We determine explicitly the Hodge ideals for the determinant hypersurface as an intersection of symbolic powers of determinantal ideals. We prove our results by studying the Hodge and weight filtrations on the mixed Hodge module \(\mathcal {O}_{\mathscr {X}}(*\mathscr {Z})\) of regular functions on the space \(\mathscr {X}\) of \(n\times n\) matrices, with poles along the divisor \(\mathscr {Z}\) of singular matrices. The composition factors for the weight filtration on \(\mathcal {O}_{\mathscr {X}}(*\mathscr {Z})\) are pure Hodge modules with underlying \(\mathcal {D}\)-modules given by the simple \({\text {GL}}\)-equivariant \(\mathcal {D}\)-modules on \(\mathscr {X}\), where \({\text {GL}}\) is the natural group of symmetries, acting by row and column operations on the matrix entries. By taking advantage of the \({\text {GL}}\)-equivariance and the Cohen–Macaulay property of their associated graded, we describe explicitly the possible Hodge filtrations on a simple \({\text {GL}}\)-equivariant \(\mathcal {D}\)-module, which are unique up to a shift determined by the corresponding weights. For non-square matrices, \(\mathcal {O}_{\mathscr {X}}(*\mathscr {Z})\) is replaced by the local cohomology modules \(H^{\bullet }_{\mathscr {Z}}(\mathscr {X},\mathcal {O}_{\mathscr {X}})\), which turn out to be pure Hodge modules. By working out explicitly the Decomposition Theorem for some natural resolutions of singularities of determinantal varieties, and using the results on square matrices, we determine the weights and the Hodge filtration for these local cohomology modules.

我们明确地将行列式超曲面的霍奇理想确定为行列式理想的符号幂的交集。我们通过研究空间\（\上常规函数的混合Hodge模块\（\ mathcal {O} _ {\ mathscr {X}}（* \ mathscr {Z}）\）上的Hodge和权重过滤来证明我们的结果mathscr {X} \）的\（N \ n次\）的矩阵，沿除数磁极\（\ mathscr {Z} \）奇异矩阵。\（\ mathcal {O} _ {\ mathscr {X}}（* \ mathscr {Z}）\）上权重过滤的组成因子是具有基础\（\ mathcal {D} \） - modules的纯Hodge模块由简单的\（{\ text {GL}} \）-等价\（\ mathcal {D} \）给出\（\ mathscr {X} \）上的-modules ，其中\（{\ text {GL}} \）是自然的对称组，通过对矩阵条目的行和列操作进行操作。通过利用\（{\ text {GL}} \）等价度及其关联的分级的Cohen–Macaulay属性，我们在简单的\（{\ text {GL}} \）上明确描述了可能的Hodge过滤-equivariant \（\ mathcal {D} \）- module，在由相应权重确定的移位之前是唯一的。对于非平方矩阵，\（\ mathcal {O} _ {\ mathscr {X}}（* \ mathscr {Z}）\）被局部同调模块\（H ^ {\ bullet} _ {{mathscr {Z}}（\ mathscr {X}，\ mathcal {O} _ {\ mathscr {X}}）\），原来是纯Hodge模块。通过明确地求出行列式奇异性的一些自然分辨率的分解定理，并使用平方矩阵的结果，我们可以确定这些局部同调模块的权重和霍奇滤波。