 1984 Publ. RIMS Kyoto Univ. 20 ^{[source pending]} — establishing the equivalence between regular holonomic D-modules on a smooth complex variety and constructible sheaves on the analytic space via the de Rham functor. The third was Alexander Beilinson, Joseph Bernstein, and Pierre Deligne's axiomatisation of perverse sheaves and t-structures in Faisceaux pervers (Astérisque 100, Société Mathématique de France 1982) ^{[source pending]}, which unified the previous developments into a coherent abelian-category framework and proved the BBD decomposition theorem as the deepest result of the new theory.

The BBD authors recognised that the support-dimension condition appearing in Goresky-MacPherson's intersection-homology construction and the regularity condition appearing in Kashiwara-Mebkhout's D-module theory could both be packaged as the heart of a single t-structure on the constructible derived category $D_{c}^{b} (X)$ , with the simples of the heart being the intersection cohomology complexes. The middle perversity is the choice of t-structure that produces self-dual cohomology under Verdier duality, equivalently the unique perversity for which the IC complex of a smooth variety is the shifted constant sheaf. The decomposition theorem (BBD §6.2.5; independently announced by Ofer Gabber in a 1981 lecture series) was proved by lifting Deligne's Frobenius purity theorem (La conjecture de Weil II, Publ. Math. IHÉS 52, 1980) ^{[source pending]} from étale cohomology of finite-field varieties to perverse sheaves, and transferring the result back to characteristic zero via spreading-out arguments. Morihiko Saito's 1988 Modules de Hodge polarisables (Publ. RIMS Kyoto Univ. 24, 849–995) ^{[source pending]} supplied the Hodge-theoretic refinement: each perverse sheaf with a Hodge structure assembles into a mixed Hodge module, and the decomposition theorem holds at the Hodge-module level.

The reconstruction theme — recovering the constructible derived category from its perverse heart — was due to Beilinson (1987 Funct. Anal. Appl. 21) ^{[source pending]} in announcement form and BBD §3.1 in full proof. The equivalence $D^{b} (Perv (X)) ≃ D_{c}^{b} (X)$ exhibits the unusual feature that the heart of the perverse t-structure determines the original triangulated category up to triangulated equivalence — a property that fails for most t-structures and that depends on the Artin-vanishing properties of constructible sheaves on algebraic varieties. The transcendental Hodge-theoretic proof of the BBD decomposition theorem was given by Mark de Cataldo and Luca Migliorini in 2005 ("The Hodge theory of algebraic maps," Ann. Sci. ÉNS 38, 693–750) using the relative hard Lefschetz theorem on cohomology of fibres, and surveyed for non-specialists in their 2009 Bulletin of the AMS 46 article "The decomposition theorem, perverse sheaves and the topology of algebraic maps" ^{[source pending]}.

The applications of perverse sheaves to representation theory and the Langlands program have been pervasive. Beilinson-Bernstein's 1981 C.R. Acad. Sci. Paris 292 ^{[source pending]} localised $g$ -modules of a complex reductive Lie algebra to D-modules on the flag variety, which through Riemann-Hilbert become perverse sheaves; the Kazhdan-Lusztig conjectures relating irreducible character values to Kazhdan-Lusztig polynomials of the Weyl group were proved by Beilinson-Bernstein (1981) and independently by Brylinski-Kashiwara (1981 Invent. Math. 64) using this perverse-sheaf machinery. George Lusztig's 1983 Astérisque 101–102 and Mirković-Vilonen's 2007 Ann. Math. 166 established the geometric Satake equivalence, identifying perverse sheaves on the affine Grassmannian of a reductive group $G$ with representations of the Langlands dual group $G^{\lor}$ . Bao Châu Ngô's 2010 Publ. Math. IHÉS 111 ^{[source pending]} used the BBD decomposition theorem applied to the Hitchin fibration to prove the Fundamental Lemma of Langlands-Shelstad, completing a programme initiated in the 1970s and earning Ngô the 2010 Fields Medal.

