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Introduction to Algebraic D-Modules Alexander Braverman Tatyana

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Introduction to Algebraic D-Modules Alexander Braverman Tatyana Introduction to algebraic D-modules Alexander Braverman Tatyana Chmutova Pavel Etingof David Yang Contents Chapter 1. D-modules on affine varieties 5 1.1. Analytic continuation of distributions with respect to a parameter and D-modules 5 1.2. Bernstein’s inequality and its applications 12 1.3. More on D(An)-modules 16 1.4. Functional dimension and homological algebra 27 1.5. Proofs of 4.7 and 4.12 31 1.6. D-modules on general affine varieties 33 1.7. Proof of Kashiwara’s theorem and its corollaries 38 1.8. Direct and inverse images preserve holonomicity 42 Chapter 2. D-modules on general algebraic varieties 48 2.1. D-modules on arbitrary varieties 48 2.2. Derived categories 52 2.3. The derived category of D-modules 58 Chapter 3. The derived category of holonomic D-modules 63 3.1. 63 3.2. Proof of Theorem 3.1.2 67 3.3. 69 Chapter 4. D-modules with regular singularities 73 4.1. Lectures 14 and 15 (by Pavel Etingof): Regular singularities and the Riemann-Hilbert correspondence for curves 73 4.2. Regular singularities in higher dimensions 77 Chapter 5. The Riemann-Hilbert correspondence and perverse sheaves 81 5.1. Riemann-Hilbert correspondence 81 Bibliography 88 3 4 0. CONTENTS This book arose from notes of a graduate course by the first author at Harvard in 2002 taken by the second author, and then edited and expanded by the first, third and fourth authors in 2013-2015. CHAPTER 1 D-modules on affine varieties In this chapter we will develop the theory of D-modules on affine varieties. Unless specified otherwise, we will work over a base field k of characteristic 0 (in most cases one can assume that k = C). 1.1. Analytic continuation of distributions with respect to a parameter and D-modules 1.1.1. An analytic problem. As a motivation for what is going to come, let us first look at the following analytic problem. Let p 2 R[x1; : : : ; xn] be a nonconstant real polynomial in n variables, viewed as a function p : Rn ! R. Let U be a connected component of fx j p(x) > 0g. Define ( p(x) if x 2 U p (x) = U 0 otherwise. λ Let us take any λ 2 C and consider the function pU (x) . λ n It is easy to see that if Reλ ≥ 0 then pU (x) makes sense as a distribution on R , i.e., R λ 1 n 1 n Rn pU (x) f(x)dx is convergent for any f 2 Cc (R ); where Cc (R ) is the space of smooth functions on Rn with compact support. R 1 λ Example 1.1.1. Let n = 1, p(x) = x and U = R+. Then 0 x f(x)dx is defined for Reλ ≥ 0. (Of course, this integral is actually well-defined for Reλ > −1, but we do not need this). We will λ denote the corresponding distribution by x+. λ It is easy to see that for Reλ ≥ 0, we have a holomorphic family of distributions λ 7! pU (x) . In fact, it is defined and holomorphic even in a slightly larger region, Reλ > −δ, for some δ > 0. However, if p has zeros then when the real part of λ decreases, at some point the integral becomes divergent. This gives rise to the following natural question. Question 1.1.2. (I. M. Gelfand, M. Sato): Can one extend this family meromorphically in λ to the whole C? λ Example 1.1.3. Let n = 1, p(x) = x, U = R+. We have a distribution x+ defined for Reλ ≥ 0. d λ+1 λ We know that dx (x+ ) = (λ + 1)x+, and the left hand side is defined for Reλ ≥ −1. Hence the expression 1 d xλ = (xλ+1) + λ + 1 dx + λ gives us an extension of x+ to Reλ ≥ −1, λ 6= −1. Continuing this process, we get 1 dn xλ = (xλ+n) + (λ + 1):::(λ + n) dxn + and thus we have 5 6 1. D-MODULES ON AFFINE VARIETIES λ Proposition 1.1.4. x+ extends to the whole of C meromorphicaly with at most simple poles 1 at the negative integers. In particular, for every f 2 Cc (R) the function Z 1 λ Jf (λ) = x f(x)dx 0 has a meromorphic continuation to the whole of C with at most simple poles at −1; −2; −3; :::. Example 1.1.5. The proposition works not only for functions f with compact support, but also