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Derived categories of coherent sheaves and triangulated categories of singularities. (English) Zbl 1200.18007
Tschinkel, Yuri (ed.) et al., Algebra, arithmetic, and geometry. In honor of Y. I. Manin on the occasion of his 70th birthday. Vol. II. Boston, MA: Birkhäuser (ISBN 978-0-8176-4746-9/hbk; 978-0-8176-4747-6/ebook). Progress in Mathematics 270, 503-531 (2009).
Summary: We establish an equivalence between the category of graded D-branes of type B in Landau-Ginzburg models with homogeneous superpotential $$W$$ and the triangulated category of singularities of the fiber of $$W$$ over zero. The main result is a theorem that shows that the graded triangulated category of singularities of the cone over a projective variety is connected via a fully faithful functor to the bounded derived category of coherent sheaves on the base of the cone. This implies that the category of graded D-branes of type B in Landau-Ginzburg models with homogeneous superpotential $$W$$ is connected via a fully faithful functor to the derived category of coherent sheaves on the projective variety defined by the equation $$W=0$$.
For the entire collection see [Zbl 1185.00042].

MSC:
 18E30 Derived categories, triangulated categories (MSC2010) 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 14B05 Singularities in algebraic geometry
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