Projective Perverse Hard Lefschetz And Decomposition

Assume AC as inherited from the ordinary constructible suppliers, and let $i:Z\hookrightarrow Y$ be a closed complex algebraic inclusion with complementary open inclusion $j:U\hookrightarrow Y$. Let $P$ be an ordinary constructible perverse sheaf over a characteristic-zero field. Set $A={}^p{H}^0{i}^{!}P$ and $B={}^p{H}^0{i}^{\ast }P$. The adjunction maps define a natural morphism $\alpha :A\to B$. It is an isomorphism if and only if $P\simeq i_{\ast }A\oplus j_{!\ast }{j}^{\ast }P\mathrm{.}$ More precisely, $i_{\ast }A$ is the largest subobject of $P$ supported on $Z$, $i_{\ast }B$ is its largest quotient supported there, and their composite equals $i_{\ast }\alpha$. If $\alpha$ is invertible, these canonical maps give the displayed splitting. For $Z=\{y\}$, $A=H^0(i_y^{!}P)$ and $B=H^0(i_y^{\ast }P)$ are finite vector spaces, so the criterion is exactly invertibility of the degree-zero costalk-to-stalk map. This criterion does not assert that that map is invertible for projective direct images.

Assume AC, let $X$ be smooth projective of complex dimension $n$, and write $H^m=H^m(X;\mathbb{R})$. Let $\eta$ be a Kähler class, and let $L$ be a real $(1,1)$ class such that $L+ϵ\eta$ is Kähler for every $ϵ>0$. Use the centered nilpotent filtration $W$ of cup product by $L$ on total cohomology, with $L:W_a\to W_{a-2}$ and $L^a:{\mathrm{Gr}}_a^W\to {\mathrm{Gr}}_{-a}^W$ an isomorphism. Assume the following three precise hypotheses, to be furnished by the preceding stages of a projective induction:

(a) $W$ is a Hodge filtration by subspaces stable under $\eta$, and $\eta :{\mathrm{Gr}}_a^W{H}^{n-2}\to {\mathrm{Gr}}_a^W{H}^n$ is injective for every $a\le 0$.

(b) For every $i\ge 0$, the form $q_i(x,y)={\int }_X{\eta }^{i+2}{L}^{i}x\wedge y$ on ${\mathrm{Gr}}_i^W{H}^{n-2-2i}$ restricts nondegenerately to the kernel of $L^{i+1}$ there. This includes well-definedness of that induced form; it concerns only nonmiddle graded primitive forms.

(c) Poincaré pairing between complementary ordinary degrees annihilates $W_a\times W_b$ when $a+b<0$.

Set $D=\ker (L:H^n\to H^{n+2})$, $R=D\cap W_{-1}{H}^n$, and $M=\eta \ker (L:H^{n-2}\to H^n)$. Then the limit $\mathrm{\Lambda }={\lim }_{ϵ\downarrow 0}\ker (L+ϵ\eta :H^n\to H^{n+2})$ exists as a Hodge subspace, and The radical of the Poincaré form on $D$ is exactly $R$. Its restriction, with the classical degree-$n$ Hodge sign, polarizes $\mathrm{\Lambda }\mathrm{/}(\mathrm{\Lambda }\cap R)$. Consequently it polarizes every Hodge subspace of $D\mathrm{/}R$ orthogonal to the image of $M$. In particular this applies to the middle double primitive subspace once that subspace's graded primitive condition identifies it with such a Hodge subspace. No polarization or semisimplicity in the middle is assumed in (a)–(c).

Assume AC. Let $f:X\to Y$ be a projective morphism of complex projective varieties, with $X$ smooth connected of dimension $n$, let $\eta$ be a very ample source class, and put $L=f^{\ast }{c}_1(B)$ for a very ample target line $B$. Write $K=R{f}_{\ast }{\mathbb{Q}}_X[n]$, $P_b={}^p{H}^{b}K$, $F_b{H}^q(X)=\mathrm{im}(H^{q-n}(Y,{}^p{\tau }_{\le b}K)\to H^q(X))$, $H_b^q=F_b{H}^q\mathrm{/}{F}_{b-1}{H}^q$, $m=\dim f(X)$ and

$r(f)={\max }_i(2i+\dim \{y:\dim f^{-1}(y)=i\}-n)\mathrm{.}$

Consider the following package: relative $\eta$ hard Lefschetz ${\eta }^i:P_{-i}\simeq P_i$; a finite normalized splitting $K\simeq {⨁}_b{P}_b[-b]$; semisimplicity of every $P_b$ with semisimple complexifications of its IC local systems; target graded hard Lefschetz $L^j:H_b^{n+b-j}\simeq H_b^{n+b+j}$; pure Hodge structures of their ordinary degree on the canonical graded groups; and polarized orthogonal double primitive decomposition. The double primitive space is

${\mathcal{P}}_{-i}^{-j}=\ker {\eta }^{i+1}\cap \ker L^{j+1}\subset H_{-i}^{n-i-j},i,j\ge 0,$

with polarization $Q_t(u,v)=(-1{)}^{t(t-1)\mathrm{/}2}{\int }_X{\eta }^i{L}^{j}u\wedge v$, $t=n-i-j$, and positivity $Q_t(Cu,\bar{u})>0$ for nonzero $u$, where $C$ acts on type $(p,q)$ by $i^{p-q}$. Filtration and primitive forms are the canonical graded ones; no primitive-compatible lift to ordinary global cohomology is assumed.

If this package holds for all smooth connected projective-source maps of lexicographically smaller pair $(r,m)$, with $r$ primary, then it holds for $f$. The image-dimension-zero base is supplied by absolute polarized Hodge theory and finite linear algebra. Finite disconnected smooth sources are handled componentwise. The statement preserves arbitrary projective fibres and requires neither semismallness nor affine paving.

The following are embedded local proof texts, not citations replacing proof. Their earlier-stage inputs are discharged in the order A–H and by the numbered proof above; only the explicitly named external local suppliers retain their own independent review obligations. Section-specific fact and step labels within a component are local to that component.

