The Fujiki relation

LetXbe a nonempty, compact, irreducible reduced complex space. Thenn := dim(X)is a natural number, and we may define a functiontXon H²(X,C)by means of the assignment

a7→

Z

Xaⁿ.

The Fujiki relation reveals how, in caseXis symplectic and withΩ²_X(_Xreg)_{of complex} dimension 1, the Beauville-Bogomolov form ofX(cf. Definition 3.2.11) relates to the functiontX. We will introduce some appropriate rigorous terminology in Definition 3.5.1 below. Note that the fact that any irreducible symplectic complex manifold satisfies the Fujiki relation is due to A. Fujiki (cf. [21]), hence the name. The main result of this section is Theorem 3.5.11 which generalizes Fujiki’s result to a wider class of certain, possibly nonsmooth, symplectic complex spaces.

Let us mention that D. Matsushita has advertised a result similar to ours in form of [55, Theorem 1.2]. However, we feel that several points of Matsushita’s line of reasoning in loc. cit. are much harder to establish than his exposition makes the reader believe. In addition to, the proof of [55, Theorem 1.2] does unfortunately rely on Y. Namikawa’s [61, Theorem 8], which we reprehend as explained at the very beginning of § 3.4. We emphasize that we do not think that Matsushita’s intermediate conclusions are wrong; on the contrary, we reckon that his argument can be amended by occasionally invoking techniques of [62]. Anyhow, our Theorem 3.5.11 is stronger than Matsushita’s in the respect that we do not require our space in question to be projective, nor do we require it to beQ-factorial (or similar).

Definition 3.5.1. — LetXbe a compact, connected, symplectic complex space such

Proposition 3.5.2. — Let X be a compact, connected, symplectic complex space of Kähler type such thatΩ²_X(Xreg)is of dimension1overC. Then the following are equivalent:

(i) X satisfies the Fujiki relation. since as Xis a compact, irreducible reduced complex space of dimension 2r, the function

(iii) implies (ii). AsXis compact, H²(X,C)is finite dimensional. Thus there exists d ∈ Nas well as aC-linear isomorphismφ:C^d → H²(X,C). Furthermore, there exist complex polynomials f andgindvariables such that fC(x) = qX(φ(x))and gC(x) = R

X(φ(x))^dim(X) for all x ∈ C^d, where fC and gC denote the polynomial functions onC^dassociated to f andg, respectively. Clearly, f andgare homogeneous of degrees 2 and dim(X), respectively. By (iii), we have Z(f) =Z(g)as, for allv∈ H²(X,C),v^dim(X)=0 in H^∗(X,C)if and only ifR

Xv^dim(X)=0. Thusp

(f) =^p(g) by elemetary algebraic geometry, whence f^j=hg. By Corollary 3.2.26,f is irreducible in the polynomial ring. Therefore, there existsλ∈ C^∗andisuch thatg= λfⁱ. By comparison of degrees,i=r. Thus, for allx∈C^d, we have

g_C(x) = (λf^r)_C(x) =λ(f_C(x))^r_. Plugging inφ⁻¹, we obtain (ii).

(ii) implies (i). AsXis a compact, reduced complex space of Kähler type, Xis of Fujiki classC. Hence, X being in addition symplectic, there exists a normed symplectic classwonXby Proposition 3.1.25, Lemma 3.2.8, Proposition 3.2.14, and

Proposition 3.2.4. By Corollary 3.2.17, we have qX(w+w) = 1. There exists a resolution of singularities f:Xe →X. Setwe:= f^∗(w). Then

f^∗((w+w)^2r) = (f^∗(w+w))^2r= (w_e+w_e)^2r.

By Proposition 3.2.14,weis a normed generically symplectic class onX. In particular,e weis the class of a global 2-differential onX. Thus we havee weⁱ =0 in H^∗(X,e C)for all

Lemma 3.5.3. — Let V be a finite dimensional complex vector space, Q a quadric inP(V), and p∈ _P(V)\Q. Denote Gpthe set of all lines inP(V)passing through p. Then there

in C[X₁, . . . ,XN]. Further, we setQ⁰ := {x ∈ H : f⁰(x) = 0}. Clearly, H is a other inclusion letx ∈Kbe arbitrary. By definition of the coneK, there existsq∈Q⁰ such thatx∈ pq, whence|pq∩Q| 6=2 again by the claim. Therefore,xis contained in the left hand side of (3.5.3.1).

