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Hard Lefschetz gives the primitive decomposition, orthogonality and the signature formula

Facts & Assumptions

Given: A finite-dimensional graded real space $H$ concentrated in one parity, with a graded non-degenerate symmetric form $⟨ -, - ⟩$ and a symmetric degree-two operator $L$ , its primitive subspaces $P_{L}^{- i}$ , Lefschetz forms $(a, b)_{L}^{- i} = ⟨ a, L^{i} b ⟩$ , the hypotheses hL and HR with the standard sign and lowest degree $- d$ , as in Primitive subspaces, hard Lefschetz and the Hodge-Riemann signs: the dictionary of hypotheses S, hL and HR.

[F1]

The form is graded and non-degenerate, so the pairing of homogeneous pieces $H^{m} \times H^{- m} \to R$ is perfect and $\dim H^{m} = \dim H^{- m}$ ; $L$ is symmetric and of degree two; hL means that $L^{i} : H^{- i} \to H^{i}$ is an isomorphism for every $i \geq 0$ , and $P_{L}^{- i} = \ker (L^{i + 1} ∣_{H^{- i}})$ (Primitive subspaces, hard Lefschetz and the Hodge-Riemann signs: the dictionary of hypotheses S, hL and HR).

[F2]

A symmetric bilinear form on a finite-dimensional space has an orthogonal basis, and its inertia $(p, q, r)$ and signature $p - q$ are independent of the diagonalizing basis; a form is non-degenerate exactly when its radical is zero (Every symmetric bilinear form on a finite-dimensional space over a field of characteristic not $2$ has an orthogonal basis, Sylvester's law of inertia: every real symmetric form is congruent to $diag (I_{p}, - I_{q}, 0_{r})$ , and $(p, q, r)$ is unique, Positive and negative definiteness, the inertia $(p, q, r)$ , rank $p + q$ , and signature $p - q$ of a real symmetric bilinear or quadratic form, The matrix, left and right radicals, rank, and nondegeneracy of a bilinear form on a finite-dimensional space).

[F3]

Rank-nullity holds for linear maps of finite-dimensional spaces (Rank-nullity: $\dim_{F} V = nullity T + rank T$ ).

[F4]

Symmetry of $L$ means $⟨ L a, b ⟩ = ⟨ a, L b ⟩$ for all $a, b$ ; hence $⟨ L^{k} a, b ⟩ = ⟨ a, L^{k} b ⟩$ for all $k \geq 0$ (Bilinear forms, and symmetric, skew-symmetric, and alternating bilinear forms, Primitive subspaces, hard Lefschetz and the Hodge-Riemann signs: the dictionary of hypotheses S, hL and HR).

Statement

Let $H$ have a graded non-degenerate symmetric form and a symmetric degree-two operator $L$ .

(a) If hL holds then $H = ⨁_{i \geq 0} ⨁_{j = 0}^{i} L^{j} P_{L}^{- i}$ ; more precisely, in each degree $- i$ one has $H^{- i} = ⨁_{k \geq 0} L^{k} P_{L}^{- (i + 2 k)}$ .

(b) In each degree $- i$ , the decomposition of (a) is orthogonal for the Lefschetz form $⟨ a, L^{i} b ⟩$ .

(d) If $H$ has one parity and lowest degree $- d$ , standard HR is equivalent to hL together with the signature formula $sig (⟨ -, L^{i} - ⟩ ∣_{H^{- i}}) = \sum_{k \geq 0} (- 1)^{(d - i - 2 k) / 2} \dim P_{L}^{- (i + 2 k)}$ for every $i \geq 0$ (zero spaces omitted).

(e) If $H$ satisfies hL, an $L$ -stable graded subspace $V \subseteq H$ with $\dim V^{- i} = \dim V^{i}$ for all $i$ inherits hL; if $H$ satisfies HR then the restricted form on $V$ is non-degenerate and $V$ inherits the ambient primitive signs, and its standard global sign agrees with that of $H$ when their lowest degrees coincide.

Proof

technique · direct; primitive-string decomposition and orthogonal signature bookkeeping

1.1givenF1algebra

One step of the decomposition. Under hL the map $L^{i + 2} : H^{- i - 2} \to H^{i + 2}$ is an isomorphism, so for every $a \in H^{- i}$ there is $b \in H^{- i - 2}$ with $L^{i + 1} a = L^{i + 2} b$ ; then $a - L b \in P_{L}^{- i}$ , and hence $H^{- i} = P_{L}^{- i} + L H^{- i - 2}$ . The sum is direct: if $L b \in P_{L}^{- i}$ , then $0 = L^{i + 1} (L b) = L^{i + 2} b$ and injectivity of $L^{i + 2}$ gives $b = 0$ ; hence $H^{- i} = P_{L}^{- i} \oplus L H^{- i - 2}$ .

2.1step 1.1F1

Iteration and the primitive decomposition (a). Iterating step 1.1 gives $H^{- i} = P_{L}^{- i} \oplus L P_{L}^{- i - 2} \oplus L^{2} P_{L}^{- i - 4} \oplus \dots = ⨁_{k \geq 0} L^{k} P_{L}^{- (i + 2 k)}$ , the sum terminating because $H^{- m} = 0$ for $m > d$ ; For positive degree $i$ , hL gives $H^{i} = L^{i} H^{- i}$ , so applying $L^{i}$ to the negative-degree decomposition supplies all positive pieces. Reindexing by the lowest primitive degree gives $H = ⨁_{i \geq 0} ⨁_{j = 0}^{i} L^{j} P_{L}^{- i}$ , since every string $L^{k} P_{L}^{- i}$ lies in degrees between $- i$ and $i$ , so $k \leq i$ for nonzero contributions. This proves (a).

