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POLARIZABLE TWISTOR D-MODULES Claude Sabbah

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POLARIZABLE TWISTOR D-MODULES Claude Sabbah POLARIZABLE TWISTOR D-MODULES Claude Sabbah Abstract. — We prove a Decomposition Theorem for the direct image of an irreducible local system on a smooth complex projective variety under a morphism with values in another smooth complex projective variety. For this purpose, we construct a category of polarized twistor D-modules and show a Decomposition Theorem in this category. Résumé (D-modules avec structure de twisteur polarisable). — Nous montrons un théorème de décomposition pour l’image directe d’un système local irréductible sur une variété projective complexe lisse par un morphisme à valeurs dans une autre variété projective complexe lisse. À cet effet, nous construisons une catégorie de D- modules avec structure de twisteur polarisée et nous montrons un théorème de dé- composition dans cette catégorie. CONTENTS Introduction ................................................... ............... 1 0. Preliminaries ................................................... ........... 11 1. Coherent and holonomic RX -modules ................................. 19 1.1. Coherent and good RX -modules ........................................ 19 1.2. The involutivity theorem ......................... ....................... 20 1.3. Examples ....................................... ......................... 23 1.4. Direct and inverse images of RX -modules .............................. 25 1.5. Sesquilinear pairings on RX -modules ................................... 29 1.6. The category R- Triples(X) ............................................. 31 2. Smooth twistor structures ............................................... 39 2.1. Twistor structures in dimension 0 .................. ..................... 39 2.2. Smooth twistor structures in arbitrary dimension . ................... 47 3. Specializable RX -modules ............................................... 55 3.1. V -filtrations ........................................ ..................... 55 3.2. Review on specializable DX -modules .................................... 64 3.3. The category S 2(X,t).................................................. 66 3.4. Localization and minimal extension across a hypersurface ............... 81 3.5. Strictly S(upport)-decomposable RX -modules .......................... 84 3.6. Specialization of a sesquilinear pairing ........... ....................... 87 3.7. Noncharacteristic inverse image .................. ....................... 98 3.8. A local computation .............................. .......................103 4. Polarizable twistor D-modules ..........................................107 4.1. Definition of a twistor D-module ........................................107 4.2. Polarization ................................... ..........................117 vi CONTENTS 5. Polarizable regular twistor D-modules on curves .....................123 5.1. A basic example .................................. .......................123 5.2. Review of some results of C.Simpson and O.Biquard ..... ..............128 5.3. Proof of Theorem 5.0.1, first part ................... ....................132 5.4. Proof of Theorem 5.0.1, second part ................. ....................145 5.A. Mellin transform and asymptotic expansions . ...................149 5.B. Some results of O.Biquard ......................... .....................151 6. The decomposition theorem for polarizable regular twistor D-modules ...................155 6.1. Statement of the main results and proof of the Main Theorems .........155 6.2. Proof of (6.1.1)(1,0) when Supp T is smooth ............................156 6.3. Proof of (6.1.1)(n,m) ⇒ (6.1.1)(n+1,m+1) .................................170 6.4. Proof of (6.1.1)(6(n−1),0) and ((6.1.1)(1,0) with Supp T smooth) ⇒ (6.1.1)(n,0) for n > 1 .................................................. ..171 6.5. Proof of Theorem 6.1.3 ............................ ......................174 7. Integrability ................................................... ............177 7.1. Integrable RX -modules and integrable triples ....................... ....177 7.2. Integrable smooth twistor structures .............. ......................181 7.3. Integrability and specialization ................. .........................185 7.4. Integrable polarizable regular twistor D-modules ........................188 Appendix. Monodromy at infinity and partial Fourier Laplace transform ............................191 A.1. Exponential twist ............................... ........................191 A.2. Partial Fourier-Laplace transform of RX -modules ......................193 A.3. Partial Fourier-Laplace transform and specialization ....................197 A.4. Partial Fourier-Laplace transform of