Signature constancy in a Lefschetz family, the L-stable subspace lemma and the weak-Lefschetz substitute

The family lemma (1). Fix $i\ge 0$ and consider the family of symmetric forms $(a,b)\mapsto ⟨a,L_t^{i}b⟩$ on the fixed space $H^{-i}$. For each $t$ it is non-degenerate, because hL makes $L_t^i:H^{-i}\to H^i$ an isomorphism and the ambient form pairs $H^i$ perfectly with $H^{-i}$; and it depends continuously on $t$ because $L_t$ does. At any parameter $t_{\ast }$, fix a real congruence diagonalizing its nonsingular matrix by [F1]. In this fixed basis all leading pivots at $t_{\ast }$ are nonzero. Symmetric elimination expresses successive pivots as continuous rational functions of the matrix entries near $t_{\ast }$; the denominators stay nonzero, and continuity of these finite sums, products and reciprocals follows by the elementary epsilon estimates for real arithmetic. By [F2], every pivot keeps its sign on a common neighborhood, so Sylvester inertia and hence signature are locally constant. A locally constant function on an interval is constant: otherwise the supremum of the initial interval on which its value agrees with the starting value contradicts local constancy at that supremum. Thus the signature is constant on $[a,b]$. At $t_0$ the primitive decomposition of $L_{t_0}$ is orthogonal for this form with definite primitive blocks, so the signature at every $t$ equals the $t_0$ signature. Moreover the primitive dimensions are the same for every $t$: $\dim P_{L_t}^{-i}=\dim H^{-i}-\dim H^{-i-2}$ for all $t$, by the string decomposition of [F1] applied to each $L_t$. Descending induction on $i$ from the top degree in the signature formula of [F1] now shows that the primitive Lefschetz forms of $L_t$ have the $t_0$-signs for every $t$: at the top nonzero degree the signature of $H^{-i}$ is the signature of the primitive form, and inductively subtract the contributions of the already-determined higher primitive spaces, whose dimensions do not depend on $t$. Orthogonality and hL make the primitive forms non-degenerate, and the signature subtraction forces their signatures to be the prescribed signs times their dimensions. Sylvester inertia then gives the standard-sign definiteness for every $t$. In the application $L_{\zeta }=L_0+\zeta M$ with $M$ the middle-position operator is polynomial in $\zeta$, hence continuous on $[0,\mathrm{\infty })$ and on every compact subinterval. This proves (1).

The weak-Lefschetz substitute, injectivity (2). For $i=0$ the map $L^0$ is the identity. Let $i\ge 1$ and $0\mathrm{\ne }a\in V^{-i}$; then $-i\le -1$, so $\phi (a)\mathrm{\ne }0$ by (a). If $L^i\phi (a)\mathrm{\ne }0$, then $L^{i}a\mathrm{\ne }0$ because $L^i\phi (a)=\phi (L^{i}a)$ (the map is $R[L]$-linear). Otherwise $L^i\phi (a)=0$, so $\phi (a)$ is a nonzero primitive vector in $W^{-i+1}$; the HR relations of $W$ give that the Lefschetz value $(\phi (a),\phi (a){)}_L^{-i+1}=⟨\phi (a),L^{i-1}\phi (a){⟩}_W$ is definite nonzero. By (b) this equals $⟨a,L^{i}a{⟩}_V$; in particular $L^{i}a\mathrm{\ne }0$. Hence $L^i$ is injective in both cases. If moreover $\dim V^{-i}=\dim V^i$ for all $i$, injectivity plus equality of finite dimensions makes each $L^i:V^{-i}\to V^i$ an isomorphism, which is hL on $V$. This proves (2).

Sign bookkeeping (3). In the situation of step 2.1 with $i=\mathrm{\ell }+1$, one has $-i+1=-\mathrm{\ell }$, the lowest degree of $W$, and the image of $V^{-(\mathrm{\ell }+1)}$ lies in $W^{-\mathrm{\ell }}$, whose $(\mathrm{\ell }+1)$-st power under $L$ is zero because $W^{\mathrm{\ell }+2}=0$. Thus the primitive branch of step 2.1 applies to every such image and shows that the Lefschetz form on $V^{-(\mathrm{\ell }+1)}$ is definite with the sign of the primitive form of $W$ in degree $-\mathrm{\ell }$, which is the sign prescribed by the standard-sign HR of $W$; the global factor of the HR statement of $V$, if any, is thereby fixed, and $V$ has the standard sign exactly when $W$ does. This proves (3).

The degenerate shifted case (4). Put $Q_L^{-d-i}:=\ker (L^{i+1}{\mathrm{\mid }}_{H^{-d-i}})\subseteq H^{-d-i}$ for $i\ge 0$, and $Q_L^{-d-i}:=0$ for $i<0$. The hypothesis that $L^i:H^{-d-i}\to H^{-d+i}$ is an isomorphism for all $i\ge 0$ gives the shifted primitive decomposition $H^m={⨁}_{j\ge 0}{L}^j{Q}_L^{m-2j}$ by the same string argument as the primitive decomposition of [F1], with the centering shifted by $d$. Fix $m\le 0$, $x\in Q_L^{m-2j}$, $y\in Q_L^{m-2k}$ with $j\ge k\ge 0$; then $(L^{j}x,L^{k}y{)}_L^m=⟨x,L^{j+k-m}y⟩$. Now $y\in Q_L^{m-2k}=\ker (L^{i+1})$ with $-d-i=m-2k$, i.e. $i=2k-m-d$, so $y$ is killed by $L^{2k-m-d+1}$; and $j+k-m\ge 2k-m-d+1$ because $j\ge k\ge k-d+1$ as $d>0$. Hence the pairing vanishes for arbitrary primitive vectors and arbitrary blocks, so the Lefschetz form on $H^m$ is identically zero. This proves (4).

The stable-subspace lemma (5). Let $V\subseteq H$ be $L$-stable with symmetric dimensions. Since $L^i:H^{-i}\to H^i$ is injective, its restriction to $V^{-i}$ is injective, and the dimension equality makes it an isomorphism onto $V^i$; so $V$ satisfies hL. The primitive subspace of $V$ is $P_V^{-i}=V^{-i}\cap P_L^{-i}$, because the kernel of $L^{i+1}$ on $V^{-i}$ is the intersection of $V^{-i}$ 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 $(-1{)}^{(d-i)\mathrm{/}2}$. By the primitive decomposition applied inside $V$ (step 2.1 of Hard Lefschetz gives the primitive decomposition, orthogonality and the signature formula gives it from hL on $V$), the restricted form on $V$ is an orthogonal sum of definite blocks, hence non-degenerate, and its signs are the ambient signs; if the lowest nonzero degree of $V$ is $-d$, the ambient standard sign is also the standard sign of $V$. This proves (5) and completes the lemma; no choice principle is used.