Mixed Hodge structure on local cohomology with support in determinantal varieties
aa r X i v : . [ m a t h . AG ] F e b MIXED HODGE STRUCTURE ON LOCAL COHOMOLOGY WITH SUPPORT INDETERMINANTAL VARIETIES
MICHAEL PERLMAN
Abstract.
We employ the inductive structure of determinantal varieties to calculate the weight filtration onlocal cohomology modules with determinantal support. We show that the weight of a simple composition factoris uniquely determined by its support and cohomological degree. As a consequence, we obtain the equivariantstructure of the Hodge filtration on each local cohomology module, and we provide a formula for its generationlevel. In the case of square matrices, we express the Hodge filtration in terms of the Hodge ideals for thedeterminant hypersurface. As an application, we describe a recipe for calculating the mixed Hodge modulestructure on any iteration of local cohomology functors with determinantal support. Introduction
Given a smooth complex variety X , and a closed subvariety Z ⊆ X , the local cohomology sheaves H jZ ( O X )are holonomic D X -modules, where D X is the sheaf of algebraic differential operators. Furthermore, the sheaves H jZ ( O X ) are functorially endowed with structures as mixed Hodge modules [Sai90], implying that they areequipped with two increasing filtrations: the Hodge filtration F • ( H jZ ( O X )), an infinite filtration by coherent O X -modules; and the weight filtration W • ( H jZ ( O X )), a finite filtration by D X -modules.When Z is a divisor, the data of the Hodge filtration on the module H Z ( O X ) is equivalent to that of Hodgeideals [MP19]. In this case, there are numerous connections between the behavior of the Hodge filtration (e.g.the jumps and generation level) and invariants of singularities arising from birational geometry, including themultiplier ideals and minimal exponent [MP20a, MP20b]. On the other hand, for higher codimension Z , littleis known about the Hodge and weight filtrations on the local cohomology modules, and it is unclear howthey detect and measure singularities. In this article, and in previous joint work with Claudiu Raicu [PR21],we aim to explicate these filtrations in the case when Z is a determinantal variety, with the hope that suchcalculations may lead to new insights regarding the mixed Hodge module structure of local cohomology.We let X = C m × n be the space of m × n generic matrices, with m ≥ n , endowed with the action of thegroup GL = GL m ( C ) × GL n ( C ) via row and column operations. For 0 ≤ q ≤ n we let Z q ⊆ X denote thedeterminantal variety of matrices of rank ≤ q . The D -module structure of the local cohomology modules H j Z q ( O X ) is well understood [RWW14, RW14, RW16, LR20]. In particular, their simple composition factorsare known [RW16, Main Theorem], which are among D , · · · , D n , where D p = L ( Z p , X ) is the intersectionhomology module associated to the trivial local system on Z p \ Z p − .We write O H X for the trivial Hodge module overlying O X . Given a D -simple composition factor M of H j Z q ( O H X ), we say that M has weight w if it underlies a summand of gr Ww H j Z q ( O H X ). Theorem 1.1.
Let ≤ p ≤ q < n ≤ m and j ≥ . If D p is a simple composition factor of H j Z q ( O H X ) , then ithas weight mn + q − p + j .In particular, the weight of each copy of D p is uniquely determined by its cohomological degree. The case q = n − In Section 2.2 we explain the choice of Hodge structure on local cohomology implicit in our discussion. Fornow we mention that it is determined functorially by pushing forward the trivial Hodge module O HU from U = X \ Z q , and so by the general theory H j Z q ( O H X ) has weight ≥ mn + j − j in this case. It would be interesting to find sharperlower bounds in general, perhaps depending on the type of singularities that the variety possesses. FromTheorem 1.1 one sees an inverse relation between the weight of a composition factor and the dimension of itssupport. A somewhat similar correlation has been observed for weights in GKZ systems [RW18, Proposition3.6(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] andTheorem 1.1. In the following examples, we express each module as a class in the Grothendieck group ofGL-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: (cid:2) H Z ( O X ) (cid:3) = [ D ] + [ D ] + [ D ] , (cid:2) H Z ( O X ) (cid:3) = [ D ] + [ D ] , (cid:2) H Z ( O X ) (cid:3) = [ D ] . Theorem 1.1 asserts that if we endow O X with the trivial pure Hodge structure, then each D p in cohomologicaldegree j above is functorially endowed with weight 18 − p + j . Thus, the weights of the above simple compositionfactors 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: (cid:2) H Z ( O X ) (cid:3) = [ D ] , (cid:2) H Z ( O X ) (cid:3) = [ D ] , (cid:2) H Z ( O X ) (cid:3) = [ D ] + [ D ] , (cid:2) H Z ( O X ) (cid:3) = (cid:2) H Z ( O X ) (cid:3) = [ D ] + [ D ] , (cid:2) H Z ( O X ) (cid:3) = (cid:2) H Z ( O X ) (cid:3) = [ D ] . If O X is endowed with the trivial pure Hodge structure, then the weights of the above simple compositionfactors are (from left to right): 43, 46, 48, 49, 51, 52, 53, 54, 56, 58.One can show that the maximal weight of a composition factor of H • Z q ( O H X ) is 2 mn − q ( q + 1), the weight of D 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 thepole order filtration of the determinant hypersurface. We let m = n , and let S = C [ x i,j ] ≤ i,j ≤ n denote thering of polynomial functions on X . The localization S det of the polynomial ring at the n × n determinantdet = det( x i,j ) is a holonomic D -module, with composition series as follows [Rai16, Theorem 1.1]:0 ( S ( h det − i D ( h det − i D ( · · · ( h det − n i D = S det , (1.1)where h det − p i D is the D -submodule of S det generated by det − p , and h det − p i D / h det − p +1 i D ∼ = D n − p . Following[LR20], we define Q n = S det , and for p = 0 , · · · , n −
1, we set Q p = S det h det p − n +1 i D . (1.2)The modules Q p constitute the indecomposable summands of local cohomology with determinantal support[LR20, Theorem 1.6]. In other words, for all 0 ≤ q ≤ n and j ≥
0, the local cohomology module H j Z q ( O X )is a direct sum of the modules Q , · · · , Q q , possibly with multiplicity. For instance, in Example 1.2 the threenonzero local cohomology modules are isomorphic as D -modules to Q , Q , and Q , respectively.Let Q Hn be the mixed Hodge module structure on Q n = S det induced by pushing forward the trivial Hodgemodule from the complement of the determinant hypersurface (see Section 2.5). For 0 ≤ p ≤ n − IXED HODGE STRUCTURE ON LOCAL COHOMOLOGY WITH SUPPORT IN DETERMINANTAL VARIETIES 3 Q Hp for the quotient of Q Hn that overlies Q p , and for k ∈ Z we denote by Q Hp ( k ) the k -th Tate twist of Q Hp . InProposition 2.2 we show that the mixed Hodge modules {Q Hp ( k ) } k ∈ Z provide a complete list of mixed Hodgemodules that may overlie Q p . As a consequence, we obtain the following result. Corollary 1.4.