Construction A: positive hyperplane Thom/cup comparison

Hyperplane restriction–Gysin composite and its exact normalization

This is the local normalization component for the joint projective package. The published proof bodies read are: lem-pullback-of-the-thom-class-along-a-transverse-section, prop-first-chern-class-of-tensor-dual-and-conjugate-lines, and the line normalization in thm-top-chern-class-equals-euler-class-of-the-underlying-real-bundle. The coherent algebraic duality item lem-smooth-projective-embedding-gysin-trace-compatibility was also read, but is not used: its coefficients are coherent sheaves and its trace is not the present analytic constant-sheaf trace. The actual analytic oriented Verdier supplier was read in full, including its positive disk generator and adjunction transposition construction.

1. Smooth divisor purity and the counit

Let M be a smooth complex manifold of dimension m and i:D→M a smooth closed complex divisor. The actual ordinary duality exchange gives

i!k_M = D_D i*D_M k_M = D_D(k_D[2m]) = k_D[-2].

Fix this isomorphism using the positive complex normal orientation. Locally choose a holomorphic coordinate z transverse to D. The compact-support normal disk cochains have the single positive generator in degree two, normalized on its disk/boundary pair. The actual extraordinary adjunction construction uses exactly this finite relative cochain evaluation; therefore its counit i_*i!k_M→k_M, after the displayed purity identification, is the positive normal Thom class morphism

i_*k_D→k_M[2].

This is a normalization of the actual counit, not a new arbitrarily chosen Gysin map. One can check equality of the two morphisms globally: adjunction identifies Hom_M(i_*k_D,k_M[2]) with Hom_D(k_D,i!k_M[2])=H^0(D,k). A morphism is determined by its locally constant scalar on each connected component. The local disk evaluation gives scalar1 on every component, so the counit and positive normal Thom morphism agree. No comparison of only global cohomology ranks is used.

Write R:k_M→i_k_D for ordinary restriction and G:i_k_D→k_M[2] for this oriented counit. Their composite is a morphism k_M→k_M[2], hence an actual class in H²(M;k). Under the relative support model it is precisely the forget-support image of the normal Thom class. Projection formula and the naturality of restriction identify, for every u∈H²(M;k),

G[2q] (i*u)^q R = u^q (G R),

as morphisms k_M→k_M[2q+2]. The even cochain degree introduces no graded interchange sign. This identity holds before passing to ordinary or perverse cohomology.

2. The divisor class is the actual line class

Now let M be smooth projective and D the smooth zero locus of a transverse holomorphic section σ of a complex line bundle B on M. Its normal derivative identifies the normal complex line with B|D, preserving the positive real rank-two orientation. The published transverse-section Thom lemma, whose scope is a closed oriented smooth manifold and is therefore exactly met here, identifies the relative normal Thom class with σ* of the bundle Thom class. Its forget-support image is e(B_R). The published complex-line normalization is c1(B)=e(B_R), so

G R = c1(B): k_M→k_M[2].

All identifications are made through the actual normalized orientation class; there is no minus sign or unproved coherent-sheaf/analytic comparison. Smooth projective manifolds have the requisite finite CW type via the actual compact controlled triangulation supplier. For a disconnected manifold apply the line-class assertion to each component; empty components contribute zero. The line tensor formula therefore applies componentwise to all source spaces below.

3. Universal incidence is a transverse zero locus

Let X be smooth projective of dimension n, with a very ample line A giving its closed embedding in P^d. Let P=(P^d)^dual and M=X×P. On M the evaluation section of

B = pr_X* A ⊗ pr_P* O_P(1)

has zero locus D={(x,H):x∈H}. This section is transverse: at every zero (x,H), evaluation in the varying hyperplane direction surjects onto the one-dimensional fiber, since some homogeneous coordinate of x is nonzero. Equivalently, on a chart where that coordinate is nonzero, the section equation has nonzero derivative in its corresponding hyperplane coefficient. Thus D is smooth of dimension n+d−1 and its normal bundle is B|D with the same complex orientation. The actual line tensor formula gives

c1(B)=pr_Xη + pr_Ph,

where η=c1(A) and h=c1(O_P(1)). This proves the universal divisor class without assuming it from a survey.

For a fixed smooth source hyperplane D_a⊂X, the identical construction with the constant section defining a gives divisor class η. For a smooth source pulled back from a target hyperplane of class L, the divisor line is fL, so its class is fL. These are the exact fixed-source and target restriction/Gysin identities used in the base-HL and nonmiddle-polarization stages. Transversality of the chosen sections is the separate generic hyperplane input; no assertion is made about singular zero loci.

4. Derived shifts and relative perverse identity

Let F=f×id_P:M→Y×P and g=F i. Set

A_0=RF_k_M[n+d]=pRf_k_X[n][d], C_0=Rg_k_D[n+d−1].

Restriction gives R:A_0→C_0[1]. The oriented trace gives G:C_0→A_0[1]. Hence for q=r−1≥0 the correctly shifted composite is

A_0 → C_0[1] → C_0[2r−1] → A_0[2r],

with middle map (ipr_Xη)^(r−1) and final map G[2r−1]. Sections1–3 prove this is cup multiplication by

(pr_Xη)^(r−1) (pr_Xη+pr_P*h)

on A_0. The maps are the same restriction and dual-oriented Gysin maps in the actual universal weak-Lefschetz supplier: that supplier obtains the dual map through the same ordinary duality adjunction and positive orientation. The adjunction triangle identities, rather than a freely chosen scalar, fix this comparison.

The pr_P*h summand is pulled back from Y×P. Its cup action is a natural transformation T:Id→[2] of the actual constructible derived category, obtained by tensoring with the class morphism k→k[2] and using the coherent projection formula. Such a degree-two natural transformation acts as zero pH^jK→pH^(j+2)K for every K. Indeed use naturality with pτ≤jK→K: its action factors through (pτ≤jK)[2], whose pH^j equals pH^(j+2)(pτ≤jK)=0, whereas pH^j(pτ≤jK)→pH^jK is an isomorphism. This is solely bounded t-structure truncation and actual tensor naturality, with no Lefschetz splitting or decomposition theorem.

Consequently after applying perverse cohomology the composite above is exactly p*(η^r)[d]. In particular R:Gysin is η in the middle case r=1, with its positive normalization. The same tensor-naturality argument removes the target class from η'+h when an absolutely very ample source class is needed for the inductive universal family. The source, target and universal restriction–Gysin interfaces are now locally constructed relative to the stated analytic orientation and published Thom/line suppliers.