for functions which are rapidly decreasing at +1 together with all derivatives. For example, we −x R 1 λ −x can take f(x) = e . In this case we have 0 x e dx = Γ(λ + 1). The proposition implies the well-known fact that Γ(λ) has a meromorphic continuation with poles at 0; −1; −2:::. It is remarkable that a similar property holds in several variables, even though it is harder to prove. Namely, we have the following theorem. λ Theorem 1.1.6. [M. Atiyah [A], J. Bernstein and S. Gelfand [BG]] The distribution pU has a meromorphic continuation to the whole of C with poles of bounded order in a finite number of semi-infinite arithmetic progressions of the form a; a − 1; a − 2; :::, a 2 C. The first proofs of Theorem 1.1.6 were based on Hironaka’s theorem on resolution of singularities. We are going to give a completely algebraic proof of Theorem 1.1.6 which is due to J. Bernstein. Let us first formulate an algebraic statement that implies Theorem 1.1.6. 1.1.2. An algebraic reformulation. Let D = D(An) denote the algebra of differential oper- n ators with polynomial coefficients acting on O = O(A ) = C[x1; : : : ; xn]. In other words, D is the @ subalgebra of End C O generated by multiplication by xi and , i = 1; :::; n. @xi Note that for any nonconstant polynomial p 2 O, we can formally apply any element of D to the formal power pλ, where λ is a variable. More precisely, the space O[p−1; λ]pλ (which is really just O[p−1; λ]) is naturally a D[λ]-module; for instance, @ pλ = λ @p pλ−1, where pλ−1 := p−1pλ, @xi @xi etc. Theorem 1.1.7. There exists L 2 D[λ] and b 2 C[λ] such that Lpλ+1 = b(λ)pλ: d Example 1.1.8. Let n = 1 and p(x) = x. Then we can take L = dx and b(λ) = λ + 1. Note that the set of possible b forms an ideal in C[λ]. As C[λ] is a principal ideal domain, this ideal must be generated by a monic polynomial. This polynomial is called the Bernstein-Sato polynomial or the b-function of p. Remark 1.1.9. It is known that the zeros of the Bernstein-Sato polynomial are at negative rational numbers (see [K]), but we will not prove it here. We claim now that Theorem 1.1.7 implies Theorem 1.1.6 (note that Theorem 1.1.7 is a completely λ+1 λ λ+1 λ algebraic statement). Indeed, suppose Lp = b(λ)p . Then LpU = b(λ)pU , where as usual for a distribution E on Rn we define @E (f) = −E( @f ). The left hand side is defined for Reλ ≥ −1, so @xi @xi the expression 1 pλ = Lpλ+1 U b(λ) U λ gives us a meromorphic continuation of pU to Reλ ≥ −1. Indeed, integrating by parts, we get Z Z λ 1 λ+1 ∗ pU (x) f(x)dx = pU (x) L f(x)dx; Rn b(λ) Rn 1.1. ANALYTIC CONTINUATION OF DISTRIBUTIONS WITH RESPECT TO A PARAMETER AND D-MODULES 7 ∗ ∗ ∗ @ @ where L is the adjoint operator of L (defined by the equalities x = xi, = − , and i @xi @xi ∗ ∗ ∗ (L1L2) = L2L1). Arguing by induction similarly to the case of one variable, we see that the distri- λ bution pU can be meromorphically extended to the whole of C, with poles at arithmetic progressions a, a − 1, a − 2, . where a is a root of b(λ). Moreover, it is clear that the orders of these poles are bounded by the degree of b. We now want to reformulate Theorem 1.1.7 once again. Set D(λ) = D ⊗C C(λ), and algebra over the field C(λ). Denote by M(p) the D(λ)-module consisting of all formal expressions qpλ−i λ−i+1 λ−i where i 2 Z and q 2 C(λ)[x1; : : : ; xn] subject to the relations qp = (qp)p (the action of −1 D(λ) is defined in the natural way). Note that M(p) is isomorphic to C(λ)[x1; ··· ; xn; p ] as a −1 λ C(λ)[x1; ··· ; xn]-module; namely, we can write M(p) = C(λ)[x1; :::; xn; p ]p . Theorem 1.1.10. M(p) is finitely generated over D(λ). Let us show that Theorem 1.1.7 and Theorem 1.1.10 are equivalent. Theorem 1.1.10 ) Theorem 1.1.7: Denote by Mi the submodule of M(p) generated by λ−i p . Then Mi ⊂ Mi+1 and [ (1.1) M(p) = Mi: i Assume that M(p) is finitely generated. Then (1.1) implies that there exists j 2 Z such that λ−j M(p) = Mj. In other words, the module M(p) is generated by p . Hence there exists Le 2 D(λ) such that Lpe λ−j = pλ−j−1.
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