Let ≤ p ≤ q < n = m and j ≥ . If Q p is a D -indecomposable summand of H j Z q ( O H X ) ,then it underlies the mixed Hodge module Q Hp (( n − q − j ) / .In particular, writing F • for the Hodge filtration, we have the following isomorphisms of S -modules: F k (cid:0) H j Z q ( O H X ) (cid:1) ∼ = q M p =0 F k − ( n − q − j ) / (cid:0) Q Hp (cid:1) ⊕ b p , for k ∈ Z , (1.3) where b p denotes the multiplicity of Q p as an indecomposable summand of H j Z q ( O X ) . Thus, in the case of square matrices, in order to determine the Hodge filtration on local cohomology, itsuffices to understand the Hodge filtration on each Q Hp . The multiplicities b p in the statement of Corollary1.4 may be calculated using the formula [LR20, Theorem 6.1].Our next result expresses the Hodge filtration on each Q Hp in terms of the Hodge ideals I k ( Z ) for thedeterminant hypersurface Z = Z n − , which combined with (1.3) will provide the first of two expressions forthe Hodge filtration on local cohomology with determinantal support in square matrices. We recall the Hodgeideals in this case, which were calculated in [PR21, Theorem 1.1]. For k ≥ I k ( Z ) = n − \ q =1 J (( n − q ) · ( k − − ( n − q ) ) q , (1.4)where J q is the determinantal ideal of q × q minors of ( x i,j ), and J ( d ) q denotes its d -th symbolic power,consisting of functions vanishing to order ≥ d on Z q − . Given a, b ≥ I a × b for GL-equivariant idealin S generated by the irreducible subrepresentation S ( b a ) C n ⊗ S ( b a ) C n ⊆ S (see Section 4.4). Theorem 1.5.
Let ≤ p ≤ n = m . For k ≥ the k -th piece of the Hodge filtration F • on Q Hp is given by: F k (cid:0) Q Hp (cid:1) = (cid:18) I k ( Z ) I ( p +1) × ( k − ( n − p )+2) ∩ I k ( Z ) (cid:19) ⊗ O X (( k + 1) · Z ) , (1.5) with the convention that I a × b = 0 if a > n and I a × b = S if b < . The ideal I a × b is the smallest GL-equivariant ideal containing the b -th powers of the a × a minors of ( x i,j ).For example, I n × b = J bn is the principal ideal generated by the b -th power of the determinant. Though notimmediately apparent from Theorem 1.5, the first nonzero level of the Hodge filtration on Q Hp is (cid:0) n − p (cid:1) , whichcoincides with the first nonzero level of the induced filtration on the composition factor D p . SpecializingTheorem 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 D p areuniquely determined by weight, and are calculated in [PR21, Theorem 3.1]. As a consequence, we obtain theHodge filtration on each local cohomology module from our knowledge of the weight filtration. The groupGL = GL m ( C ) × GL n ( C ) acts on X , preserving each determinantal variety, and inducing the structure of aGL-representation on each piece of the Hodge filtration on local cohomology with determinantal support. Assuch, we express the GL-equivariant structure of the Hodge filtration F • via multisets W ( F k ( H j Z q ( O X ))) of dominant weights λ = ( λ ≥ · · · ≥ λ n ) ∈ Z n , encoding the irreducible representations that appear and theirmultiplicities (see Section 2.4). In the following statement, we write c p = codim Z p = ( m − p )( n − p ). MICHAEL PERLMAN
Corollary 1.6.
For k ∈ Z and ≤ p ≤ n we let D pk = (cid:8) λ ∈ Z n dom : λ p ≥ p − n, λ p +1 ≤ p − m, λ p +1 + · · · + λ n ≥ − k − c p (cid:9) . (1.6) Let ≤ q < n ≤ m and j ≥ . The k -th piece of the Hodge filtration on local cohomology is determined by thefollowing multiset of dominant weights: W (cid:0) F k (cid:0) H j Z q (cid:0) O H X (cid:1) (cid:1)(cid:1) = q G p =0 (cid:16) D pk − ( c p + p − q − j ) / (cid:17) ⊔ a p , (1.7) where a p is the multiplicity of D p as a simple composition factor of H j Z q ( O H X ) .If s is minimal such that a s = 0 in (1.7), then the generation level of the Hodge filtration on H j Z q ( O H X ) is ( c s + s − q − j ) / . If m = n , then s = 0 , so the generation level is ( n − q − j ) / . In Section 4, we elaborate on this result and explain how to deduce it from Theorem 1.1, [PR21], and [LR20].For now we discuss what one can immediately conclude about the Hodge filtration. Each D pk in (1.7) arisesfrom the induced Hodge filtration on a composition factor D p , and Corollary 1.6 asserts that the filtration oneach D p in cohomological degree j is the same. It follows from (1.6) and (1.7) that the first nonzero level ofHodge filtration on each D p is ( c p + p − q − j ) /
2. In particular, as cohomological degree increases, the startinglevel of each D p decreases. As the sets D pk and D rd are disjoint for p = r , there is no ambiguity in (1.7), andit completely describes the Hodge filtration on a local cohomology module. The case q = n − Example 1.7.
Continuing Example 1.2, for k ∈ Z we have W (cid:0) F k (cid:0) H Z (cid:0) O H X (cid:1)(cid:1)(cid:1) = D k ⊔ D k − ⊔ D k − , W (cid:0) F k (cid:0) H Z (cid:0) O H X (cid:1)(cid:1)(cid:1) = D k − ⊔ D k − , W (cid:0) F k (cid:0) H Z (cid:0) O H X (cid:1)(cid:1)(cid:1) = D k − . The induced filtrations on each of the simple composition factors start in the following levels (from left toright): 0, 2, 5, 1, 4, 3, and the generation level of the Hodge filtration on each of the three nonzero localcohomology modules is 5, 4, 3, respectively.Theorem 1.1 is a special case of our main result, Theorem 3.1, which describes the weight filtration onlocal cohomology of any pure Hodge module overlying a simple module D p . As a consequence, we provide aprocedure to calculate the Hodge and weight filtrations on any iteration of local cohomology functors. Strategy and Organization.
We summarize our strategy to prove Theorem 3.1 (and Theorem 1.1). Theproof proceeds by induction on n ≥ Step 1.
In Section 3.2 we employ the inductive structure of determinantal varieties to relate the mixed Hodgemodule structure of local cohomology on X with support in Z q to that on smaller matrices C ( m − × ( n − with support matrices of rank ≤ q −
1. By inductive hypothesis, this allows us to reduce the proof of Theorem3.1 to the problem of verifying that each D 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 weightsin some Grothendieck spectral sequences for local cohomology.In Section 4 we discuss how to deduce the Hodge filtration on each local cohomology module from Theorem3.1 and [PR21]. As an application, we determine the generation level.
IXED HODGE STRUCTURE ON LOCAL COHOMOLOGY WITH SUPPORT IN DETERMINANTAL VARIETIES 5 Preliminaries
In this section we establish notation, and review some relevant background regarding functors on Hodgemodules, 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 d X ,with sheaf of algebraic differential operators D X . We write D bh ( D X ) for the bounded derived category ofholonomic D X -modules, and we write MHM( X ) for the category of algebraic mixed Hodge modules on X ,with D b MHM( 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 Z reg ⊆ Z . For example, L ( X, X ) = O X .We write IC HZ for the pure Hodge module associated to the trivial variation of Hodge structure on Z reg [HTT08, Section 8.3.3], which has weight d Z . For a mixed Hodge module M = ( M, F • , W • ) and k ∈ Z , wewrite M ( k ) = ( M, F •− k , W • +2 k ) for its k -th Tate twist . For example, IC HZ ( k ) has weight d Z − k . The modulesIC HZ ( k ) provide a complete list of pure Hodge modules that may overlie the D -module L ( Z, X ).Let f : X → Y be a morphism between smooth complex varieties, and let M ∈ D bh ( D X ) and N ∈ D bh ( D Y ).We write the following for the direct and inverse image functors for D -modules: f + ( M ) := R f ∗ ( D Y ← X ⊗ L M ) , and f † ( N ) := D X → Y ⊗ L f − N [ d X − d Y ] . These functors induce functors on the corresponding bounded derived categories of mixed Hodge modules,which we denote as follows: f ∗ : D b MHM( X ) → D b MHM( Y ) , and f ! : D b MHM( Y ) → D b MHM( X ) . Given
M ∈ D b MHM( X ), we say that M is mixed of weight ≤ w (resp. ≥ w ) if gr Wi ( H j ( 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 be a closed subvariety. We write R H Z ( − ) for the functor on D bh ( D X ) of sections with support in Z , whosecohomology functors H iZ ( − ) are the local cohomology functors with support in Z .We set U = X \ Z with open immersion j : U → X . Given M ∈ D bh ( D X ), there is a distinguished trianglein D bh ( D X ) [HTT08, Proposition 1.7.1(i)]: R H Z ( M ) −→ M −→ j + j † ( M ) +1 −→ . (2.1)If M underlies M ∈ D b MHM( X ), then j + j † ( M ) underlies j ∗ j ! ( M ) ∈ D b MHM( X ), so this triangle endows R H Z ( M ) with the structure of an object in D b MHM( X ). In particular, if M ∈
MHM( X ) then we have anexact sequence of mixed Hodge modules:0 −→ H Z ( M ) −→ M −→ H ( j ∗ j ! ( M )) −→ H Z ( M ) −→ , and isomorphisms H qZ ( M ) ∼ = H q − ( j ∗ j ! ( M )) for q ≥ X is an affine space, we identify all of the above sheaves with their global sections, and view everythingas a module over the Weyl algebra D = Γ( C N , D C N ). For ease of notation throughout, we write R Γ Z ( − ) := R Γ Z ( C N , − ) , and H jZ ( − ) := H jZ ( C N , − ) . We store the following lemma, which will serve as the base case of our inductive proof of Theorem 3.1.
