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Hodge Theory and Deformations of Maps Compositio Mathematica, Tome 97, No 3 (1995), P COMPOSITIO MATHEMATICA ZIV RAN Hodge theory and deformations of maps Compositio Mathematica, tome 97, no 3 (1995), p. 309-328 <http://www.numdam.org/item?id=CM_1995__97_3_309_0> © Foundation Compositio Mathematica, 1995, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Compositio Mathematica 97: 309-328, 1995. 0 1995 Kluwer Academic Publishers. Printed in the Netherlands. 309 Hodge theory and deformations of maps ZIV RAN* Department of Mathematics, University of California, Riverside, CA 92521 Received 23 January 1992; accepted in revised form 30 June 1993; final revision June 1995. The purpose of this paper is to establish and apply a general principle (Theorem 1.1, Section 1) which serves to relate Hodge theory and defor- mation theory. This principle, which we state in a somewhat technical abstract-nonsense framework, takes on two dual forms which say roughly, respectively, that: (0.1) for a functorial homomorphism from a Hodge-type group to an infinitesimal deformation group, im( 1") consists of unobstructed deformations; (0.2) for a functorial homomorphism from an obstruction group to a Hodge-type group, "actual" obstructions lie in ker( n). A familiar example of (0.1) is the case of deformations of manifolds with trivial canonical bundle, treated from a similar viewpoint in [Rl] ; a familiar example of (0.2) is Bloch’s semi-regularity map [B], treated from a similar viewpoint in [R3]. Our principle may be viewed as an amalga- mation and abstraction of parts of [R1] and [R3]. We will apply the above principle to deformations of maps of complex manifolds. After recalling and developing in Section 2 some general formalism concerning such deformations, we develop, in Sections 3 and 4, two instances of a map 03C0 as in (0.2) in the respective cases of generic immersions and fibre spaces. It may be noted that even in the case of an *Supported in part by NSF. DMS 9202050. 310 embedding our map 03C0 does not reduce to Bloch’s semi-regularity map but rather has the latter as a component (cf. Remark 3.2.1); in particular our map has smaller kernel than Bloch’s, leading to a better consequence of (0.2) (cf. Remark 3.4.1). Some applications given in Sections 3 and 4 are to the question of stability, i.e. of identifying, given a generic immersion f : Y - X or a fibre space f: X - x the locus of deformations of X to which f extends. We will show, in particular, that under favorable conditions the latter locus may be identified Hodge-theoretically (Corollaries 3.2, 4.2); in the case of an embedding f this may be considered as a special case of an infinitesimal form of the Generalized Hodge Conjecture. A particularly nice case is that of q-Lagrangian submanifolds of a q-symplectic manifold, in which we give a result (Corollary 3.4) generalizing work of Voisin [V]. In the final Section 5 we systematically apply the foregoing to the problem of moving curves on a manifold, a problem made popular in recent years by Mori theory. We will give a variety of conditions under which a curve is forced to move. As a final application, we describe the structure of manifolds X such that Qi is spanned off a finite set: namely we prove (Theorem 5.3) that either X is fibred by tori or the canonical bundle Kx is ample. This is in complete analogy with the known case of 1-forms (Ueno’s conjecture, cf. [R4]). It is interesting to note that the proof involves applications of both (0.1) and (0.2) (at two different points). We begin in Section 1 by setting up an abstract categorical framework for deformation theory, one which seems best suited for stating (let alone proving) our general principle; it is shown in Section 2 that deformations of maps of manifolds fit into this framework, which is a variant of the usual one (cf. [S]); perhaps the main novelty here over [S] is the systematic use of "canonical elements" to describe infinitesimal deformations, in general- ization of the method of [Rl] and [R2]. For a different application of the results of Section 1, see [R6]. Convention. In this paper ’Kahler manifold’ should be understood in the cohomological sense, i.e. a manifold whose Hodge-de Rham spectral sequence degenerates at E 1. 