Mixed Hodge Structure on Local Cohomology with Support in Determinantal Varieties 3

Home , Local cohomology, Mixed Hodge structure

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In Section 2.2 we explain the choice of Hodge structure on local cohomology implicit in our discussion. For H now we mention that it is determined functorially by pushing forward the trivial Hodge module OU from U = X \Z , and so by the general theory Hj (OH ) has weight ≥ mn+j−1 [Sai89, Proposition 1.7]. Theorem q Zq X 1.1 demonstrates that this bound is not sharp for any j in this case. It would be interesting to find sharper lower bounds in general, perhaps depending on the type of singularities that the variety possesses. From Theorem 1.1 one sees an inverse relation between the weight of a composition factor and the dimension of its support. A somewhat similar correlation has been observed for weights in GKZ systems [RW18, Proposition 3.6(2)], though weight is not completely determined by support dimension in that case. One may calculate the weight filtration on any local cohomology module using [RW16, Main Theorem] and Theorem 1.1. In the following examples, we express each module as a class in the Grothendieck group of GL-equivariant holonomic D-modules (see Section 2.3). Example 1.2. Let m = n = 4 and q = 2. The three nonzero local cohomology modules are: 4 6 8 HZ2 (OX ) = [D2] + [D1] + [D0], HZ2 (OX ) = [D1] + [D0], HZ2 (OX ) = [D0]. Theorem 1.1 asserts that if we endow OX with the trivial pure Hodge structure, then each Dp in cohomological degree j above is functorially endowed with weight 18−p+j. Thus, the weights of the above simple composition factors are (from left to right): 20, 21, 22, 23, 24, 26. We carry out a larger example on non-square matrices. Example 1.3. Let m = 7, n = 5, and q = 3. The seven nonzero local cohomology modules are: 8 10 12 HZ3 (OX ) = [D3], HZ3 (OX ) = [D2], HZ3 (OX ) = [D2] + [D1], 14 16 18 20 HZ3 (OX ) = HZ3 (OX ) = [D1] + [D0], HZ3 (OX ) = HZ3 (OX ) = [D0]. If OX is endowed with the trivial pure Hodge structure, then the weights of the above simple composition factors are (from left to right): 43, 46, 48, 49, 51, 52, 53, 54, 56, 58. • H One can show that the maximal weight of a composition factor of HZq (OX ) is 2mn−q(q +1), the weight of D0 in the largest degree in which it appears. This upper bound on weight resembles the case of GKZ systems, where weight is bounded above by twice the dimension of the relevant torus [RW18, Proposition 3.6(1)]. In the case of square matrices, we rephrase Theorem 1.1 in terms of mixed Hodge modules arising from the pole order filtration of the determinant hypersurface. We let m = n, and let S = C[xi,j]1≤i,j≤n denote the ring of polynomial functions on X . The localization Sdet of the polynomial ring at the n × n determinant det = det(xi,j) is a holonomic D-module, with composition series as follows [Rai16, Theorem 1.1]: −1 −2 −n 0 ( S ( hdet iD ( hdet iD ( · · · ( hdet iD = Sdet, (1.1) −p −p −p −p+1 ∼ where hdet iD is the D-submodule of Sdet generated by det , and hdet iD/hdet iD = Dn−p. Following [LR20], we define Qn = Sdet, and for p = 0, · · · ,n − 1, we set

Sdet Qp = p−n+1 . (1.2) hdet iD

The modules Qp constitute the indecomposable summands of local cohomology with determinantal support j [LR20, Theorem 1.6]. In other words, for all 0 ≤ q ≤ n and j ≥ 0, the local cohomology module H (OX ) Zq is a direct sum of the modules Q0, · · · ,Qq, possibly with multiplicity. For instance, in Example 1.2 the three nonzero local cohomology modules are isomorphic as D-modules to Q2, Q1, and Q0, respectively. H Let Qn be the mixed Hodge module structure on Qn = Sdet induced by pushing forward the trivial Hodge module from the complement of the determinant hypersurface (see Section 2.5). For 0 ≤ p ≤ n − 1 we write MIXED HODGE STRUCTURE ON LOCAL COHOMOLOGY WITH SUPPORT IN DETERMINANTAL VARIETIES 3

H H H H Qp for the quotient of Qn that overlies Qp, and for k ∈ Z we denote by Qp (k) the k-th Tate twist of Qp . In H Proposition 2.2 we show that the mixed Hodge modules {Qp (k)}k∈Z provide a complete list of mixed Hodge modules that may overlie Qp. As a consequence, we obtain the following result. Corollary 1.4. j H Let 0 ≤ p ≤ q

n−1 n−q ((n−q)·(k−1)−( 2 )) Ik(Z )= Jq , (1.4) q\=1 (d) where Jq is the determinantal ideal of q × q minors of (xi,j), and Jq denotes its d-th symbolic power, consisting of functions vanishing to order ≥ d on Zq−1. Given a, b ≥ 0 we write Ia×b for GL-equivariant ideal n n in S generated by the irreducible subrepresentation S(ba)C ⊗ S(ba)C ⊆ S (see Section 4.4). H Theorem 1.5. Let 0 ≤ p ≤ n = m. For k ≥ 0 the k-th piece of the Hodge ﬁltration F• on Qp is given by: Z H Ik( ) F Q = ⊗ OX ((k + 1) · Z ), (1.5) k p I ∩ I (Z ) (p+1)×(k−(n−p)+2) k with the convention that Ia×b = 0 if a>n and Ia×b = S if b< 0.