It remains to show that rk(f⁰) = rk(f)−1. Denote byAandA⁰ the symmetric coefficient matrices associated to f and f⁰, respectively. The rows of Aare numbered starting with 0. When first multiplying each of the rows 1 toNofAby−a0and then adding, fori=1, . . . ,N,b_0i-times the 0-th row to thei-th row, one obtains the matrix

Corollary 3.5.4. — Let V be a finite dimensional complex vector space, Q a quadric of rank

≥2inP(V), and p∈P(V)\Q. Then there exists a nonempty, Zariski-open subset U of P(V)such that p∈/U and, for all q∈U, we have|L∩Q|=2, where L denotes the line in P(V)joining p and q.

Proof. — By Lemma 3.5.3 there exist a hyperplaneHinP(V)and a quadricQ⁰inH such thatp ∈/H, rk(Q⁰) =rk(Q)−1, and (3.5.3.1) holds, whereGpandKstand for the set of all lines inP(V)passing throughpand the cone with baseQ⁰and apexp, respectively. SetU:=P(V)\K. Then asKis an algebraic set inP(V),Uis Zariski-open inP(V). Since rk(Q) ≥ 2 by assumption, we have rk(Q⁰) = rk(Q)−1≥ 1, whenceH\Q⁰ 6=_{∅. Thus}U6=_∅asH\Q⁰ ⊂U. SinceQ⁰ 6=_∅,p ∈K; thusp∈/U.

Now letqbe an arbitrary element ofU. LetLbe the line inP(V)joiningpandq. Then if|L∩Q| 6=2, we would haveL⊂Kby (3.5.3.1) and thereforeq∈K, which is clearly not the case looking at the definition ofU. In conclusion,|L∩Q|=2.

Lemma 3.5.5. — Let A be a commutativeC-algebra, V a finite dimensionalC-vector sub-space of A, r ∈N\ {0}, and Q a quadric of rank≥ 2inP(V). Assume that there exists c∈V such that c^2r 6=0in A. Moreover, assume that

Q⊂R:={p∈_P(V):(∀x∈ p)x^r+1=0 inA}. Then we have:

Q=R=S:={p∈P(V):(∀x∈p)x^2r =0 inA}.

Proof. — Obviously we haveQ⊂R⊂Ssincer+1≤2r(thus for allx∈ A,x^r+1=0 inAimpliesx^2r =_{0 in}A). Hence, it suffices to show thatS⊂Q. For that matter, letp be an arbitrary element ofS. Assume thatpis not an element ofQ. Then by Corollary 3.5.4, there exists a nonempty, Zariski-open subsetUofP(V)such thatp ∈/Uand, for allq∈U, one has|L∩Q|=2, whereLstands for the line inP(V)joiningpandq.

Observe thatSis an algebraic set inP(V). Since there existsc∈Vsuch thatc^2r6=0 in A,P(V)\S6=_{∅. Thus}U\S6=_∅and there existsq∈U\S. Denote the line inP(V) joiningp andqbyL. Then|L∩Q| = 2, i.e., there exist p⁰ andq⁰such that p⁰ 6=q⁰ andL∩Q={p⁰,q⁰}. SinceQ⊂R, we havep⁰,q⁰ ∈R, whence there existx∈ p⁰and y∈q⁰such thatx^r+1=y^r+1=0 inA. As the multiplication ofAis commutative, we infer using the binomial theorem that, for allλ,µ∈C:

(3.5.5.1) (λx+µy)^2r =

∑

2r j=0

2r j

(λx)^2r−j(µy)^j = 2r

r

(λ^rµ^r)x^ry^r.

Letv ∈ pandw∈ q. Sincexandyspan the lineL, there areλ,µ,λ₁,µ₁ ∈ Csuch thatv =λx+µyandw=λ1x+µ1y. Sinceq∈/S, we havew^2r 6=0 inA. Therefore, substitutingλ₁andµ₁forλandµ, respectively, in (3.5.5.1), we see thatx^ry^r6=0 inA.

As p∈ S, we havev^2r =0. Thus now, (3.5.5.1) yields thatλ^rµ^r =0. In turn,λ=₀ and consequentlyp=q⁰or elseµ=0 and consequentlyp =p⁰. Either way,p ∈Q.