1. Abstract nonsense Let A be the category of Artin local k-algebras with residue field k. In order to use geometric language, we may, by the assignment R - S = Spec(R), identify the opposite category dOP with a category of (1-point) schemes over k. We may also, from time to time, refer to an element R ~ A as (R, M) or (R, M, I) where M is the maximal ideal and 311 I = Ann(M) ci R. To category such as A we may associate 2 other categories: W-.,Yom whose objects are morphisms of A and whose morphism are commutative squares in A; and A-Mod, whose objects are pairs (R, A) where R ~ A, and A is an R-module; and whose morphisms are defined by Now consider a category AIW, i.e. a small category together with a contravariant functor 03A6:B - .91; we will sometimes write an object X of 11 as X/R or X/S indicating that R = O(X) or S = Spec(03A6(X)). We will assume that 11 has an unique object X °/k. Define another category éé as the fibre product We will say that the category 14/.91 is base-changing if it comes equipped with a functor À - e-Yeom such that the following diagram commutes where q’(X2 ~ X 1) = X1 and 03A6’ is the evident functor induced by 03A6; more concretely, what this means is that any X1/S1 and S2 ~ SI may be functorially completed to a "base change" diagram = = In this case we will use the notation X 2/S2 X 1 S1 S2, X21R, X1 XR1 R2, Si = Spec(Ri). 312 Note that to a base-changing category B/A we may associate a (covariant) functor and we will say that A is a realization of F. In what follows we will assume that F admits a hull in the sense of Schlessinger [S], in which case we will say that A is hullable. Let à = {(X/R, R’):X/R~B, R = R’/FR’ = R’/F’}. Now given B/A as above, by an admissible B-module, we mean a functor L: à -+ A-Mod compatible with 03A6, together with the data, for all (R, M, I) E d and (X/R, R’) ~ , of a functorial exact diagram where the top row is the obvious thing and g is the map induced by base-change, plus a group L(X’) with functorial isomorphisms, for all (R, M, I = M) ~ A, L(X0/k, R) = L(X0) (D M. L is said to be right (resp. left)- exact if g is always surjective (resp. f is always injective). By a ô-pair of A-modules we mean a pair (LI, L2) of admissible A-modules together with the data, for all (R, M, I)~A and X/R~B, of a functorial exact sequence L1(XO) (D 1 ~ L1(X, R’) ~ L1(X x (R/1), R) ~ L2(X0) O 1 ~ L2(X, R’) ~ L2(X x (R/1), R). By a linearization of PÃ / .91 we mean a ô-pair (L’, L2) of admissible A-modules together with the data, for all (R, M, I) ~ A, J = Ann(M/I) c R and X/(R/1) ~ B, of a canonical element with the property that X comes via base-change from some X/R iff a lies in the image of the canonical map ’Vote that given a linearized base-changing category B/A, an admissible module L’ c LI gives rise to a subcategory é8’ c: A defined "inductively" by 313 the conditions that X0/K~B’ and that given X/R ~ B such that X x R (R/1) G é3’, X/R ~ B’ iff the canonical element Next, by an exact triple of the linearized base-changing categories over si we mean a triple (Bi, Li , L2i) of such, together with functors an extension of (L11, L12) as ô-pair to PÃ2, still denoted (L11, L21), and an exact sequence of PÃ2-modules where (L13, L23) are viewed as e2-modules in the obvious way, such that the respective canonical a-elements are compatible. Now we can finally state our result: THEOREM 1.1. Let (Bi, Ll, L?), i = 1, 2, 3 be an exact triple of linearized base-changing .91 -categories, realizing respectively the functors Fi: A ~ Yets. Let Z,, i = 1, 2, 3, be a formal scheme which is a hull for Fi and Z1 ~ Z2 ~ Z3 the induced maps. Also put lji = dimk(Lji(X0i)) where X0i/k~Bi is the unique object. (i) Suppose K is a right-exact B3-module with a natural transformation and put Q = rkk(03C4: K(X03) ~ L13(X03)). Then there is a smooth a-dimensional formal subscheme Zo c Z3 corresponding to im(r), such that Zo:= 03B2-12(Z03) has embedding dimension 6 + l11 and is defined by at most 11 equations.
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