The ideal Ia×b is the smallest GL-equivariant ideal containing the b-th powers of the a × a minors of (xi,j). b For example, In×b = Jn is the principal ideal generated by the b-th power of the determinant. Though not H n−p immediately apparent from Theorem 1.5, the first nonzero level of the Hodge filtration on Qp is 2 , which coincides with the first nonzero level of the induced filtration on the composition factor Dp. Specializing Theorem 1.5 to the case p = n recovers the defining equality for the Hodge ideals [MP19, Section A]. We now return to the setting m ≥ n. The possible Hodge filtrations on each simple module Dp are uniquely determined by weight, and are calculated in [PR21, Theorem 3.1]. As a consequence, we obtain the Hodge filtration on each local cohomology module from our knowledge of the weight filtration. The group GL = GLm(C) × GLn(C) acts on X , preserving each determinantal variety, and inducing the structure of a GL-representation on each piece of the Hodge filtration on local cohomology with determinantal support. As j such, we express the GL-equivariant structure of the Hodge filtration F via multisets W(F (H (OX ))) of • k Zq n dominant weights λ = (λ1 ≥···≥ λn) ∈ Z , encoding the irreducible representations that appear and their multiplicities (see Section 2.4). In the following statement, we write cp = codim Zp = (m − p)(n − p). 4 MICHAEL PERLMAN

Corollary 1.6. For k ∈ Z and 0 ≤ p ≤ n we let Dp n k = λ ∈ Zdom : λp ≥ p − n, λp+1 ≤ p − m, λp+1 + · · · + λn ≥−k − cp . (1.6) Let 0 ≤ q

Strategy and Organization. We summarize our strategy to prove Theorem 3.1 (and Theorem 1.1). The proof proceeds by induction on n ≥ 1. Step 1. In Section 3.2 we employ the inductive structure of determinantal varieties to relate the mixed Hodge (m−1)×(n−1) module structure of local cohomology on X with support in Zq to that on smaller matrices C with support matrices of rank ≤ q −1. By inductive hypothesis, this allows us to reduce the proof of Theorem 3.1 to the problem of verifying that each D0 in cohomological degree j is endowed with the desired weight. Step 2. In Section 3.3 we proceed by induction on q ≥ 0, completing the inductive step by examining weights in some Grothendieck spectral sequences for local cohomology. In Section 4 we discuss how to deduce the Hodge ﬁltration on each local cohomology module from Theorem 3.1 and [PR21]. As an application, we determine the generation level. MIXED HODGE STRUCTURE ON LOCAL COHOMOLOGY WITH SUPPORT IN DETERMINANTAL VARIETIES 5

2. Preliminaries In this section we establish notation, and review some relevant background regarding functors on Hodge modules, local cohomology, and equivariant D-modules. All of our D-modules are left D-modules.

2.1. D-modules, Hodge modules, and functors. Let X be a smooth complex variety of dimension dX , b with sheaf of algebraic diﬀerential operators DX . We write Dh(DX ) for the bounded derived category of holonomic DX -modules, and we write MHM(X) for the category of algebraic mixed Hodge modules on X, with DbMHM(X) the corresponding bounded derived category. Given an irreducible closed subvariety Z ⊆ X, we write L(Z, X) for the intersection homology D-module associated to the trivial local system on the regular locus Zreg ⊆ Z. For example, L(X, X)= OX . H We write ICZ for the pure Hodge module associated to the trivial variation of Hodge structure on Zreg [HTT08, Section 8.3.3], which has weight dZ . For a mixed Hodge module M = (M, F•,W•) and k ∈ Z, we H write M(k) = (M, F•−k,W•+2k) for its k-th Tate twist. For example, ICZ (k) has weight dZ −2k. The modules H ICZ (k) provide a complete list of pure Hodge modules that may overlie the D-module L(Z, X). b b Let f : X → Y be a morphism between smooth complex varieties, and let M ∈ Dh(DX ) and N ∈ Dh(DY ). We write the following for the direct and inverse image functors for D-modules: L † L −1 f+(M) := Rf∗(DY ←X ⊗ M), and f (N) := DX→Y ⊗ f N[dX − dY ]. These functors induce functors on the corresponding bounded derived categories of mixed Hodge modules, which we denote as follows: b b ! b b f∗ : D MHM(X) → D MHM(Y ), and f : D MHM(Y ) → D MHM(X). b W j Given M ∈ D MHM(X), we say that M is mixed of weight ≤ w (resp. ≥ w) if gri (H (M)) = 0 for i > j + w (resp. i < j + w). We say M is pure of weight w if it is mixed of weight ≤ w and ≥ w.

2.2. Local cohomology as a mixed Hodge module. Let X be a smooth complex variety and let Z ⊆ X 0 b be a closed subvariety. We write RHZ(−) for the functor on Dh(DX ) of sections with support in Z, whose i cohomology functors HZ (−) are the local cohomology functors with support in Z. b We set U = X \ Z with open immersion j : U → X. Given M ∈ Dh(DX ), there is a distinguished triangle b in Dh(DX ) [HTT08, Proposition 1.7.1(i)]: 0 † +1 RHZ(M) −→ M −→ j+j (M) −→ . (2.1) b † ! b If M underlies M ∈ D MHM(X), then j+j (M) underlies j∗j (M) ∈ D MHM(X), so this triangle endows 0 b RHZ(M) with the structure of an object in D MHM(X). In particular, if M ∈ MHM(X) then we have an exact sequence of mixed Hodge modules: 0 0 ! 1 0 −→ HZ(M) −→ M −→ H (j∗j (M)) −→ HZ(M) −→ 0, q ∼ q−1 ! and isomorphisms HZ (M) = H (j∗j (M)) for q ≥ 2. When X is an aﬃne space, we identify all of the above sheaves with their global sections, and view everything N as a module over the Weyl algebra D = Γ(C , DCN ). For ease of notation throughout, we write N j j N RΓZ (−) := RΓZ (C , −), and HZ(−) := HZ (C , −). We store the following lemma, which will serve as the base case of our inductive proof of Theorem 3.1. N H Lemma 2.1. If Z ⊆ C is deﬁned by homogeneous equations, then RΓ{0}(ICZ ) is pure of weight dZ . Equiv- j H H alently, each H{0}(ICZ ) is a direct sum of copies of IC{0}((−dZ − j)/2). 6 MICHAEL PERLMAN