Bibliography Master

@article{BBD1982,
  author    = {Beilinson, Alexander A. and Bernstein, Joseph and Deligne, Pierre},
  title     = {Faisceaux pervers},
  journal   = {Ast{\'e}risque},
  volume    = {100},
  year      = {1982},
  publisher = {Soci{\'e}t{\'e} Math{\'e}matique de France},
  note      = {Proceedings of the 1981 CIRM colloquium on the analysis and topology on singular spaces}
}

@article{GoreskyMacPherson1980,
  author    = {Goresky, Mark and MacPherson, Robert},
  title     = {Intersection homology theory},
  journal   = {Topology},
  volume    = {19},
  number    = {2},
  pages     = {135--162},
  year      = {1980}
}

@article{GoreskyMacPherson1983,
  author    = {Goresky, Mark and MacPherson, Robert},
  title     = {Intersection homology {II}},
  journal   = {Inventiones Mathematicae},
  volume    = {72},
  pages     = {77--129},
  year      = {1983}
}

@article{Kashiwara1984,
  author    = {Kashiwara, Masaki},
  title     = {The {Riemann-Hilbert} problem for holonomic systems},
  journal   = {Publications of the Research Institute for Mathematical Sciences, Kyoto University},
  volume    = {20},
  number    = {2},
  pages     = {319--365},
  year      = {1984}
}

@phdthesis{Mebkhout1980,
  author    = {Mebkhout, Zoghman},
  title     = {Sur le probl{\`e}me de {Hilbert-Riemann}},
  school    = {Universit{\'e} Paris VII},
  year      = {1980}
}

@article{Saito1988,
  author    = {Saito, Morihiko},
  title     = {Modules de {Hodge} polarisables},
  journal   = {Publications of the Research Institute for Mathematical Sciences, Kyoto University},
  volume    = {24},
  number    = {6},
  pages     = {849--995},
  year      = {1988}
}

@article{DeligneWeilII,
  author    = {Deligne, Pierre},
  title     = {La conjecture de {Weil}. {II}},
  journal   = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume    = {52},
  pages     = {137--252},
  year      = {1980}
}

@article{Beilinson1987,
  author    = {Beilinson, Alexander A.},
  title     = {On the derived category of perverse sheaves},
  journal   = {Funktsional'nyi Analiz i Ego Prilozheniya},
  volume    = {21},
  year      = {1987},
  note      = {English translation in Functional Analysis and Its Applications; the reconstruction theorem $D^b(\mathrm{Perv}(X)) \simeq D^b_c(X)$}
}

@article{deCataldoMigliorini2005,
  author    = {de Cataldo, Mark Andrea A. and Migliorini, Luca},
  title     = {The {Hodge} theory of algebraic maps},
  journal   = {Annales Scientifiques de l'{\'E}cole Normale Sup{\'e}rieure},
  volume    = {38},
  number    = {5},
  pages     = {693--750},
  year      = {2005}
}

@article{deCataldoMigliorini2009,
  author    = {de Cataldo, Mark Andrea A. and Migliorini, Luca},
  title     = {The decomposition theorem, perverse sheaves and the topology of algebraic maps},
  journal   = {Bulletin of the American Mathematical Society},
  volume    = {46},
  number    = {4},
  pages     = {535--633},
  year      = {2009}
}

@book{HottaTakeuchiTanisaki2008,
  author    = {Hotta, Ryoshi and Takeuchi, Kiyoshi and Tanisaki, Toshiyuki},
  title     = {D-Modules, Perverse Sheaves, and Representation Theory},
  series    = {Progress in Mathematics},
  volume    = {236},
  publisher = {Birkh{\"a}user},
  year      = {2008}
}

@book{KiehlWeissauer2001,
  author    = {Kiehl, Reinhardt and Weissauer, Rainer},
  title     = {Weil Conjectures, Perverse Sheaves and l'adic Fourier Transform},
  series    = {Ergebnisse der Mathematik und ihrer Grenzgebiete},
  volume    = {42},
  publisher = {Springer-Verlag},
  year      = {2001}
}

