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The nilpotent Lefschetz filtration is canonical and self-dual

Statement

Let $N$ be a nilpotent endomorphism of a finite-dimensional vector space $H$ over a field $F$ (Nilpotent endomorphisms and their nilpotency index). An admissible filtration is an increasing finite filtration $W$ of $H$ by linear subspaces (Linear subspace of a vector space): $W_{i} \subseteq W_{i + 1}$ for all $i$ , $W_{i} = 0$ for all sufficiently negative $i$ , $W_{i} = H$ for all sufficiently positive $i$ , such that

$N (W_{i}) \subseteq W_{i - 2} for all i, N^{r} : {Gr}_{r}^{W} ⟶ {Gr}_{- r}^{W} is an isomorphism for every r \geq 0,$

where ${Gr}_{i}^{W} : = W_{i} / W_{i - 1}$ and the second map is the one induced by $N^{r}$ , which is well defined because $N^{r} (W_{r}) \subseteq W_{- r}$ and $N^{r} (W_{r - 1}) \subseteq W_{- r - 1}$ . There is a unique admissible filtration $W$ , and it is given intrinsically by

$W_{i} = \sum_{a, b \geq 0, a - b = i} \ker N^{a + 1} \cap im N^{b} .$

If in addition $S$ is a nondegenerate bilinear form on $H$ with $S (N v, w) + S (v, N w) = 0$ for all $v, w$ (Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms, The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space), and $U^{⊥} : = {x \in H : S (u, x) = 0 for every u \in U}$ for a subspace $U$ , then $(W_{i})^{⊥} = W_{- i - 1}$ for every $i$ , and consequently $S$ induces a perfect pairing between ${Gr}_{i}^{W}$ and ${Gr}_{- i}^{W}$ ; moreover for $i \geq 0$ the form $S_{i} ([v], [w]) : = S (v, N^{i} w)$ on ${Gr}_{i}^{W}$ is well defined and nondegenerate.

Facts & Assumptions

Given: A field $F$ , a finite-dimensional $F$ -vector space $H$ , a nilpotent endomorphism $N$ of $H$ , an admissible filtration $W$ as in the Statement, and, where used, a nondegenerate bilinear form $S$ with $S (N v, w) + S (v, N w) = 0$ ; the right-orthogonal complement $U^{⊥} = {x : S (u, x) = 0 for all u \in U}$ .

[F1]

Jordan string basis: there is a basis of $H$ that is the concatenation of strings $v_{1}, \dots, v_{l}$ with $N v_{1} = 0$ and $N v_{t} = v_{t - 1}$ for $t \geq 2$ ; for $H = 0$ this is the empty family (Every finite-dimensional nilpotent endomorphism has a basis of Jordan strings).

[F2]

Graded hard Lefschetz: let $G^{j}$ be objects of an abelian category, zero for all but finitely many $j$ , and $λ^{j} : G^{j} \to G^{j + 2}$ morphisms with $λ^{r} : G^{- r} \to G^{r}$ an isomorphism for every $r \geq 0$ ; with $π^{- r} : = \ker λ^{r + 1}$ one has $G^{- r} = ⨁_{j \geq 0} λ^{j} π^{- r - 2 j}$ and $G^{r} = ⨁_{j \geq 0} λ^{r + j} π^{- r - 2 j}$ , each restricted power being a monomorphism on its primitive source (Graded hard Lefschetz has a canonical primitive decomposition). The ambient category here is the category of $F$ -vector spaces, which is abelian (Modules over a ring form an abelian category).

[F3]

Annihilator dimension: for a subspace $U$ of a finite-dimensional space $V$ , the dual dimension is $\dim V^{*} = \dim V$ (The dual family of a finite basis is a basis of the dual space, with the same dimension), the restriction $V^{*} \to U^{*}$ , $f \mapsto f ∣_{U}$ , is surjective because a basis of $U$ extends to a basis of $V$ (If $\dim_{F} V = n$ and $U$ is a linear subspace of $V$ , then $U$ is finite-dimensional, $\dim_{F} U \leq n$ , and $\dim_{F} U = n$ if and only if $U = V$ ), its kernel is the annihilator $U^{\circ} = {f : f ∣_{U} = 0}$ (The annihilator $U^{\circ} \leq V^{*}$ of $U \leq V$ and the preannihilator $^{\circ} S \leq V$ of $S \leq V^{*}$ ), and rank-nullity (Rank-nullity: $\dim_{F} V = nullity T + rank T$ ) therefore gives $\dim U^{\circ} = \dim V - \dim U$ . If $U \subseteq U^{'}$ are subspaces of equal finite dimension, then $U = U^{'}$ (If $\dim_{F} V = n$ and $U$ is a linear subspace of $V$ , then $U$ is finite-dimensional, $\dim_{F} U \leq n$ , and $\dim_{F} U = n$ if and only if $U = V$ ).