This argument shows that indeed, for allp ∈ S, we havep ∈ Q. In other words, S⊂Q, quod erat demonstrandum.

Proposition 3.5.6. — Let X be a compact, connected, symplectic complex space of Kähler type such thatΩ²_X(Xreg)is of dimension1overC. Assume that

(3.5.6.1) QX ⊂ {l∈Gr(1, H²(X,C)):(∀v∈l)v^r+1=0 in H^∗(X,C)}, where r:=¹/²dim(X). Then X satisfies the Fujiki relation.

Proof. — DefineAto be the complex subalgebra H^2∗(X,C)of H^∗(X,C)andVto be theC-submodule ofAwhich is given by the canonical image of H²(X,C)inA. Then Ais a commutativeC-algebra andVis a finite dimensionalC-submodule ofA(asX is compact). Evidently,ris a natural number different from 0. Set

Q:={p∈P(V):(∀x∈p)qX(ψ(x)) =0},

whereψdenotes the inverse function of the canonical isomorphism H²(X,C)→V.

ThenQis a quadric of rank≥2 inP(V)by Corollary 3.2.26. By Proposition 3.1.25, there exists a symplectic classwonX. We identifywwith its image inV. As in the proof of Proposition 3.5.2, we calculate inA(or H^∗(X,C)):

(w+w)^2r = 2r

r

w^rw^r. By Proposition 3.2.4,R

Xw^rw^r >0; in particular,(w+w)^2r 6=0 inA. By (3.5.6.1), we see that

Q⊂ {p∈P(V):(∀x∈ p)x^r+1=0 inA}. Hence by Lemma 3.5.5,

Q={p∈P(V):(∀x∈ p)x^2r =0 inA}.

This implies that (iii) of Proposition 3.5.2 holds. Applying Proposition 3.5.2, one infers thatXsatisfies the Fujiki relation.

Lemma 3.5.7. — Let f:X→S be a proper, equidimensional morphism of complex spaces and t ∈ S such that Xtis connected and symplectic,Ω²_Xt((Xt)reg)is of dimension1over the field of complex numbers, S is simply connected, and the fibers of f are of Fujiki classC. Moreover, assume that, for all i∈N, the sheafHⁱ(f,C)(cf.(A.3.11)) is locally constant on Stop. SetP:=_P^2,2_t (f)_MHS. Suppose that the mapping

P:S−→G:=Gr(1, H²(Xt,C))

is holomorphic and there exists a neighborhood W ofP(t)in G such that Q_Xt∩W⊂_P(S).

Then Xtsatisfies the Fujiki relation.

Proof. — Definerto be the unique natural number such that 2r=dim(Xt). Further on, introduce the following notation, where we exponentiate in the ring H^∗(Xt,C), into which H²(Xt,C)embeds canonically, and the “0” shall be considered the zero element of the ring H^∗(Xt,C):

Rt:={l∈ |G|:(∀v∈l)v^r+1=0}, St:={l∈ |G|:(∀v∈l)v^2r =₀}.

As f is locally topologically trivial attand the topological spaceStopis locally con-nected, there exists a connected open neighborhoodVoftinStopsuch that the sheaf H^∗(_f,C)is constant onStopoverV. In particular, for alls∈V, the canonical maps

(H^∗(f,C))(V)−→(H^∗(f,C))_s are bijective.

Let nows∈Vbe arbitrary andv ∈_P(s). Putρ:=ρ²(f,C). SinceStopis simply connected, there exists a unique elementain(_Π(Stop))₁(s,t). Putφs,t:= (ρ₁(s,t))(a) and recall that thus

φs,t: H²(Xs,C)−→H²(Xt,C)

is an isomorphism of complex vector spaces. Therefore, there exists a unique element u ∈ H²(Xs,C) such thatφs,t(u) = v; note thatP(s) ⊂ H²(Xt,C), whence v ∈ H²(Xt,C). By the definition ofP^2,2_t (f)_MHS, we have

P(s) =φs,t[F²H²(Xs)],

which tells us thatu∈ F²H²(Xs). Since f is equidimensional, we have dim(Xs) = dim(Xt) =2r. Consequently, we haveu^r+1=0 in H^∗(Xs,C)since the cohomological cup product is filtered with respect to the Hodge filtrations on the cohomology ofXs. As the morphism f is proper, the base change map

(H²(f,C))_t−→H²(Xt,C)

is a bijection. Therefore, there exists one, and only one,γ∈(_H²(f,C))(V)_{which is} sent tovby the composition of functions:

(3.5.7.1) (H²(f,C))(V)−→(H²(f,C))_t−→H²(Xt,C).