@article{BeilinsonBernstein1981,
  author    = {Beilinson, Alexander A. and Bernstein, Joseph},
  title     = {Localisation de $g$-modules},
  journal   = {Comptes Rendus de l'Acad{\'e}mie des Sciences de Paris S{\'e}rie I},
  volume    = {292},
  pages     = {15--18},
  year      = {1981}
}

@article{BrylinskiKashiwara1981,
  author    = {Brylinski, Jean-Luc and Kashiwara, Masaki},
  title     = {{Kazhdan-Lusztig} conjecture and holonomic systems},
  journal   = {Inventiones Mathematicae},
  volume    = {64},
  number    = {3},
  pages     = {387--410},
  year      = {1981}
}

@article{MirkovicVilonen2007,
  author    = {Mirkovi{\'c}, Ivan and Vilonen, Kari},
  title     = {Geometric {Langlands} duality and representations of algebraic groups over commutative rings},
  journal   = {Annals of Mathematics},
  volume    = {166},
  number    = {1},
  pages     = {95--143},
  year      = {2007}
}

@article{Ngo2010,
  author    = {Ng{\^o}, Bao Ch{\^a}u},
  title     = {Le lemme fondamental pour les alg{\`e}bres de {Lie}},
  journal   = {Publications Math{\'e}matiques de l'IH{\'E}S},
  volume    = {111},
  pages     = {1--169},
  year      = {2010}
}

@article{Lusztig1983,
  author    = {Lusztig, George},
  title     = {Singularities, character formulas, and a $q$-analog of weight multiplicities},
  journal   = {Ast{\'e}risque},
  volume    = {101--102},
  pages     = {208--229},
  year      = {1983}
}

@article{Gabber1981,
  author    = {Gabber, Ofer},
  title     = {Purity of the decomposition theorem},
  note      = {Lecture series 1981, IHÉS; results announced and used in BBD 1982},
  year      = {1981}
}

Prerequisites

04.03.18
04.03.16

Tier anchors

beginner: de Cataldo-Migliorini 2009 *Bull. AMS* 46 §1 informal — perverse sheaves are constructible complexes whose support dimensions are constrained by an inequality dual to Verdier duality, and they form an abelian category sitting inside the constructible derived category
intermediate: Beilinson-Bernstein-Deligne *Faisceaux pervers* (*Astérisque* 100, 1982) §§2.1–2.2; Kiehl-Weissauer *Weil Conjectures, Perverse Sheaves and l'adic Fourier Transform* Ch. III; Hotta-Takeuchi-Tanisaki *D-Modules, Perverse Sheaves, and Representation Theory* Ch. 8; de Cataldo-Migliorini 2009 *Bull. AMS* 46
master: Beilinson-Bernstein-Deligne *Faisceaux pervers* (*Astérisque* 100, 1982) §§2.1, 4.0, 5.0–6.2 (perverse t-structure, intermediate extensions, decomposition theorem with weights); Goresky-MacPherson *Intersection Homology I* (*Topology* 19, 1980) and *Intersection Homology II* (*Invent. Math.* 71/72, 1983); Kashiwara 1984 *Publ. RIMS* 20 + Mebkhout 1980 thesis (Riemann-Hilbert correspondence); Saito 1988 *Publ. RIMS* 24 (mixed Hodge modules); de Cataldo-Migliorini 2005 *Ann. Sci. ÉNS* 38 (Hodge-theoretic proof of decomposition theorem); Hotta-Takeuchi-Tanisaki *D-Modules, Perverse Sheaves, and Representation Theory* (Birkhäuser 2008); Ngô 2010 *Publ. Math. IHÉS* 111

