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Register a new userGrothendieck--Lefschetz for ample subvarieties
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We establish a Grothendieck--Lefschetz theorem for smooth ample subvarieties of smooth projective varieties over an algebraically closed field of characteristic zero and, more generally, for smooth subvarieties whose complement has small cohomological dimension. A weaker statement is also proved in a more general context and in all characteristics. Several applications are included.
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Under some positivity assumptions, extension properties of rationally connected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold X inducing a covering family on a submanifold Y with ample normal bundle in X, the main results relate, under suitable conditions, the associated rational connected fiber structures on X and on Y. Applications of these results include an extension theorem for Mori contractions of fiber type and a classification theorem in the case Y has a structure of projective bundle or quadric fibration.
Using deformation theory of rational curves, we prove a conjecture of Sommese on the extendability of morphisms from ample subvarieties when the morphism is a smooth (or mildly singular) fibration with rationally connected fibers. We apply this result in the context of Fano fibrations and prove a classification theorem for projective bundle and quadric fibration structures on ample subvarieties.
We prove a trace formula in stable motivic homotopy theory over a general base scheme, equating the trace of an endomorphism of a smooth proper scheme with the Euler characteristic integral of a certain cohomotopy class over its scheme of fixed points. When the base is a field and the fixed points are etale, we compute this integral in terms of Morels identification of the ring of endomorphisms of the motivic sphere spectrum with the Grothendieck-Witt ring. In particular, we show that the Euler characteristic of an etale algebra corresponds to the class of its trace form in the Grothendieck-Witt ring.
A field $K$ is called ample if for every geometrically integral $K$-variety $V$ with a smooth $K$-point, $V(K)$ is Zariski-dense in $V$. A field $K$ is virtually ample if some finite extension of $K$ is ample. We prove that there exists a virtually ample field that is not ample.
We announce recent results on a connection between factorization statistics of polynomials over a finite field and the structure of the cohomology of configurations in $mathbb{R}^3$ as a representation of the symmetric group. This connection parallels a result of Church, Ellenberg, and Farb relating factorization statistics of squarefree polynomials and the cohomology of configurations in $mathbb{R}^2$.
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