Lemma 2.1. If Z ⊆ C N is defined by homogeneous equations, then R Γ { } (IC HZ ) is pure of weight d Z . Equiv-alently, each H j { } (IC HZ ) is a direct sum of copies of IC H { } (( − d Z − j ) / . MICHAEL PERLMAN
Proof.
Let i : { } ֒ → C N be the closed immersion of the origin. By [HTT08, Proposition 1.7.1(iii)] we havethat i ∗ i ! = R Γ Z ( − ) on the category D b MHM( C N ), in a manner compatible with our fixed choice of mixedHodge module structure on local cohomology. Since i is the inclusion of the origin, Kashiwara’s equivalenceand [HTT08, Section 8.3.3(m13)] imply that the functor i ∗ preserves pure complexes and their weight, soit suffices to show that i ! (IC HZ ) ∈ D b MHM( { } ) is pure of weight d Z . Let D denote the duality functor onthe bounded derived category of mixed Hodge modules. Since Z is the affine cone over a projective variety,[HTT08, Lemma 13.2.10] implies that D i ! D (IC HZ ) is pure of weight d Z , and thus i ! (IC HZ ) is pure of weight d Z ,as desired. The second assertion follows from the first, and the fact that the pure Hodge modules IC H { } ( k )provide a complete list of those supported on the origin. (cid:3) -equivariant D -modules on C m × n . We let X = C m × n be the space of m × n generic matrices,with m ≥ n . This space is endowed with an action of the group GL = GL m ( C ) × GL n ( C ) via row andcolumn operations, and the orbits of stratify X by matrix rank. We let D denote the Weyl algebra on X ofdifferential operators with polynomial coefficients.All D -modules considered in this work are objects in the category mod GL ( D ) of GL-equivariant holonomic D -modules. The simple objects in this category are the modules D , D , · · · , D n , where D p = L ( Z p , X ) is the intersection homology D -module associated to Z p .Given M ∈ mod GL ( D ) and 0 ≤ q ≤ n , the local cohomology modules H j Z q ( M ) are also objects of mod GL ( D ),and thus have composition factors among D , · · · , D n . When m = n , the category mod GL ( D ) is semi-simple[LW19, Theorem 5.4(b)], so each local cohomology module decomposes as a D -module into a direct sum ofits 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 mod GL ( D ) is not semi-simple [LW19, Theorem5.4(a)]. For instance, each simple module D p admits a unique extension by D p +1 . The modules Q p from theIntroduction are built by such extensions. We recall that Q n = S det , and Q p = Q n h det p − n +1 i D , for p = 0 , · · · , n − , so by (1.1) each Q p has composition factors D , · · · , D p , each with multiplicity one.We denote by add( Q ) the additive subcategory of mod GL ( D ) consisting of modules that are isomorphic toa direct sum of the modules Q , · · · , Q n . By [LR20, Theorem 1.6] each local cohomology module of the form H j Z q ( D p ) and H j Z q ( Q p ) belongs to add( Q ). Though we will not use them here, there are formulas to calculatethe multiplicity of each Q p in local cohomology [LR20, Theorem 1.6, Theorem 6.1].2.4. Subrepresentations of equivariant D -modules. For N ≥
1, the irreducible representations of thegeneral linear group GL N ( C ) are in one-to-one correspondence with dominant weights λ = ( λ ≥ λ ≥ · · · ≥ λ N ) ∈ Z N . We write Z N dom for the set of dominant weights, and S λ C N for the irreducible representation corresponding toa dominant weight λ , where S λ is a Schur functor . For b ∈ Z and a ≥ b a ) = ( b, · · · , b, , · · · ,
0) forthe dominant weight with b repeated a times.For 0 ≤ p ≤ n the module D p decomposes into irreducible GL-representations as follows [Rai16, Section 5]: D p = M λ ∈ W p S λ ( p ) C m ⊗ S λ C n , (2.2) IXED HODGE STRUCTURE ON LOCAL COHOMOLOGY WITH SUPPORT IN DETERMINANTAL VARIETIES 7 where λ ( p ) = ( λ , · · · , λ p , ( p − n ) m − n , λ p +1 + ( m − n ) , · · · , λ n + ( m − n )), and W p := (cid:8) λ ∈ Z n dom : λ p ≥ p − n, λ p +1 ≤ p − m (cid:9) . If N is a subrepresentation of a GL-equivariant holonomic D -module, then it is a subrepresentation of a finitedirect sum of the modules D , · · · , D n . Thus N has a GL-decomposition of the following form: N = M ≤ p ≤ n M λ ∈ W p (cid:0) S λ ( p ) C m ⊗ S λ C n (cid:1) ⊕ b λ . We encode the equivariant structure of such an N via a multiset of dominant weights W ( N ) = { ( λ, b λ ) : λ ∈ Z n dom } . Since the sets W p are pairwise disjoint, and λ ( p ) is uniquely determined by λ , the multiset W ( N ) completelydescribes the equivariant structure of N . Weights of direct sums are described by disjoint unions: W ( N ⊕ N ) = W ( N ) ⊔ W ( N ) = (cid:8) ( λ, b λ + b λ ) : ( λ, b λ ) ∈ W ( N ) , ( λ, b λ ) ∈ W ( N ) (cid:9) . When N is multiplicity free, we simply write W ( N ) as a set of dominant weights. For example, W ( D p ) = W p for p = 0 , · · · , n . Since Q p has simple composition factors D , · · · , D p , each with multiplicity one, we have W ( Q p ) = W ( D ⊕ · · · ⊕ D p ) = W ⊔ · · · ⊔ W p . In particular, W ( Q n ) = Z n dom . Further, by definition of Q p we have W (cid:0) h det p − n +1 i D (cid:1) = W ( Q n ) \ W ( Q p ) = W p +1 ⊔ · · · ⊔ W n , (2.3)which is equal to the following set of dominant weights { λ ∈ Z n dom : λ p +1 ≥ p − n + 1 } .2.5. Hodge modules on C m × n . We write IC H Z p for the pure Hodge module associated to the trivial variationof Hodge structure on Z p \ Z p − , which has weight d p = dim Z p = p ( m + n − p ), and overlies the simple D -module D p . Given k ∈ Z , the k -th Tate twist of IC H Z p is pure of weight d p − k .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 D , · · · , D n , the graded pieces of the weight filtration on M aredirect sums of the intersection cohomology pure Hodge modules defined above. If D p underlies a summandof gr Ww ( M ) for some w ∈ Z , then D p underlies IC H Z p (( d p − w ) / D p has weight w .For the remainder of this section we consider only the case of square matrices. We recall the possible mixedHodge module structures on Q n , and classify the possible mixed Hodge module structures on each Q p .We let Z = Z n − denote the determinant hypersurface, and we write U = X \ Z with open immersion j : U → X . We define Q Hn := j ∗ O HU , where O HU is the trivial pure Hodge module on U . Up to a Tate twist, Q Hn is the unique mixed Hodge module that may overlie Q n [PR21, Section 4.1].The weight filtration W • on Q Hn is described as follows: if w < n or w > n + n , then gr Ww Q Hn = 0, andgr Wn + n − p Q Hn = IC H Z p (cid:18) − (cid:18) n − p + 12 (cid:19)(cid:19) , for p = 0 , · · · , n . (2.4)In other words, the copy of D p in Q Hn has weight n + n − p . Using (2.4) we define a mixed Hodge modulestructure on each Q p for 0 ≤ p ≤ n − −→ W n + n − p − ( Q Hn ) −→ Q Hn −→ Q Hn /W n + n − p − ( Q Hn ) −→ . (2.5)By (1.2) and (2.4), it follows that Q Hn /W n + n − p − ( Q Hn ) is a Hodge module overlying Q p . We define Q Hp = Q Hn W n + n − p − ( Q Hn ) , for p = 0 , · · · , n − . (2.6) MICHAEL PERLMAN
Proposition 2.2.