Acceptance and metadata boundary

This component supplies the formerly open exact cup/trace interface, including its local disk scalar, degree shifts and base-class-zero argument. Required additional direct published dependencies for the joint core are lem-pullback-of-the-thom-class-along-a-transverse-section and prop-first-chern-class-of-tensor-dual-and-conjugate-lines; the latter already explicitly carries the line c1=e normalization, so the separately read top-Chern theorem is not needed as a new direct edge. Root owns their native plan integration after writers drain. The refined-fibre intersection and moving-family components remain separate necessary geometry. No native mathematical acceptance or complete projective package is claimed here.

Construction B: fixed hyperplane interfaces

Actual fixed hyperplane interfaces for the simultaneous induction

This component supplies the generic transversality and weak-Lefschetz maps used in the base/nonmiddle stage. The published thm-generic-smoothness-characteristic-zero and thm-bertini-smooth-hyperplane-section bodies were read; Bertini explicitly covers finite-dimensional basepointfree systems, not only a very ample embedding.

1. A general parameter slice is genuinely noncharacteristic

Take the universal source-hyperplane map g:D→Y×P and its bounded constructible direct image C. Choose a finite complex algebraic Whitney stratification on Y×P compatible with its finitely many ordinary/perverse cohomology objects and the maps and kernels needed in the current calculation, using the actual finite algebraic stratification supplier. This is a finite collection, not an arbitrary cyclic diagram.

For each smooth locally closed stratum T, apply the published generic-smoothness theorem to its projection q_T:T→P, separately on its finitely many irreducible components. On components not dominating P avoid the proper closed image closure; on dominant components take the dense target open on which the whole inverse image is smooth. Intersect these finitely many nonempty opens. Over the resulting parameter open, dq_T is onto at every point. Therefore the slice Y×{a} is transverse to every stratum it meets. The actual noncharacteristic lemma gives perverse-exact a*[-dimP], with its ordinary proper base-change comparison.

Intersect this open also with the generic smooth-source and defect-drop opens. Proper base change identifies aC[-dimP] with Rf_ak_(X_a)[n−1], while apP_b[dimP]=P_b. Thus every restriction/Gysin isomorphism, monomorphism, epimorphism and canonical primitive kernel of the universal weak-Lefschetz supplier specializes with the exact fixed-source shifts. The fixed-source smoothness is the actual Bertini clause, and its defect decrease is the already proved dimension argument. Exact pullback preserves kernels, images and the finite primitive decompositions. Applying lower-defect semisimplicity makes the middle injections/epimorphisms split; no current middle semisimplicity is assumed. Repeat with each new smooth source for the required finite sequence of cuts only while its defect is positive. Strict source-defect drop is invoked only for r>0 and at most r successive nonempty cuts; a defect-zero map cannot drop below zero. Its nonzero perverse grades and negative eta primitive indices already vanish, and its middle stage is handled by the target/image or point base. Thus no source cut after the defect reaches zero is needed. Perverse degrees outside the defect amplitude vanish, so cuts beyond that amplitude are unnecessary.

2. General target hyperplanes are transverse to every required stratum

For a smooth algebraic stratum S⊂Y in a projective embedding, its universal hyperplane incidence I_S⊂S×P is a smooth projective-space bundle over S. Explicit local evaluation in a nonzero homogeneous coordinate shows the defining section is transverse, exactly as in the universal Thom component. Projection I_S→P is a morphism from a smooth variety. Applying the same published generic-smoothness theorem on each component gives a common parameter open where its differential is onto at every point. The kernel of that differential is T_yS∩T_yH. The smooth bundle dimension is dimS+dimP−1, so the kernel has dimension dimS−1; hence S and H are transverse. Nondominant or zero-dimensional incidence components are avoided. Finite intersection over the strata gives one parameter open; no assertion of simultaneous avoidance of all positive-dimensional fibres is used.

Apply published Bertini also to the globally generated pullback of the target hyperplane system on X. It is basepointfree because a nonzero homogeneous coordinate at f(x) provides a section nonzero at x. Thus on a common nonempty parameter open its whole zero scheme X^1 is smooth, even though this line need not be ample. The original stratification of Y includes the fibre-dimension loci, so the chosen target section lowers every positive-dimensional such locus by one and avoids the zero-dimensional loci. The source has dimension n−1 when nonempty; the new defect contributions are

2i+(dimY_i−1)−(n−1)=2i+dimY_i−n.

Thus defect does not increase. Image dimension decreases by one on a positive-dimensional projective image, by the actual projective hypersurface dimension-drop supplier. For image dimension zero the full induction already has its point base case. These facts put every nonempty target-cut map strictly below the original pair (defect,image dimension).

The actual noncharacteristic pullback identifies K^1=iK[-1], and its perverse cohomology with iP_b[-1]. It commutes with the eta kernels and finite heart projectors. This supplies the target primitive compatibility used in the base-HL and nonmiddle calculations.

3. Target weak Lefschetz from actual affine vanishing

Let P be any ordinary perverse object on the projective target Y, H a chosen noncharacteristic target hyperplane, i:Y_1→Y, and j:U→Y its affine complement. Put P_1=iP[-1]. The ordinary localization triangle is j!jP→P→i_P_1[1]. Global compact-support Artin gives H_c^q(U,jP)=0 for q<0. Its long exact sequence makes restriction

H^ell(Y,P)→H^(ell+1)(Y_1,P_1)

an isomorphism for ell<−1 and a monomorphism at ell=−1. Dualize with the actual projective finite cohomology pairing and complex codimension-one orientation to obtain

H^(ell−1)(Y_1,P_1)→H^ell(Y,P)

isomorphic for ell>1 and surjective at ell=1. The index in this Gysin inequality is the TARGET index. The two maps are the actual restriction and oriented adjunction trace; their composite is target cup L by the independently constructed positive Thom/line component. Apply these bounds successively for general target cuts and to the primitive summands of P, since each is again perverse and the pullback is exact.

Inputs and current boundary

This completes the exact source-parameter specialization, target transversality/basepointfree smoothness, target defect bound and weak-Lefschetz mechanisms consumed in the earlier local base-HL/nonmiddle deduction. Direct core prerequisites should explicitly include the published generic-smoothness and Bertini theorems and the actual finite algebraic stratification supplier; all already occur in its existing transitive context. The source and target cup identities are the hyperplane-Thom component. The projective refined-intersection and moving-family stage remains separately owned construction; this component confers no native mathematical acceptance.