Proof. Let i : {0} ֒→ CN be the closed immersion of the origin. By [HTT08, Proposition 1.7.1(iii)] we have ! b N that i∗i = RΓZ(−) on the category D MHM(C ), in a manner compatible with our fixed choice of mixed Hodge module structure on local cohomology. Since i is the inclusion of the origin, Kashiwara’s equivalence and [HTT08, Section 8.3.3(m13)] imply that the functor i∗ preserves pure complexes and their weight, so ! H b it suffices to show that i (ICZ ) ∈ D MHM({0}) is pure of weight dZ . Let D denote the duality functor on the bounded derived category of mixed Hodge modules. Since Z is the affine cone over a projective variety, ! H ! H [HTT08, Lemma 13.2.10] implies that Di D(ICZ ) is pure of weight dZ , and thus i (ICZ ) is pure of weight dZ , H as desired. The second assertion follows from the first, and the fact that the pure Hodge modules IC{0}(k) provide a complete list of those supported on the origin.

2.3. GL-equivariant D-modules on Cm×n. We let X = Cm×n be the space of m × n generic matrices, with m ≥ n. This space is endowed with an action of the group GL = GLm(C) × GLn(C) via row and column operations, and the orbits of stratify X by matrix rank. We let D denote the Weyl algebra on X of diﬀerential operators with polynomial coeﬃcients. All D-modules considered in this work are objects in the category modGL(D) of GL-equivariant holonomic D-modules. The simple objects in this category are the modules

D0, D1, · · · , Dn, where Dp = L(Zp, X ) is the intersection homology D-module associated to Zp. Given M ∈ mod (D) and 0 ≤ q ≤ n, the local cohomology modules Hj (M) are also objects of mod (D), GL Zq GL and thus have composition factors among D0, · · · , Dn. When m 6= n, the category modGL(D) is semi-simple [LW19, Theorem 5.4(b)], so each local cohomology module decomposes as a D-module into a direct sum of its simple composition factors. For instance, each local cohomology module in Example 1.3 is semi-simple. On the other hand, for square matrices, the category modGL(D) is not semi-simple [LW19, Theorem 5.4(a)]. For instance, each simple module Dp admits a unique extension by Dp+1. The modules Qp from the Introduction are built by such extensions. We recall that Qn = Sdet, and

Qn Qp = p−n+1 , for p = 0, · · · ,n − 1, hdet iD

so by (1.1) each Qp has composition factors D0, · · · , Dp, each with multiplicity one. We denote by add(Q) the additive subcategory of modGL(D) consisting of modules that are isomorphic to a direct sum of the modules Q0, · · · ,Qn. By [LR20, Theorem 1.6] each local cohomology module of the form Hj (D ) and Hj (Q ) belongs to add(Q). Though we will not use them here, there are formulas to calculate Zq p Zq p the multiplicity of each Qp in local cohomology [LR20, Theorem 1.6, Theorem 6.1].

2.4. Subrepresentations of equivariant D-modules. For N ≥ 1, the irreducible representations of the general linear group GLN (C) are in one-to-one correspondence with dominant weights N λ = (λ1 ≥ λ2 ≥···≥ λN ) ∈ Z . N N We write Zdom for the set of dominant weights, and SλC for the irreducible representation corresponding to a a dominant weight λ, where Sλ is a Schur functor. For b ∈ Z and a ≥ 0 we write (b ) = (b, · · · , b, 0, · · · , 0) for the dominant weight with b repeated a times. For 0 ≤ p ≤ n the module Dp decomposes into irreducible GL-representations as follows [Rai16, Section 5]:

m n Dp = Sλ(p)C ⊗ SλC , (2.2) λM∈W p MIXED HODGE STRUCTURE ON LOCAL COHOMOLOGY WITH SUPPORT IN DETERMINANTAL VARIETIES 7

m−n where λ(p) = (λ1, · · · , λp, (p − n) , λp+1 + (m − n), · · · , λn + (m − n)), and p n W := λ ∈ Zdom : λp ≥ p − n, λp+1 ≤ p − m . If N is a subrepresentation of a GL-equivariant holonomic D-module, then it is a subrepresentation of a finite direct sum of the modules D0, · · · , Dn. Thus N has a GL-decomposition of the following form: m n ⊕bλ N = Sλ(p)C ⊗ SλC . 0≤Mp≤n λM∈W p We encode the equivariant structure of such an N via a multiset of dominant weights W n (N)= {(λ, bλ) : λ ∈ Zdom} . Since the sets W p are pairwise disjoint, and λ(p) is uniquely determined by λ, the multiset W(N) completely describes the equivariant structure of N. Weights of direct sums are described by disjoint unions: W W W 1 2 1 W 2 W (N1 ⊕ N2)= (N1) ⊔ (N2)= (λ, bλ + bλ) : (λ, bλ) ∈ (N1), (λ, bλ) ∈ (N2) . p When N is multiplicity free, we simply write W(N) as a set of dominant weights. For example, W(Dp)= W for p = 0, · · · ,n. Since Qp has simple composition factors D0, · · · , Dp, each with multiplicity one, we have 0 p W(Qp)= W(D0 ⊕···⊕ Dp)= W ⊔···⊔ W . W n In particular, (Qn)= Zdom. Further, by definition of Qp we have p−n+1 p+1 n W hdet iD = W(Qn) \ W(Qp)= W ⊔···⊔ W , (2.3) n which is equal to the following set of dominant weights {λ ∈ Zdom : λp+1 ≥ p − n + 1}. m×n H 2.5. Hodge modules on C . We write ICZp for the pure Hodge module associated to the trivial variation of Hodge structure on Zp \ Zp−1, which has weight dp = dim Zp = p(m + n − p), and overlies the simple H D-module Dp. Given k ∈ Z, the k-th Tate twist of ICZp is pure of weight dp − 2k. Let M be a GL-equivariant holonomic D-module, and let M be a mixed Hodge module that it underlies. Since the composition factors of M are among D0, · · · , Dn, the graded pieces of the weight filtration on M are direct sums of the intersection cohomology pure Hodge modules defined above. If Dp underlies a summand W H of grw (M) for some w ∈ Z, then Dp underlies ICZp ((dp − w)/2). In this case we say that Dp has weight w. For the remainder of this section we consider only the case of square matrices. We recall the possible mixed Hodge module structures on Qn, and classify the possible mixed Hodge module structures on each Qp. We let Z = Zn−1 denote the determinant hypersurface, and we write U = X \ Z with open immersion X H H H j : U → . We define Qn := j∗OU , where OU is the trivial pure Hodge module on U. Up to a Tate twist, H Qn is the unique mixed Hodge module that may overlie Qn [PR21, Section 4.1]. H 2 2 W H The weight filtration W• on Qn is described as follows: if wn + n, then grw Qn = 0, and