Up to a Tate twist, Q Hp is the only mixed Hodge module overlying Q p . The proof of Proposition 2.2 is identical to the proof for the case Q n in [PR21, Section 4.1], except that Q p for p < n does not have full support, so [PR21, Equation (2.11)] cannot be used. However, this is remediedin the following lemma. We write δ p = (( p − n ) n ) ∈ W ( D p ). Lemma 2.3.
Let ≤ p ≤ n and consider a mixed Hodge module overlying Q p with Hodge filtration F • . Given ≤ r ≤ p − , if δ r ∈ W ( F l ( Q p )) for some l ∈ Z , then δ r + (1 r +1 ) ∈ W ( F l ( Q p )) .Proof. Suppose that δ r ∈ W ( F l ( Q p )), and let m be a nonzero element of the corresponding isotypic componentof Q p . We let V r +1 C n ⊗ V r +1 C n be the subspace of the polynomial ring S spanned by the ( r +1)-minors. Since F l ( Q p ) is an S -submodule of Q p , it suffices to show to show that the subspace N = ( V r +1 C n ⊗ V r +1 C n ) · m of Q p is nonzero. Suppose for contradiction that N = 0. Since V r +1 C n ⊗ V r +1 C n is the space of definingequations of Z r , it follows that the S -submodule of Q p generated by m has support contained in Z r . Thus, H Z r ( Q p ) = 0, which implies that Q p has a D -submodule with support contained in Z r . Since r ≤ p −
1, thisis impossible, as D p has support Z p and is the socle of Q p [LR20, Lemma 6.3]. (cid:3) We store the following immediate consequence for use in the proof of Theorem 3.1.
Corollary 2.4.
Let ≤ p ≤ n , and let N be a mixed Hodge module overlying Q p . If for some ≤ r ≤ p and w ∈ Z , the copy of D r in N has weight w − r , then for all ≤ q ≤ p , the copy of D q in N has weight w − q . The weight filtration
Let X = C m × n be the space of m × n generic matrices, with m ≥ n . Theorem 1.1 is a special case ofour main result, which addresses the weight filtration on local cohomology of any D p with support in anydeterminantal variety. We let D p underly the pure Hodge module IC H Z p , corresponding to the trivial variationof Hodge structure on Z p \ Z p − , and we write IC H Z p ( k ) for its Tate twists, each of weight d p − k , where d p = dim Z p = p ( m + n − p ). The goal of this section is to prove the following. Theorem 3.1.
Let ≤ r ≤ q < p ≤ n ≤ m and j ≥ . If D r is a simple composition factor of H j Z q (IC H Z p ) ,then it has weight d p + q − r + j . In other words, D r underlies IC H Z r (( d r − d p + r − q − j ) / . As Z n \ Z n − is dense, we have O H X = IC H Z n , so Theorem 1.1 is the case p = n of Theorem 3.1. Ourstrategy for proving Theorem 3.1 is explained in the Introduction. We discuss how one may deduce the Hodgefiltration on these local cohomology modules from Theorem 3.1 and [PR21] in Section 4.We recall the formula for the simple composition factors in this general setting, which we express as agenerating function with coefficients in the Grothendieck group of GL-equivariant holonomic D -modules.Given integers 0 ≤ q < p ≤ n ≤ m we have the identity [LR20, Theorem 3.1]: X j ≥ (cid:2) H j Z q ( D p ) (cid:3) · t j = q X r =0 [ D r ] · t ( p − q ) +( p − r ) · ( m − n ) · (cid:18) n − rp − r (cid:19) t · (cid:18) p − r − q − r (cid:19) t , (3.1)where (cid:0) ab (cid:1) t is a Gaussian binomial coefficient , defined as follows. For a ≥ b we write: (cid:18) ab (cid:19) t = (1 − t a ) · (1 − t a − ) · · · (1 − t a − b +1 )(1 − t b ) · (1 − t b − ) · · · (1 − t ) , with the convention that (cid:0) ab (cid:1) t = 0 if a < b , and (cid:0) a (cid:1) t = (cid:0) aa (cid:1) t = 1. Specializing to p = n recovers the formula[RW16, Main Theorem] for local cohomology of the polynomial ring with support in a determinantal variety.Surprisingly, to prove Theorem 3.1, we only require one piece of information from (3.1). IXED HODGE STRUCTURE ON LOCAL COHOMOLOGY WITH SUPPORT IN DETERMINANTAL VARIETIES 9
Lemma 3.2.
Let ≤ q < p ≤ n ≤ m and j ≥ . The module D appears as a simple composition factor of H j Z q ( D p ) only if j ≡ ( p − q ) + p ( m − n ) (mod 2) .Proof. By (3.1) the smallest degree in which D appears is ( p − q ) + p ( m − n ). The result then followsimmediately from the fact that the Gaussian binomial coefficients in (3.1) are supported in even degrees. (cid:3) The inductive structure of determinantal varieties.
We now establish the inductive setup. Ourtreatment follows [LR20, Section 2H], and we refer the reader there for more details.We choose coordinates ( x i,j ) ≤ i ≤ m, ≤ j ≤ n on the space of matrices X , and we let X denote the open subsetof X defined by non-vanishing of the top left coordinate x , . By performing row and column operations toeliminate entries in the first row and column of the generic matrix, we obtain an isomorphism: X ∼ = X ′ × C m − × C n − × C ∗ , where X ′ is isomorphic to the space of matrices C ( m − × ( n − , with coordinates x ′ i,j = x i,j − x i, · x ,j x , . (3.2)The copy of C ∗ above corresponds to the coordinate x , , and the spaces C m − and C n − correspond to theremaining entries of the first column and row of X , respectively.For p = 0 , · · · n −
1, we write Z ′ p ⊆ X ′ for the determinantal variety of matrices of rank ≤ p , with dimension d ′ p = p ( m + n − p − D ′ p = L ( Z ′ p , X ′ ) for the intersection homology module associated to Z ′ p .Let φ : X → X be the open immersion of X into X , and let π : X → X ′ be the projection map, notingthat these are both smooth morphisms. For ease of notation we set X ∼ = X ′ × T , so that π has relativedimension d T = m + n −
1. We note that d p = d ′ p + d T .Since Z p is defined by the vanishing of the ( p + 1)-minors of the matrix of variables ( x i,j ), one can showusing (3.2) that φ − ( Z p ) = π − ( Z ′ p − ) for 1 ≤ p ≤ n . Further, if we write φ ∗ , π ∗ for the (non-shifted) inverseimage functors of D -modules, then we have the following isomorphisms: φ ∗ ( D p ) ∼ = π ∗ ( D ′ p − ) , for p = 1 , · · · , n. (3.3)In addition, for all 1 ≤ q ≤ p , we have the following: φ ∗ (cid:0) H j Z q (cid:0) D p (cid:1)(cid:1) ∼ = π ∗ (cid:0) H j Z ′ q − (cid:0) D ′ p − (cid:1)(cid:1) , for j ≥ . (3.4)3.2. Hodge modules and the inductive setting.