Construction C: relative hard Lefschetz

Relative hard Lefschetz stage of the projective induction

This is the relative hard Lefschetz stage of the joint-package construction: a conditional deduction whose exact remaining geometric input is stated below, not assumed under the word “standard.” Sources read: de Cataldo–Migliorini §5.1.1–5.1.3 with §4.7.8–4.7.9, and the BBD maximal-subobject argument in full.

Let K=Rf_Q_X[n], X smooth projective of dimension n, and let eta be a very ample source class. For the universal hyperplane family g, write C=g_Q_univ[n+d−1], where d=dim|eta| and p:Y×P^d→Y. The completed defect calculation gives r(g)<r(f) when r(f)>0. By the induction hypothesis the universal-family direct image has relative HL and semisimple perverse cohomology.

Exact geometric interface required

The actual universal-hyperplane supplier already gives weak Lefschetz maps R and G between p^*pH^b(K)[d] and pH^(b+1)(C), with R an isomorphism for b<-1 and a monomorphism at b=-1, and the dual G isomorphisms above the middle and epimorphism at the middle. Its maximal-base-pullback assertion gives the largest pullback subobject and quotient of pH^0(C).

For the present proof one must additionally establish the compatible restriction/Gysin cup identity on the actual source: G eta'^(r−1) R is p^*(eta^r)[d] on the corresponding perverse groups. Here eta'=i^*p'^*eta, and the universal divisor has line bundle eta_bundle⊠O(1). Its actual Thom trace gives divisor class c1(eta_bundle)+h_parameter. The parameter summand is pulled back from the target and therefore acts by zero on perverse cohomology. The exact available published proof route uses lem-pullback-of-the-thom-class-along-a-transverse-section and prop-first-chern-class-of-tensor-dual-and-conjugate-lines, whose full bodies have been read. Those potential additional supplier edges have been reported to root before actual core acceptance; they are not silently declared supplied by a generic six-operation interface. That precisely typed identity is now locally constructed in semisimplicity-hyperplane-thom-cup-proof.md, including the actual counit scalar, transverse evaluation-section line, shifted restriction/Gysin composite and tensor base-class-zero calculation. It remains conditional on its named upstream orientation/Thom suppliers and focused review; no new native acceptance is inferred.

The universal source class is usable for induction

The class eta' is only relatively ample for g, but eta'+h_parameter is absolutely very ample on the universal source: it is the restriction of O(1,1) under the closed embedding into X×P^d and the Segre embedding. The parameter class is pulled back from the target. For any target class t of degree2, its natural transformation K→K[2] induces zero pH^b(K)→pH^(b+2)(K): by naturality restrict to pτ≤bK, whose pH^(b+2) is zero. Therefore the induced relative operator of eta'+h_parameter equals that of eta'. The inductive HL for the absolute class is exactly the required relative HL for eta'. No general nonprojective ampleness or compactification theorem is used.

Hard Lefschetz in the middle two perverse degrees

Let A=p^*pH^(-1)(K)[d] and B=p^*pH^1(K)[d]. The preceding interface gives a monomorphism A→C_0=pH^0(C), an epimorphism C_0→B, and identifies their composite with p^eta[d]. Suppose the composite has a nonzero kernel T⊂A. Semisimplicity of C_0 splits the chain T⊂ker(C_0→B)⊂C_0, yielding C_0=T⊕R_0⊕S with S→B an isomorphism. The projection C_0→T⊕S is then a quotient consisting of pullbacks from Y. It strictly contains the maximal pullback quotient B, because T is nonzero. This contradicts the maximality assertion. Thus the composite is a monomorphism. The proper direct image K is self-dual, and eta is self-dual, by smooth complex orientation and the proper Verdier comparison; dualizing makes it an epimorphism too. Hence it is an isomorphism. Smooth surjective p^[d] is exact and conservative, so eta:pH^(-1)K→pH^1K is an isomorphism on Y.

For r≥2 both outside restriction/Gysin maps are isomorphisms by weak Lefschetz, and the inside eta'^(r−1) is an isomorphism by induction. Their exact composite therefore proves eta^r:pH^(-r)K→pH^rK is an isomorphism. The r=0 assertion is the identity. The already locally proved bounded-t hard-Lefschetz splitting lemma now gives the finite derived splitting of K. It does not invoke a stronger primitive-compatible global lift.

Nonzero perverse semisimplicity

Weak Lefschetz puts p^pH^bK[d], b<0, inside a semisimple perverse object from the lower-defect universal family (isomorphically below the first degree and as a subobject in that degree). Its finite heart decomposition has orthogonal endomorphism projectors. Degree-zero heart full faithfulness of p^[d] lifts those projectors, and exactness splits them in the abelian heart on Y. Simplicity is reflected by exact conservative pullback: a nonzero proper subobject downstairs would yield one upstairs. Thus pH^bK is semisimple. Positive b follows by duality. The same argument preserves the package’s complexified-local-system clause: ordinary finite coefficient comparison commutes with six operations, and support/dual-support descriptions make flat field extension perverse-exact. Uniqueness of t-truncations makes it commute with perverse cohomology, and exactness makes it commute with the image defining intermediate extension. Thus the above direct-summand maps complexify to direct summands of the inductively semisimple complexified family objects. The same heart projector and simplicity argument applies over $\mathbb{C}$, proving semisimplicity of the resulting complexified local systems. This is an ordinary formal deduction from the already declared ordinary six-operation/perverse suppliers, not an import of equivariant splitting. This proves the exact nonzero-perverse claim without assuming pH^0K semisimple for the original map.

Defect-zero boundary

When r(f)=0, K is already perverse. Proper base change gives stalk cohomology H^(n+q)(f^−1(y)); on the fiber-dimension-i locus it vanishes for n+q>2i. The defect bound gives dimension(Y_i)≤n−2i. Therefore a nonzero q-th cohomology stalk occurs on a locus of dimension≤−q, proving the upper perverse support condition. The normalized smooth-source constant sheaf and its proper pushforward are self-dual, so the lower condition follows. Relative HL and derived splitting are then tautological. Semisimplicity of this middle perverse object remains the later fiber-intersection obligation; it is not proved by this boundary argument.