W H H n − p + 1 gr 2 Q = ICZ − , for p = 0, · · · ,n. (2.4) n +n−p n p 2 H 2 In other words, the copy of Dp in Qn has weight n + n − p. Using (2.4) we deﬁne a mixed Hodge module structure on each Qp for 0 ≤ p ≤ n − 1 as follows. Consider the short exact sequence H H H H 0 −→ Wn2+n−p−1(Qn ) −→ Qn −→ Qn /Wn2+n−p−1(Qn ) −→ 0. (2.5) H H By (1.2) and (2.4), it follows that Qn /Wn2+n−p−1(Qn ) is a Hodge module overlying Qp. We deﬁne H H Qn Qp = H , for p = 0, · · · ,n − 1. (2.6) Wn2+n−p−1(Qn ) 8 MICHAEL PERLMAN

H Proposition 2.2. Up to a Tate twist, Qp is the only mixed Hodge module overlying Qp.

The proof of Proposition 2.2 is identical to the proof for the case Qn in [PR21, Section 4.1], except that Qp for p

Lemma 2.3. Let 1 ≤ p ≤ n and consider a mixed Hodge module overlying Qp with Hodge filtration F•. Given r r r+1 1 ≤ r ≤ p − 1, if δ ∈ W(Fl(Qp)) for some l ∈ Z, then δ + (1 ) ∈ W(Fl(Qp)). r Proof. Suppose that δ ∈ W(Fl(Qp)), and let m be a nonzero element of the corresponding isotypic component r+1 n r+1 n of Qp. We let C ⊗ C be the subspace of the polynomial ring S spanned by the (r+1)-minors. Since r+1 n r+1 n Fl(Qp) is an SV-submoduleV of Qp, it suffices to show to show that the subspace N = ( C ⊗ C ) · m r+1 n r+1 n of Qp is nonzero. Suppose for contradiction that N = 0. Since C ⊗ C Vis the spaceV of defining equations of Zr, it follows that the S-submodule of Qp generatedV by m hasV support contained in Zr. Thus, 0 Z HZr (Qp) 6= 0, which implies that Qp has a D-submodule with support contained in r. Since r ≤ p − 1, this is impossible, as Dp has support Zp and is the socle of Qp [LR20, Lemma 6.3]. We store the following immediate consequence for use in the proof of Theorem 3.1.

Corollary 2.4. Let 0 ≤ p ≤ n, and let N be a mixed Hodge module overlying Qp. If for some 0 ≤ r ≤ p and w ∈ Z, the copy of Dr in N has weight w − r, then for all 0 ≤ q ≤ p, the copy of Dq in N has weight w − q. 3. The weight filtration Let X = Cm×n be the space of m × n generic matrices, with m ≥ n. Theorem 1.1 is a special case of our main result, which addresses the weight ﬁltration on local cohomology of any Dp with support in any H determinantal variety. We let Dp underly the pure Hodge module ICZp , corresponding to the trivial variation Z Z H of Hodge structure on p \ p−1, and we write ICZp (k) for its Tate twists, each of weight dp − 2k, where dp = dim Zp = p(m + n − p). The goal of this section is to prove the following. Theorem 3.1. Let 0 ≤ r ≤ q

Lemma 3.2. Let 0 ≤ q

Lemma 3.4. Let 1 ≤ p ≤ n. We have the following isomorphims in MHM(X1) for all k ∈ Z:

! H ! H ∼ ′ φ ICZp (k) = π ICZ (k) [−dT ] . (3.6) p−1 Proof. It suﬃces to verify the case k = 0. Both sides of (3.6) correspond to a variation of Hodge structure on Z ′ Z ′ the trivial local system on ( p−1 \ p−2) × T . Thus, we need to show that their respective weights match. By [Sch16, Theorem 9.3], given a smooth morphism f : X → Y , and a pure Hodge module N of weight v on dY −dX ! ! H Y , the inverse image H (f (N )) is pure of weight v + dX − dY . It follows that φ (ICZp ) is pure of weight ! H ′ ′ dp, and π (ICZ ′ [−dT ]) is pure of weight d + dT . Since dp = d + dT , the result follows. p−1 p−1 p−1

b Proposition 3.5. Let 1 ≤ q

! H ! H Z ∼ ′ ′ φ RΓ q ICZp = π RΓZ −1 ICZ [−dT ] . q p−1 X Z ′ X ′ Z ′ −1 −1 ′ Proof. Let U = \ p, U = \ p−1, and U1 = φ (U)= π (U ), and consider the following commutative diagram: ˜ ′ π˜ φ U U1 U

i j k (3.7) X ′ X X π 1 φ

where the vertical arrows are the open immersions, andπ ˜, φ˜ are the maps on U1 induced by π, φ. Using the distinguished triangle (2.1) and Lemma 3.4, it suﬃces to show that