[F4]

Bilinear correspondence: for any bilinear form $B$ the map $B^{♭} : v \mapsto B (v, -)$ is linear in $v$ , and $B \mapsto B^{♭}$ is a linear bijection onto $L (V, V^{*})$ (Bilinear forms on $V$ correspond linearly and bijectively to linear maps $V \to V^{*}$ ); for a nondegenerate form on a finite-dimensional space, $B^{♭}$ is an isomorphism (The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space). Apply the same correspondence to the transpose form $S^{o p} (v, w) : = S (w, v)$ to obtain $S^{♯} (x) : = S (-, x)$ ; its kernel is the right radical, so it is an isomorphism by rank-nullity and equality of the finite dimensions of $H$ and $H^{*}$ . Thus the right-orthogonal complement is $U^{⊥} = (S^{♯})^{- 1} (U^{\circ})$ .

Proof

technique · direct, with finite ascending inductions on filtration indices

1.1givenF1construct

Fix a Jordan string basis of $H$ as in [F1], with strings $v_{c, 1}, \dots, v_{c, l_{c}}$ indexed by $c$ . Give $v_{c, t}$ the weight $2 t - l_{c} - 1$ , so that the weights on a string of length $l_{c}$ are $- l_{c} + 1, - l_{c} + 3, \dots, l_{c} - 1$ , and let $W_{i}$ be the span of all $v_{c, t}$ with weight at most $i$ .

1.2givenalgebra

On a Jordan string $v_{1}, \dots, v_{l}$ , one has $\ker N^{a + 1} = ⟨ v_{t} : 1 \leq t \leq \min (a + 1, l) ⟩$ and $im N^{b} = ⟨ v_{t} : 1 \leq t \leq l - b ⟩$ , since $N^{j} v_{t} = v_{t - j}$ for $t > j$ and is zero otherwise. For $1 \leq t \leq l$ , the vector $v_{t}$ belongs to a summand $\ker N^{a + 1} \cap im N^{b}$ with $a - b = i$ exactly when some integer $b$ satisfies $\max (0, - i, t - i - 1) \leq b \leq l - t$ . Such a $b$ exists exactly when $2 t - l - 1 \leq i$ : necessity follows from $t - i - 1 \leq l - t$ , and sufficiency follows because $l - t \geq 0$ and $2 t - l - 1 \leq i$ also gives $- i \leq l - t$ . Thus the sum of these nested spans consists precisely of the string vectors of weight at most $i$ , including the zero and whole-string cases.

1.3F2givenalgebra

Consequences of [F2] for an admissible filtration $W$ . The graded pieces ${Gr}_{s} = W_{s} / W_{s - 1}$ are $F$ -vector spaces, zero for all but finitely many $s$ , and $N$ induces $\overset{ˉ}{N} : {Gr}_{s} \to {Gr}_{s - 2}$ . Putting $G^{j} : = {Gr}_{- j}$ and $λ^{j} : = \overset{ˉ}{N} : G^{j} \to G^{j + 2}$ , the hypothesis says that $λ^{r} : G^{- r} \to G^{r}$ , which is ${\overset{ˉ}{N}}^{r} : {Gr}_{r} \to {Gr}_{- r}$ , is an isomorphism for every $r \geq 0$ . [F2] therefore gives, with $P_{m} : = \ker ({\overset{ˉ}{N}}^{m + 1} : {Gr}_{m} \to {Gr}_{- m - 2})$ for $m \geq 0$ , the decompositions ${Gr}_{s} = ⨁_{m \geq ∣ s ∣, m \equiv s (2)} {\overset{ˉ}{N}}^{(m - s) / 2} P_{m}$ for every $s$ (the first display of [F2] for $s \geq 0$ , the second for $s < 0$ ), each restricted power being a monomorphism. Moreover for $t \leq - q$ the map ${\overset{ˉ}{N}}^{q} : {Gr}_{t + 2 q} \to {Gr}_{t}$ is surjective: with $m^{'} : = - t \geq q$ each summand ${\overset{ˉ}{N}}^{m^{'} + j} P_{m^{'} + 2 j}$ of the displayed decomposition of ${Gr}_{t}$ equals ${\overset{ˉ}{N}}^{q} ({\overset{ˉ}{N}}^{m^{'} + j - q} P_{m^{'} + 2 j})$ , and ${\overset{ˉ}{N}}^{m^{'} + j - q} P_{m^{'} + 2 j} \subseteq {Gr}_{m^{'} + 2 j - 2 (m^{'} + j - q)} = {Gr}_{t + 2 q}$ .

