D-modules in birational geometry Contents Introduction vii Chapter 1. Background on filtered D-modules1 1.1. Generalities on D-modules1 1.2. Filtered D-modules4 1.3. Push-forward of D-modules7 1.4. Induced D-modules and filtered differential morphisms 11 1.5. Push-forward of filtered D-modules 18 1.6. Strictness 22 Chapter 2. The Bernstein-Sato polynomial 25 2.1. Push-forward via a graph embedding 25 2.2. Definition, existence, and examples 28 2.3. Quasi-homogeneous singularities 33 2.4. Analytic continuation of the archimedean zeta function 38 2.5. Kashiwara's rationality theorem 42 2.6. Log resolutions, log canonical thresholds, multiplier ideals 47 2.7. Log canonical threshold and jumping numbers as roots 55 Chapter 3. The V -filtration, and more on Bernstein-Sato polynomials 59 3.1. V -filtration: the smooth case 59 3.2. V -filtration: the general case 63 3.3. Nearby and vanishing cycles 69 3.4. Sabbah's description of the V -filtration 73 3.5. Lichtin's theorem and generalizations 78 3.6. Multiplier ideals vs. V -filtration on OX 85 3.7. Minimal exponent 88 Chapter 4. Filtered D-modules and Hodge D-modules 93 4.1. F -filtration and V -filtration 93 4.2. D-modules with Q-structure 97 4.3. Pure Hodge modules 101 4.4. Mixed Hodge modules 105 4.5. Saito's vanishing theorem 109 Chapter 5. Applications to holomorphic forms and families of varieties 113 5.1. Singular loci of morphisms 113 5.2. Zeros of holomorphic 1-forms 115 5.3. Families of varieties of general type 121 v vi CONTENTS Chapter 6. Localization and Hodge ideals 123 6.1. Definition and basic properties 123 6.2. Local properties 135 6.3. Topological vanishing 136 6.4. Vanishing theorem for Hodge ideals 137 6.5. Applications to hypersurfaces in Pn 143 6.6. Applications to singularities of theta divisors 145 Bibliography 149 INTRODUCTION vii Introduction These are notes written for a course I taught at Northwestern during the Winter and Spring quarters of 2020, as well as for a course I am currently (Spring semester 2021) teaching at Harvard. They cover various topics on D-modules, Hodge modules, and especially their connections to birational geometry. At the moment they are limited to the material I was able to cover during those two quarters. My intention is however to gradually add more material, especially on various properties and applications of Hodge modules. Among the topics to be added: vanishing and positivity package for Hodge mod- ules; applications of Hodge modules to generic vanishing, holomorphic forms, families of varieties; Hodge ideals and applications; Hodge filtration on local cohomology. What these notes are not meant to be is a careful introduction to D-modules, or to mixed Hodge modules (or to birational geometry for that matter). For general D-module theory there are several excellent references (as well as lots of online lecture notes). Among these, the main reference that covers essentially all the basic facts needed here is [HTT]. Nevertheless as these notes develop I will include more precise references, and also an introductory chapter with some background on D-modules, as well as on constructions in birational geometry. As for the theory of mixed Hodge modules, the fundamental references are Saito's papers [Sa1] and [Sa2]. There are also his more recent improvement [Sa7] and survey [Sa8]. Schnell's overview of the theory [Sch] is an excellent quick introduction to the topic, while the developing MHM project [MHM] of Sabbah and Schnell is meant to gradually become a comprehensive reference. CHAPTER 1 Background on filtered D-modules This chapter contains a very brief review of basic definitions and facts from the theory of D-modules. The main source for this material, including proofs and further details, is the book [HTT]. For more specialized facts regarding the theory of filtered D-modules, the main source is [Sa1]. 