! ! H ! ! H ∼ ′ φ k∗k ICZp = π i∗i ICZ [−dT ] . (3.8) p−1 −1 −1 ′ ! ! Since U1 = φ (U) = π (U ), the base change theorem [HTT08, Theorem 1.7.3] implies that φ k∗ = j∗φ˜ ! ! ! ! ! ! ! ! ! ! and π i∗ = j∗π˜ . Commutativity of (3.7) implies φ˜ k = j φ andπ ˜ i = j π , so we obtain

! ! H ! ! H ! ! H ! ! H ∼ ′ ∼ ′ φ k∗k ICZp = j∗j φ ICZp , and π i∗i ICZ [−dT ] = j∗j π ICZ [−dT ] . p−1 p−1 The desired isomorphism (3.8) then follows from (3.6). As a corollary, we translate (3.4) to the level of mixed Hodge modules.

Corollary 3.6. Let 1 ≤ q

! j H ! j H ∼ ′ ′ φ HZ ICZp = π HZ ICZ [−dT ] . (3.9) q q−1 p−1 Proof. Since φ is smooth of relative dimension zero, we have

j ! H ! j H R Z H φ Γ q ICZp = φ HZq ICZp . On the other hand, since π is smooth of relative dimension dT , we have j ! H ! j H ′ ′ ′ ′ H π RΓZ −1 ICZ [−dT ] = π HZ ICZ [−dT ] . q p−1 q−1 p−1 Thus, (3.9) is the consequence of taking cohomology of the isomorphism stated in Proposition 3.5. Finally, we prove Proposition 3.3. MIXED HODGE STRUCTURE ON LOCAL COHOMOLOGY WITH SUPPORT IN DETERMINANTAL VARIETIES 11

′ Proof of Proposition 3.3. Let 1 ≤ r ≤ q

(dr − w)/2 = (e + r − q − j)/2, ′ ′ so w = dr − e + q − r + j. Since dr − dr−1 = dT and dp−1 + dT = dp, we have w = dp + q − r + j. 3.3. Proof of Theorem 3.1. In this section we complete the proof of Theorem 3.1. Let X = Cm×n be the space of m × n generic matrices, with m ≥ n. We write Zp ⊆ X for the determinantal variety of matrices of rank ≤ p, with dp = dim Zp = p(m + n − p). We restate Theorem 3.1 for convenience of the reader. Theorem 3.1. j H Let 0 ≤ r ≤ q

Proof. Since Zp ⊆ X is deﬁned by the (homogeneous) (p+1)×(p+1) minors of the m×n matrix of variables (xi,j)1≤i≤m,1≤j≤n, and Z0 is the origin, this follows from Lemma 2.1. As the base case is complete, we assume that n ≥ 2, and that Theorem 3.1 holds for n′

Claim 3.9. Assuming inductive hypothesis on n, it suﬃces to prove Theorem 3.1 for copies of D0 that are direct summands of local cohomology. Proof. If m 6= n, this statement has no content, as the local cohomology modules in this case are semi-simple. For the case m = n, let M be a Hodge module overlying a copy of D in Hj (D ) that is not a direct 0 Zq p summand. We will show that M has the desired weight dp + q + j, proving Theorem 3.1 in this case. Since M is not a direct summand, and Hj (D ) is an object in add(Q), it follows that M is a quotient of a Zq p Hodge module N overlying Qi for some 1 ≤ i ≤ q. By Proposition 2.2, the possible Hodge module structures on each Qi are unique up to a Tate twist. By Claim 3.8(1), if r ≥ 1, and Dr is a D-simple composition factor of N , then Dr has weight dp + q − r + j. By Corollary 2.4, it follows that M has weight dp + q + j. 12 MICHAEL PERLMAN

We now proceed by by induction on q ≥ 0, the base case being Lemma 3.7. Going forward we ﬁx 1 ≤ q < p ≤ n ≤ m and j ≥ 0. The inductive hypothesis on q will not be invoked until Claim 3.13. Consider a copy of D that is a direct summand of Hj (D ), and let M be the pure Hodge module in 0 Zq p 0 Hj (ICH ) that it underlies. By Claim 3.9, to prove Theorem 3.1, it suﬃces to verify the following. Zq Zp

Goal 3.10. The pure Hodge module M0 has weight dp + q + j. To this end, we examine weights in the following Grothendieck spectral sequence: Es,t = Hs Ht ICH =⇒ Hs+t ICH . (3.10) 2 Zq−1 Zq Zp Zq−1 Zp For u ≥ 2, the diﬀerentials on the u-th page are written s,t s,t s+u,t−u+1 du : Eu −→ Eu . (3.11)