1.4F3F4givenalgebra

For a subspace $U \subseteq H$ , [F4] gives $U^{⊥} = (S^{♯})^{- 1} (U^{\circ})$ , hence $\dim U^{⊥} = \dim H - \dim U$ by [F3]. Define the left-orthogonal complement $^{⊥} U : = {v : S (v, u) = 0 for all u \in U}$ . Using $S^{♭}$ instead gives $\dim^{⊥} U = \dim H - \dim U$ . The inclusion $U \subseteq^{⊥} (U^{⊥})$ follows directly from $S (U, U^{⊥}) = 0$ ; both sides have the same dimension, so $^{⊥} (U^{⊥}) = U$ by [F3]. Also $H^{⊥} = 0$ by nondegeneracy and ${0}^{⊥} = H$ . No symmetry of $S$ is used.

1.5givenalgebra

Skew-adjointness iterated: $S (N^{r} v, w) = (- 1)^{r} S (v, N^{r} w)$ for all $r \geq 0$ , by induction on $r$ from $S (N v, w) = - S (v, N w)$ .

2.1step 1.1algebra

The filtration of step 1.1 is admissible: it is increasing and finite by construction, and $N (W_{i}) \subseteq W_{i - 2}$ because $N v_{c, t} = v_{c, t - 1}$ has weight $2 (t - 1) - l_{c} - 1 = (2 t - l_{c} - 1) - 2$ while $N v_{c, 1} = 0$ .

2.2step 1.1step 1.2algebra

The filtration of step 1.1 satisfies the intrinsic formula of the Statement: both $W_{i}$ and the sum $\sum_{a - b = i} \ker N^{a + 1} \cap im N^{b}$ are direct sums over the strings of their chainwise parts, because $N$ preserves each string span and the spans of the strings form a direct sum decomposition of $H$ ; by the chainwise computation of step 1.2 these parts are the same span of weights at most $i$ .

2.3step 1.1algebra

The filtration of step 1.1 has $N^{r} : {Gr}_{r} \to {Gr}_{- r}$ an isomorphism for every $r \geq 0$ : chainwise, ${Gr}_{r}$ is spanned by the classes of the $v_{c, t}$ of weight exactly $r$ , and $N^{r}$ sends such a class to the class of $v_{c, t - r}$ of weight $- r$ , which is an isomorphism on each string and hence on the direct sum of the stringwise graded pieces.

2.4step 1.3algebra

For any admissible $W$ the graded surjectivity of step 1.3 upgrades to $N^{m + 1} (W_{m - 1}) = W_{- m - 3}$ for every $m \geq 0$ . Indeed $N^{m + 1} (W_{m - 1}) \subseteq W_{- m - 3}$ , and for surjectivity one shows $W_{s} \subseteq N^{m + 1} (W_{m - 1})$ by ascending induction on $s$ : for $x \in W_{s}$ apply step 1.3 with $q = m + 1$ and $t = s \leq - m - 3$ to get $y \in W_{s + 2 m + 2}$ with $[N^{m + 1} y] = [x]$ in ${Gr}_{s}$ ; then $N^{m + 1} y - x \in W_{s - 1}$ , which lies in the image by the induction hypothesis, so $x$ does too; the induction starts with $W_{s} = 0$ for $s$ very negative and runs up to $s = - m - 3$ . Note $s + 2 m + 2 \leq m - 1$ , so $y$ may be taken in $W_{m - 1}$ .

2.5step 1.4step 1.5givenalgebra

Define $W_{i}^{'} : = (W_{- i - 1})^{⊥}$ for every $i$ . Then $W^{'}$ is an increasing finite filtration: $i \leq j$ gives $W_{- i - 1} \supseteq W_{- j - 1}$ and hence $W_{i}^{'} \subseteq W_{j}^{'}$ ; for $i$ very negative, $W_{- i - 1} = H$ and $W_{i}^{'} = H^{⊥} = 0$ ; for $i$ very positive, $W_{- i - 1} = 0$ and $W_{i}^{'} = {0}^{⊥} = H$ . Moreover $N (W_{i}^{'}) \subseteq W_{i - 2}^{'}$ : for $x \in (W_{- i - 1})^{⊥}$ and $w \in W_{- i + 1}$ one has $S (w, N x) = - S (N w, x) = 0$ because $N w \in W_{- i - 1}$ , so $N x \in (W_{- i + 1})^{⊥} = W_{i - 2}^{'}$ .

