We develop linear discretization of complex analysis, originally introduced by R. Isaacs, J. Ferrand, R. Duffin, and C. Mercat. We prove convergence of discrete period matrices and discrete Abelian integrals to their continuous counterparts. We also prove a discrete counterpart of the Riemann–Roch theorem. The proofs use energy estimates inspired by electrical networks.
We define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective variety X, a graded vector space which controls the restriction morphism from the Hochschild cohomology of X to the Hochschild cohomology of the orthogonal complement of this admissible subcategory. When the subcategory is generated by an exceptional collection, we define a new invariant (the height) and show that the orthogonal to an exceptional collection of height h in the derived category of a smooth projective variety X has the same Hochschild cohomology as X in degrees up to h - 2. We use this to describe the second Hochschild cohomology of quasiphantom categories in the derived categories of some surfaces of general type. We also give necessary and sufficient conditions for the fullness of an exceptional collection in terms of its height and of its normal Hochschild cohomology.
Мы строим исчисление Шуберта для разрешений Ботта-Самельсона в кольце алгебраических кобордизмов многообразия полных флагов G/B.
We show that infinitely many Gorenstein weakly-exceptional quotient singularities exist in all dimensions, we prove a weak-exceptionality criterion for five-dimensional quotient singularities, and we find a sufficient condition for being weakly-exceptional for six-dimensional quotient singularities. The proof is naturally linked to various classical geometrical constructions related to subvarieties of small degree in projective spaces, in particular Bordiga surfaces and Bordiga threefolds.