1.1. Generalities on D-modules We consider a smooth complex variety X of dimension n. We denote by DX the sheaf of differential operators on X; this is the sheaf of C-subalgebras of EndC(OX ) generated by OX and TX , where OX acts by multiplication by functions, and we think of the tangent sheaf TX as being the sheaf of derivations DerC(OX ) ⊂ EndC(OX ). Locally in algebraic coordinates x1; : : : ; xn on an affine neighborhood U around a point x 2 X, it can be described as follows. Consider the corresponding local basis @1;:::;@n of vector fields, dual to dx1; : : : ; dxn. They satisfy [@i;@j] = 0 and [@i; xj] = δi;j n α α1 αn for all i and j. If we denote α = (α1; : : : ; αn) 2 N and @ = @1 ··· @n , then we have M α DX (U) = OX (U)@ : α2Nn P α Therefore locally we will write sections of DX as finite sums P = α aα@ , with aα 2 OX (U). The order of P is the maximal jαj = α1 + ··· + αn appearing in this sum. n Remark 1.1.1 (Affine space). When X = A , we can take x1; : : : ; xn to be global coordinates, and we have DX (X) = An, the n-th Weyl algebra. In this case the theory is essentially equivalent to the theory of modules over An; an excellent introduction, including some special features in this case, can be found in [Co]. Definition 1.1.2. The order filtration on DX is defined (locally) by X α FkDX := f aα@ j jαj ≤ k g: α It can be easily checked that this induces a global filtration on X, and FkDX is called the sheaf of differential operators on X of order at most k. It satisfies the following properties: (1) F•DX is an increasing, exhaustive filtration by locally free sheaves of finite rank. (2) F0DX = OX and F1DX ' OX ⊕ TX . (3) FkDX · F`DX ⊆ Fk+`DX for all k and `. (4) If P 2 FkDX and Q 2 F`DX , then [P; Q] 2 Fk+`−1DX . 1 2 1. BACKGROUND ON FILTERED D-MODULES We use the notation F F M F grk DX := FkDX =Fk−1DX and gr• DX := grk DX k2N for the individual, respectively total, associated graded objects. The natural extension to differential operators of the assignment TX (U) −! OX (U)[y1; : : : ; yn];@i 7! yi induces isomorphisms F k F • grk DX ' Sym TX and gr• DX ' Sym TX ; • where Sym TX is the symmetric algebra of the tangent sheaf of X; see [HTT, x1.1]. In particular, if π : T ∗X −! X is the cotangent bundle of X, since by the standard correspondence between vector bundles locally free sheaves (see e.g. [Ha, Ch.II, Exer. ∗ • 5.17, 5.18]) we have T X ' Spec(Sym TX ), it follows that F gr• DX ' π∗OT ∗X : Definition 1.1.3. A sheaf of OX -modules M is a left DX -module if for every open set U ⊆ X, M(U) has a left DX (U)-module structure, compatible with restrictions. Informally, M admits an action by differentiation. An analogous definition can be made for right DX -modules. It is well known that the data of a DX -module structure on M is equivalent to the data of a C-linear map 1 r: M ! M ⊗ ΩX satisfying the properties of an integrable connection. Concretely, for every open set U ⊆ X, and f 2 OX (U) and s 2 M(U) we have (1.1.1) r(fs) = fr(s) + s ⊗ df; and in addition r ◦ r = 0. See for instance [HTT, Lemma 1.2.1]. This interpretation leads to one of the key objects associated to a DX -module: Definition 1.1.4. The de Rham complex of M is the complex h 1 n i DR(M) = M! ΩX ⊗ M ! · · · ! ΩX ⊗ M : with C-linear differentials induced by iterating r. We consider it to be placed in degrees −n; : : : ; 0.1 We next give a few first examples of DX -modules of a geometric nature. They will play an important role throughout. 1We will work with this convention, even though strictly speaking as such it is usually considered to be the de Rham complex associated to the corresponding right D-module. 1.1. GENERALITIES ON D-MODULES 3 Example 1.1.5. (1) The structure sheaf OX is a left DX -module, via the usual differentiation of functions. It is sometimes called the trivial DX -module. (2) More generally, a vector bundle E on X endowed with an integrable (or flat) connection 1 r: E ! E ⊗ ΩX is an example of a DX -module according to the discussion above. Example (1) is the 1 special case corresponding to the standard differential d: OX ! ΩX . It turns out that this is the only way in which a DX -module can be coherent as an OX -module; see [HTT, Theorem 1.4.10]: Theorem 1.1.6. If M is a DX -module which is coherent as an OX -module, then M is locally free, hence an integrable connection. We now turn to examples that are not coherent as OX -modules. (3) Let D be an effective divisor on X, and consider the quasi-coherent OX -module of rational functions which are regular away from D and have with poles of arbitrary order along D, i.e [ OX (∗D) = OX (kD): k≥0 This is sometimes called the localization of OX along D; indeed if U = Spec(R) is an affine open set in X in which D = Z(f) with f 2 R, then OX (∗D) = Rf , the localization of R at f.