References

Beilinson-Bernstein-Deligne — Faisceaux pervers · *Astérisque* 100, Société Mathématique de France 1982; §2.1 (perversities and the middle-perversity t-structure on the bounded constructible derived category $D^b_c(X)$); §2.2 (the heart $\mathrm{Perv}(X)$ and its abelian-category structure); §4.0–4.3 (intermediate extension $j_{!*}$, simple perverse sheaves $\mathrm{IC}_{\overline{Z}}(\mathcal{L})$, classification of simples); §5.1–5.4 (weights, pure perverse sheaves, Frobenius weight filtration in the $\ell$-adic setting); §6.2 (the BBD decomposition theorem $Rf_* \mathrm{IC} \cong \bigoplus \mathrm{IC}[n]$ for proper $f$)
Goresky-MacPherson — Intersection Homology I and II · I: *Topology* 19 (1980), 135–162; II: *Inventiones Mathematicae* 71/72 (1983), 77–129 + 169–215; the topological definition of intersection homology $IH_*^{\bar p}(X)$ for a pseudomanifold with stratification, the perversity function $\bar p$ controlling allowed cycles meeting strata, and the topological invariance of $IH_*$ under refinement of stratification (Goresky-MacPherson + later Bredon); the bridge from chain-level intersection homology to sheaf-theoretic $\mathrm{IC}_X$ in the bounded constructible derived category
Kashiwara — The Riemann-Hilbert problem for holonomic systems · *Publ. RIMS Kyoto Univ.* 20 (1984), 319–365; the equivalence of categories between regular holonomic $\mathcal{D}_X$-modules on a smooth complex analytic manifold $X$ and perverse sheaves on $X^{\mathrm{an}}$ with $\mathbb{C}$-coefficients via the de Rham functor $\mathrm{DR}(\mathcal{M}) = \mathrm{Sol}(\mathcal{M}^*) = \Omega^\bullet_X \otimes_{\mathcal{D}_X}^L \mathcal{M}$
Mebkhout — Sur le problème de Hilbert-Riemann · thesis, Université Paris VII, 1980; independent proof of the Riemann-Hilbert correspondence for regular holonomic $\mathcal{D}$-modules using the functor $\mathbb{R}\mathcal{H}om_{\mathcal{D}_X}(\mathcal{M}, \mathcal{O}_X)$ and the analytic-de Rham comparison
Saito — Modules de Hodge polarisables · *Publ. RIMS Kyoto Univ.* 24 (1988), 849–995; the construction of mixed Hodge modules $\mathrm{MHM}(X)$, an abelian category fibred over $\mathrm{Perv}(X, \mathbb{Q})$ that assigns a polarised mixed Hodge structure to each perverse sheaf; functoriality of the six operations on $\mathrm{MHM}$; the decomposition theorem with Hodge-theoretic weights
de Cataldo-Migliorini — The decomposition theorem, perverse sheaves and the topology of algebraic maps · *Bulletin of the American Mathematical Society* 46 (2009), 535–633; survey of the BBD decomposition theorem, intersection cohomology, perverse sheaves, and the Hodge-theoretic proof; intended as an introduction for non-specialists
Hotta-Takeuchi-Tanisaki — D-Modules, Perverse Sheaves, and Representation Theory · Progress in Mathematics 236 (Birkhäuser 2008); Ch. 8 (perverse sheaves: definition via support-dimension condition, the heart as abelian category, intermediate extensions); Ch. 7 (Riemann-Hilbert correspondence); Ch. 9–12 (applications to representation theory, Beilinson-Bernstein localisation, Kazhdan-Lusztig)
Kiehl-Weissauer — Weil Conjectures, Perverse Sheaves and l'adic Fourier Transform · Ergebnisse der Mathematik 42 (Springer 2001); Ch. III (the perverse t-structure on $D^b_c(X, \overline{\mathbb{Q}}_\ell)$ for an algebraic variety $X$ over a finite or algebraically closed field, including the support-dimension definition, the heart, intermediate extensions, and the decomposition theorem in the $\ell$-adic setting)
Ngô — Le lemme fondamental pour les algèbres de Lie · *Publications Mathématiques de l'IHÉS* 111 (2010), 1–169; the geometric proof of the Fundamental Lemma via the BBD decomposition theorem applied to the Hitchin fibration, using a support theorem for direct images of perverse sheaves on the elliptic part of the Hitchin base

Estimated time

beginner: 18m
intermediate: 45m
master: 80m