Università degli studi di Modena e Reggio Emilia

We investigate effectiveness and ampleness of adjoint divisors of the form aL + bKX, where L is a suitably positive line bundle on a smooth projective variety X and a, b are positive integers.

We compute the rational cohomology groups of the smooth Brill–Noether varieties Gr d .C /, parametrizing linear series of degree d and dimension exactly r on a general curve C. As an application, we determine the whole intersection cohomology of the singular Brill–Noether loci W r d .C /, parametrizing complete linear series on C of degree d and dimension at least r.

We prove several results concerning the intersection cohomology and the perverse filtration associated with a Lagrangian fibration of an irreducible symplectic variety. We first show that the perverse numbers only depend on the deformation equivalence class of the ambient variety. Then we compute the border of the perverse diamond, which further yields a complete description of the intersection cohomology of the Lagrangian base and the invariant cohomology classes of the fibers. Lastly, we identify the perverse and Hodge numbers of intersection cohomology when the irreducible symplectic variety admits a symplectic resolution. These results generalize some earlier work by the second and third authors in the nonsingular case.

We establish P=W and PI=WI conjectures for character varieties with structural group GLn and SLn which admit a symplectic resolution, i.e., for genus 1 and arbitrary rank, and genus 2 and rank 2. We formulate the P=W conjecture for a resolution, and prove it for symplectic resolutions. We exploit the topology of birational and quasi-étale modifications of Dolbeault moduli spaces of Higgs bundles. To this end, we prove auxiliary results of independent interest, like the construction of a relative compactification of the Hodge moduli space for reductive algebraic groups, and the projectivity of the compactification of the de Rham moduli space. In particular, we study in detail a Dolbeault moduli space which is a specialization of the singular irreducible holomorphic symplectic variety of type O’Grady 6. Résumé (Les conjectures P=W pour les variétés de caractères ayant une résolution symplectique) On établit les conjectures P=W et PI=WI pour les variétés de caractères avec groupe structurel GLn et SLn qui admettent une résolution symplectique, c’est-à-dire pour le genre 1 en rang arbitraire, et le genre 2 en rang 2. On formule la conjecture P=W pour une résolution et on la prouve pour les résolutions symplectiques. Pour la démonstration on fait appel à la topologie des modifications birationnelles et quasi-étales des espaces de modules de fibrés de Higgs. Pour cela, on démontre des résultats auxiliaires d’intérêt indépendant, comme la construction d’une compactification relative de l’espace de modules de Hodge pour les groupes algébriques réductifs, ou la théorie de l’intersection de certains cycles lagrangiens singuliers. En particulier, on étudie en détail un espace de modules des fibrés de Higgs qui est une spécialisation de la variété symplectique holomorphe irréductible singulière de type O’Grady 6.

Consider a family of integral complex locally planar curves. We show that under some assumptions on the base, the relative nested Hilbert scheme is smooth. In this case, the decomposition theorem of Beilinson, Bernstein and Deligne asserts that the pushforward of the constant sheaf on the relative nested Hilbert scheme splits as a direct sum of shifted semisimple perverse sheaves. We will show that no summand is supported in positive codimension.