Completion boundary

The preceding deduction closes the relative-HL/nonzero-semiperversity stage once the exact hyperplane maps and their Thom cup composite are constructed with compatible orientations. The actual joint core still requires the base graded HL, nonmiddle restriction, refined intersection and moving-family geometric stages, as well as its independent upstream Morse construction. Root owns any metadata correction needed before integration. No native acceptance is claimed.

Construction D: base hard Lefschetz and nonmiddle forms

Base hard Lefschetz and nonmiddle polarizations: exact induction deduction

This is a local proof component of the owned joint-package construction. It is conditional on the stated hyperplane map interface, not a claim that the whole projective induction has already been proved. No new ID, item home or metadata edge is added. Full final dcM5.2.1–5.3 and the corresponding fixed source/target hyperplane clauses4.7.6–4.7.7 have actually been read.

Exact geometric interface

Take f:X→Y, X smooth projective of dimension n, source very ample eta, and L=f^*A for a target very ample class A. The preceding relative-HL stage supplies the finite perverse splitting, relative eta strings, and semisimplicity outside perverse degree0.

For k general source hyperplanes, write X_k and f_k. Restriction maps pH^(b−k)K→pH^bK_k are isomorphisms for b<0 and split monomorphisms for b=0; Gysin maps pH^bK_k→pH^(b+k)K are isomorphisms for b>0 and split epimorphisms for b=0. Their composite is eta^k. The fixed-general defect calculation gives r(f_k)≤r(f)−k for1≤k≤r(f). The perverse amplitude is inside[−r(f),r(f)]: the fiber-dimension support computation used for the defect-zero case gives K∈pD≤r(f), and self-duality gives the lower bound. Thus source cuts outside that range are unnecessary.

For j general target hyperplanes, write X^j and f^j. The source is smooth projective by basepointfree Bertini, and defect does not increase while image dimension decreases by j. The exact noncharacteristic pullback is i^*K[−j]=K^j and commutes with eta primitive kernels. The restriction/Gysin maps on the degree0 graded cohomology have the following precisely indexed one-step projective perverse weak-Lefschetz bounds. For a perverse object P on Y and P_1=i^*P[−1], R:H^ell(Y,P)→H^(ell+1)(Y_1,P_1) is an isomorphism for ell<−1 and a monomorphism at ell=−1. The dual G:H^(ell−1)(Y_1,P_1)→H^ell(Y,P) is an isomorphism for ell>1 and an epimorphism at ell=1. Thus the index in the Gysin bound is its target index, not its source index. Their compatible composite is L^j.

These are the precise geometric inputs, not additional consequences asserted of an unspecified “standard Lefschetz theorem.” Absolute Artin plus the open/closed triangle gives target weak Lefschetz because the complement of a target hyperplane is affine, and the noncharacteristic supplier gives the shifted restriction. Fixed-source relative weak Lefschetz comes from the universal-family affine complement and a general noncharacteristic parameter slice. Generic slices must be chosen transverse to all finitely many relevant algebraic strata; the actual strata supplier and Bertini/generic-smoothness clauses must justify this choice. The source divisor Thom/Gysin identities and tensor-line Chern normalization are the explicit pending comparison identified in the separate relative-HL note.

Base hard Lefschetz outside perverse degree0

For b=−k<0, the source-cut restriction identifies H_b^m(X) with a direct summand of H_0^m(X_k). All its projection maps live on Y, so they commute with the target cup class L by naturality. The lower-defect inductive package gives L-HL on X_k about the center dimX_k=n−k=n+b. The restriction and its inverse preserve this direct summand, giving L^j:H_b^(n+b−j)→H_b^(n+b+j) isomorphically. For b=k>0, use the dual source-cut Gysin identification with the same summand shifted in ordinary degree by2k. Its center becomes n−k+2k=n+b. Degrees outside the perverse defect amplitude vanish, so this proves every b≠0 case.

In particular ker(L:H^(n−1)→H^(n+1)) is contained in F_0H^(n−1). Indeed the finite derived splitting makes L block diagonal on perverse grades. In each b>0 block the ordinary degree n−1 lies strictly below its center n+b, so the already proved L-HL makes its first power injective. There is therefore no positive-grade component in the kernel. The same block decomposition makes L strict for the perverse filtration: kernels and images are direct sums of the respective graded kernels and images. This is finite-dimensional filtration strictness, not a hidden mixed-Hodge strictness assertion.

Degree0: powers at least2

Choose one general target hyperplane and its smooth source X^1, of dimension n−1. The image dimension is lower and defect no larger, so its full package is an induction hypothesis. For j≥2, target weak Lefschetz makes the restriction H_0^(n−j)(X)→H_0^(n−j)(X^1) and the Gysin H_0^(n+j−2)(X^1)→H_0^(n+j)(X) isomorphisms. The inductive middle map is L^(j−1) about the center n−1. Its composite with restriction/Gysin is precisely L^j, proving the desired isomorphism.

Degree0: the first power

Let alpha∈H_0^(n−1)(X) satisfy L alpha=0. The canonical eta primitive decomposition writes alpha=Σ_(t≥0)eta^t alpha_t, with alpha_t in the eta-primitive perverse grade−2t and ordinary degree n−1−2t. Each projection commutes with L, so L alpha_t=0. For t>0, its ordinary degree is one below its L-Lefschetz center n−2t. The nonzero-grade case just proved makes that first L map an isomorphism; hence alpha_t=0. Thus alpha is eta-primitive.

Filtration strictness and kerL⊂F_0 give an actual lift a∈kerL⊂H^(n−1) of alpha. Decompose a into its ordinary Hodge types using classical Hodge theory on X. Every type component is still in kerL because L has type(1,1), and hence in F_0 by the preceding kernel bound. If alpha were nonzero, at least one component would have a nonzero class in H_0^(n−1); replace a by it and alpha by its class. No Hodge structure on the original perverse filtration has been assumed at this point. The argument just given again makes this alpha eta-primitive.

