Lecture 22 bertinis theorem, coherent sheves on curves. Txis a closed conical subset associated to f such that ssf c for c ndimensional irreducible components. Varieties over algebraically closed elds 3 these are notes from both class 39 and class 40. What is the sheaf of differentials of projective space.
How can i compute the cotangent sheaf for a projective variety. In differential geometry, one can attach to every point of a smooth or differentiable manifold, a vector space called the cotangent space at. The qtwisted sheaf f is torsionfree if fis torsionfree. A di erential, or dually, a tangent vector, should be something like the data of a point in a scheme. Math 256b notes university of california, berkeley. Our approach to object of this study is the cotangent map that is a morphism of the projectivized cotangent sheaf to the projective space of dimension q1. We next examine the differentials of projective space pn k. Computing with sheaves and sheaf cohomology in algebraic geometry.
Vector space structure on the tangent bundle of a scheme and relation to the tangent sheaf. A gentle introduction to homology, cohomology, and sheaf. Why is the cotangent space of complex projective space not. A normal projective variety over c with ample tangent sheaf is isomorphic to the projective space. For instance, using this formula it is di cult to determine whether the. Phys 500 southern illinois university oneforms and cotangent space. This causes problems when attempting to define left derived functors of a right exact. Eudml twisted cotangent sheaves and a kobayashiochiai.
We will then discuss varieties with ample cotangent bundle, and. For this we need the notion of projective space, which we introduced in projective geometry. These techniques have been developped later by nadel to produce elegant examples of hyperbolic surfaces of low degree in projective 3 space. One using taylor maps, incidence schemes, jet bundles and generalized verma modules. Y of an irreducible hermitian symmetric space y of compact type is stable. They are being developed for lectures i am giving at the arizona winter school in tucson, march 1115, 2006. If ais qcartier qdivisoron xwewrite f forthetwistof fbythenumericalclassof a. Questions about tangent and cotangent bundle on schemes.
On the characteristic cycle of an etale sheaf takeshi saito 2526 septembre 2014, a lihes. Twisted cotangent sheaves and a kobayashiochiai theorem. The cotangent sheaf on a projective space is related to the tautological line bundle o1 by the following exact sequence. Given a ringed space x, o, if f is an osubmodule of o, then it is called the sheaf of ideals or ideal sheaf of o, since for each open subset u of x, fu is an ideal of the ring ou. Space of surjective morphisms between projective varietiestalk at amc2005, singaporejunmuk hwang. The tangent bundle and projective bundle let us give the first. The definition of the sheaf is done on the base of open sets of principal open sets dp, where p varies over the set. We study here surfaces of general type where the cotangent sheaf is generated by his global sections and with an irregularity q at least equal to 4. In this talk we will focus on computing these structure constants for the cotangent bundle to projective space, first computing them directly using definitions from maulik and okounkov, and then putting forth a conjectural positive formula which uses a variant of knutsontao puzzles. This is the case, for example, when looking at the category of sheaves on projective space in the zariski topology. Twisted cotangent sheaves and a kobayashiochiai theorem for foliations article in annales institut fourier 646 july 20 with 19 reads how we measure reads. There is a standard way to construct the tangent and cotangent bundles on projective space. The sheaf of regular functions on a variety is also called the structure sheaf on and is written. Puzzles and cohomology of the cotangent bundle on projective.
Are the global sections of a vector bundle a projective module. Resolution of singularities of the cotangent sheaf of a. The basis one forms gdxk are the di erentials of the coordinate functions xkp. Geometric interpretation of the exact sequence for the. The construction shows in particular that the cotangent sheaf is quasicoherent. Let x be a delignemumford stack with quasi projective coarse moduli space and which has the resolution.
Introduction to algebraic geometry, class 24 ravi vakil contents 1. As the ber of the cotangent sheaf is canonically isomorphic to the zariski tangent space at closed points done earlier. In fact, on any smooth projective variety, the dualising sheaf is precisely the canonical sheaf. We know that quasicoherent sheaves over an a ne variety correspond to the modules over its coordinate ring. Moduli spaces of sheaves on k3 surfaces and symplectic stacks ziyu zhang abstract. On the computation of the dimensions of the cohomology groups. The above definition means that the cotangent sheaf on x is the restriction to x of the conormal sheaf to the diagonal embedding of x. We view a moduli space of semistable sheaves on a k3 surface as a global quotient stack, and compute its cotangent complex in terms of the universal sheaf on the quot scheme using the classical and reduced atiyah classes.