By the definition ofρ²(f,C), we know that γ is sent tou by the composition of functions (3.5.7.1), where we replacetbys. Clearly, the following diagram of sets and functions commutes:

(3.5.7.2) (H²(f,C))(V) ^//

(H²(f,C))_s ^//

H²(Xs,C)

(H^∗(f,C))(V) ^//(H^∗(f,C))_s ^//H^∗(Xs,C)

Thus the canonical image ofγin(_H^∗(f,C))(V)is mapped to the canonical image of uin H^∗(Xs,C)by the composition of the two functions in the bottom row of (3.5.7.2).

In addition, the composition of the two functions in the bottom row of (3.5.7.2) is a morphism of rings from(H^∗(f,C))(V)to H^∗(Xs,C). Thus the composition of the two functions in the bottom row of (3.5.7.2) sendsγ^r+1tou^r+1, whereγandudenote the canonical images ofγanduin(H^∗(f,C))(V)and H^∗(Xs,C), respectively. Employing the fact that the composition of the two functions in the bottom row of (3.5.7.2) is one-to-one (it is a bijection actually), we further infer thatγ^r+1=0 in(H^∗(f,C))(V). Playing the same game as before withtin place ofs, we see thatγ^r+1is sent tov^r+1 by the composition of functions

(H^∗(f,C))(V)−→(H^∗(f,C))_t−→H^∗(Xt,C),

whencev^r+1=0 in H^∗(Xt,C). Asvwas an arbitrary element ofP(s)andswas an arbitrary element ofV, we obtain thatP(V)⊂Rt.

AsP:S →Gis a holomorphic mapping,Sis a smooth and connected complex space,Rtis a closed analytic subset ofG, andVis an open subset ofS, the identity theorem for holomorphic mappings yieldsP(S) ⊂ Rt. We know there exists an open neighborhood W of P(t)in G such that QX_t∩W ⊂ P(S). In consequence, QXt∩W⊂Rt. AsP(t)∈QXt, we know thatQXt∩Wis nonempty. Thus sinceQXt

is an irreducible closed analytic subset ofGandRtis a closed analytic subset ofG, we haveQX_t ⊂Rt. Consulting Proposition 3.5.6, it follows thatXtsatisfies the Fujiki relation.

Definition 3.5.8. — a) Let f:E→Bbe a morphism of topological spaces andb∈ B. f is said to be a(topological) fiber bundle at twhen there exist an open neighborhood VofbinB, a topological spaceF, and a homeomorphism

h:X|f⁻¹(V)−→(B|V)×F such that f =pr₀◦h, where pr₀denotes the projection toB|V.

b) Let f:X → Sbe a morphism of ringed spaces (or else complex spaces) and t∈S. Then f is said to belocally topologically trivial at twhen ftop:Xtop→Stopis a fiber bundle att.

Lemma 3.5.9. — Let f:X→S be a proper, flat morphism of complex spaces and t∈S such that Xtis connected and symplectic,Ω²_Xt((Xt)_reg)is of dimension1over the field of complex numbers, S is smooth and simply connected, and the fibers of f have rational singularities, are of Kähler type, and have singular loci of codimension≥4. Denote g:Y→S the submersive share of f and setP⁰ :=P^2,2_t (g). Assume that the tangent map

Tt(P⁰): TS(t)−→T_Gr(H2(Yt))(F²H²(Yt))

is an injection with1-dimensional cokernel and f is locally topologically trivial at t. Then Xt

satisfies the Fujiki relation.

Proof. — As Sis smooth (att) and f is locally topologically trivial att, there ex-ists an open neighborhoodVoftinSsuch thatS|Vis isomorphic to the d-dimen-sional complex unit ball for somed∈Nand(fV)topis isomorphic to the projection (S|V)_top×F→(S|V)_topfor some topological spaceF. This last condition implies that, for alli∈N, the sheaf Hⁱ(fV,C)is a constant sheaf onS|V. Moreover, fV:XV →S|V is a proper, flat morphism of complex spaces,S|Vis smooth and simply connected, and the fibers of fV have rational singularities, are of Kähler type, and have singular loci of codimension≥4;(XV)_tis connected and symplectic and

dimC

Ω²_(X

V)t(((XV)_t)_reg)=1.

