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We initiate and develop a framework to handle the specialisation morphism as a filtered morphism for the perverse, and for the perverse Leray filtration, on the cohomology with constructible coefficients of varieties and morphisms parameterised by a curve. As an application, we use this framework to carry out a detailed study of filtered specialisation for the Hitchin morphisms associated with the compactification of Dolbeault moduli spaces in [de 2018].
We prove that the direct image complex for the
Hitchin fibration is determined by its restriction to the elliptic locus, where the spectral curves are integral. The analogous result for
is due to Chaudouard and Laumon. Along the way, we prove that the Tate module of the relative Prym group scheme is polarizable, and we also prove
-regularity results for some auxiliary weak abelian fibrations.
Mark Andrea A. de Cataldo, Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794, USA, firstname.lastname@example.org,
Luca Migliorini, Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy, email@example.com
The former is to give an introduction to our earlier work and more generally to some of the main themes of the theory of perverse sheaves and to some of its geometric applications. Particular emphasis is put on the topological properties of algebraic maps.
The latter is to prove a motivic version of the decomposition theorem for the resolution of a threefold Y. This result allows to define a pure motive whose Betti realization is the intersection cohomology of Y.
We assume familiarity with Hodge theory and with the formalism of derived categories. On the other hand, we provide a few explicit computations of perverse truncations and intersection cohomology complexes which we could not find in the literature and which may be helpful to understand the machinery. We discuss in detail the case of surfaces, threefolds and fourfolds. In the surface case, our “intersection forms” version of the decomposition theorem stems quite naturally from two well-known and widely used theorems on surfaces, the Grauert contractibility criterion for curves on a surface and the so called “Zariski Lemma,” cf.
The following assumptions are made throughout the paper
Assumption 3.1.1.We work with varieties over the complex numbers. A map f : X → Y is a proper morphism of varieties. We assume that X is smooth. All (co)homology groups are with rational coefficients.
By means of an ad hoc modification of the so-called “Castelnuovo-Harris analysis” we derive an upper bound for the genus of integral curves on the three dimensional nonsingular quadric which lie on an integral surface of degree 2/c, as a function of k and the degree d of the curve. In order to obtain this we revisit the Uniform Position Principle to make its use computation-free. The curves which achieve this bound can be conveniently characterized.
We make precise the structure of the first two reduction morphisms associated with codimension two non-singular subvarieties of non-singular quadrics Qn, n ≥ 5. We give a coarse classification of the same class of subvarieties when they are assumed not to be of log-general-type.
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