3.1givenF3step 1.3step 2.4chooseconstruct

Fix an admissible $W$ . Only finitely many $P_{m}$ in step 1.3 are nonzero, and each is finite-dimensional by [F3]; choose a finite basis of each using the extension theorem in [F3]. For each of these finitely many basis classes choose a representative $v \in W_{m}$ . Primitivity gives $N^{m + 1} v \in W_{- m - 3}$ , since its class in ${Gr}_{- m - 2}$ vanishes. By step 2.4 choose $u \in W_{m - 1}$ with $N^{m + 1} u = N^{m + 1} v$ and replace $v$ by $v - u$ . The representative now satisfies $N^{m + 1} v = 0$ . Its string $v, N v, \dots, N^{m} v$ lies in $W_{m}, W_{m - 2}, \dots, W_{- m}$ and has graded classes ${\overset{ˉ}{N}}^{i} [v]$ of weight $m - 2 i$ . All selections are finite.

3.2step 1.4step 2.5F3algebra

The pairing ${Gr}_{- i}^{W} \times {Gr}_{i}^{W^{'}} \to F$ , $([v], [x]) \mapsto S (v, x)$ , is well defined: $x \in W_{i}^{'} = (W_{- i - 1})^{⊥}$ kills errors in $W_{- i - 1}$ in the first variable, and errors in $W_{i - 1}^{'} = (W_{- i})^{⊥}$ kill $v \in W_{- i}$ in the second. If $S (v, W_{i}^{'}) = 0$ , then $v \in^{⊥} ((W_{- i - 1})^{⊥}) = W_{- i - 1}$ by step 1.4, so the left kernel vanishes. If $S (W_{- i}, x) = 0$ , then $x \in W_{i - 1}^{'}$ , so the right kernel vanishes. The pairing induces injections from each graded space to the dual of the other; [F3] then forces their dimensions to agree, and rank-nullity makes both injections isomorphisms. Thus the pairing is perfect.

4.1step 1.3step 3.1algebra

The graded classes of the string vectors of step 3.1 are linearly independent and span every ${Gr}_{s}$ : in the decomposition of step 1.3, ${Gr}_{s}$ is the direct sum over $m \geq ∣ s ∣$ , $m \equiv s (2)$ of the images of the monomorphisms ${\overset{ˉ}{N}}^{(m - s) / 2} : P_{m} \to {Gr}_{s}$ , so the images of the chosen bases form a basis of ${Gr}_{s}$ . Hence the string vectors are linearly independent, since a nontrivial finite combination would have a nonzero class in the largest weight occurring. They span $H$ by ascending induction on $s$ : if every element of $W_{s - 1}$ is a combination of string vectors of weight at most $s - 1$ and $x \in W_{s}$ , then either $[x] = 0$ in ${Gr}_{s}$ and $x \in W_{s - 1}$ , or $[x]$ is a combination of string classes of weight exactly $s$ , and subtracting the corresponding string vectors leaves an element of $W_{s - 1}$ ; the induction starts where $W_{s} = 0$ . This proves $W_{s} \subseteq ⟨ string vectors of weight \leq s ⟩$ for every $s$ , and the reverse inclusion is clear; consequently the string vectors form a basis of $H$ and $W_{i}$ is exactly the span of those of weight at most $i$ .

4.2givenstep 1.5step 3.2F3algebra

For any admissible $W$ the $W^{'}$ -graded maps ${\overset{ˉ}{N}}^{r} : {Gr}_{r}^{W^{'}} \to {Gr}_{- r}^{W^{'}}$ are isomorphisms. Let $ψ : {Gr}_{- r}^{W} \to ({Gr}_{r}^{W^{'}})^{*}$ and $ϕ : {Gr}_{r}^{W} \to ({Gr}_{- r}^{W^{'}})^{*}$ be the isomorphisms induced by the perfect pairings of step 3.2, and let $T : = {\overset{ˉ}{N}}^{r} : {Gr}_{r}^{W^{'}} \to {Gr}_{- r}^{W^{'}}$ . For $y \in W_{r}$ and $x \in W_{r}^{'}$ steps 1.5 and the definitions give $ψ ({\overset{ˉ}{N}}^{r} [y]) ([x]) = S (N^{r} y, x) = (- 1)^{r} S (y, N^{r} x) = (- 1)^{r} (T^{*} ϕ [y]) ([x])$ , so $T^{*} = (- 1)^{r} ψ {\overset{ˉ}{N}}^{r} ϕ^{- 1}$ is an isomorphism because ${\overset{ˉ}{N}}^{r} : {Gr}_{r}^{W} \to {Gr}_{- r}^{W}$ is an isomorphism by the admissibility of $W$ ; a linear map between finite-dimensional spaces whose transpose is an isomorphism is itself an isomorphism: surjectivity of $T^{*}$ forces $\ker T = 0$ because every functional on the source then factors through $T$ , and bijectivity of $T^{*}$ and [F3] give equality of the source and target dimensions, so rank-nullity makes $T$ surjective as well.

