Compactifiable submersive morphisms

Theorem 2.4.1(Ohsawa’s criterion). — Let X be a compact, locally pure dimensional complex space of Kähler type and A a closed analytic subset of X such thatSing(_X)⊂ A.

Then the Frölicher spectral sequence of X\A degenerates in entries I:={(p,q)∈Z×Z: p+q+2≤codim(A,X)}

at sheet1inMod(C). Moreover, for all(p,q)∈I, we have H ^p,q(X\A)∼=H^q,p(X\A) inMod(C).

Proof. — WhenXis pure dimensional, the assertions follow from [65, Theorem 1].

WhenXis only locally pure dimensional,Xhas finitely many connected components X0, . . . ,Xb−1. For allν<b, we know thatX_νis pure dimensional, so that the Frölicher spectral sequence ofYν:=Xν\(A∩Xν)degenerates in entriesIat sheet 1. Therefore, since the Frölicher spectral sequence ofX\Ais isomorphic to the direct sum of the Frölicher spectral sequences of theY_ν, wheresνranges throughb, it degenerates in entries Iat sheet 1 also. Similarly, the Hodge symmetry is traced back to the pure dimensional case.

Question 2.4.2. — LetXandAbe as in Theorem 2.4.1,nan integer such that n+2≤codim(A,X).

Does the filtration(_F^p_Hⁿ(X\A))_p∈ZonHⁿ(X\A)coincide with the Hodge filtra-tion of the mixed Hodge structure ofn-th cohomology associated to the compactifica-tionX\A→Xin the spirit of Deligne and Fujiki (cf. [20, 12, 13])?

Proposition 2.4.3. — Let f: X→S be a flat morphism of complex spaces with locally pure dimensional base S. Then the following are equivalent:

(i) f is locally equidimensional.

(ii) X is locally pure dimensional.

Proof. — Since the morphism f is flat, Theorem 3.3.11 tells that (2.4.3.1) dimx(X) =dimx(X_f(x)) +dim_f(x)(S)

for allx∈X. By assumption, the complex spaceSis locally pure dimensional so that the function given by the assignment

s7−→dims(S)

is locally constant on S. Thus by the continuity of f, the function given by the assignment

x7−→dim_f(x)(S)

is a locally constant function onX. Therefore, by (2.4.3.1) we see that dimx(X_f(x)) is a locally constant function ofxonXif and only if dimx(X)is a locally constant function ofxonX. This implies that (i) holds if and only if (ii) holds.

Proposition 2.4.4. — Let f:X→Y be a morphism of complex spaces and p∈X. Then dimp(X)−dimp(X_f(p))≤dim_f(p)(S).

Proof. — See [18, 3.9, Proposition (∗)].

Definition 2.4.5. — Let f:X→Sbe a morphism of complex spaces andAa closed analytic subset ofX. Then we define:

(2.4.5.1) codim(A,f):=inf{codim(A∩ |Xs|,Xs):s∈S},

whereXsdenotes the fiber of f overs. We call codim(A,f)therelative codimensionof Awith respect to f. Note that the set in (2.4.5.1) over which the infimum is taken is a subset ofN∪ {ω}. We agree on taking the infimum with respect to the canonical (strict) partial order onN∪ {ω}given by the∈-relation.

Proposition 2.4.6. — Let f:X→S be a morphism of complex spaces, A a closed analytic subset of X.

a) For all p∈A, when f is flat in p, we have

codimp(A∩ |X_f(p)|,X_f(p))≤codimp(A,X). b) When f is flat, we have

codim(A,f)≤codim(A,X).

Proof. — a). Letp∈Aand assume that f is flat inp. Putt:= f(p). Then we have dimp(Xt) +dimt(S) =dimp(X)

by Theorem 3.3.11. By abuse of notation we denoteAalso the closed analytic subspace ofXinduced onA. Writei: A→Xfor the corresponding inclusion morphism and setg:= f ◦i. Then Proposition 2.4.4, applied tog, implies that

dimp(A)−dimp(At)≤dimt(S). On the whole, we obtain:

codimp(A∩ |Xt|,Xt) =dimp(Xt)−dimp(At)

≤(dimp(X)−dimt(S)) + (dimt(S)−dimp(A))

=_codimp(A,X)_.