Target weak Lefschetz injects alpha into H_0^(n−1)(X^1), and noncharacteristic primitive compatibility makes the restricted class eta-primitive and L-primitive there. It is a nonzero class in the lower-induction middle double primitive space. It is of pure Hodge type: ordinary holomorphic restriction preserves type, and the lower-induction perverse filtration is Hodge. The lower-induction polarization therefore makes its integral against its conjugate nonzero, up to the specified nonzero Weil/sign factor. But the exact target-divisor Gysin formula gives

∫(X^1) a|(X^1)∧bar a|_(X^1)=∫_X L a∧bar a=0,

a contradiction. Thus the first L map is injective. Perverse self-duality gives equal finite dimensions in degrees n−1 and n+1, so it is an isomorphism. This completes graded L-HL with no use of the original middle primitive positivity.

The Hodge perverse filtration

The now proved L-HL identifies the total perverse filtration with the centered nilpotent filtration. Its intrinsic kernel/image formula is Hodge degree by degree because powers of L have fixed Hodge type. This is the full formal argument recorded in the graded-middle-inputs proof. Therefore the original perverse graded spaces are now pure of their ordinary cohomological degrees. It is legitimate to use their Hodge structures in the remaining nonmiddle restriction argument; doing so in the preceding first-power contradiction would have been circular.

Nonmiddle polarization when the eta index is positive

Let u belong to P_(−i)^(−j), with i≥1,j≥0. Restrict to i general source hyperplanes, obtaining an injective map into degree n−i−j on X_i, in perverse grade0. The injection on cohomology follows from the split monomorphism of perverse sheaves; its maps commute with L. It preserves Hodge type because the original and lower-induction perverse filtrations are now Hodge and ordinary restriction is a Hodge map.

The restricted class is eta-primitive. Indeed apply Gysin to eta times its restriction. The compatible cup identity gives eta^(i+1)u=0. Gysin on the positive perverse degree2 of X_i is an isomorphism by relative weak Lefschetz, so eta of the restricted class is zero. It remains L-primitive because restriction commutes with L and L^(j+1)u=0. Thus it lands in the lower-induction double primitive space of indices(0,j). The lower map f_i has smaller defect and its full polarization package is already known. Its form pulls back exactly to

∫(X_i) L^j u|(X_i)∧v|_(X_i)=∫_X eta^i L^j u∧v.

Consequently the required nonmiddle primitive form is a positive polarization on the original Hodge subspace, with the exact inherited degree sign. It is nondegenerate. No ordinary globally eta-primitive lift is claimed: the argument uses canonical graded primitive kernels and the perverse Gysin isomorphism.

Nonmiddle polarization when the eta index is0

For j≥1, restrict a double primitive class u of degree n−j to j general target hyperplanes. The successive target weak-Lefschetz restrictions are isomorphisms until the last step, which is a monomorphism, so the resulting map into H_0^(n−j)(X^j) is injective. Its source map f^j has smaller image dimension and no greater defect. Noncharacteristic pullback preserves eta-primitive kernels. The restricted class is also L-primitive: Gysin of L times that restriction is L^(j+1)u=0, and the successive Gysin maps from degree2 above the lower middle are isomorphisms. Hence it lands in the lower-induction middle double primitive space. Its polarization pulls back by the compatible divisor identity to ∫_X L^j u∧v. This proves the remaining nonmiddle forms. If j is larger than m=dim f(X), make only m general target cuts. Every successive restriction in this case is an isomorphism, because its source hypercohomological index remains below−1 until the last step. The final index is m−j<0 on a perverse object supported on the finite zero-dimensional image. That object is a finite sum of point-supported heart objects and has hypercohomology only in degree0. Thus the original class space was zero. For m=0 this is already the point-image base case. This proves the large-j boundary directly from the stated hyperplane maps and ordinary point perverse heart, without adding an unproved perverse amplitude estimate.

Completion boundary

This proves the ordered conditional deductions for graded base HL, Hodge purity of the perverse filtration and nonmiddle primitive polarizations. Their exact geometric restriction/Gysin/transversality inputs remain necessary and are explicitly stated. Together with the completed graded-middle-input verification and middle-limit lemma, these stages would supply all primitive polarizations before the refined fiber intersection and moving-family semisimplicity stage. The local Morse product and source Thom/Gysin comparisons are not silently replaced by citation.

Construction E: exact middle inputs

Local verification of the middle-limit lemma's earlier-stage inputs

This is a conditional proof component of the joint projective induction; it does not claim that the earlier geometric induction has been completed. Its inputs are exactly: relative perverse hard Lefschetz, graded base-class hard Lefschetz, the constructed graded primitive decomposition and centered nilpotent-filtration lemmas, classical Poincaré/Hodge theory, and the earlier nonmiddle primitive polarizations. No new IDs, homes or metadata edges are introduced.

Let m denote ordinary cohomological degree and b the perverse grade. Write H_b^m for the canonical graded perverse cohomology, and F_bH^m for its filtration. Earlier-stage relative HL and abstract splitting identify H_b^m with the cohomology of the corresponding perverse sheaf and give the natural graded eta primitive decomposition. Assume the earlier-stage base HL L^j:H_b^{n+b-j}→H_b^{n+b+j} is actually proved. Then the filtration

W_aH^m = F_(m−n+a)H^m

on total cohomology is the centered filtration of the nilpotent operator L: its degree-j isomorphism is exactly the displayed base HL with b=m−n+j. Thus uniqueness in the local centered-filtration lemma identifies it intrinsically with the kernel/image formula. The following calculations verify all formal inputs of the new middle-limit lemma.

Its Kähler hypothesis also holds. Embed $Y$ using $B$. By [F4] applied to the ambient projective space with $\mathcal{O}(1)$, choose a positive closed $(1,1)$-form ${\omega }_{\mathrm{F}\mathrm{S}}$ representing $c_1(\mathcal{O}(1))$. The composite $X\to Y\hookrightarrow {\mathbb{P}}^N$ is holomorphic, so its pullback ${\omega }_L$ is closed, of type $(1,1)$, and semipositive: ${\omega }_L(v,Jv)={\omega }_{\mathrm{F}\mathrm{S}}(df(v),Jdf(v))\ge 0$. By [F4], $\eta$ has a positive closed representative ${\omega }_{\eta }$. Hence ${\omega }_L+ϵ{\omega }_{\eta }$ is positive for every $ϵ>0$ and represents $L+ϵ\eta$, as required.