5.1F1step 1.2step 2.2step 4.1

Every admissible $W$ equals the intrinsic formula and hence the filtration of step 1.1; in particular the admissible filtration is unique. The strings of step 3.1 form a basis of $H$ and satisfy $N^{m + 1} v = 0$ , so each is a Jordan string for $N$ in the sense of [F1]; the chainwise computation of step 1.2 applied to this basis says that $\sum_{a - b = i} \ker N^{a + 1} \cap im N^{b}$ is exactly the span of the string vectors of weight at most $i$ , which is $W_{i}$ by step 4.1. This formula depends only on $N$ , and the filtration of step 1.1 satisfies it too by step 2.2; hence every admissible $W$ equals it.

6.1step 2.5step 4.2step 5.1

The filtration $W^{'}$ of step 2.5 is admissible: it is an increasing finite filtration with $N (W_{i}^{'}) \subseteq W_{i - 2}^{'}$ by step 2.5, and its graded maps are isomorphisms by step 4.2. By the uniqueness of step 5.1, $W^{'} = W$ .

7.1step 3.2step 6.1

Consequently $(W_{i})^{⊥} = W_{- i - 1}$ for every $i$ : $W^{'} = W$ reads $(W_{- i - 1})^{⊥} = W_{i}$ , and reindexing $j : = - i - 1$ gives the claim. Moreover the pairing ${Gr}_{i}^{W} \times {Gr}_{- i}^{W} \to F$ , $([v], [w]) \mapsto S (v, w)$ , is perfect: it is the pairing of step 3.2 read with $W^{'} = W$ .

8.1givenstep 7.1

The forms $S_{i}$ are well defined and nondegenerate. For the restriction of $S$ to a pairing on ${Gr}_{i}^{W}$ : an error $v^{'} \in W_{i - 1}$ in the first argument contributes $S (v^{'}, N^{i} w) = 0$ because $N^{i} w \in W_{- i}$ and $(W_{i - 1})^{⊥} = W_{- i}$ by step 7.1, and an error $w^{'} \in W_{i - 1}$ in the second contributes $S (v, N^{i} w^{'}) = 0$ because $N^{i} w^{'} \in W_{- i - 1} = (W_{i})^{⊥}$ by admissibility and step 7.1. Thus $S_{i}$ is well defined on ${Gr}_{i}^{W} \times {Gr}_{i}^{W}$ , and it is the composite of the perfect pairing $⟨ [v], [x] ⟩ : = S (v, x)$ of step 7.1 with the second variable $[w] \mapsto N^{i} [w]$ , which is an isomorphism ${Gr}_{i}^{W} \to {Gr}_{- i}^{W}$ by the admissibility of $W$ ; hence the induced map ${Gr}_{i}^{W} \to ({Gr}_{i}^{W})^{*}$ , $[v] \mapsto S_{i} ([v], -)$ is an isomorphism: $S_{i}$ is nondegenerate.

9.1step 2.1step 2.2step 2.3step 5.1step 6.1step 7.1step 8.1∎

Steps 2.1, 2.2 and 2.3 exhibit the admissible filtration and identify it with the intrinsic formula; step 5.1 proves that any admissible filtration equals that formula, which is uniqueness; steps 6.1, 7.1 and 8.1 prove the self-duality, the perfect pairing and the descended forms $S_{i}$ . The zero space is included: its empty string basis of [F1] gives the zero filtration $W_{i} = 0 = H$ , which is admissible and agrees with the intrinsic formula, which is zero for every $i$ .

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Sources

M. A. A. de Cataldo and L. Migliorini, «The Hodge theory of algebraic maps», arXiv:math/0306030v2 (complete arXiv PDF) (standard reference, not scraped)