Drinfeld, quantization of hitchins integrable system and hecke eigensheaves. Space of surjective morphisms between projective varieties. X of a projective variety x starting from the cotangent bundle of the ambient projective space. As projective space is covered by afne open sets of the form an, on which the differential form a rank n locally free sheaf, pn kk is also a rank n locally free sheaf. The tangent complex and hochschild cohomology of e nrings john francis abstract. Apr 20, 2017 in this talk we will focus on computing these structure constants for the cotangent bundle to projective space, first computing them directly using definitions from maulik and okounkov, and then putting forth a conjectural positive formula which uses a variant of knutsontao puzzles. Relationship between tangent bundle and tangent sheaf. A is generically nef with respect to the polarisation a. Then the tangent sheaf of x is the dual of the cotangent sheaf and the canonical sheaf is the nth exterior power determinant of. In order to maximize the range of applications of the theory of manifolds it is necessary to generalize the concept. Cohomology of projective space let us calculate the cohomology of projective space. We prove a generalization of a conjecture of kontsevich, that there is a ber sequence an 1. A normal projective variety over c with ample tangent sheaf is isomorphic to the complex projective space.
Full pdf abstract top let x be a normal projective variety, and let a be an ample cartier divisor on x. Chapter 6 manifolds, tangent spaces, cotangent spaces, vector. Computing with sheaves and sheaf cohomology in algebraic. Canonical surfaces with big cotangent bundle xavier roulleau, erwan rousseau. Assume from now on that cis projective and still nonsingular. In this work, we study the deformation theory of enrings and the en analogue of the tangent complex, or topological andr equillen cohomology. The positivity of projective space is manifested in the amplitude of its tangent bundle and the normal bundles of all smooth subvarieties.
In the following application of theorem 2, the singularities of the. On the computation of the dimensions of the cohomology groups of coherent sheaves on a projective space july 14, 2015, pnu math forum. In addition, we show that smooth quasi projective varieties with. Then for a generic hyperplane h, y x\his again smooth. Show that x sys is isomorphic to the direct sum of the pullbacks of xs and ys, i. We show that certain line bundles on the cotangent bundle of a grassmannian arising from. It is coherent if s is noetherian and f is of finite type. On the computation of the dimensions of the cohomology groups of coherent sheaves on a projective space july 14, 2015, pnu math forum graduate school of mathematics, kyushu university, momonari kudo. Introduction to algebraic geometry, class 24 contents. The above definition means that the cotangent sheaf on x is the restriction to x of the conormal sheaf to the diagonal embedding of x over s. A is generically nef with respect to the polarisation a unless x is a projective space. As an application we prove a kobayashiochiai theorem for foliations. On the positivity of the logarithmic cotangent bundle. For the module of the tangent sheaf, i cant think of anything easier than just taking the dual of m.
For example, the structure sheaf o x of the subvariety xwould be represented by the module ringpn1idealx. A di erential, or dually, a tangent vector, should be something like the data of a point in a scheme, together with an \in nitesimal direction vector at that point. Let x be a normal projective variety, and let a be an ample cartier divisor on x. February 28, 2006 1 introduction these notes are still in a preliminary form. We want to construct the relative cotangent sheaf associated to a mor phism f. A gentle introduction to homology, cohomology, and sheaf cohomology jean gallier and jocelyn quaintance department of computer and information science. Manifolds, tangent spaces, cotangent spaces, vector fields, flow, integral curves 6. Twisted cotangent sheaves and a kobayashiochiai theorem for. The cotangent sheaf wont be locally free, but it will still be a quasicoher.
How can i compute the cotangent sheaf for a projective. The cotangent sheaf on a projective space is related to the tautological line bundle o1 by the following exact. Bertinis theorem, coherent sheves on curves lets consider some ways to construct smooth varieties. In this post we will give some more important kinds of sheaves. Thus the grothendieckriemannroch theorem gives a formula for the virtual canonical bundle. We show that all subvarieties of a quasi projective variety with positive log cotangent bundle are of log general type. Chapter i lays out the basic definitions of schemes, sheaves, and mor.
In the situations we are interested in fwill either be a torsionfree sheaf on a normal. Given a vector bundle e, the projective bundle pe is the space of 1 dimensional quotients of the fibers of e. Introduction in this paper we give a proof for the following theorem. One of the points in the talk is that, people accept some results but whose proofs are. Then we earlier identi ed the cotangent space to xat pwith mm2. On the computation of the dimensions of the cohomology. Lecture 14 quasicoherent sheaves on projective spaces.
Sommers to professor shoji, on the occasion of his 60th birthday. For future use we start fixing the notation we are to adopt when working on ordinary projective spaces p n. Chapter 6 manifolds, tangent spaces, cotangent spaces. Intro to algebraic geometry, problem set 12 line bundles. Exercises, algebraic geometry ii week 3 exercise 10.