Note that implicitly we have used that the closed complex subspace ofXtinduced on A∩ |Xt|is canonically isomorphic toAt, which is by definition the closed complex subspace ofAinduced ong⁻¹({t}).

b). Noticing that

codim(A,f) =_inf{codimp(A∩X_f(p),X_f(p))_:p∈A}, the desired inequality follows immediately from part a).

Proposition 2.4.7. — Let f: X → S be a proper, flat morphism of complex spaces, A a closed analytic subset of X such thatSing(f)⊂ A, and t∈S. Assume that S is smooth, f is locally equidimensional, and Xtis of Kähler type. Define g to be the composition of the canonical morphism Y:= X\A→X and f . Then, for all ordered pairs of integers(p,q) such that

(2.4.7.1) p+q+2≤c:=codim(A,f), yet not(p,q+2) = (0,c), the following assertions hold:

a) The moduleH ^p,q(g)is locally finite free on S in t.

b) The Hodge base change map

(H ^p,q(g))(t)−→H ^p,q(Yt) is an isomorphism inMod(C).

Proof. — Let(p,q)be an ordered pair of integers as above. Whenp<0, assertions a) and b) are trivially fulfilled. So, assume that bothp≥0. We claim that

(2.4.7.2) (q+₁) +₂≤c=_codim(A,f)_.

Indeed, when p = 0, we haveq+2 ≤ c by (2.4.7.1), but also q+2 6= c, so that q+2<c. Whenp>0, (2.4.7.2) follows directly from (2.4.7.1).

Using (2.4.7.2) in conjunction with Proposition 2.4.6 b), we obtain:

q+2≤(q+1) +2≤codim(A,X).

Moreover, by Proposition 2.4.3, the complex spaceXis locally pure dimensional.

Thus by Proposition 2.1.14, we see thatH^p,q+1(g)andH ^p,q(g)are coherent modules onS.

Next, we claim that, for all integersνandµsuch thatν+µ= p+q, the Hodge moduleH^ν,µ(Yt)is of finite type overC. Whenν<0, this is clear. Whenν≥0, we have

µ+2≤ν+µ+2=p+q+2≤c≤codim(A∩ |Xt|,Xt).

In addition, the complex spaceXtis compact and locally pure dimensional. Thus H^ν,µ(Xt\(A∩ |Xt|))is of finite type overCby Proposition 2.1.14, whenceH^ν,µ(Yt) is of finite type overCgiven that

Yt∼=Xt\(A∩ |Xt|) as complex spaces.

Theorem 2.4.1 implies that for all integersνandµsuch thatν+µ = p+qthe Frölicher spectral sequence ofYtdegenerates in(ν,µ)at sheet 1. Therefore, we deduce a) and b) from a) and b) of Proposition 2.3.7, respectively.

Question 2.4.8. — Let f,A, andtbe as in Proposition 2.4.7. Definegandc accord-ingly, and assume thatA∩ |Xt| 6= ∅(which implies thatc ∈ N). Do a) and b) of Proposition 2.4.7 hold for(p,q) = (0,c−2)?

Proposition 2.4.9. — Let X be a Cohen-Macaulay complex space, A a closed analytic subset of X, F a locally finite free module on X.

a) For all integers q such that q+1≤codim(A,X), we haveH^q_A(X,F)∼=0.

b) Denote j:X\A→X the inclusion morphism. When2≤codim(A,X), the canoni-cal morphism

(2.4.9.1) F−→R⁰j∗(j^∗(F))

of modules on X is an isomorphism. Moreover, for all integers q 6= 0such that q+2 ≤ codim(A,X), we haveR^qj∗(j^∗(F))∼=0.