In this subsection we determine how the mixed Hodgestructure of local cohomology modules on X is related to that on X ′ . The main result here is the following. Proposition 3.3.
Let ≤ r ≤ q < p ≤ n ≤ m and let j ≥ . If Theorem 3.1 holds for the parameters ( r − , q − , p − , n − , m − , j ) , then it holds for the parameters ( r, q, p, n, m, j ) . We emphasize that this result cannot be used to prove Theorem 3.1 for r = 0.To prove Proposition 3.3 above, we determine versions of the isomorphisms (3.3) and (3.4) in the categoryof mixed Hodge modules. The analogue of (3.4) that we formulate must respect our choice of mixed Hodgestructure on local cohomology (see Section 2.2). For this reason, we work in the derived categories of mixedHodge modules on X , X , and X ′ , and thus use the (cohomologically shifted) inverse image functors φ ! and π ! , which lift the D -module functors φ † = φ ∗ and π † = π ∗ [ d T ] respectively (see Section 2.1).Changing the functors in (3.3), we immediately obtain the following isomorphisms of D -modules: φ † ( D p ) ∼ = π † (cid:0) D ′ p − [ − d T ] (cid:1) , for p = 1 , · · · , n. (3.5)We start by translating these isomorphisms to the level of Hodge modules. Lemma 3.4.
Let ≤ p ≤ n . We have the following isomorphims in MHM( X ) for all k ∈ Z : φ ! (cid:16) IC H Z p ( k ) (cid:17) ∼ = π ! (cid:16) IC H Z ′ p − ( k ) [ − d T ] (cid:17) . (3.6) Proof.
It suffices to verify the case k = 0. Both sides of (3.6) correspond to a variation of Hodge structure onthe trivial local system on ( Z ′ p − \ Z ′ p − ) × 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 Y , the inverse image H d Y − d X ( f ! ( N )) is pure of weight v + d X − d Y . It follows that φ ! (IC H Z p ) is pure of weight d p , and π ! (IC H Z ′ p − [ − d T ]) is pure of weight d ′ p − + d T . Since d p = d ′ p − + d T , the result follows. (cid:3) Proposition 3.5.
Let ≤ q < p ≤ n . We have an isomorphism in D b MHM( X ) : φ ! (cid:16) R Γ Z q (cid:16) IC H Z p (cid:17)(cid:17) ∼ = π ! (cid:16) R Γ Z ′ q − (cid:16) IC H Z ′ p − [ − d T ] (cid:17)(cid:17) . Proof.
Let U = X \ Z p , U ′ = X ′ \ Z ′ p − , and U = φ − ( U ) = π − ( U ′ ), and consider the following commutativediagram: U ′ U U X ′ X X i ˜ π ˜ φj kπ φ (3.7)where the vertical arrows are the open immersions, and ˜ π , ˜ φ are the maps on U induced by π , φ . Using thedistinguished triangle (2.1) and Lemma 3.4, it suffices to show that φ ! k ∗ k ! (cid:16) IC H Z p (cid:17) ∼ = π ! i ∗ i ! (cid:16) IC H Z ′ p − [ − d T ] (cid:17) . (3.8)Since U = 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 ! and ˜ π ! i ! = j ! π ! , so we obtain φ ! k ∗ k ! (cid:16) IC H Z p (cid:17) ∼ = j ∗ j ! φ ! (cid:16) IC H Z p (cid:17) , and π ! i ∗ i ! (cid:16) IC H Z ′ p − [ − d T ] (cid:17) ∼ = j ∗ j ! π ! (cid:16) IC H Z ′ p − [ − d T ] (cid:17) . The desired isomorphism (3.8) then follows from (3.6). (cid:3)
As a corollary, we translate (3.4) to the level of mixed Hodge modules.
Corollary 3.6.
Let ≤ q < p ≤ n and j ≥ . We have an isomorphism in MHM( X ) : φ ! (cid:16) H j Z q (cid:16) IC H Z p (cid:17)(cid:17) ∼ = π ! (cid:16) H j Z ′ q − (cid:16) IC H Z ′ p − (cid:17) [ − d T ] (cid:17) . (3.9) Proof.
Since φ is smooth of relative dimension zero, we have H j (cid:16) φ ! (cid:16) R Γ Z q (cid:16) IC H Z p (cid:17)(cid:17)(cid:17) = φ ! (cid:16) H j Z q (cid:16) IC H Z p (cid:17)(cid:17) . On the other hand, since π is smooth of relative dimension d T , we have H j (cid:16) π ! (cid:16) R Γ Z ′ q − (cid:16) IC H Z ′ p − [ − d T ] (cid:17)(cid:17)(cid:17) = π ! (cid:16) H j Z ′ q − (cid:16) IC H Z ′ p − (cid:17) [ − d T ] (cid:17) . Thus, (3.9) is the consequence of taking cohomology of the isomorphism stated in Proposition 3.5. (cid:3)
Finally, we prove Proposition 3.3.
IXED HODGE STRUCTURE ON LOCAL COHOMOLOGY WITH SUPPORT IN DETERMINANTAL VARIETIES 11
Proof of Proposition 3.3.
Let 1 ≤ r ≤ q < p ≤ n ≤ m and let j ≥
0. The hypothesis is: if D ′ r − is a simplecomposition factor of H j Z ′ q − (IC H Z ′ p − ), then it has weight d ′ p − + q − r + j . Equivalently, D ′ r − underlies thepure Hodge module IC H Z ′ r − (( d ′ r − − d ′ p − + r − q − j ) / e = d ′ r − − d ′ p − .Suppose that D r is a simple composition factor of H j Z q (IC H Z p ), underlying M = IC H Z r (( d r − w ) /
2) for some w . We want to show that w = d p + q − r + j . By Corollary 3.6, φ ! ( M ) is isomorphic to π ! ( M ′ [ − d T ]) forsome pure Hodge module M ′ that is a composition factor of H j Z ′ q − (IC H Z ′ p − ). By (3.5), M ′ overlies D ′ r − , soby hypothesis we have M ′ = IC H Z ′ r − (( e + r − q − j ) / d r − w ) / e + r − q − j ) / , so w = d r − e + q − r + j . Since d r − d ′ r − = d T and d ′ p − + d T = d p , we have w = d p + q − r + j . (cid:3) Proof of Theorem 3.1.
In this section we complete the proof of Theorem 3.1. Let X = C m × n be thespace of m × n generic matrices, with m ≥ n . We write Z p ⊆ X for the determinantal variety of matrices ofrank ≤ p , with d p = dim Z p = p ( m + n − p ). We restate Theorem 3.1 for convenience of the reader. Theorem 3.1.
Let 0 ≤ r ≤ q < p ≤ n ≤ m and j ≥
0. If D r is a simple composition factor of H j Z q (IC H Z p ),then it has weight d p + q − r + j . In other words, D r underlies IC H Z r (( d r − d p + r − q − j ) / n ≥
1. The following implies the case n = 1, and will serve as base case whenwe later induct on q ≥ Lemma 3.7. If ≤ p ≤ n ≤ m , then R Γ Z (IC H Z p ) is pure of weight d p . Equivalently, for all j ≥ , if D isa D -simple composition factor of H j Z (IC H Z p ) , then it has weight d p + j .Proof. Since Z p ⊆ X is defined by the (homogeneous) ( p + 1) × ( p + 1) minors of the m × n matrix of variables( x i,j ) ≤ i ≤ m, ≤ j ≤ n , and Z is the origin, this follows from Lemma 2.1. (cid:3) As the base case is complete, we assume that n ≥
2, and that Theorem 3.1 holds for n ′ < n . We summarizewhat we may conclude from inductive hypothesis on n , and the inductive structure of determinantal varieties. Claim 3.8.
