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.

For $G = \mathrm {GL}_2, \mathrm {SL}_2, \mathrm {PGL}_2$ we compute the intersection E-polynomials and the intersection Poincaré polynomials of the G-character variety of a compact Riemann surface C and of the moduli space of G-Higgs bundles on C of degree zero. We derive several results concerning the P=W conjectures for these singular moduli spaces.

We extend the infinitesimal Torelli theorem for smooth hypersurfaces to nodal hypersurfaces.

Given an $n$-dimensional variety $Z$ with rational singularities, we conjecture that if $f:Y\rightarrow Z$ is a resolution of singularities whose reduced exceptional divisor $E$ has simple normal crossings, then

$$\begin{eqnarray}\displaystyle R^{n-1}f_{\ast }\unicode[STIX]{x1D6FA}_{Y}(\log E)=0. & & \displaystyle \nonumber\end{eqnarray}$$

$Z$

$D$

$n$

$X$

$\mathscr{O}_{X}(\ast D)$

$n-3$

The dimensions of the graded quotients of the cohomology of a plane curve complement $U\,=\,{{\mathbb{P}}^{2}}\,\backslash \,C$ with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail. We also give a precise numerical estimate for the difference between the Hodge filtration and the pole order filtration on ${{H}^{2}}\left( U,\,\mathbb{C} \right)$.

Using a recent result by S. Papadima and A. Suciu, we show that the equivariant Poincaré– Deligne polynomial of the Milnor fiber of a projective line arrangement having only double and triple points is combinatorially determined.

Given a mixed Hodge module $\mathcal{N}$ and a meromorphic function $f$ on a complex manifold, we associate to these data a filtration (the irregular Hodge filtration) on the exponentially twisted holonomic module $\mathcal{N}\otimes \mathcal{E}^{f}$, which extends the construction of Esnault et al. ($E_{1}$-degeneration of the irregular Hodge filtration (with an appendix by Saito), J. reine angew. Math. (2015), doi:10.1515/crelle-2014-0118). We show the strictness of the push-forward filtered ${\mathcal{D}}$-module through any projective morphism ${\it\pi}:X\rightarrow Y$, by using the theory of mixed twistor ${\mathcal{D}}$-modules of Mochizuki. We consider the example of the rescaling of a regular function $f$, which leads to an expression of the irregular Hodge filtration of the Laplace transform of the Gauss–Manin systems of $f$ in terms of the Harder–Narasimhan filtration of the Kontsevich bundles associated with $f$.

Consider a smooth quasi-projective variety $X$ equipped with a $\mathbb{C}^{\ast }$-action, and a regular function $f:X\rightarrow \mathbb{C}$ which is $\mathbb{C}^{\ast }$-equivariant with respect to a positive weight action on the base. We prove the purity of the mixed Hodge structure and the hard Lefschetz theorem on the cohomology of the vanishing cycle complex of $f$ on proper components of the critical locus of $f$, generalizing a result of Steenbrink for isolated quasi-homogeneous singularities. Building on work by Kontsevich and Soibelman, Nagao, and Efimov, we use this result to prove the quantum positivity conjecture for cluster mutations for all quivers admitting a positively graded nondegenerate potential. We deduce quantum positivity for all quivers of rank at most 4; quivers with nondegenerate potential admitting a cut; and quivers with potential associated to triangulations of surfaces with marked points and nonempty boundary.

We endow certain GKZ-hypergeometric systems with a natural structure of a mixed Hodge module, which is compatible with the mixed Hodge module structure on the Gauß–Manin system of an associated family of Laurent polynomials. As an application we show that the underlying perverse sheaf of a GKZ-system with rational parameter has quasi-unipotent local monodromy.

The order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.

It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over ℚ, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.

We construct a hyperplane arrangement defined over ℚ, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

Let X be an irreducible complex analytic space with j: U ↪ X an immersion of a smooth Zariski-open subset, and let 𝕍 be a variation of Hodge structure of weight n over U. Assume that X is compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent, IHk (X, 𝕍) is known to carry a pure Hodge structure of weight k+n, while Hk (U, 𝕍) carries a mixed Hodge structure of weight at least k +n. In this note it is shown that the image of the natural map IHk (X, 𝕍) → Hk (U, 𝕍) is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complement X — U is not a hypersurface.