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We show that the Friedlander-Mazur conjecture holds for a sequence of products of projective varieties such as the product of a smooth projective curve and a smooth projective surface, the product of two smooth projective surfaces, the product of arbitrary number of smooth projective curves. Moreover, we show that the Friedlander-Mazur conjecture is stable under a surjective map. As applications, we show that the Friedlander-Mazur conjecture holds for the Jacobian variety of smooth projective curves, uniruled threefolds and unirational varieties up to certain range.
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For a Grothendieck category C which, via a Z-generating sequence (O(n))_{n in Z}, is equivalent to the category of quasi-coherent modules over an associated Z-algebra A, we show that under suitable cohomological conditions taking quasi-coherent modules defines an equivalence between linear deformations of A and abelian deformations of C. If (O(n))_{n in Z} is at the same time a geometric helix in the derived category, we show that restricting a (deformed) Z-algebra to a thread of objects defines a further equivalence with linear deformations of the associated matrix algebra.
In this paper, we prove that for the oper stratification of the de Rham moduli space $M_{mathrm{dR}}(X,r)$, the closed oper stratum is the unique minimal stratum with dimension $r^2(g-1)+g+1$, and the open dense stratum consisting of irreducible flat bundles with stable underlying vector bundles is the unique maximal stratum.
Using the work of Guillen and Navarro Aznar we associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on Borel-Moore homology with Z/2 coefficients an analog of the weight filtration for complex algebraic varieties.
We construct nontrivial L-equivalence between curves of genus one and degree five, and between elliptic surfaces of multisection index five. These results give the first examples of L-equivalence for curves (necessarily over non-algebraically closed fields) and provide a new bit of evidence for the conjectural relationship between L-equivalence and derived equivalence. The proof of the L-equivalence for curves is based on Kuznetsovs Homological Projective Duality for Gr(2,5), and L-equivalence is extended from genus one curves to elliptic surfaces using the Ogg--Shafarevich theory of twisting for elliptic surfaces. Finally, we apply our results to K3 surfaces and investigate when the two elliptic L-equivalent K3 surfaces we construct are isomorphic, using Neron--Severi lattices, moduli spaces of sheaves and derived equivalence. The most interesting case is that of elliptic K3 surfaces of polarization degree ten and multisection index five, where the resulting L-equivalence is new.
This paper deals with two aspects of the theory of characteristic classes of star products: first, on an arbitrary Poisson manifold, we describe Morita equivalent star products in terms of their Kontsevich classes; second, on symplectic manifolds, we describe the relationship between Kontsevichs and Fedosovs characteristic classes of star products.
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