Assuming inductive hypothesis on n , Theorem 3.1 holds for all r ≥
1. In particular(1) for all j ≥
0, all copies of D , · · · , D q , in H j Z q (IC H Z p ) have the desired weight, and thus(2) it suffices to prove Theorem 3.1 for r = 0. Proof.
Let r ≥ D r is a D -simple composition factor of H j Z q (IC H Z p ). By inductive hypothesisand Proposition 3.3, it follows that D r has the desired weight d p + q − r + j . (cid:3) Next, we make a reduction that simplifies the case of square matrices.
Claim 3.9.
Assuming inductive hypothesis on n , it suffices to prove Theorem 3.1 for copies of D that aredirect summands of local cohomology. Proof. If m = 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 H j Z q ( D p ) that is not a directsummand. We will show that M has the desired weight d p + q + j , proving Theorem 3.1 in this case.Since M is not a direct summand, and H j Z q ( D p ) is an object in add( Q ), it follows that M is a quotient of aHodge module N overlying Q i for some 1 ≤ i ≤ q . By Proposition 2.2, the possible Hodge module structureson each Q i are unique up to a Tate twist. By Claim 3.8(1), if r ≥
1, and D r is a D -simple composition factorof N , then D r has weight d p + q − r + j . By Corollary 2.4, it follows that M has weight d p + q + j . (cid:3) We now proceed by by induction on q ≥
0, the base case being Lemma 3.7. Going forward we fix 1 ≤ q
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 H j Z q ( D p ), and let M be the pure Hodge module in H j Z q (IC H Z p ) that it underlies. By Claim 3.9, to prove Theorem 3.1, it suffices to verify the following. Goal 3.10.
The pure Hodge module M has weight d p + q + j .To this end, we examine weights in the following Grothendieck spectral sequence: E s,t = H s Z q − (cid:0) H t Z q (cid:0) IC H Z p (cid:1)(cid:1) = ⇒ H s + t Z q − (cid:0) IC H Z p (cid:1) . (3.10)For u ≥
2, the differentials on the u -th page are written d s,tu : E s,tu −→ E s + u,t − u +1 u . (3.11)We will use this spectral sequence and inductive hypothesis to prove the existence of a morphism from M to a pure Hodge module of the desired weight.Since M is supported on the origin, we have H Z q − ( M ) = M , and H s Z q − ( M ) = 0 for s ≥
1. Because M overlies a direct summand of local cohomology, it follows that M overlies a direct summand of E ,j .By (3.10) we have E s,t = 0 for s < t <
0, so it follows from (3.11) that there are no nonzerodifferentials to E ,ju for all u ≥
2. Thus, M ⊆ E ,j ∞ unless d ,jv ( M ) = 0 for some v ≥ Claim 3.11.
There exists v ≥ d ,jv ( M ) = 0. Proof.
Since M overlies a copy of D that is a simple composition factor of H j Z q ( D p ), Lemma 3.2 impliesthat j ≡ ( p − q ) + p ( m − n ) (mod 2). Suppose for contradiction that the claim is false. By the discussionabove, it follows that M ⊆ E ,j ∞ , so M overlies a copy of D that is a composition factor of H j Z q − ( D p ).Again using Lemma 3.2, we have j ≡ ( p − q + 1) + p ( m − n ) (mod 2). However,( p − q ) + p ( m − n ) − (( p − q + 1) + p ( m − n )) = − p + 2 q − , which is odd, yielding a contradiction. (cid:3) Going forward, we write v for the v that satisfies Claim 3.11. It will turn out that the precise value of v isirrelevant for our purposes. Using the inductive hypothesis, we will show that d ,jv ( M ) has weight d p + q + j .To do so, we examine weights in the codomain E v,j − v +1 v of d ,jv . We first consider E v,j − v +12 . Claim 3.12.
The mixed Hodge module E v,j − v +12 is equal to H v Z q − ( N ), where N is a direct sum of pureHodge modules of weight d p + j − v + 1, each overlying a copy of D q . Proof. If m = n , then the local cohomology module H j − v +1 Z q ( D p ) is semi-simple, and its summands areamong D , · · · , D q , possibly with multiplicity. For i ≤ q −
1, each D i is supported in Z q − , so we have that H Z q − ( D i ) = D i and H s Z q − ( D i ) = 0 for s ≥
1. Therefore, since v ≥
2, it follows that E v,j − v +12 overlies adirect sum of copies of H v Z q − ( D q ). By Claim 3.8(1) applied to these copies of D q , it follows that each of themunderlies a pure Hodge module of weight d p + j − v + 1.If m = n , then H j − v +1 Z q ( D p ) is an object in add( Q ). For i ≤ q −
1, each Q i is supported in Z q − , so wehave H Z q − ( Q i ) = Q i and H s Z q − ( Q i ) = 0 for s ≥
1. Therefore, since v ≥
2, it follows that E v,j − v +12 overlies adirect sum of copies of H v Z q − ( Q q ). By (1.2) the module Q q is an extension of Q q − by D q . Since v ≥
2, theassociated long exact sequence of local cohomology yields that H v Z q − ( Q q ) is isomorphic to H v Z q − ( D q ). Thus, IXED HODGE STRUCTURE ON LOCAL COHOMOLOGY WITH SUPPORT IN DETERMINANTAL VARIETIES 13 E v,j − v +12 overlies a direct sum of copies of H v Z q − ( D q ). Again, by Claim 3.8(1) applied to these copies of D q ,it follows that each of them underlies a pure Hodge module of weight d p + j − v + 1. (cid:3) We now consider E v,j − v +1 v , which is a subquotient of E v,j − v +12 . Claim 3.13. If D is a D -subquotient of E v,j − v +1 v , then D underlies a pure Hodge module of weight d p + q + j . Proof.
By Claim 3.12, it follows that E v,j − v +1 v is a subquotient of H v Z q − ( N ), where N is a direct sum of pureHodge modules of weight d p + j − v + 1, each overlying a copy of D q . In this case, N must be a direct sumof copies of IC H Z q (( d q − d p + v − j − / D is a D -subquotient of E v,j − v +1 v . Then D is a D -simple composition factor of one of thecopies of H v Z q − (IC H Z q (( d q − d p + v − j − / q that D underliesIC H Z (( − d q − ( q − − v ) /
2) ( d q − d p + v − j − /
2) = IC H Z (( − d p − q − j ) / , and therefore has weight d p + q + j . (cid:3) Conclusion of proof of Theorem 3.1.
Since d ,jv ( M ) = 0, there is a nonzero map d ,jv : M → E v,j − v +1 v .Because M overlies D , its image overlies D , and is a submodule of E v,j − v +1 v . Claim 3.13 implies that d ,jv ( M ) has weight d p + q + j . Because morphisms in the category of mixed Hodge modules are strict withrespect to the weight filtration, it follows that M has weight d p + q + j , as required to complete Goal 3.10,and thus the proof of Theorem 3.1. (cid:3) In the case of square matrices, we may rephrase Theorem 3.1 in terms of the modules Q p . The followingimplies Theorem 1.4. We recall c p = codim Z p = ( n − p ) in the case of square matrices. Corollary 3.14.
Let ≤ r ≤ q ≤ p < n = m . If Q r is a D -indecomposable summand of H j Z q (IC H Z p ) then itunderlies Q Hr (( c p + n − q − j ) / .Proof. By Proposition 2.2, if Q r is a D -indecomposable summand of H j Z q (IC H Z p ), then in underlies Q Hr ( k ) forsome k ∈ Z . By (2.4) and (2.6) the copy of D in Q Hr ( k ) has weight n + n − k . Thus, by Theorem 3.1 wehave n + n − k = d p + q + j , so k = ( c p + n − q − j ) /
2, as required. (cid:3)
Weight filtrations on iterations of local cohomology.
