4,372 research outputs found
Differential Chow Form for Projective Differential Variety
In this paper, a generic intersection theorem in projective differential
algebraic geometry is presented. Precisely, the intersection of an irreducible
projective differential variety of dimension d>0 and order h with a generic
projective differential hyperplane is shown to be an irreducible projective
differential variety of dimension d-1 and order h. Based on the generic
intersection theorem, the Chow form for an irreducible projective differential
variety is defined and most of the properties of the differential Chow form in
affine differential case are established for its projective differential
counterpart. Finally, we apply the differential Chow form to a result of linear
dependence over projective varieties given by Kolchin.Comment: 17 page
Semipurity of tempered Deligne cohomology
In this paper we define the formal and tempered Deligne cohomology groups,
that are obtained by applying the Deligne complex functor to the complexes of
formal differential forms and tempered currents respectively. We then prove the
existence of a duality between them, a vanishing theorem for the former and a
semipurity property for the latter. The motivation of these results comes from
the study of covariant arithmetic Chow groups. The semi-purity property of
tempered Deligne cohomology implies, in particular, that several definitions of
covariant arithmetic Chow groups agree for projective arithmetic varieties
Rota-Baxter algebras, singular hypersurfaces, and renormalization on Kausz compactifications
We consider Rota-Baxter algebras of meromorphic forms with poles along a
(singular) hypersurface in a smooth projective variety and the associated
Birkhoff factorization for algebra homomorphisms from a commutative Hopf
algebra. In the case of a normal crossings divisor, the Rota-Baxter structure
simplifies considerably and the factorization becomes a simple pole
subtraction. We apply this formalism to the unrenormalized momentum space
Feynman amplitudes, viewed as (divergent) integrals in the complement of the
determinant hypersurface. We lift the integral to the Kausz compactification of
the general linear group, whose boundary divisor is normal crossings. We show
that the Kausz compactification is a Tate motive and that the boundary divisor
and the divisor that contains the boundary of the chain of integration are
mixed Tate configurations. The regularization of the integrals that we obtain
differs from the usual renormalization of physical Feynman amplitudes, and in
particular it may give mixed Tate periods in some cases that have non-mixed
Tate contributions when computed with other renormalization methods.Comment: 35 pages, LaTe
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