Proof. — a). Whenqis an integer such thatq+1≤codim(A,X), then we have q+1+dimx(A)≤dimx(X) =dim(OX,x) =prof_X,x(F)

for all x ∈ A. Thus H^q_A(X,F) ∼= 0 by [9, II, Theorem 3.6, (b) ⇒(c)]; see also II, Corollary 3.9 inloc. cit.

b). When 2≤codim(A,X), we deduce that H⁰_A(X,F)∼=0 and H¹_A(X,F)∼=0 from part a). Besides, by Proposition 2.1.5, there exists an exact sequence

0−→H⁰_A(X,F)−→F−−→^can R⁰j∗(j^∗(F))−→H¹_A(X,F)−→0

of sheaves of modules onX. Thus the canonical morphism (2.4.9.1) is an isomorphism.

Letqbe an integer6= 0 such thatq+2 ≤ codim(A,X). Whenq < 0, we have R^qj∗(j^∗(F))∼=0 in Mod(X)trivially. Whenq>0, we have

R^qj∗(j^∗(F))∼=H^q+1_A (X,F)

according to Proposition 2.1.5, yet H^q+1_A (X,F)∼=0 by means of a).

Proposition 2.4.10. — Let f:X→S be a morphism of complex spaces, A a closed analytic subset of X and F a locally finite free module on X. Assume that X is Cohen-Macaulay.

Denote j:X\A→ X the inclusion morphism and set g:= f ◦j. Then, for all integers q such that q+2≤codim(A,X), the canonical morphism

(2.4.10.1) R^qf∗(F)−→R^qg∗(j^∗(F)) of modules on S is an isomorphism.

Proof. — Letqbe an integer as above. Whenq<0, our assertion is clear. So assume thatq≥0. DenoteEthe Grothendieck spectral sequence associated to the triple

Mod(X\A) ^{j∗ //}Mod(X) ^{f∗ //}Mod(S)

of categories and functors and the objectj^∗(F)of Mod(X\A). Then, for all integers νandµ, we have

E^ν,µ₂ ∼=R^νf∗(R^µj∗(j^∗(F))).

In particular, Proposition 2.4.9 b) implies thatE₂^ν,µ∼=0 wheneverµ6=0 andµ+2≤ codim(A,X). Of course, we also have E₂^ν,µ ∼= 0 whenever ν < 0. Therefore, E degenerates in entries of total degreeqat sheet 1, and the edge morphism

R^qf∗(R⁰j∗(j^∗(F)))−→R^q(f∗◦j∗)(j^∗(F))

is an isomorphism in Mod(S). Now f∗◦j∗ = g∗, and the edge morphism is the canonical one, so that the canonical morphism (2.4.10.1) factors through the edge morphism via the morphism

R^qf∗(F)−→R^qf∗(R⁰j∗(j^∗(F)))

which is obtained from (2.4.9.1) by applying the functor R^qf∗. By Proposition 2.4.9 b), the morphism (2.4.9.1) is an isomorphism in Mod(X)since

2≤q+2≤codim(A,X).

Hence we conclude that (2.4.10.1) is an isomorphism in Mod(S).

Proposition 2.4.11. — Let f:X →S be a proper, flat morphism of complex spaces and t∈S such that Xtis a reduced complex space. Then the moduleR⁰f∗(OX)is locally finite free on S in t and the base change map

C⊗_OS,t(_R⁰f∗(_OX))_t−→H⁰(Xt,OXt) is an isomorphism of complex vector spaces.

Proof. — See [9, III, Proposition 3.12].

Proposition 2.4.12. — Let f:X→S be a proper, flat morphism of complex spaces and A a closed analytic subset of X such thatSing(f)⊂ A. Assume that S is smooth and the fibers of f are Cohen-Macaulay and of Kähler type. Define g to be the composition of the canonical morphism Y:=X\A→X and f . Then for all integers p and q such that

(2.4.12.1) p+q+2≤c:=codim(A,f) the following assertions hold:

a) The moduleH ^p,q(g)is locally finite free on S.

b) For all s∈S, the Hodge base change map

(_H ^p,q(g))(s)−→_H ^p,q(Ys) is an isomorphism inMod(C).