(a) Hodge stability and eta injectivity

The intrinsic kernel/image formula for W, degree by degree, consists of kernels and images of powers of cup product by L, of Hodge type(j,j), within the appropriate ordinary degree. Hence it consists of Hodge subspaces; finite sums and intersections stay Hodge. Since eta commutes with L, it preserves every kernel and image and therefore W.

For a≤0, Gr_a^W H^(n−2)=H_(a−2)^(n−2), while Gr_a^W H^n=H_a^n. The eta map between them is the first power in the relative eta string from perverse degree a−2. Relative hard Lefschetz makes eta^(2−a) an isomorphism on that string. If the first power kills a class, so does that isomorphism; therefore the class was zero. This proves the exact eta-injectivity in hypothesis(a), rather than assuming a stronger primitive-compatible splitting in the derived category.

(c) Centered-filtration pairing

On total cohomology define a nondegenerate bilinear form S(u,v)=(-1)^floor(m/2) ∫X u∧v when u∈H^m,v∈H^(2n−m), and zero for other ordinary-degree pairs. Poincaré duality makes it nondegenerate. It also has the exact symmetry required by the self-dual-filtration supplier: S(v,u)=(-1)^n S(u,v) on every complementary-degree pair. For m=2k the degree sign ratio is(-1)^(n−2k) and the cup-interchange sign is+1; for m=2k+1 the ratio is(-1)^(n−2k−1) and the cup-interchange sign is−1. Their products both equal(-1)^n. Thus S is symmetric for even n and skew-symmetric for odd n. Since the sign changes under m→m+2 and the even cup class L is self-adjoint for the ordinary integral, S(Lu,v)+S(u,Lv)=0. The actual local self-dual-centered-filtration lemma therefore gives W_a^⊥=W(−a−1). Its restriction to any complementary-degree pair differs from the ordinary integral only by a nonzero sign. Hence the ordinary integral annihilates W_a×W_b when a+b<0. This is exactly hypothesis(c). No Hodge weight theorem or decomposition theorem is used in this argument.

(b) The nonmiddle form q_i

Fix i≥0 and put m_i=n−2−2i, A=i+2. Then Gr_i^W H^(m_i)=H_(−A)^(m_i). The heart-level eta primitive decomposition and its cohomology give the direct sum

H_(−A)^(m_i)= ⊕_(r≥0) eta^r U_r,

where U_r is the eta-primitive part of H_(−A−2r)^(m_i−2r). This is the locally constructed canonical primitive decomposition: no chosen global Deligne lifts are used. Its projections commute with L because the heart-level decomposition is natural and L is the target cup class on cohomology. Therefore the kernel K_i of L^(i+1) decomposes as ⊕ eta^r V_r, where V_r=ker L^(i+1) on U_r.

The L-Lefschetz center on U_r is n−A−2r. Its degree m_i−2r equals that center minus i. Thus V_r is precisely the double primitive part with first primitive index A+2r=i+2+2r and second index i. Every one of these first indices is positive, so its nondegenerate form is supplied by the previously established NONMIDDLE polarizations.

Choose representatives u∈W_i H^(m_i−2r) and v∈W_i H^(m_i−2s). The form q_i on their eta powers is

q_i(eta^r u,eta^s v)=∫_X eta^(A+r+s) L^i u∧v.

This is well-defined on the graded spaces: replacing either representative by a W_(i−1) vector gives, after L^i, a W_(−i−1) vector paired with W_i, which vanishes by(c). If r>s, then A+r+s≥A+2s+1. The eta-primitive condition on v makes eta^(A+2s+1)v belong to W_(i−1) in its new ordinary degree. Apply the remaining eta powers and L^i, and pair with u; again(c) gives zero. If s>r, interchange the arguments, changing only the Poincaré graded sign, and obtain the same zero. Therefore the direct sum is orthogonal for q_i.

For r=s its form is exactly ∫ eta^(A+2r)L^i u∧v, the previously polarized double primitive form of indices(i+2+2r,i), up to its specified nonzero Hodge normalization. It is nondegenerate on V_r. A finite orthogonal sum of these nondegenerate forms is nondegenerate on K_i. This proves the exact hypothesis(b). It also supplies the graded-form well-definedness, which cannot be omitted merely because a published source cites a double Lefschetz decomposition.

Identification of the middle double primitive subspace

The natural map from D=ker(L:H^n→H^(n+2)) to ker(L:Gr_0^W H^n→Gr_(−2)^W H^(n+2)) is surjective, with kernel R=D∩W_(−1)H^n. Indeed lift a graded kernel vector to x∈W_0H^n. Its error Lx lies in W_(−3)H^(n+2). The homogeneous centered Jordan strings make L:W_(−1)H^n→W_(−3)H^(n+2) surjective: a target of weight≤−3 has a predecessor of weight≤−1. Subtract that predecessor to obtain an actual lift in D. All maps here are Hodge maps, so this identifies D/R as a Hodge space with the indicated graded kernel.

The middle double primitive space T is the simultaneous kernel of this graded L map and eta:Gr_0^W H^n→Gr_0^W H^(n+2). For any representative x∈D of a class in T, eta x lies in W_(−1)H^(n+2). Every z∈ker(L:H^(n−2)→H^n) lies in W_0 by the centered string weights. Therefore

∫_X x∧eta z = ∫_X eta x∧z =0

by(c). Thus T is a Hodge subspace of D/R orthogonal to the image of M=eta ker L. This is precisely the hypothesis consumed by the conclusion of lem-middle-lefschetz-primitive-polarization-follows-from-nonmiddle-polarizations. That local six-step theorem now gives its positive middle polarization. No middle positivity has been used to establish any of(a)–(c) or this identification.

Double primitive decomposition and orthogonality

For $a,b\ge 0$, first apply the canonical $\eta$ decomposition and then the $L$ decomposition on each $\eta$-primitive summand. The latter is legitimate because the $\eta$ projectors commute with $L$, so the already proved $L$ hard Lefschetz restricts to their images. This gives the finite decomposition $H_{-a}^{n-a-b}={⨁}_{r,s\ge 0}{\eta }^r{L}^s{\mathcal{P}}_{-a-2r}^{-b-2s}\mathrm{.}$ All other grades and degrees follow by the corresponding hard-Lefschetz isomorphisms. On the displayed space the form ${\int }_X{\eta }^a{L}^b(-)\wedge (-)$ is well defined: its representatives have $W$-weight at most $b$, and an error in $W_{b-1}$ pairs, after $L^b$, with $W_b$ to zero by (c).