We will use this spectral sequence and inductive hypothesis to prove the existence of a morphism from M0 to a pure Hodge module of the desired weight. 0 s Since M0 is supported on the origin, we have HZq−1 (M0)= M0, and HZq−1 (M0)=0 for s ≥ 1. Because 0,j M0 overlies a direct summand of local cohomology, it follows that M0 overlies a direct summand of E2 . s,t By (3.10) we have E2 = 0 for s < 0 and for t < 0, so it follows from (3.11) that there are no nonzero 0,j 0,j 0,j differentials to Eu for all u ≥ 2. Thus, M0 ⊆ E∞ unless dv (M0) 6= 0 for some v ≥ 2. 0,j Claim 3.11. There exists v ≥ 2 such that dv (M0) 6= 0. Proof. Since M overlies a copy of D that is a simple composition factor of Hj (D ), Lemma 3.2 implies 0 0 Zq p that j ≡ (p − q)2 + p(m − n) (mod 2). Suppose for contradiction that the claim is false. By the discussion above, it follows that M ⊆ E0,j, so M overlies a copy of D that is a composition factor of Hj (D ). 0 ∞ 0 0 Zq−1 p Again using Lemma 3.2, we have j ≡ (p − q + 1)2 + p(m − n) (mod 2). However, (p − q)2 + p(m − n) − ((p − q + 1)2 + p(m − n)) = −2p + 2q − 1, which is odd, yielding a contradiction. Going forward, we write v for the v that satisfies Claim 3.11. It will turn out that the precise value of v is 0,j irrelevant for our purposes. Using the inductive hypothesis, we will show that dv (M0) has weight dp + q + j. v,j−v+1 0,j v,j−v+1 To do so, we examine weights in the codomain Ev of dv . We first consider E2 . Claim 3.12. v,j−v+1 v The mixed Hodge module E2 is equal to HZq−1 (N ), where N is a direct sum of pure Hodge modules of weight dp + j − v + 1, each overlying a copy of Dq. Proof. If m 6= n, then the local cohomology module Hj−v+1(D ) is semi-simple, and its summands are Zq p among D0, · · · , Dq, possibly with multiplicity. For i ≤ q − 1, each Di is supported in Zq−1, so we have that H0 (D ) = D and Hs (D ) = 0 for s ≥ 1. Therefore, since v ≥ 2, it follows that Ev,j−v+1 overlies a Zq−1 i i Zq−1 i 2 v direct sum of copies of HZq−1 (Dq). By Claim 3.8(1) applied to these copies of Dq, it follows that each of them underlies a pure Hodge module of weight dp + j − v + 1. If m = n, then Hj−v+1(D ) is an object in add(Q). For i ≤ q − 1, each Q is supported in Z , so we Zq p i q−1 0 s v,j−v+1 have HZq−1 (Qi)= Qi and HZq−1 (Qi)=0 for s ≥ 1. Therefore, since v ≥ 2, it follows that E2 overlies a v direct sum of copies of HZq−1 (Qq). By (1.2) the module Qq is an extension of Qq−1 by Dq. Since v ≥ 2, the v v associated long exact sequence of local cohomology yields that HZq−1 (Qq) is isomorphic to HZq−1 (Dq). Thus, MIXED HODGE STRUCTURE ON LOCAL COHOMOLOGY WITH SUPPORT IN DETERMINANTAL VARIETIES 13

v,j−v+1 v E2 overlies a direct sum of copies of HZq−1 (Dq). Again, by Claim 3.8(1) applied to these copies of Dq, it follows that each of them underlies a pure Hodge module of weight dp + j − v + 1. v,j−v+1 v,j−v+1 We now consider Ev , which is a subquotient of E2 . v,j−v+1 Claim 3.13. If D0 is a D-subquotient of Ev , then D0 underlies a pure Hodge module of weight dp +q+j. v,j−v+1 v Proof. By Claim 3.12, it follows that Ev is a subquotient of HZq−1 (N ), where N is a direct sum of pure Hodge modules of weight dp + j − v + 1, each overlying a copy of Dq. In this case, N must be a direct sum H of copies of ICZq ((dq − dp + v − j − 1)/2). v,j−v+1 Suppose that D0 is a D-subquotient of Ev . Then D0 is a D-simple composition factor of one of the v H copies of HZq−1 (ICZq ((dq − dp + v − j − 1)/2)). Since Tate twists commute with local cohomology functors, it follows from inductive hypothesis on q that D0 underlies H H ICZ0 ((−dq − (q − 1) − v)/2) (dq − dp + v − j − 1)/2) = ICZ0 ((−dp − q − j)/2), and therefore has weight dp + q + j. 0,j 0,j v,j−v+1 Conclusion of proof of Theorem 3.1. Since dv (M0) 6= 0, there is a nonzero map dv : M0 → Ev . v,j−v+1 Because M0 overlies D0, its image overlies D0, and is a submodule of Ev . Claim 3.13 implies that 0,j dv (M0) has weight dp + q + j. Because morphisms in the category of mixed Hodge modules are strict with respect to the weight ﬁltration, it follows that M0 has weight dp + q + j, as required to complete Goal 3.10, and thus the proof of Theorem 3.1.

In the case of square matrices, we may rephrase Theorem 3.1 in terms of the modules Qp. The following 2 implies Theorem 1.4. We recall cp = codim Zp = (n − p) in the case of square matrices. Corollary 3.14. Let 0 ≤ r ≤ q ≤ p

H n − p + 1 H H 0 −→ ICZ − −→ Q −→ Q −→ 0. p 2 p p−1 14 MICHAEL PERLMAN

By [LR20, Equation (6.14)] the associated long exact sequence of local cohomology with support in Zq splits into the following short exact sequences: for j ≥ 0 we have

j−1 H j H n − p + 1 j H 0 −→ H (Q ) −→ H ICZ − −→ H (Q ) −→ 0. Zq p−1 Zq p 2 Zq p

In particular, Hj (QH ) is a quotient of Hj ICH − n−p+1 for all j ≥ 0. By Corollary 3.14, if Q is an Zq p Zq Zp 2 r indecomposable summand of the latter, then it underlies n − p + 1 QH ((c + n − q − j)/2) − = QH ((p − q − j)/2). r p 2 r

Thus, if Q is an indecomposable summand of Hj (QH ), then it underlies QH ((p − q − j)/2). r Zq p r

Via iterated application of Corollary 3.14, Theorem 3.15, and [LR20, Theorem 1.6, Theorem 6.1], one may H H calculate the weight ﬁltration on any iteration of local cohomology functors applied to ICZp or Qp . Example 3.16. Let m = n = 4. By Corollary 3.14 the local cohomology modules in Example 1.2 are

