1,861 research outputs found
An open problem in complex analytic geometry arising in harmonic analysis
In this paper, an open problem in the multidimensional complex analysis is
pesented that arises in the investigation of the regularity properties of
Fourier integral operators and in the regularity theory for hyperbolic partial
differential equations. The problem is discussed in a self-contained elementary
way and some results towards its resolution are presented. A conjecture
concerning the structure of appearing affine fibrations is formulated.Comment: 8 page
Lefschetz fibrations, complex structures and Seifert fibrations on S^1 X M^3
We consider product 4--manifolds S^1 X M, where M is a closed, connected and
oriented 3-manifold. We prove that if S^1 X M admits a complex structure or a
Lefschetz or Seifert fibration, then the following statement is true: S^1 X M
admits a symplectic structure if and only if M fibers over S^1, under the
additional assumption that M has no fake 3-cells. We also discuss the
relationship between the geometry of M and complex structures and Seifert
fibrations on S^1 X M.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-24.abs.htm
Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems
Let X -> Y be a fibration whose fibers are complete intersections of two
quadrics. We develop new categorical and algebraic tools---a theory of relative
homological projective duality and the Morita invariance of the even Clifford
algebra under quadric reduction by hyperbolic splitting---to study
semiorthogonal decompositions of the bounded derived category of X. Together
with new results in the theory of quadratic forms, we apply these tools in the
case where X -> Y has relative dimension 1, 2, or 3, in which case the fibers
are curves of genus 1, Del Pezzo surfaces of degree 4, or Fano threefolds,
respectively. In the latter two cases, if Y is the projective line over an
algebraically closed field of characteristic zero, we relate rationality
questions to categorical representability of X.Comment: 43 pages, changes made and some material added and corrected in
sections 1, 4, and 5; this is the final version accepted for publication at
Journal de Math\'ematiques Pures et Appliqu\'ee
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