As a consequence of Theorem 3.1, one maycalculate the Hodge and weight filtrations on any iteration of local cohomology functors with determinantalsupport, applied to an intersection cohomology Hodge module: H • Z i (cid:0) H • Z i (cid:0) · · · H • Z it (cid:0) IC H Z p (cid:1) · · · (cid:1)(cid:1) . Indeed, if m = n then repeated application of Theorem 3.1 to the composition factors of each iteration oflocal cohomology functors yields the weight filtration (Tate twists commute with local cohomology functors).On the other hand, if m = n , to calculate the weights of an iteration one needs to understand the weightfiltration on local cohomology of a module Q p , which is addressed in the following. Theorem 3.15.
Let ≤ r ≤ q ≤ p < n = m . If Q r is an D -indecomposable summand of H j Z q ( Q Hp ) then itunderlies Q Hr (( p − q − j ) / .Proof. We consider the short exact sequence induced by (2.6):0 −→ IC H Z p (cid:18) − (cid:18) n − p + 12 (cid:19)(cid:19) −→ Q Hp −→ Q Hp − −→ . By [LR20, Equation (6.14)] the associated long exact sequence of local cohomology with support in Z q splitsinto the following short exact sequences: for j ≥ −→ H j − Z q ( Q Hp − ) −→ H j Z q (cid:18) IC H Z p (cid:18) − (cid:18) n − p + 12 (cid:19)(cid:19)(cid:19) −→ H j Z q ( Q Hp ) −→ . In particular, H j Z q ( Q Hp ) is a quotient of H j Z q (cid:16) IC H Z p (cid:0) − (cid:0) n − p +12 (cid:1)(cid:1)(cid:17) for all j ≥
0. By Corollary 3.14, if Q r is anindecomposable summand of the latter, then it underlies Q Hr (( c p + n − q − j ) / (cid:18) − (cid:18) n − p + 12 (cid:19)(cid:19) = Q Hr (( p − q − j ) / . Thus, if Q r is an indecomposable summand of H j Z q ( Q Hp ), then it underlies Q Hr (( p − q − j ) / (cid:3) Via iterated application of Corollary 3.14, Theorem 3.15, and [LR20, Theorem 1.6, Theorem 6.1], one maycalculate the weight filtration on any iteration of local cohomology functors applied to IC H Z p or Q Hp . Example 3.16.
Let m = n = 4. By Corollary 3.14 the local cohomology modules in Example 1.2 are H Z ( O H X ) = Q H ( − , H Z ( O H X ) = Q H ( − , H Z ( O H X ) = Q H ( − . We calculate the mixed Hodge module structure of the compositions H i Z H j Z ( O H X ) for i, j ≥
0. UsingTheorem 3.15 and [LR20, Theorem 1.6] the nonzero modules are H Z H Z ( O H X ) = H Z H Z ( O H X ) = Q H ( − , H Z H Z ( O H X ) = H Z H Z ( O H X ) = Q H ( − ,H Z H Z ( O H X ) = Q H ( − , H Z H Z ( O H X ) = H Z H Z ( O H X ) = Q H ( − . The Hodge filtration
In this section we explain how to deduce the Hodge filtration on each local cohomology module fromTheorem 3.1 and the previous work [PR21] regarding the Hodge filtration on the intersection cohomologypure Hodge modules. As an application, we determine the generation level. In Section 4.3 we carry outanother 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 , andlet F • denote the Hodge filtration on the pure Hodge module IC H Z p ( k ). The simple D -module D p underlyingIC H Z p ( k ) is a representation of the group GL = GL m ( C ) × GL m ( C ), and the filtered pieces F • (IC H Z p ( k )) areGL-subrepresentations of D p . We write W p = W ( D p ) and c p = codim Z p = ( m − p )( n − p ). For l ∈ Z thedominant weights of the pieces of the Hodge filtration on IC H Z p ( k ) are given by [PR21, Theorem 3.1]: W (cid:0) F l (cid:0) IC H Z p ( k ) (cid:1)(cid:1) = D pl − c p − k , where D pd := (cid:8) λ ∈ W p : λ p +1 + · · · + λ n ≥ − d − c p (cid:9) . (4.1)Thus, the filtration is determined by the sum of the last n − p entries of λ ∈ W p . It follows that F l (IC H Z p ( k )) = 0 for l < c p + k , and F c p + k (IC H Z p ( k )) = 0 . (4.2)One observes the general phenomenon that Tate twisting by k amounts to a shift of the filtration by k . IXED HODGE STRUCTURE ON LOCAL COHOMOLOGY WITH SUPPORT IN DETERMINANTAL VARIETIES 15
The Hodge filtration on a local cohomology module.
In this subsection we combine the resultsfrom 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 thecategory 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 ≤ r ≤ n and k ∈ Z , let a kr denote the multiplicity of IC H Z r ( k ) as a composition factor of M . Then for l ∈ Z , thedominant weights of the l -th piece of the Hodge filtration on M are given by W (cid:0) F l (cid:0) M (cid:1)(cid:1) = n G r =0 G k ∈ Z (cid:0) D rl − c p − k (cid:1) ⊔ a kr . 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 cohomologywith determinantal support of a pure Hodge module. Theorem 4.2.
Let ≤ q < p ≤ n ≤ m and j ≥ . For ≤ r ≤ q let a r denote the multiplicity of D r as acomposition factor of H j Z q ( D p ) . The Hodge filtration is encoded by the following multiset of dominant weights W (cid:0) F k (cid:0) H j Z q (IC H Z p ) (cid:1)(cid:1) = q G r =0 (cid:0) D rk − ( c r + c p + r − q − j ) / (cid:1) ⊔ a r , for k ∈ Z . (4.3)Specializing Theorem 4.2 to the case p = n yields Corollary 1.6. Since the induced Hodge filtration on eachcopy of D r is the same, and the sets D rd and D se are disjoint for r = s , there is no ambiguity in (4.3), and itcompletely describes the Hodge filtration on each local cohomology module.In the case of square matrices, we package the Hodge filtration in terms of the Hodge filtration on theHodge modules Q Hp . For 1 ≤ p ≤ n and k ∈ Z we define Q pk = W ( F k ( Q Hp )). By (2.4), (2.6), and Lemma 4.1the multiset Q pk is described via the following disjoint union: Q pk = p G r =0 D rk − ( n − r ) . (4.4)If M is a mixed Hodge module overlying an object in add( Q ), then W ( F l ( M )) is a disjoint union ofmultisets of the form Q pk . The next result follows immediately from Corollary 3.14 and Theorem 3.15. Theorem 4.3.
Let ≤ q < p ≤ n = m and j ≥ . (1) Let a r denote the multiplicity of Q r as a D -indecomposable summand of H j Z q (IC H Z p ) . The Hodgefiltration is encoded by the following multiset of dominant weights W (cid:0) F k (cid:0) H j Z q (IC H Z p ) (cid:1)(cid:1) = q G r =0 (cid:0) Q rk − ( c p + n − q − j ) / (cid:1) ⊔ a r , for k ∈ Z . (4.5)(2) Let b r denote the multiplicity of Q r as a D -indecomposable summand of H j Z q ( Q Hp ) . The Hodge filtrationis encoded by the following multiset of dominant weights W (cid:0) F k (cid:0) H j Z q ( Q Hp ) (cid:1)(cid:1) = q G r =0 (cid:0) Q rk − ( p − q − j ) / (cid:1) ⊔ b r , for k ∈ Z . (4.6)As a consequence, using [LR20, Theorem 1.6, Theorem 6.1], one may calculate the Hodge filtration on anyiteration of local cohomology functors applied to the Hodge modules IC H Z p and Q Hp , and their Tate twists. Generation level.