LetgV:YV → S|Vbe the submersive share of fV and setP⁰_V := P^2,2_t (gV). Let it:(YV)t →Ytbe the canonical morphism of complex spaces, which is induced by the fact thatY_Vis an open complex subspace ofY. Write

i^∗_t:H²(Yt)−→_H²((YV)_t)

for the associated morphism taking algebraic de Rham cohomology. As it is an isomorphism of complex spaces,i^∗_t is an isomorphism in Mod(C). Hence,i^∗_t gives rise to an isomorphism of complex spaces

Gr(_H²(Yt))−→Gr(_H²((Y_V)_t)) that we sloppily denotei^∗_t, too. Given this notation we find that

P_V⁰ =i^∗_t ◦_P⁰|V.

Thus as Tt(_P⁰)is an injection with 1-dimensional cokernel, Tt(_PV⁰ )is an injection with 1-dimensional cokernel. DefineP:=_P^2,2_t (fV)_MHSin the sense of

P: S|V−→G:=Gr(H²((X_V)_t,C))

being a morphism of complex spaces. Then by Theorem 3.4.11, there exists a mor-phism of complex spacesP:S|V→Q_(XV)t such thatP=i◦P, whereidenotes the inclusion morphismQ_(XV)t → G, andPis biholomorphic att. In particular, there exist open neighborhoodsV⁰oftinS|VandW⁰ofP(t)inQ_(XV)t such thatPinduces an isomorphism from(S|V)|V⁰ to Q_(XV)t|W⁰. Specifically, we have P(V⁰) = W⁰. Since(Q_(XV)t)_topis a topological subspace ofGtop, there exists an open subsetWof Gtopsuch thatW⁰ = Q_(XV)t∩W. Therefore, exploiting the fact thatPandPagree set-theoretically,Wis an open neighborhood ofP(t)inGtopand

Q_(XV)t∩W=_P(V⁰)⊂_P(S).

Now from Lemma 3.5.7 (applied to fVandt), we infer that(XV)_tsatisfies the Fujiki relation. As(XV)_t∼=XtinAn, we deduce thatXtsatisfies the Fujiki relation.

Theorem 3.5.10. — For all compact complex spaces X there exist a proper, flat morphism of complex spaces f:X→ S and t ∈ S such that X ∼= XtinAnand the morphism f is semi-universal in t and complete in s for all s∈S.

Proof. — The statement is proven as “Hauptsatz” in [25, § 5]⁽²⁾.

Theorem 3.5.11. — Let X be a compact, connected, symplectic complex space of Kähler type such thatdimC(Ω²_X(Xreg)) =1andcodim(Sing(X),X)≥4. Then X satisfies the Fujiki relation.

Proof. — By Theorem 3.5.10, asXis compact complex space, there exist a proper, flat morphism of complex spaces f0:X0→S0andt∈S0such that f0is semi-universal in tandX∼= (X0)_tinAn. By Theorem 3.3.18, the complex spaceS0is smooth att. By Corollary 3.3.19 (or else Theorem 3.3.17), there exists a neighborhoodV₀oftinS₀such that, for alls∈V0, the complex space(_X0)_sis connected, symplectic, of Kähler type, and with codimension of its singular locus not deceeding 4. By the smoothness ofS0

att, there exists an open neighborhoodV₀⁰oftinS0such thatV₀⁰ ⊂V0andS0|V₀⁰is isomorphic inAnto some complex unit disk. Define f₁:X1→S₁to be the morphism of complex spaces obtained from f0by shrinking the base toV₀⁰. Then by Theorem 3.4.4, the tangent map Tt(P⁰₁)is an injection with cokernel of dimension 1, where P⁰₁ := P^2,2_t (g1)andg1denotes the submersive share of f1. SinceP⁰₁is a morphism between two complex manifolds, there exists an open neighborhoodV1oftinS1such that Ts(_P1⁰)is an injection with cokernel of dimension 1 for alls∈V₁. LetV₁⁰ be an open neighborhood oftinS₁such thatV₁⁰ ⊂ V₁andS₁|V₁⁰ is isomorphic inAnto some complex unit disk, and define f:X→Sto be the morphism of complex spaces obtained from f1by shrinking the base toV₁⁰.

