The present issue of Compositio Mathematica is dedicated to Jozef Steenbrink on the occasion of his sixtieth birthday.

We prove the following theorem characterizing Du Bois singularities. Suppose that $Y$ is smooth and that $X$ is a reduced closed subscheme. Let $\pi : \tilde{Y} \rightarrow Y$ be a log resolution of $X$ in $Y$ that is an isomorphism outside of $X$. If $E$ is the reduced pre-image of $X$ in $\tilde{Y}$, then $X$ has Du Bois singularities if and only if the natural map $\mathcal{O}_X \rightarrow R \pi_* \mathcal{O}_E$ is a quasi-isomorphism. We also deduce Kollár's conjecture that log canonical singularities are Du Bois in the special case of a local complete intersection and prove other results related to adjunction.

The purpose of this paper is to develop the theory of equisingular deformations of plane curve singularities in arbitrary characteristic. We study equisingular deformations of the parametrization and of the equation and show that the base space of its semiuniversal deformation is smooth in both cases. Our approach through deformations of the parametrization is elementary and we show that equisingular deformations of the parametrization form a linear subfunctor of all deformations of the parametrization. This gives additional information, even in characteristic zero, the case which was treated by J. Wahl. The methods and proofs extend easily to good characteristic, that is, when the characteristic does not divide the multiplicity of any branch of the singularity. In bad characteristic, however, new phenomena occur and we are naturally led to consider weakly trivial (respectively, weakly equisingular) deformations, that is, deformations which become trivial (respectively, equisingular) after a finite and dominant base change. The semiuniversal base space for weakly equisingular deformations is, in general, not smooth but becomes smooth after a finite and purely inseparable base extension. The proof of this fact is more complicated and we introduce new constructions which may have further applications in the theory of singularities in positive characteristic.

We present relations between cycles with rational coefficients modulo algebraic equivalence on the Jacobian of a curve. These relations depend on the linear systems that the curve admits. They are obtained in the tautological ring, the smallest subspace containing (an embedding of) the curve and closed under the basic operations of intersection, Pontryagin product and the pullback and pushdown induced by homotheties.

By using the Grothendieck–Riemann–Roch theorem we derive cycle relations modulo algebraic equivalence in the Jacobian of a curve. The relations generalize the relations found by Colombo and van Geemen and are analogous to but simpler than the relations recently found by Herbaut. In an appendix by Zagier, it is shown that these sets of relations are equivalent.

The morphic Abel–Jacobi map is the analogue of the classical Abel–Jacobi map one obtains by using Lawson and morphic (co)homology in place of the usual singular (co)homology. It thus gives a map from the group of $r$-cycles on a complex variety that are algebraically equivalent to zero to a certain ‘Jacobian’ built from the Lawson homology groups viewed as inductive limits of mixed Hodge structures. In this paper, we define the morphic Abel–Jacobi map, establish its foundational properties, and then apply these results to the study of algebraic cycles. In particular, we show the classical Abel–Jacobi map (when restricted to cycles algebraically equivalent to zero) factors through the morphic version, and show that the morphic version detects cycles that cannot be detected by its classical counterpart; that is, we give examples of cycles in the kernel of the classical Abel–Jacobi map that are not in the kernel of the morphic version. We also investigate the behavior of the morphic Abel–Jacobi map on the torsion subgroup of the Chow group of cycles algebraically equivalent to zero modulo rational equivalence.

This paper addresses several questions related to the Hodge conjecture. First of all we consider the question, asked by Maillot and Soulé, whether the Hodge conjecture can be reduced to the case of varieties defined over number fields. We show that this is the case for the Hodge classes whose corresponding Hodge locus is defined over a number field. We also give simple criteria for this last condition to be satisfied. Finally we discuss the relation between this condition and the notion of absolute Hodge class.

Applying the main results of the author's previous work (J. Wildeshaus, The boundary motive: definition and basic properties, Compositio Math. 142 (2006), 631–656), we identify the motives occurring as cones in the co-localization and localization filtrations of the boundary motive of a Shimura variety.

We compute the rational cohomology of the moduli space of non-singular complex plane quartic curves with two marked points. This allows us to calculate the rational cohomology of the moduli space of non-singular complex projective curves of genus three with two marked points.

We recover the Newton diagram (modulo a natural ambiguity) from the link for any surface hypersurface singularity with non-degenerate Newton principal part whose link is a rational homology sphere. As a corollary, we show that the link determines the embedded topological type, the Milnor fibration, and the multiplicity of such a germ. This proves (an even stronger version of) Zariski's conjecture about the multiplicity for such a singularity.

Allcock, Carlson and Toledo defined a period map for cubic threefolds which takes values in a ball quotient of dimension 10. A theorem of Voisin implies that this is an open embedding. We determine its image and show that on the algebraic level this amounts to identification of the algebra of $\operatorname{SL}(5,\mathbb{C})$-invariant polynomials on the representation space $\operatorname{Sym}^3(\mathbb{C}^5)^*$ with an explicitly described algebra of meromorphic automorphic forms on the complex 10-ball.

We prove that certain roots of the Bernstein–Sato polynomial (i.e. $b$-function) are jumping coefficients up to a sign, showing a partial converse of a theorem of L. Ein, R. Lazarsfeld, K. E. Smith, and D. Varolin. We also prove that certain roots are determined by a filtration on the Milnor cohomology, generalizing a theorem of B. Malgrange in the isolated singularity case. This implies a certain relation with the spectrum which is determined by the Hodge filtration, because the above filtration is related to the pole order filtration. For multiplier ideals we prove an explicit formula in the case of locally conical divisors along a stratification, generalizing a formula of Mustaţ̌a in the case of hyperplane arrangements. We also give another proof of a formula of U. Walther on the $b$-function of a generic hyperplane arrangement, including the multiplicity of $-1$.

We show that a line arrangement in the complex projective plane supports a nontrivial resonance variety if and only if it is the underlying arrangement of a ‘multinet’, a multi-arrangement with a partition into three or more equinumerous classes which have equal multiplicities at each inter-class intersection point, and satisfy a connectivity condition. We also prove that this combinatorial structure is equivalent to the existence of a pencil of plane curves, also satisfying a connectivity condition, whose singular fibers include at least three products of lines, which comprise the arrangement. We derive numerical conditions which impose restrictions on the number of classes, and the line and point multiplicities that can appear in multinets, and allow us to detect whether the associated pencils yield nonlinear fiberings of the complement.