We compute the Jantzen filtration of a $\mathcal {D}$-module on the flag variety of $\operatorname {\mathrm {SL}}_2(\mathbb {C})$. At each step in the computation, we illustrate the $\mathfrak {sl}_2(\mathbb {C})$-module structure on global sections to give an algebraic picture of this geometric computation. We conclude by showing that the Jantzen filtration on the $\mathcal {D}$-module agrees with the algebraic Jantzen filtration on its global sections, demonstrating a famous theorem of Beilinson and Bernstein.

For a reduced hyperplane arrangement, we prove the analytic Twisted Logarithmic Comparison Theorem, subject to mild combinatorial arithmetic conditions on the weights defining the twist. This gives a quasi-isomorphism between the twisted logarithmic de Rham complex and the twisted meromorphic de Rham complex. The latter computes the cohomology of the arrangement’s complement with coefficients from the corresponding rank one local system. We also prove the algebraic variant (when the arrangement is central), and the analytic and algebraic (untwisted) Logarithmic Comparison Theorems. The last item positively resolves an old conjecture of Terao. We also prove that: Every nontrivial rank one local system on the complement can be computed via these Twisted Logarithmic Comparison Theorems; these computations are explicit finite-dimensional linear algebra. Finally, we give some $\mathscr {D}_{X}$-module applications: For example, we give a sharp restriction on the codimension one components of the multivariate Bernstein–Sato ideal attached to an arbitrary factorization of an arrangement. The bound corresponds to (and, in the univariate case, gives an independent proof of) M. Saito’s result that the roots of the Bernstein–Sato polynomial of a non-smooth, central, reduced arrangement live in $(-2 + 1/d, 0).$

Let $C\; : \;y^2=f(x)$ be a hyperelliptic curve of genus $g\geq 1$, defined over a complete discretely valued field $K$, with ring of integers $O_K$. Under certain conditions on $C$, mild when residue characteristic is not $2$, we explicitly construct the minimal regular model with normal crossings $\mathcal{C}/O_K$ of $C$. In the same setting we determine a basis of integral differentials of $C$, that is an $O_K$-basis for the global sections of the relative dualising sheaf $\omega _{\mathcal{C}/O_K}$.

We prove a Tannakian form of Drinfeld's lemma for isocrystals on a variety over a finite field, equipped with actions of partial Frobenius operators. This provides an intermediate step towards transferring V. Lafforgue's work on the Langlands correspondence over function fields from $\ell$-adic to $p$-adic coefficients. We also discuss a motivic variant and a local variant of Drinfeld's lemma.

We study the Hodge and weight filtrations on the localization along a hypersurface, using methods from birational geometry and the V-filtration induced by a local defining equation. These filtrations give rise to ideal sheaves called weighted Hodge ideals, which include the adjoint ideal and a multiplier ideal. We analyze their local and global properties, from which we deduce applications related to singularities of hypersurfaces of smooth varieties.

We develop the theory of relative regular holonomic $\mathcal {D}$-modules with a smooth complex manifold $S$ of arbitrary dimension as parameter space, together with their main functorial properties. In particular, we establish in this general setting the relative Riemann–Hilbert correspondence proved in a previous work in the one-dimensional case.

We define Bernstein–Sato polynomials for meromorphic functions and study their basic properties. In particular, we prove a Kashiwara–Malgrange-type theorem on their geometric monodromies, which would also be useful in relation with the monodromy conjecture. A new feature in the meromorphic setting is that we have several b-functions whose roots yield the same set of the eigenvalues of the Milnor monodromies. We also introduce multiplier ideal sheaves for meromorphic functions and show that their jumping numbers are related to our b-functions.

We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions and derive applications regarding the local cohomological dimension, the Du Bois complex, local vanishing and reflexive differentials associated to Z.

In this paper we construct an action of the affine Hecke category (in its ‘Soergel bimodules’ incarnation) on the principal block of representations of a simply connected semisimple algebraic group over an algebraically closed field of characteristic bigger than the Coxeter number. This confirms a conjecture of G. Williamson and the second author, and provides a new proof of the tilting character formula in terms of antispherical $p$-Kazhdan–Lusztig polynomials.

Let $G$ be a split connected reductive group over a finite field of characteristic $p > 2$ such that $G_\text {der}$ is absolutely almost simple. We give a geometric construction of perverse $\mathbb {F}_p$-sheaves on the Iwahori affine flag variety of $G$ which are central with respect to the convolution product. We deduce an explicit formula for an isomorphism from the spherical mod $p$ Hecke algebra to the center of the Iwahori mod $p$ Hecke algebra. We also give a formula for the central integral Bernstein elements in the Iwahori mod $p$ Hecke algebra. To accomplish these goals we construct a nearby cycles functor for perverse $\mathbb {F}_p$-sheaves and we use Frobenius splitting techniques to prove some properties of this functor. We also prove that certain equal characteristic analogues of local models of Shimura varieties are strongly $F$-regular, and hence they are $F$-rational and have pseudo-rational singularities.