3.1step 2.1F1F4algebra

Orthogonality (b). Let $p \in P_{L}^{- (i + 2 k)}$ and $q \in P_{L}^{- (i + 2 l)}$ with $k < l$ and consider $a = L^{k} p$ , $b = L^{l} q$ in $H^{- i}$ . Using symmetry of $L$ repeatedly, $(a, b)_{L}^{- i} = ⟨ L^{k} p, L^{i + l} q ⟩ = ⟨ p, L^{i + k + l} q ⟩$ , and the exponent satisfies $i + k + l \geq i + 2 k + 1$ ; since $L^{i + 2 k + 1} p = 0$ , the pairing vanishes. Hence distinct strings are orthogonal for the Lefschetz form, as claimed.

4.1step 3.1F1F3

Non-degeneracy of the Lefschetz forms is equivalent to hL (c). For fixed $i$ , the Lefschetz form at degree $- i$ is the composite of the linear map $L^{i} : H^{- i} \to H^{i}$ with the perfect pairing $H^{i} \times H^{- i} \to R$ given by the form; the latter is an isomorphism $H^{i} \to (H^{- i})^{*}$ by non-degeneracy of $⟨ -, - ⟩$ [F1]. Hence the Lefschetz form is non-degenerate if and only if $L^{i}$ is an isomorphism; this holds simultaneously for all $i \geq 0$ exactly when hL holds, and for $i > d$ both sides are zero by [F1] ( $\dim H^{i} = \dim H^{- i} = 0$ ). This proves (c).

5.1step 3.1step 4.1F2

Signature formula (d). Assume hL and the standard-sign definiteness on primitives. The orthogonal decomposition of step 3.1 expresses the form on $H^{- i}$ as an orthogonal sum of the forms $(p, q) \mapsto ⟨ p, L^{i + 2 k} q ⟩$ on the primitive spaces $P_{L}^{- (i + 2 k)}$ ; each of these is definite of sign $(- 1)^{(d - i - 2 k) / 2}$ by HR, and its own inertia is therefore $(\dim P, 0, 0)$ or $(0, \dim P, 0)$ according to that sign. Sylvester's law [F2] makes the signature of the orthogonal sum the sum of the summand signatures, giving the displayed formula. Conversely, assume hL and the signature formula for all $i$ ; descending from the bottom degree, the formula determines the signature of the primitive form on $P_{L}^{- (i + 2 k)}$ after the signatures of the higher-degree primitive spaces are known, the orthogonal decomposition and step 4.1 make each primitive form non-degenerate, and the subtraction forces its signature to be the prescribed sign times its dimension. Its inertia therefore has only positive or only negative entries by [F2], so each primitive form is definite with the stated sign; this recovers HR. This proves (d).

6.1step 2.1step 4.1step 5.1F1F2∎

Stable subspaces (e). Let $V \subseteq H$ be $L$ -stable with $\dim V^{- i} = \dim V^{i}$ . Since $L^{i}$ is injective on $H$ (hL), its restriction to $V$ is injective, and the dimension symmetry makes it an isomorphism onto $V^{i}$ ; hence $V$ satisfies hL for the restricted operator and form. Its primitive spaces are $P_{V}^{- i} = V^{- i} \cap P_{L}^{- i}$ , because the kernel of $L^{i + 1}$ on $V^{- i}$ is the intersection with the kernel on $H^{- i}$ ; the restricted Lefschetz form on $P_{V}^{- i}$ is the restriction of the ambient definite primitive form, hence definite with the ambient sign $(- 1)^{(d - i) / 2}$ . The string construction in steps 1.1–2.1 uses only hL, so it applies to $V$ before non-degeneracy of its restricted form is known. This primitive decomposition then shows, exactly as in step 5.1, that the restricted form on $V$ is an orthogonal sum of definite blocks, hence non-degenerate, and that its signs are the ambient signs; if the lowest nonzero degree of $V$ equals $- d$ , then the ambient sign formula is also the standard sign of $V$ . This proves (e) and completes the lemma.

Remarks

The dimension input $\dim P_{L}^{- i} = \dim H^{- i} - \dim H^{- i - 2}$ follows from step 1.1 and [F3]; it is the string count used implicitly in (a) and (d).
Part (e) does not assert that an arbitrary $L$ -stable subspace with symmetric dimensions is a direct summand of $H$ ; it uses only the restricted form, which is shown non-degenerate by the orthogonal-string decomposition.

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Sources

Elias-Williamson, The Hodge theory of Soergel bimodules (arXiv:1212.0791v2, final version; Ann. of Math. 180 (2014) 1089-1136) (standard reference, not scraped)
Riche, La theorie de Hodge des bimodules de Soergel (d'apres Soergel et Elias-Williamson), Seminaire Bourbaki no. 1139 (arXiv:1711.02464, Octobre 2017) (standard reference, not scraped)