Define g:Y → Sto be the submersive share of f and setP⁰ := P^2,2_t (g). Theng equals the morphism obtained fromg₁by shrinking the base toV₁⁰. In consequence, P⁰is isomorphic toP₁⁰|V₁⁰ as an arrow underS = S₁|V₁⁰inAn. In particular, for all s∈ |S|=V₁⁰, as Ts(P₁⁰)is an injection with cokernel of dimension 1, the map Ts(P⁰₁|V₁⁰) is an injection with cokernel of dimension 1, hence Ts(P⁰)is an injection with cokernel of dimension 1. Note thatP^2,2s (g)is isomorphic toP^2,2_t (g)as an arrow underS in Anfor alls∈ S. Thus, Ts(P^2,2s (g))is an injection with cokernel of dimension 1 for alls ∈ S. Note that on account of Proposition 3.1.17, for alls ∈ S, as the complex spaceXs ∼= (X0)_sis symplectic,Xshas rational singularities. Therefore, by Corollary 2.4.15, the moduleH^2,0(g)is locally finite free onSand, for alls∈S, the Hodge base change

(_H^2,0(g))(s)−→_H^2,0(_Ys)

(2)One should note that in [25] Grauert calls “versell” what we call “semi-universal”; for the modern reader this might be confusing given that today people use the (English) word “versal” as a synonym for

“complete”, which is a condition strictly weaker than that of semi-universality. So, Grauert’s (German)

“versell” is not equivalent to but strictly stronger than the contemporary (English) “versal”.

is an isomorphism in Mod(C). Observe that, for alls∈S, we haveYs∼= (Xs)_regdue theory and Thom’s First Isotopy Lemma, for f being proper, there exists a connected open subsetTofSsuch thatt ∈ Tand f is locally topologically trivial atsfor all s∈T. Employing Lemma 3.5.9, we obtain that, for alls∈T, the spaceXssatisfies the Fujiki relation.

Now leta∈H²(_Xt,C). Since f is proper, the base change map (H²(f,C))_t−→H²(_Xt,C)

is a surjection (a bijection, in fact). Therefore, there exists a connected open neighbor-hoodVoftinSandea∈(H²(f,C))(V)such thateais sent toaby the composition of canonical functions

(3.5.11.1) (H²(f,C))(V)−→(H²(f,C))t−→H²(Xt,C).

Ast∈Tand the topological spaceStopis first-countable att, there exists anN-indexed sequence(s_α)of elements ofT∩Vsuch that(s_α)converges totinStop. Whens∈V, let us writeasfor the image ofeaunder the composition of functions (3.5.11.1), where we replacetbys. Setr:=¹_/2dim(_Xt)(which makes sense as the complex spaceXtis

Then φ is a locally constant function from Stop|V to C. In particular, we have lim(φ(sα)) = φ(t). In a separate proof below we show that lim(ψ(sα)) = ψ(t).

Accordingly, remarking that by the definition ofatwe havea=at, we obtain:

Z

Proof thatlim(ψ(s_α)) =ψ(t). — Assume that a = 0 in H²(_Xt,C). Then we have a bijection. Therefore, there exists a uniquel-tuplevof elements ofH⁰(S)which is pushed forward tob⁰by the composition of evident functions:

(3.5.11.3) H⁰(S)−→(H⁰)t−→b_t^∗(H²(g))−→H²((Xt)_reg)−→H²((Xt)_reg,C).

Defineb⁰_s to be the pushforward ofvunder the composition of functions (3.5.11.3), where one replacestbys. Set: that s ∈ T. Then i_s^∗ is an isomorphism ofC-vector spaces; thus there exists one, and only one,l-tupleb_s of elements of H²(Xs,C)such thati^∗_s ◦b_s = b⁰_s. We have constant onS; as Tis a connected open subset ofS, the latter functions are even

constant. AsT 6=∅, there exists, for allν,ν⁰ ∈N^l, a unique complex numberC_ν,ν⁰ such that, for alls∈T, we haveR

Xs(b_s)^ν(b_s)^ν0=C_ν,ν⁰.