Given a filtered D -module ( M, F • ), we say that F • is generated in level q if F l ( D ) · F q ( M ) = F q + l ( M ) for all l ≥ , 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 ≤ q < p ≤ n ≤ m and j ≥ . Let s be minimal such that D s is a simple compositionfactor of H j Z q ( D p ) . Then the generation level of the Hodge filtration on H j Z q (IC H Z p ) is ( c s + c p + s − q − j ) / .If m = n , then s = 0 , so the generation level is ( n + c p − q − j ) / .Proof. Let g = ( c s + c p + s − q − j ) /
2. We first show that the generation level of the Hodge filtration isat most g . To do so, it suffices to verify that the generation level of the induced Hodge filtrations on eachsimple composition factor is at most g . By Theorem 3.1 we are interested IC H Z r (( d r − d p + r − q − j ) /
2) for0 ≤ r ≤ q . The Hodge filtration on IC H Z r (( d r − d p + r − q − j ) /
2) has generation level ( c r + c p + r − q − j ) / g ≥ ( c r + c p + r − q − j ) / r ≥ s , we conclude thatthe 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,the induced Hodge filtration on D s starts in level g , so it suffices to show that D s is a quotient of H j Z q ( D p ).When m = n , H j Z q ( D p ) is a semi-simple D -module, so D s is necessarily a quotient. If m = n , then H j Z q ( D p )is an object of add( Q ), so D is a quotient and s = 0. (cid:3) Example 4.5.
Let m = 5, n = 3, p = 2, and q = 1. Using (3.1) the nonzero local cohomology modules areexpressed in the Grothendieck group of GL-equivariant holonomic D -modules as follows: (cid:2) H Z ( D ) (cid:3) = [ D ] , (cid:2) H Z ( D ) (cid:3) = [ D ] + [ D ] , (cid:2) H Z ( D ) (cid:3) = (cid:2) H Z ( D ) (cid:3) = [ D ] . If D underlies the pure Hodge module IC H Z , Theorem 3.1 implies that the simple composition factors aboveare 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 (cid:0) F k (cid:0) H Z (cid:0) IC H Z (cid:1)(cid:1)(cid:1) = D k − , W (cid:0) F k (cid:0) H Z (cid:0) IC H Z (cid:1)(cid:1)(cid:1) = D k − ⊔ D k − , W (cid:0) F k (cid:0) H Z (cid:0) IC H Z (cid:1)(cid:1)(cid:1) = D k − , W (cid:0) F k (cid:0) H Z (cid:0) IC H Z (cid:1)(cid:1)(cid:1) = D k − . The induced filtrations on each of the simple composition factors start in the following levels (from leftto right): 4, 3, 6, 5, 4, and the generation level of the Hodge filtration on each of the four nonzero localcohomology modules is 4, 6, 5, 4, respectively.4.4.
The Hodge filtrations on the modules Q p . Let S = C [ x i,j ] ≤ i,j ≤ n be the ring of polynomial functionson the space of n × n matrices. Specializing (2.2) to the case p = n recovers Cauchy’s Formula : S = M λ =( λ ≥···≥ λ n ≥ S λ C n ⊗ S λ C n . For a, b ≥ I a × b to the be ideal in S generated by the subrepresentation S ( b a ) C n ⊗ S ( b a ) C n . Wehave the following description of the dominant weights of I a × b . W ( I a × b ) = { λ ∈ Z n dom : λ a ≥ b, λ n ≥ } . (4.7)Let Z = Z n − denote the determinant hypersurface. For k ≥ I k ( Z ) is defined via thefollowing equality F k ( Q Hn ) = I k ( Z ) ⊗ O X (( k + 1) · Z ) . (4.8)Since Q p = Q n / h det p − n +1 i D , Theorem 1.5 is equivalent to the following. IXED HODGE STRUCTURE ON LOCAL COHOMOLOGY WITH SUPPORT IN DETERMINANTAL VARIETIES 17
Proposition 4.6.
Let ≤ p ≤ n , and consider h det p − n +1 i D ⊆ S det endowed with the mixed Hodge modulestructure induced from being a D -submodule of Q Hn . Then F k (cid:0) h det p − n +1 i D (cid:1) = ( I ( p +1) × ( k − ( n − p )+2) ∩ I k ( Z )) ⊗ O X (( k + 1) · Z ) . Proof.
Since h det p − n +1 i D is a mixed Hodge submodule of Q Hn we have λ ∈ W ( F k (cid:0) h det p − n +1 i D (cid:1)(cid:1) ⇐⇒ λ ∈ W (cid:0) F k (cid:0) Q Hn (cid:1)(cid:1) ∩ W (cid:0) h det p − n +1 i D (cid:1) . By (4.8) we have λ ∈ W (cid:0) F k (cid:0) Q Hn (cid:1)(cid:1) if and only if λ + (( k + 1) n ) ∈ W ( I k ( Z )), and by (2.3) we have λ ∈ W (cid:0) h det p − n +1 i D (cid:1) ⇐⇒ λ p +1 ≥ p − n + 1 ⇐⇒ λ p +1 + ( k + 1) ≥ k − ( n − p ) + 2 . Thus, λ ∈ W ( F k (cid:0) h det p − n +1 i D (cid:1)(cid:1) if and only if λ + (( k + 1) n ) ∈ W ( I k ( Z )) ∩ W ( I ( p +1) × ( k − ( n − p )+2) ). (cid:3) Acknowledgments
We are grateful to Claudiu Raicu for numerous valuable conversations, especially during the early stagesof 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, Birkh¨auser Boston, Inc., Boston, MA, 2008. Translated from the 1995 Japaneseedition by Takeuchi.[LR20] Andr´as C. L˝orincz and Claudiu Raicu, Iterated local cohomology groups and Lyubeznik numbers for determinantal rings ,Algebra & Number Theory (2020), no. 9, 2533–2569.[LW19] Andr´as C L˝orincz and Uli Walther, On categories of equivariant D -modules , Adv. Math. (2019), 429–478.[MP19] Mircea Mustat¸˘a and Mihnea Popa, Hodge ideals , Mem. Amer. Math. Soc. (2019), no. 1268.[MP20a] ,
Hodge ideals for Q -divisors, V -filtration, and minimal exponent , Forum Math. Sigma (2020), Paper No. e19,41.[MP20b] , Hodge filtration, minimal exponent, and local vanishing , Invent. Math. (2020), no. 2, 453–478.[PR21] Michael Perlman and Claudiu Raicu,
Hodge ideals for the determinant hypersurface , Selecta Mathematica New Series (2021), no. 1, 1–22.[Rai16] Claudiu Raicu, Characters of equivariant D -modules on spaces of matrices , Compositio Mathematica (2016), no. 9,1935–1965.[RW14] Claudiu Raicu and Jerzy Weyman, Local cohomology with support in generic determinantal ideals , Algebra & NumberTheory (2014), no. 5, 1231–1257.[RW16] , Local cohomology with support in ideals of symmetric minors and Pfaffians , Journal of the London Mathemat-ical Society (2016), no. 3, 709–725.[RWW14] Claudiu Raicu, Jerzy Weyman, and Emily E. Witt, Local cohomology with support in ideals of maximal minors andsub-maximal Pfaffians , Adv. Math. (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 , Ast´erisque (1989), 10, 145–162. Actes du Colloque deTh´eorie de Hodge (Luminy, 1987).[Sai90] ,
Mixed Hodge modules , Publ. Res. Inst. Math. Sci. (1990), no. 2, 221–333.[Sch16] Christian Schnell, On Saito’s vanishing theorem , Math. Res. Lett. (2016), no. 2, 499–527. Department of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada, K7L 3N6
Email address ::