Proof. — Letpandqbe integers satisfying (2.4.12.1). Whenp<0 orq<0, assertions a) and b) are trivially fulfilled. So, assume that bothpandqare≥0. Then,

2≤p+q+₂≤codim(A,f)≤codim(A,X)≤codim(_Sing(X)_,X)_, where we have used that

Sing(X)⊂Sing(f)⊂A.

Moreover, given that f:X→Sis flat and fiberwise Cohen-Macaulay with a smooth baseS, the complex spaceXis Cohen-Macaulay. ThusXis normal, whence locally pure dimensional by Proposition 3.3.13. By Proposition 2.4.3, the morphism f is locally equidimensional. Therefore, as long as(p,q+2)6= (0,c), assertions a) and b) are implied by the corresponding assertions of Proposition 2.4.7.

Now assume that (p,q+2) = (0,c) (and stillq ≥ 0). We claim that R^qf∗(OX) is locally finite free on S and compatible with base change. In case q = _{0, this} follows from Proposition 2.4.11. So, beq >0. Lets ∈ Sbe arbitrary. ThenXs is a Cohen-Macaulay complex space andA∩ |Xs|is a closed analytic subset ofXssuch that

q+2≤codim(A,f)≤codim(A∩ |Xs|,Xs).

Hence by Proposition 2.4.10 (applied toaXs: Xs→C), the canonical morphism H^q(Xs,OXs)−→H^q(Ys,OYs) =H^0,q(Ys)

is an isomorphism in Mod(C). By Theorem 2.4.1, we have:

H^0,q(Ys)∼=H^q,0(Ys).

By what we have already proven, the moduleH^q,0(g)is locally finite free onSand compatible with base change, so that, in particular, (abandoning our fixation ofs) the function

s7−→dimC(_H^q,0(Ys)) is locally constant onS. As a consequence, the function

s7−→dimC(H^q(Xs,OXs))

is a locally constant function onStoo, so that Grauert’s continuity theorem yields our claim.

We allowq=0 again. By Proposition 2.4.10, the canonical morphism R^qf∗(OX)−→R^qg∗(OY)

is an isomorphism in Mod(S). Therefore R^qg∗(_OY) =_H ^p,q(g)is locally finite free on S. Furthermore, by the functoriality of base changes, the diagram

(R^qf∗(OX))(s) ^{∼ //}

BC

(R^qg∗(OY))(s)

BC

H^q(Xs,OXs) _∼ ^//H^q(Ys,OYs)

commutes in the category of complex vector spaces for alls ∈ S. As the left base change in the diagram is an isomorphism, the right one is as well.

Theorem 2.4.13. — Let f:X →S be a proper, flat morphism of complex spaces and A a closed analytic subset of X such thatSing(f)⊂A. Assume that S is smooth and the fibers of f are Cohen-Macaulay as well as of Kähler type. Define g to be the composition of the inclusion Y:=X\A→X and f and denote E the Frölicher spectral sequence of g.

DefineKto be the kernel of the morphism of modules onS, Rⁿg∗(i^≥pΩq

g): Rⁿg∗(σ^≥pΩq

g)−→Rⁿg∗(_Ωg^q).

Lett∈Sbe arbitrary. Adopt the notation of Setup 2.2.1 for infinitesimal neighbor-hoods (forg:Y→Sin place of f:X→S). Fix a natural numbermand writeKmfor

of modules onSm. Then by the functoriality of base change maps the diagram b^∗_m(Rⁿg∗(σ^≥pΩq

commutes in Mod(Sm). Thus, there exists one, and only one,α_mrendering commuta-tive in Mod(Sm)the following diagram:

b^∗_m(K) ^b

By Theorem 2.4.1, the Frölicher spectral sequence ofY₀=Ytdegenerates in entries of total degreenat sheet 1, so that the Frölicher spectral sequence ofYmdegenerates

in entries of total degreenat sheet 1 by means of Theorem 2.2.9 am). In particular, the Frölicher spectral sequence ofYm degenerates from behind in(p,q)at sheet 1.