We establish some cohomological bounds in $D$-module theory that are known in the holonomic case and folklore in general. The method rests on a generalization of the $b$-function lemma for non-holonomic $D$-modules.

We introduce irregular constructible sheaves, which are ${\mathbb {C}}$-constructible with coefficients in a finite version of the Novikov ring $\Lambda$ and special gradings. We show that the bounded derived category of cohomologically irregular constructible complexes is equivalent to the bounded derived category of holonomic ${\mathcal {D}}$-modules by a modification of D’Agnolo and Kashiwara's irregular Riemann–Hilbert correspondence. The bounded derived category of cohomologically irregular constructible complexes is equipped with the irregular perverse $t$-structure, which is a straightforward generalization of usual perverse $t$-structure, and we prove that its heart is equivalent to the abelian category of holonomic ${\mathcal {D}}$-modules. We also develop the algebraic version of the theory.

We study log $\mathscr {D}$-modules on smooth log pairs and construct a comparison theorem of log de Rham complexes. The proof uses Sabbah’s generalized b-functions. As applications, we deduce a log index theorem and a Riemann-Roch type formula for perverse sheaves on smooth quasi-projective varieties. The log index theorem naturally generalizes the Dubson-Kashiwara index theorem on smooth projective varieties.

We show that the only summands of the theta divisor on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the powers of the curve and the Fano surface of lines on the threefold. The proof only uses the decomposition theorem for perverse sheaves, some representation theory and the notion of characteristic cycles.

We compute the Hodge ideals of $\mathbb{Q}$-divisors in terms of the $V$-filtration induced by a local defining equation, inspired by a result of Saito in the reduced case. We deduce basic properties of Hodge ideals in this generality, and relate them to Bernstein–Sato polynomials. As a consequence of our study we establish general properties of the minimal exponent, a refined version of the log canonical threshold, and bound it in terms of discrepancies on log resolutions, addressing a question of Lichtin and Kollár.

Following the work of Mustaţă and Bitoun, we recently developed a notion of Bernstein–Sato roots for arbitrary ideals, which is a prime characteristic analogue for the roots of the Bernstein–Sato polynomial. Here, we prove that for monomial ideals the roots of the Bernstein–Sato polynomial (over $\mathbb{C}$) agree with the Bernstein–Sato roots of the mod $p$ reductions of the ideal for $p$ large enough. We regard this as evidence that the characteristic-$p$ notion of Bernstein–Sato root is reasonable.

We consider the space $X=\bigwedge ^{3}\mathbb{C}^{6}$ of alternating senary 3-tensors, equipped with the natural action of the group $\operatorname{GL}_{6}$ of invertible linear transformations of $\mathbb{C}^{6}$. We describe explicitly the category of $\operatorname{GL}_{6}$-equivariant coherent ${\mathcal{D}}_{X}$-modules as the category of representations of a quiver with relations, which has finite representation type. We give a construction of the six simple equivariant ${\mathcal{D}}_{X}$-modules and give formulas for the characters of their underlying $\operatorname{GL}_{6}$-structures. We describe the (iterated) local cohomology groups with supports given by orbit closures, determining, in particular, the Lyubeznik numbers associated to the orbit closures.

We will establish a nearby and vanishing cycle formalism for the arithmetic $\mathscr{D}$-module theory following Beilinson’s philosophy. As an application, we define smooth objects in the framework of arithmetic $\mathscr{D}$-modules whose category is equivalent to the category of overconvergent isocrystals.

We determine the irregular Hodge filtration, as introduced by Sabbah, for the purely irregular hypergeometric ${\mathcal{D}}$-modules. We obtain, in particular, a formula for the irregular Hodge numbers of these systems. We use the reduction of hypergeometric systems from GKZ-systems as well as comparison results to Gauss–Manin systems of Laurent polynomials via Fourier–Laplace and Radon transformations.

Let $f$ be a quasi-homogeneous polynomial with an isolated singularity in $\mathbf{C}^{n}$. We compute the length of the ${\mathcal{D}}$-modules ${\mathcal{D}}f^{\unicode[STIX]{x1D706}}/{\mathcal{D}}f^{\unicode[STIX]{x1D706}+1}$ generated by complex powers of $f$ in terms of the Hodge filtration on the top cohomology of the Milnor fiber. When $\unicode[STIX]{x1D706}=-1$ we obtain one more than the reduced genus of the singularity ($\dim H^{n-2}(Z,{\mathcal{O}}_{Z})$ for $Z$ the exceptional fiber of a resolution of singularities). We conjecture that this holds without the quasi-homogeneous assumption. We also deduce that the quotient ${\mathcal{D}}f^{\unicode[STIX]{x1D706}}/{\mathcal{D}}f^{\unicode[STIX]{x1D706}+1}$ is nonzero when $\unicode[STIX]{x1D706}$ is a root of the $b$-function of $f$ (which Saito recently showed fails to hold in the inhomogeneous case). We obtain these results by comparing these ${\mathcal{D}}$-modules to those defined by Etingof and the second author which represent invariants under Hamiltonian flow.