4 H H 6 H H 8 H H HZ2 (OX )= Q2 (−1),HZ2 (OX )= Q1 (−2),HZ2 (OX )= Q0 (−3).

i j H We calculate the mixed Hodge module structure of the compositions HZ0 HZ2 (OX ) for i, j ≥ 0. Using Theorem 3.15 and [LR20, Theorem 1.6] the nonzero modules are

8 4 H 5 6 H H 10 4 H 7 6 H H HZ0 HZ2 (OX )= HZ0 HZ2 (OX )= Q0 (−4),HZ0 HZ2 (OX )= HZ0 HZ2 (OX )= Q0 (−5),

12 4 H H 3 6 H 0 8 H H HZ0 HZ2 (OX )= Q0 (−6),HZ0 HZ2 (OX )= HZ0 HZ2 (OX )= Q0 (−3).

4. The Hodge filtration In this section we explain how to deduce the Hodge ﬁltration on each local cohomology module from Theorem 3.1 and the previous work [PR21] regarding the Hodge ﬁltration on the intersection cohomology pure Hodge modules. As an application, we determine the generation level. In Section 4.3 we carry out another example. Finally, in Section 4.4 we prove Theorem 1.5.

4.1. The Hodge filtrations on the equivariant pure Hodge modules. Let 0 ≤ p ≤ n and k ∈ Z, and H let F• denote the Hodge filtration on the pure Hodge module ICZp (k). The simple D-module Dp underlying H H ICZp (k) is a representation of the group GL = GLm(C) × GLm(C), and the filtered pieces F•(ICZp (k)) are p GL-subrepresentations of Dp. We write W = W(Dp) and cp = codim Zp = (m − p)(n − p). For l ∈ Z the H dominant weights of the pieces of the Hodge filtration on ICZp (k) are given by [PR21, Theorem 3.1]: W H Dp Dp p Fl ICZp (k) = l−cp−k, where d := λ ∈ W : λp+1 + · · · + λn ≥−d − cp . (4.1) Thus, the filtration is determined by the sum of the last n − p entries of λ ∈ W p. It follows that

H H Fl(ICZp (k))=0 for l < cp + k, and Fcp+k(ICZp (k)) 6= 0. (4.2) One observes the general phenomenon that Tate twisting by k amounts to a shift of the ﬁltration by k. MIXED HODGE STRUCTURE ON LOCAL COHOMOLOGY WITH SUPPORT IN DETERMINANTAL VARIETIES 15

4.2. The Hodge filtration on a local cohomology module. In this subsection we combine the results from Sections 3 and 4.1 to obtain the Hodge filtration on local cohomology with determinantal support. We begin with a consequence of (4.1), which follows immediately from the fact that morphisms in the category of mixed Hodge modules are strict with respect to the Hodge filtration. Lemma 4.1. Let M be a mixed Hodge module overlying a GL-equivariant holonomic D-module. For 0 ≤ r ≤ n k H and k ∈ Z, let ar denote the multiplicity of ICZr (k) as a composition factor of M. Then for l ∈ Z, the dominant weights of the l-th piece of the Hodge filtration on M are given by n k W Dr ⊔ar Fl M = l−cp−k . rG=0 kG∈Z Therefore, we obtain the Hodge filtration on M from knowledge of the weight filtration.

As a consequence of Theorem 3.1 and Lemma 4.1, we calculate the Hodge ﬁltration F• on local cohomology with determinantal support of a pure Hodge module.

Theorem 4.2. Let 0 ≤ q

If M is a mixed Hodge module overlying an object in add(Q), then W(Fl(M)) is a disjoint union of Qp multisets of the form k. The next result follows immediately from Corollary 3.14 and Theorem 3.15. Theorem 4.3. Let 0 ≤ q

4.3. Generation level. Given a ﬁltered D-module (M, F•), we say that F• is generated in level q if

Fl(D) · Fq(M)= Fq+l(M) for all l ≥ 0, where F•(D) denotes the order ﬁltration on D. The generation level of (M, F•) is deﬁned to be the minimal q such that F• is generated in level q. We determine the generation level of each local cohomology module.

Proposition 4.4. Let 0 ≤ q

Proof. Let g = (cs + cp + s − q − j)/2. We first show that the generation level of the Hodge filtration is at most g. To do so, it suffices to verify that the generation level of the induced Hodge filtrations on each H simple composition factor is at most g. By Theorem 3.1 we are interested ICZr ((dr − dp + r − q − j)/2) for H 0 ≤ r ≤ q. The Hodge filtration on ICZr ((dr − dp + r − q − j)/2) has generation level (cr + cp + r − q − j)/2, the first nonzero level [PR21, Section 4.2]. Since g ≥ (cr + cp + r − q − j)/2 for all r ≥ s, we conclude that the generation level is at most g, as desired. It remains to show that the generation level of the Hodge filtration is at least g. By (4.2) and Theorem 4.2, j the induced Hodge filtration on Ds starts in level g, so it suffices to show that Ds is a quotient of HZq (Dp). When m 6= n, Hj (D ) is a semi-simple D-module, so D is necessarily a quotient. If m = n, then Hj (D ) Zq p s Zq p is an object of add(Q), so D0 is a quotient and s = 0. Example 4.5. Let m = 5, n = 3, p = 2, and q = 1. Using (3.1) the nonzero local cohomology modules are expressed in the Grothendieck group of GL-equivariant holonomic D-modules as follows: 3 5 7 9 HZ1 (D2) = [D1], HZ1 (D2) = [D1] + [D0], HZ1 (D2) = HZ1 (D2) = [D0]. H If D2 underlies the pure Hodge module ICZ2 , Theorem 3.1 implies that the simple composition factors above are functorially endowed with the following weights (from left to right): 15, 17, 18, 20, 22. From this, we compute the equivariant structure of the Hodge filtrations. For k ∈ Z we have W 3 H D1 W 5 H D1 D0 Fk HZ1 ICZ2 = k−4, Fk HZ1 ICZ2 = k−3 ⊔ k−6, W 7 H D0 W 9 H D0 Fk HZ1 ICZ2 = k−5, Fk HZ1 ICZ2 = k−4. The induced filtrations on each of the simple composition factors start in the following levels (from left to right): 4, 3, 6, 5, 4, and the generation level of the Hodge filtration on each of the four nonzero local cohomology modules is 4, 6, 5, 4, respectively.