Therefore, by Proposition 1.7.9 a), there existsψmsuch that the diagram Rⁿgm∗(σ^≥pΩq

commutes in Mod(Sm). Thus the composition of the two arrows in the bottom row of the following diagram is zero:

By Proposition 2.3.7 b), the Hodge base change map which is the rightmost vertical arrow in the diagram in (2.4.13.2) is an isomorphism. Moreover, by the functoriality of base change maps and the definition ofα_m, the diagram in (2.4.13.2) commutes.

In consequence, we see that the composition of the two arrows in the top row of (2.4.13.2) equals zero. Sincemwas an arbitrary natural naumber, we deduce that the composition of stalk maps

Kt−→(_Rⁿg∗(σ^≥pΩq

g))_t−→(_Rⁿg∗(σ^=pΩq

g))_t

equals zero. Astwas an arbitrary element ofS, we deduce further that the composi-tion

K−→Rⁿg∗(σ^≥pΩq

g)−→Rⁿg∗(σ^=pΩq

g)

of morphisms of sheaves of modules onSequals zero. Thus there existsψsuch that the diagram

commutes in Mod(S)and, in turn, Proposition 1.7.9 a) tells that the Frölicher spectral sequence ofgdegenerates from behind in(p,q)at sheet 1.

b). Not that by a), the spectral sequenceEcertainly degenerates in entries {(p,q)∈Z×Z:p+q+3≤codim(A,f)}

at sheet 1. Therefore, our assertions holds in caseA=_{∅. In case}A6=∅, our assertion follows readily from Proposition 1.7.9 b).

Proposition 2.4.14. — Let f: X→S and A be as in Theorem 2.4.13. Define g:Y →S accordingly. Assume that A6=∅and put n:=codim(A,f)−2. Moreover, assume that, for all s∈S, the canonical mapping

(2.4.14.1) Hⁿ(Xs,C)−→H^0,n(Xs)

is a surjection. Then the canonical morphism(2.4.13.1)is an epimorphism inMod(S). Proof. — Lets∈Sbe arbitrary. As f is proper, the base change map

(Rⁿf∗(CX))_s −→Hⁿ(Xs,C)

is a bijection. By assumption, the canonical map (2.4.14.1) is a surjection. AsXs is Cohen-Macaulay andAs :=A∩ |Xs|is a closed analytic subset ofXssuch that

n+2=codim(A,f)≤codim(As,Xs), the morphism

H^0,n(Xs)−→_H^0,n(Xs\As)

induced by the inclusionXs\As →Xs of complex spaces is a bijection. Now the morphism Ys → Xs of complex spaces which is induced on fibers over sby the inclusionY→Xis isomorphic toXs\As →Xsin the overcategoryAn_/Xs, hence by the functoriality ofH^0,n, the morphism

H^0,n(Xs)−→H^0,n(Ys)

induced byYs → Xs is a bijection. Combining these results, one infers that the composition of functions

(2.4.14.2) (Rⁿf∗(CX))_s−→Hⁿ(Xs,C)−→_H^0,n(Xs)−→_H^0,n(Ys) is a surjection.

The diagram

(2.4.14.3) (Rⁿf∗(C_X))_s ^//

(Rⁿg∗(C_Y))_s ^//

(_H ⁿ(g))(s)

xx

Hⁿ(Xs,C) ^//

Hⁿ(Ys,C) ^//

Hⁿ(Ys)

wwnnnnnnnnnnnn

H^0,n(Xs) ^//_H^0,n(Ys)^oo (H^0,n(g))(s)

commutes in Mod(C) by the functoriality of the various base changes appearing in it. By the commutativity of the diagram in (2.4.14.3) and the surjectivity of the composition (2.4.14.2), we deduce that the composition