For primitive representatives $u,v$ in summands $(r,s),(r^{\mathrm{\prime }},s^{\mathrm{\prime }})$, their pairing is ${\int }_X{\eta }^{a+r+r^{\mathrm{\prime }}}{L}^{b+s+s^{\mathrm{\prime }}}u\wedge v$. If $r>r^{\mathrm{\prime }}$, move ${\eta }^{a+2r^{\mathrm{\prime }}+1}$ onto $v$; its graded primitive condition lowers its $W$-weight by at least one. The remaining powers and (c) make the pairing zero. If $r=r^{\mathrm{\prime }}$ and $s>s^{\mathrm{\prime }}$, use $L^{b+2s^{\mathrm{\prime }}+1}v$ in the same way. Interchanging arguments handles the opposite inequalities. Thus distinct summands are orthogonal. On a diagonal summand the form is exactly the primitive form of indices $(a+2r,b+2s)$, transported by the injective Hodge map ${\eta }^r{L}^s$. Its primitive degree is $t=n-a-b-2r-2s$; the sign $(-1{)}^{t(t-1)\mathrm{/}2}$ gives its polarization by the positivity already proved. Relative to the sign for degree $n-a-b$, this adds $(-1{)}^{r+s}$. This establishes the claimed orthogonal polarized decomposition and the raw-cup orthogonality used in Construction F.

Completion boundary

These calculations complete the FORMAL verification of the middle-limit theorem's inputs, conditional on the previously ordered relative HL, base graded HL and nonmiddle polarization stages. The actual geometric proofs of those earlier stages, refined fiber intersection-map invertibility and moving hyperplane family, plus the local Morse product/sheaf transport, remain required components of the joint projective core. This note does not mark the joint-package supplier complete or grant native acceptance.

Construction F: refined fibre intersection

Refined fibre intersection and the costalk-to-stalk map

This is a component of the same simultaneous projective induction. It proves the refined-intersection stage from the exact already-ordered induction inputs below; it does not assert that those earlier stages have independent verification.

Statement

Let $f:X\to Y$ be a projective morphism of complex projective varieties, with $X$ smooth projective of pure complex dimension $n$, $K=R{f}_{\ast }{\mathbb{Q}}_X[n]$, $P_b={}^p{H}^{b}K$, and $P=P_0$. Choose a very ample source class $\eta$, a very ample target line bundle $A$, and set $L=f^{\ast }{c}_1(A)$. Let $y$ be a zero-dimensional stratum, $i:\{y\}\hookrightarrow Y$, and $Z=f^{-1}(y)$. Set $A_y=H^0(i^{!}P)$ and $B_y=H^0(i^{\ast }P)$. Under the ordered inputs below, the canonical map $I_y:A_y\to B_y$ is the degree-zero graded refined intersection map $H_{n,0}^{\mathrm{B}\mathrm{M}}(Z)\to H_0^n(Z)$ and is invertible. Its factorization through $H_0^n(X)$ is the class map followed by restriction. The class image is a pure Hodge substructure stable under the canonical perverse $\eta$ primitive projectors. The raw graded cup form restricts nondegenerately to it, although that raw form need not be a polarization of the entire middle graded group.

Facts & Assumptions

Given: The projective map, zero stratum and full AC convention, together with the following genuinely earlier-stage inputs. The argument uses finite $p$-splitness produced by the relative-HL stage and the local abstract splitting lemma; it does not assume semisimplicity of the current $P_0$, the refined intersection theorem, or a primitive-compatible lift to ordinary global cohomology.

[F1] Ordinary constructible coherent six operations supply proper base change for $i^{\ast }$ and $i^{!}$, compatible adjunction/trace, Verdier duality, localization and projection formula. Smooth complex orientation gives ${\omega }_X={\mathbb{Q}}_X[2n]$, hence ${\omega }_X[-n]={\mathbb{Q}}_X[n]$. Proper duality therefore gives $DK=K$, and perverse duality gives $DP=P$. The perverse heart supplies $i^{\ast }{P}_b\in D^{\le 0}(\{y\})$, $i^{!}{P}_b\in D^{\ge 0}(\{y\})$ and finite perfect dual costalk/stalk pairings. The exact suppliers are lem-analytic-constructible-sheaves-admit-coherent-six-operations and thm-ordinary-middle-perverse-t-structure-glues-and-has-simple-intersection-complexes.

[F2] The earlier relative-HL stage gives a finite normalized isomorphism $K\simeq {⨁}_b{P}_b[-b]$ inducing the identity on each $P_b$. Its heart-level $\eta$ primitive decomposition is canonical and functorial for target cup operators. The previously proved base-HL/Hodge/middle-polarization stages give pure Hodge perverse graded cohomology, its self-dual filtration, and the orthogonal double primitive polarizations, including $P_0^0$. These are the exact earlier clauses of the same joint induction; their local proofs remain owned by the semisimplicity author.

[F3] Artin affine vanishing is the actual local theorem lem-artin-affine-vanishing-for-constructible-perverse-sheaves. The weight-image equality for smooth $U^{\mathrm{\prime }}\subset X$ and proper $Z\to U^{\mathrm{\prime }}$ is thm-mixed-hodge-weights-control-proper-cohomological-restriction-images. The properly typed class map $H_i^{\mathrm{B}\mathrm{M}}(Z)\to H^{2n-i}(X)(n)$ is a mixed Hodge morphism by thm-proper-cohomology-has-functorial-mixed-hodge-weights-at-most-degree.

[F4] On primitive ordinary degree $t$ use $Q_t=ϵ(t)\int {\eta }^i{L}^j(-)\wedge (-)$, $ϵ(t)=(-1{)}^{t(t-1)\mathrm{/}2}$, and $C=i^{p-q}$, with positive Hermitian form $Q_t(Cu,\bar{v})$. The polarized sub-Hodge restriction lemma is lem-polarized-hodge-structures-have-orthogonal-semisimple-substructures. This is the preserved classical convention, not dcM's placement of $C$ in the second slot.