Fibrations of flux backgrounds by supersymmetric cycles are investigated. For an internal sixmanifold M with static SU(2) structure and mirror M̂ , it is argued that the product M×M̂ is doubly fibered by supersymmetric three-tori, with both sets of fibers transverse to M and M̂ . The mirror map is then realized by T-dualizing the fibers. Mirror-symmetric properties of the fluxes, both geometric an...

In this paper it is proved that the volumes of the moduli spaces of polarized Calabi-Yau manifolds with respect to Weil-Petersson metrics are rational numbers. Mumford introduce the notion of a good metric on vector bundle over a quasi-projective variety in [10]. He proved that the Chern forms of good metrics define classes of cohomology with integer coefficients on the compactified quasi-proje...

In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. The method of rigidity was modified to work over general commutative base rings in our paper [YZ5]. In the present paper we obtain many of the important l...

Let f : X → Y be a Cohen-Macaulay map of finite type between Noetherian schemes, and g : Y ′ → Y a base change map, with Y ′ Noetherian. Let f ′ : X → Y ′ be the base change of f under g and g : X → X the base change of g under f . We show that there is a canonical isomorphism θ g : g ∗ ωf ≃ ωf ′ , where ωf and ωf ′ are the relative dualizing sheaves. The map θ g f is easily described when f is...

Abstract Let X be a semistable curve and L line bundle whose multidegree is uniform, i.e., in the range between those of structure sheaf dualizing . We establish an upper bound for $$h^0(X,L)$$ h 0 ( X , L</mml:m...

Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical counterparts, these dimensions rarely come with practical and robust criteria for finiteness, even over commutative noetherian local rings. Indeed, over such a ring ...

We introduce a flexible and graphically intuitive framework that constructs complex quantum error correction codes from simple or states, generalizing code concatenation. More specifically, we represent the constructions as tensor networks built tensors of states in modular fashion. Using set local moves known operator pushing, one can derive properties more codes, such transversal non-Clifford...