(Hⁿ(g))(s)−→(H^0,n(g))(s)−→H^0,n(Ys) is a surjection. As the base change map

(_H^0,n(g))(s)−→_H^0,n(Ys)

is a bijection by Proposition 2.4.12 b), we see that

(_Hⁿ(g))(s)−→(_H^0,n(g))(s)

is a surjection. By Proposition 2.4.12 a), the module(_H^0,n(g))_sis finite free overOS,s, whence Nakayama’s lemma tells that

(Hⁿ(g))_s −→(H^0,n(g))_s

is a surjection. Taking into account that while conducting this argument,swas an arbitrary point ofS, we conclude that the canonical morphism (2.4.13.1) of sheaves of modules onSis an epimorphism.

Corollary 2.4.15. — Let f:X→S be a proper, flat morphism of complex spaces such that S is smooth and the fibers of f have rational singularities, are of Kähler type, and have singular loci of codimension≥ 4. Define g:Y → S to be the submersive share of f (cf. Notation 2.1.13). Set

I:={(ν,µ)∈Z×Z:ν+µ≤2}. a) For all(p,q)∈ I, the moduleH^p,q(g)is locally finite free on S.

b) For all(p,q)∈I and all s∈S, the Hodge base change map (2.4.15.1) (_H ^p,q(g))(s)−→H ^p,q(Ys) is an isomorphism inMod(_C).

c) The Frölicher spectral sequence of g degenerates in entries I at sheet1inMod(S). Proof. — SetA:=Sing(f). ThenAis a closed analytic subset ofXwhich contains Sing(f). Moreover, according to Notation 2.1.13, we haveY=X\Aandgequals the composition of the canonical morphismY→Xandf. Since any complex space which has rational singularities is Cohen-Macaulay, the fibers of f are Cohen-Macaulay. Let (p,q)∈I. Then

p+_q+₂≤4≤codim(_A,f)_.

Thus by Proposition 2.4.12, the Hodge moduleH ^p,q(g)is locally finite free onSand, for alls∈S, the Hodge base change map (2.4.15.1) is an isomorphism in Mod(C). As (p,q)was an arbitrary element ofI, this proves a) and b).

For proving c), we distinguish two cases: Firstly, suppose that codim(A,f)≥5.

Then clearly

I⊂ {(p,q)∈Z×Z:p+q+3≤codim(A,f)}.

So, c) is implied by a) of Theorem 2.4.13. Secondly, suppose that codim(A,f) <5.

Then as codim(A,f)≥ 4, we have codim(A,f) =4. In particular, A6=∅. For all s∈Sthe complex spaceXshas rational singularities, whence the canonical mapping

H²(Xs,C)−→_H^0,2(Xs)

is a surjection. Therefore, c) is implied by Proposition 2.4.14 in conjunction with b) of Theorem 2.4.13.

SYMPLECTIC COMPLEX SPACES

In this chapter we study symplectic complex spaces. The main results are Theorem 3.4.4 (Local Torelli) and Theorem 3.5.11 (Fujiki Relation).

In § 3.1, we define what we mean by a symplectic complex space; moreover, we establish certain fundamental properties of such spaces. In § 3.2, we generalize the notion of the Beauville-Bogomolov form, which has proven a pivotal tool in the study of irreducible symplectic manifolds, to the context of compact, connected, symplectic complex spaces Xwhose Ω²_X(Xreg) is 1-dimensional over the field of complex numbers. § 3.3 is consecrated to the (local) deformation theory of compact, Kähler, symplectic complex spaces whose singular loci have codimension≥ 4. In

§ 3.4, we derive our version of a local Torelli theorem for symplectic complex spaces as well as several corollaries thereof. Eventually, in § 3.5, we prove that compact, Kähler, symplectic complex spacesXwith 1-dimensionalΩ²_X(Xreg)and singular locus of codimension≥4 satisfy the Fujiki relation.

