Once we dene cohomology of quasicoherent sheaves, we will quickly prove that if f is a coherent sheaf on pn k, then pn. Anghel quantum sheaf cohomology on surfaces of general type needs a so called virtual fundamental class which is used to give a rigorous meaning for the cup p roduct on m x. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. The theory is presented in a way that is meant to be read and appreciated by a broad audience. On the push forwards for motivic cohomology theories with invertible stable hopf element article pdf available in manuscripta mathematica 15012. Curry march, 20 abstract this note advertises the theory of cellular sheaves and cosheaves, which are devices for conducting linear algebra parametrized by a cell complex. Lecture notes on motivic cohomology carlo mazza, vladimir voevodsky, charles a.
And we have a powerful tool known as sheaf cohomology. Look up push forward in wiktionary, the free dictionary. In section 5, we discuss derived push forward algorithms. Vanishing theorems for higher cohomology groups of the structural sheaf on certain complex topological vector spaces article in complex variables and elliptic equations 512. Cohomology groups hax, e are defined, where x is a topological space and. Fiberwise integral, the direct image of a differential form or cohomology by a smooth map, defined by integration on the fibres transfer operator, the pushforward on the space of measurable functions.
Related we have an equivalence to a category given by ext i which is just a derived functor. We first prove a simple lemma that relates pushforward of sheaves of modules for a morphism of algebraic spaces to pushforward of sheaves of modules for a morphism of schemes. A characterization of ordinary abelian varieties by the frobenius push forward of the structure sheaf ii. F into x and therefore we can consider f to be a sheaf on x. Pushforward differential, the differential of a smooth map between manifolds, and the pushforward operations it defines. Push forward of the structure sheaf along covering. But that cohomology vanishing will imply that derived categories form a stack on the acyclic site. More generally we claim that for any f, the cech complex is acyclic on v s 2i u. The notion of pushforward in mathematics is dual to the notion of pullback, and can mean a number of different but closely related things. From varieties to sheaf cohomology university of utah. The relation of this more conceptual and more general point of view on abelian sheaf cohomology to the original definition was originally clarified in.
These groups generalize the ordinary cohomology groups of x with coefficients in an abelian sheaf, as well as the. Indeed for the categories we are looking at, if we derive from the functor of global sections, we get the sheaf cohomology functor, so to determine ext i we can work. If f is a coherent sheaf on x, show that f f is a coherent sheaf. In this case, the cohomology sheaf hrs is the sheaf kerdimd. If x2xis a point, then locally xis cut out by a maximal ideal m x of functions vanishing on x. Sheaf cohomology course notes, spring 2010 the basic. Reduced, integral, noetherian schemes, mor phisms of finite type, finite morphisms, closed immersions, dimension, fibre product 4. The sheaf of holomorphic functions or holomorphic di erential forms on a complex manifold. Pdf on the pushforwards for motivic cohomology theories. Intersection cohomology sheaf as an example of a perverse sheaf. Cohomology, derived categories, and derived functors in the di erential graded setting.
It is wellknown that this functor admits a left adjoint f, which can be geometrically described in terms of the espace etale as follows. For example, if we take the constant sheaf on y then we get the result that h. In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. There is a natural way to push forward currents by duality with the pullback of smooth forms. Direct image sheaf, the pushforward of a sheaf by a map. Affine schemes, proj, structure sheaf, scheme associated to variety 3. In this thesis we study quantum sheaf cohomology of bundles on smooth projective toric varieties. In 1904 schur studied a group isomorphic to h2g,z, and this group is known as the schur multiplier of g. Fiber integration or pushforward is a process that sends generalized cohomology classes on a bundle e b e \to b of manifolds to cohomology classes on the base b b of the bundle, by evaluating them on each fiber in some sense.
Topics in quantum sheaf cohomology zhentao lu a dissertation. In fact, on any smooth projective variety, the dualising sheaf is precisely the canonical sheaf. Homology, cohomology, and sheaf cohomology university of. Equivalently, the complex of stalks at xis acyclic for all x2v.
Topics in quantum sheaf cohomology zhentao lu ron donagi quantum sheaf cohomology generalizes the theory quantum cohomology, in the sense that it deals with a class of more general sheaves rather than the tangent bundle. A result by ramanan claims that the paracompactness assumption may be removed, but ramanans proof in fact implicitly relies on the assumption of paracompactness and does not work in the stated generality. Properties of the category of perverse sheaves, including the classi cation of simples. Push forward of line bundle and of the associated divisor. How should one think about pushforward in cohomology. A categorical introduction to sheaves daping weng abstract.
Cohomology, derived categories, and derived functors in the differential graded. The sheaf of rvalued continuous functions on a topological space x, given by fu ff. X is any closed subset and if f is a sheaf on y we can push it forward i. Mathematical notes here are several short notes, written mainly for my benefit or for the benefit of my students. Sheaf cohomology on a grothendieck site github pages. Therefore, you can push forward a smooth class considering it a current class, and using then the previous natural isomorphism. Less obvious examples include the push forward fs of a complex by a continuous, smooth, or algebraic mapping f. Quantum sheaf cohomology i eric sharpe, virginia tech banff 0,2 meeting, march 812, 2010 based on hepth0406226, 0502064, 0605005, 0704. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally when it can be solved locally. Sheaves on topological spaces constructible sheaves on. Thus taking cohomology of a complex of amodules commutes.
The perverse tstructure on the constructible derived category. By the long exact sequence on cohomology this implies that hra. There are several ways to build up sheaf theory with di erent axioms. In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Algebraic groups, grassmannians, and flag varieties. X is its sheaf of rings, and an o xmodule is a sheaf which is a. Vanishing theorems for higher cohomology groups of the. Pushing around sheaves, especially by the frobenius. Characterization of ordinary abelian varieties by the. Shy, that sends a sheaf fof abelian groups on xto the push forward f fon y. These notes began with a course given at brown university in the.
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