Sincebis a basis for H²(_Xt,C), we may writevtas linear combination of the(b_i), i∈k. Applyingi^∗_t to this linear combination, comparing with the definition ofv⁰_tand noting thatb⁰_t=b⁰, we see that

is a surjection (indeed, it is even a bijection), there exists a connected open neighbor-hoodW oftinSas well as ak-tupleuof elements of(H²(f,C))(W)such that the pushforward ofuunder the composition of functions

(3.5.11.4) (H²(f,C))(W)−→(H²(f,C))_t−→H²(_Xt,C)

equalsb. Sincet ∈ T, there existss^∗ ∈ W∩T. It is an easy matter to verify that composinguwith the composition of functions (3.5.11.4), where one replacestbys^∗, we obtainb_s∗|k. Therefore, for anyζ,η∈N^ksuch that|ζ|+|η|=2r, we have: wsis a normed symplectic class onXs. Defineµto be the unique function onSwhich sendss∈Sto thel-tupleµssuch that, for alli∈l, we have:

(µs)_i= (^2r√

Is)⁻¹λ_i(s).

Then, for alls∈T,

+ (1−r)









∑

ν,ν⁰∈N^k

|ν|=r−1,|ν⁰|=r

r−1 ν

r ν⁰

(µ_t)^ν(µ_t)^ν0Cν+(1,0,...,0),ν⁰









·









∑

ν,ν⁰∈N^k

|ν|=r,|ν⁰|=r−1

r ν

r−1 ν⁰

(µt)^ν(µt)^ν0Cν+(1,0,...,0),ν⁰









=q_Xt(at).

This shows that lim(ψ(s_α)) =ψ(t).

FOUNDATIONS AND CONVENTIONS

This appendix serves the purpose of increasing the rigor, the consistency, and the comprehensibility of the bulk of our text, i.e., of Chapters 1, 2, and 3. More concretely, in what follows, we will lay out certain conventions of speech, certain notational conventions, as well as certain conventions concerning the axiomatic foundations.

Disclaimer: We do not intend to explain (in a textbook sort of way) any of the concepts that we are going to address in the subsequent. Quite on the contrary, will we assume the reader’s familiarity with the mentioned concepts so that we merely clarify our points of view. For instance, when, in the realm of Definition A.2.1, we define what a category is, our intention is not to explain the notion of a category to a reader who has not heard of what a category is beforehand; rather would we like to tell the reader who is familiar with the concept of categories which of the various possible definitions (a single collection of morphisms versus a collection of hom-sets, small versus large, etc.) we adopt.

A.1. Set Theory

A.1.1. Our exposition is formally based upon the Zermelo-Fraenkel axiomatic set theory with axiom of choice, “ZFC” (cf. e.g., [41, Chapter I.1]). All statements we make can be formulated in ZFC. All of our proofs can be executed in ZFC.

A.1.2. Note that occasionally we do work with classes. For instance, do we consider the large categories Mod(X) of modules on a given ringed space X or the large categoryAnof complex analytic spaces. We feel that our reader has two choices of how to deal with the occurrences of these concepts: The first possibility is to strictly stick to ZFC and interpret classes not as objects of the theory but as metaobjects, i.e., at each occurrence of a class, the reader replaces the class by a formula in the language of ZFC which describes the class. The second possibility would be to use a conservative extension of ZFC, such as the von Neumann-Bernay-Gödel set theory (NBG), which

can deal with classes as an additional type of the theory, as an overall foundation for our text. This second approach has the advantage that classes can be quantified over, so that, for instance, “for all (large) abelian categories the five lemma holds” or

“for all (large) abelian categoriesC,D, andE, all functorsf:C→Dandg: D→E, and all objectsXofCsuch that (. . . ) there exists a Grothendieck spectral sequence”

become valid statements. Some sources we refer to make (implicitly or explicitly) use of statements where classes are quantified over.

We point out that we do not presuppose any sort of universe axiom like e.g., the existence of Grothendieck universes. Mind that, even though many authors in modern algebraic geometry seem to have forgotten about this, the standard construction of a (bounded) derived category fails for arbitrary large abelian categories since hom-sets, and not even hom-classes, do exist. Therefore, we cannot (and won’t) talk about the D⁺of Mod(X), for instance. Yet, we do fine without it.

We point out that we do not presuppose any sort of universe axiom like e.g., the existence of Grothendieck universes. Mind that, even though many authors in modern algebraic geometry seem to have forgotten about this, the standard construction of a (bounded) derived category fails for arbitrary large abelian categories since hom-sets, and not even hom-classes, do exist. Therefore, we cannot (and won’t) talk about the D⁺of Mod(X), for instance. Yet, we do fine without it.

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