3.1. Symplectic structures on complex spaces

We intend to define a notion of symplecticity for complex spaces. Our definition (cf. Definition 3.1.12 below) is inspired by Y. Namikawa’s notions of a “projective symplectic variety” and a “symplectic variety” in [60] and [61], respectively (note that the two definitions differ slightly), as well as by A. Beauville’s notion of “symplectic singularities”, cf. [3, Definition 1.1]. These concepts rely themselves on the concept of symplectic structures on complex manifolds. For the origins of symplectic structures on complex manifolds, we refer our readers to the works [6] and [7] of F. Bogomolov, where the term “Hamiltonian” is used instead of “symplectic”, as well as to Beauville’s [2, “Définition” in § 4].

In Definition 3.1.8, we coin the new term of a “generically symplectic structure” on a complex manifold—not with the intention of specifically studying the geometry of spaces possessing these structures, but merely as a tool to define and study the

Beauville-Bogomolov form for (possibly nonsmooth) complex spaces in § 3.2. Further-more, we introduce notions of “symplectic classes”, which seem new in the literature too. Apart from giving definitions, we state several easy or well-known consequences of the fact that a complex space is symplectic. Proposition 3.1.17 (mildness of sin-gularities) and Proposition 3.1.21 (purity of the mixed Hodge structure H²(X)) are of particular importance and fundamental for the theory developed in subsequent sections.

Let us point out that our view on symplectic structures is purely algebraic in the sense that, to begin with, our candidates for symplectic structures are elements of the sheaf of Kähler 2-differentials on a complex space. That way, our ideas and terminology may be translated effortlessly into the framework of, say, (relative) schemes, even though we refrain from realizing this translation here. We start by defining nondegeneracy for a global 2-differential on a complex manifold. The definition might appear a little unusual, yet we like it for its algebraic nature.

Definition 3.1.1(Nondegeneracy). — LetXbe a complex manifold andσ∈Ω²_X(X). Defineφto be the composition of the following morphisms in Mod(X):

(3.1.1.1) ΘX−→_ΘX⊗_OX −−−→^id⊗σ _ΘX⊗_Ω²_X −→_Ω¹_X.

Here, the first and last arrows stand for the inverse of the right tensor unit forΘXon Xand the contraction morphismγ²_X(_Ω¹_X), cf. Notation 1.3.10, respectively, and

σ:OX −→Ω²_X

denotes, by abuse of notation, the unique morphism of modules onXsending the 1 ofOX(X)to the actualσ∈Ω²_X(X).

a) Letp∈X. Thenσis callednondegenerateonXatpwhenφis an isomorphism of modules onXatp, i.e., when there exists an open neighborhoodUofpinXsuch thati^∗(φ)is an isomorphism in Mod(X|U), wherei:X|U→Xdenotes the evident inclusion morphism.

b) σis callednondegenerateonXwhenσis nondegenerate onXatpfor allp∈X.

c) σis calledgenerically nondegenerateonXwhen there exists a thin subsetAofX such thatσis nondegenerate onXatpfor allp∈X\A.

Observe that allowingXto be an arbitrary complex space in Definition 3.1.1 would not cause any problems. However, as we do not see whether the thereby obtained more general concept possesses intriguing meaning—especially at the singular points ofX—, we desisted from admitting the further generality.

Remarks 3.1.2. — LetXandσbe as in Definition 3.1.1. Defineφaccordingly.

a) By general sheaf theory, we see thatσis nondegenerate onXif and only if φ: ΘX−→Ω¹_X

is an isomorphism in Mod(X).

b) Let p ∈ X. Thenσ is nondegenerate onXatpif and only if there exists an open neighbohoodUofpinXsuch that the image ofσunder the canonical mapping Ω²_X(X)→Ω²_X|U(U)is nondegenerate onX|U.

c) Define

D:={p∈ X:σis not nondegenerate onXatp}

to be thedegeneracy locusofσonX. ThenDis a closed analytic subset ofX. Moreover, σis generically nondegenerate onXif and only ifDis thin inX;σis nondegenerate onXif and only ifD=∅.