4.4. The Hodge ﬁltrations on the modules Qp. Let S = C[xi,j]1≤i,j≤n be the ring of polynomial functions on the space of n × n matrices. Specializing (2.2) to the case p = n recovers Cauchy’s Formula: n n S = SλC ⊗ SλC .

λ=(λ1≥···≥M λn≥0) n n For a, b ≥ 0 we deﬁne Ia×b to the be ideal in S generated by the subrepresentation S(ba)C ⊗ S(ba)C . We have the following description of the dominant weights of Ia×b. W n (Ia×b)= {λ ∈ Zdom : λa ≥ b, λn ≥ 0} . (4.7)

Let Z = Zn−1 denote the determinant hypersurface. For k ≥ 0 the Hodge ideal Ik(Z ) is deﬁned via the following equality H Z Z Fk(Qn )= Ik( ) ⊗ OX ((k + 1) · ). (4.8) p−n+1 Since Qp = Qn/hdet iD, Theorem 1.5 is equivalent to the following. MIXED HODGE STRUCTURE ON LOCAL COHOMOLOGY WITH SUPPORT IN DETERMINANTAL VARIETIES 17

p−n+1 Proposition 4.6. Let 0 ≤ p ≤ n, and consider hdet iD ⊆ Sdet endowed with the mixed Hodge module H structure induced from being a D-submodule of Qn . Then p−n+1 Z Z Fk hdet iD = (I(p+1)×(k−(n−p)+2) ∩ Ik( )) ⊗ OX ((k + 1) · ). p−n+1 H Proof. Since hdet iD is a mixed Hodge submodule of Qn we have W p−n+1 W H W p−n+1 λ ∈ (Fk hdet iD ⇐⇒ λ ∈ Fk Qn ∩ hdet iD . W H n W Z By (4.8) we have λ ∈ Fk Qn if and only if λ + ((k + 1) ) ∈ (Ik( )), and by (2.3) we have p−n+1 λ ∈ W hdet iD ⇐⇒ λp+1 ≥ p − n + 1 ⇐⇒ λp+1 + (k + 1) ≥ k − (n − p) + 2. p−n+1 n Z Thus, λ ∈ W(Fk hdet iD if and only if λ + ((k + 1) ) ∈ W(Ik( )) ∩ W(I(p+1)×(k−(n−p)+2)). Acknowledgments We are grateful to Claudiu Raicu for numerous valuable conversations, especially during the early stages of this project while work on [PR21] was in progress. References [HTT08] Ryoshi Hotta, Kiyoshi Takeuchi, and Toshiyuki Tanisaki, D-modules, perverse sheaves, and representation theory, Progress in Mathematics, vol. 236, Birkhäuser Boston, Inc., Boston, MA, 2008. Translated from the 1995 Japanese edition by Takeuchi. [LR20] András C. L˝orincz and Claudiu Raicu, Iterated local cohomology groups and Lyubeznik numbers for determinantal rings, Algebra & Number Theory 14 (2020), no. 9, 2533–2569. [LW19] András C L˝orincz and Uli Walther, On categories of equivariant D-modules, Adv. Math. 351 (2019), 429–478. [MP19] Mircea Mustat¸˘aand Mihnea Popa, Hodge ideals, Mem. Amer. Math. Soc. 262 (2019), no. 1268. [MP20a] , Hodge ideals for Q-divisors, V -filtration, and minimal exponent, Forum Math. Sigma 8 (2020), Paper No. e19, 41. [MP20b] , Hodge filtration, minimal exponent, and local vanishing, Invent. Math. 220 (2020), no. 2, 453–478. [PR21] Michael Perlman and Claudiu Raicu, Hodge ideals for the determinant hypersurface, Selecta Mathematica New Series 27 (2021), no. 1, 1–22. [Rai16] Claudiu Raicu, Characters of equivariant D-modules on spaces of matrices, Compositio Mathematica 152 (2016), no. 9, 1935–1965. [RW14] Claudiu Raicu and Jerzy Weyman, Local cohomology with support in generic determinantal ideals, Algebra & Number Theory 8 (2014), no. 5, 1231–1257. [RW16] , Local cohomology with support in ideals of symmetric minors and Pfaffians, Journal of the London Mathemat- ical Society 94 (2016), no. 3, 709–725. [RWW14] Claudiu Raicu, Jerzy Weyman, and Emily E. Witt, Local cohomology with support in ideals of maximal minors and sub-maximal Pfaffians, Adv. Math. 250 (2014), 596–610. [RW18] Thomas Reichelt and Uli Walther, Weight filtrations on GKZ-systems, arXiv preprint arXiv:1809.04247 (2018). [Sai89] Morihiko Saito, Introduction to mixed Hodge modules, Astérisque 179-180 (1989), 10, 145–162. Actes du Colloque de Théorie de Hodge (Luminy, 1987). [Sai90] , Mixed Hodge modules, Publ. Res. Inst. Math. Sci. 26 (1990), no. 2, 221–333. [Sch16] Christian Schnell, On Saito’s vanishing theorem, Math. Res. Lett. 23 (2016), no. 2, 499–527.

Department of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada, K7L 3N6 Email address: [email protected]