regular twistor D-modules ........198 Bibliography ................................................... ...............201 Notation ................................................... ...................207 INTRODUCTION Let X be a smooth complex projective manifold and let F be a locally constant sheaf of C-vector spaces of finite dimension on X. We assume that F is semisimple, i.e., a direct sum of irreducible locally constant sheaves on X. Then it is known that, given any ample line bundle on X, the corresponding Hard Lefschetz Theorem holds for the cohomology of X with values in F : if F is constant, this follows from Hodge theory; for general semisimple local systems, this was proved by C. Simpson [63] using the existence of a harmonic metric [13]. The existence of such a metric also allows him to show easily that the restriction of F to any smooth subvariety of X remains semisimple. In this article, we extend to such semisimple local systems other properties known to be true for the constant sheaf, properties usually deduced from Hodge theory. These properties will concern the behaviour with respect to morphisms. They were first proved for the constant sheaf (cf. [15, 17, 60, 66, 3, 28]) and then, more generally, for local systems underlying a polarizable Hodge Module, as a consequence of the work of M. Saito [56]. Given a local system F of finite dimensional C-vector spaces on a complex man- ifold X, it will be convenient to denote by pF the associated perverse complex F [dim X], i.e., the complex having F as its only nonzero term, this term being in degree − dim X. The proof of the following results will be given in § 6.1. Main Theorem 1 (Decomposition Theorem). — Let X be a smooth complex projective variety and let F be a semisimple local system of finite dimensional C-vector spaces on X. Let U be an open set of X and let f : U → Y be a proper holomorphic mapping in a complex manifold Y . Fix an ample line bundle on X. Then (1) the relative Hard Lefschetz Theorem holds for the perverse cohomology sheaves p i p H (Rf∗ F|U ) of the direct image; p (2) the direct image complex Rf∗ F|U decomposes (maybe non canonically) as the direct sum of its perverse cohomology sheaves: p p i p Rf∗ F|U ≃⊕ H (Rf∗ F|U )[−i]; i 2 INTRODUCTION p i p (3) each perverse cohomology sheaf H (Rf∗ F|U ) decomposes as the direct sum of intersection complexes supported on closed irreducible analytic subsets Z of Y , i.e., of the form IC•(pL ), where L is a local system on a smooth open dense set Z r Z′, with Z′ closed analytic in Z; (4) if moreover U = X and Y is projective, then each perverse cohomology sheaf p i p H (Rf∗ F ) is semisimple, i.e., the local systems L are semisimple. Main Theorem 2 (Vanishing cycles). — Let X be a smooth complex projective variety and let F be a semisimple local system on X. Let U be an open set of X and let f : U → C be a holomorphic function on U which is proper. Then, for any ℓ ∈ Z, Mp p Mp p the perverse complexes grℓ ψf F and grℓ φf F , obtained by grading with respect to the monodromy filtration the perverse complexes of nearby or vanishing cycles, are semisimple perverse sheaves on f −1(0). Remarks (1) We note that (1) ⇒ (2) in Main Theorem 1 follows from an argument of Deligne [15]. (2) The nearby and vanishing cycles functors ψf and φf defined by Deligne [19] are shifted by −1, so that they send perverse sheaves to perverse sheaves. They are p p denoted by ψf and φf , following M. Saito [56]. (3) It is known that the Main Theorem 1 implies the local invariant cycle theorem for the cohomology with coefficients in F (cf. [3, Cor.6.2.8 and 6.2.9], see also [57, Cor. 3.6 and 3.7]). If for instance Y = C then, for any k > 0 and for t 6= 0 small enough, there is an exact sequence T − Id Hk(f −1(0), F ) −→ Hk(f −1(t), F ) −−−−−−→ Hk(f −1(t), F ), where T denotes the monodromy. It also implies the exactness of the Clemens-Schmid sequence. (4) Owing to the fact that, if FQ is a perverse complex of Q-vector spaces on a complex analytic manifold, then FQ is semisimple if and only if FC = C ⊗Q FQ is so, the previous results apply as well to Q-local systems, giving semisimple Q-perverse complexes as a result. (5) It would be possible to define a category of perverse complexes “of smooth origin”, obtained after iterating various operations starting from a semisimple local system on a smooth complex projective variety, e.g., taking perverse cohomology of a projective direct image, taking monodromy-graded nearby or vanishing cycles relative to a projective holomorphic function, taking sub-quotients of such objects. The perverse complexes in this category are semisimple. (6) A conjecture of M. Kashiwara [34] —which was the main motivation for this work— asserts in particular that these results should hold when F is any semisimple perverse sheaf (with coefficients in C) on X. In the complex
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