We briefly digress in order to establish, for later use, a typical characterization of the nondegeneracy of a Kähler 2-differentialσon a complex manifoldXemploying the wedge powers ofσ, cf. Proposition 3.1.7. The quick reader may well skip this discussion and head on to Definition 3.1.8 immediately.

Proposition 3.1.3. — Let X be a complex manifold,σ ∈ _Ω²_X(X), and p∈X. Let n be a natural number and z:U→Cⁿan n-dimensional (holomorphic) chart on X at p. Then the following are equivalent:

(i) σis nondegenerate on X at p;

(ii) φp:ΘX,p→Ω¹_X,pis an isomorphism inMod(OX,p), whereφdenotes the composi-tion(3.1.1.1)inMod(X)(just as in Definition 3.1.1);

(iii) when A is the unique alternating n×n-matrix with values in the ringOX(U)such that

σ|U=

∑

i<j

A_ij·dzⁱ∧dz^j,

then A(p) ∈ GLn(C), where A(p)denotes the composition of A with the evaluation of sections inOXover U at p.

Proof. — (i) implies (ii) since for any open neighborhoodVof pin Xthe stalk-at-p functor Mod(X) → Mod(OX,p) on X factors over i^∗: Mod(X) → Mod(X|V), wherei:X|V→Xdenotes the obvious inclusion morphism. That (ii) implies (i) is due to the fact thatΘX andΩ¹_X both are coherent modules onX. Now letAbe a matrix as in (iii). Then, essentially by the definition of the contraction morphism, the matrix associated with the morphism ofOX(U)-modulesφ(U):ΘX(U) →Ω¹_X(U) relative to the bases (∂_z0, . . . ,∂_zn−1)and(dz⁰, . . . , dzⁿ⁻¹)is the transpose A^> of A.

Thus the matrix associated with the morphism ofOX,p-modulesφp: ΘX,p →Ω¹_X,p relative to the bases ((∂_z0)_p, . . . ,(∂_zn−1)_p)and ((dz⁰)_p, . . . ,(dzⁿ⁻¹)_p)is (A^>)_p, by which we mean the matrix of germs ofA^>. Hence we have (ii) if and only if(A^>)_p∈ GLn(OX,p). Yet(A^>)_p∈ GLn(OX,p)if and only if(A^>)(p)∈GLn(C)if and only if A(p)∈GLn(C), the latter equivalence being true for(A^>)(p) = (A(p))^>.

Corollary 3.1.4. — Let X be a complex manifold,σ∈_Ω²_X(X), and p∈ X such thatσis nondegenerate on X at p. Thendimp(X)is even.

Proof. — Setn:=dimp(X). Then there exists ann-dimensional holomorphic chart z:U → Cⁿ on Xat p. There exists an alternatingn×n-matrix Awith values in the ringOX(U)such thatσ|U = _∑i<jAij·dzⁱ∧dz^j. By Proposition 3.1.3, asσ is nondegenerate onXatp, we haveA(p)∈GLn(_C). Yet the existence of an invertible, alternating complexn×n-matrix implies thatnis even (see, e.g., [48, XV, Theorem 8.1]).

Remark 3.1.5(Pfaffians). — LetRbe a commutative ring. We define theR-Pfaffian PfRas a function on the set of alternating (quadratic) matrices of arbitrary size over R; the function Pf_Ris to take values inR. Concretely, whenAis an alternatingn× n-matrix overR, wherenis some natural number, we set Pf_R(A):=0 in casenis odd;

in casenis even, we set Pf_R(A):=

∑

π∈Π

sgn(π)A_π(0),π(1)A_π(2),π(3)·. . .·Aπ(n−2),π(n−1), whereΠdenotes the set of all permutationsπofnsuch that we have

π(0)<π(1), π(2)<π(3), . . . , π(n−2)<π(n−1) and

π(0)<π(2)<· · ·<π(n−2).

TheR-Pfaffian enjoys property that, for any alternatingn×n-matrix overR, we have

TheR-Pfaffian enjoys property that, for any alternatingn×n-matrix overR, we have

Im Dokument Irreducible symplectic complex spaces (Seite 123-0)