They discuss the riemannroch theorem and serre duality. Precise connection between poincare duality and serre duality. Complex manifolds and kahler geometry prof joyce 16 mt. We construct a complex manifold x, dim x 3, which is an increasing union of 1, 1 convexconcave open subsets having the same fixed convex boundary, and a holomorphic line bundle l on x, such. Well first of all its just very useful for computations. An version of the serre duality on domains in complex manifolds involving duality of hilbert space realizations of the operator is established. Let x be a complex manifold with finitely many ends such that each end is either qconcave or nqconvex. Special cases of serre duality using l received by the editors june 15, 2010. Because of the adjoint property of f 1, we have a morphismf 1o y. Some examples of complete cr manifolds are presented. Hodge and lefschetz decompositions on cohomology, kodaira. A complex manifold is a differentiable manifold endowed with the additional datum of a complex structure which is much more rigid than the geometrical structures in differential geometry. This duality is used to study the solution of the equation with prescribed support. Riemannroch for compact complex manifolds of arbitrary dimensions.
This course will be an introduction to the basic theory of riemann surfaces. In algebraic geometry, a branch of mathematics, serre duality is a duality present on nonsingular projective algebraic varieties v of dimension n and in greater generality for vector bundles and further, for coherent sheaves. I have heard tangential statements from some people that there is indeed some connection between the two. Simms school of mathematics, trinity college, dublin, ireland received october 4, 1976 let f be a polarisation, in the sense of kostant, of a 2ndimensional symplectic manifold m, and let l be a complex line bundle with flat connection along f. To translate this statement into mathematical rigor, we just take the direct limit over all neighborhoods of x. A fundamental result in the theory of co mplex manifolds is serres duality theorem.
Vector bundles on riemann surfaces sabin cautis contents 1. In this note we extend kodairas method to noncompact hermitian manifolds to obtain an l 2analog of the serre duality. It shows that a cohomology group h i is the dual space of another one, h n. If q co mplex manifolds is serre s duality theorem. Due to this rigidity, one is often able to describe the geometry of complex manifolds in very explicit terms. Complex manifolds lecture notes, winter term 201819 janko b ohm february 17, 2019. People had versions of serre duality on complex compact manifolds before serre in 55 gave the modern statement of the theorem. This generalizes to suitable nonsingular projective algebraic varieties over other base rings. Grothendieck extends sheaf theory in line with the needs of algebraic geometry, introducing. Serre duality in complex analytic geometry is the duality induced by the hodge star operator on the dolbeault complex. It is based on earlier works in several complex variables, however, we will observe it algebraically. Complex manifolds and deformation of complex structures, springer, 1986. It is proved that the hodge decomposition and serre duality hold on a noncompact weighted cr manifold with negligible boundary. This duality induces the generalized serre functor on t, which is a linear triangle equivalence between two thick triangulated subcategories of t.
Special cases of complex manifolds include riemann surfaces and smooth algebraic varieties, so the theory of complex manifolds is much more rigid than di erential geometry, and the tools used are quite di erent. Applying the methods of serre duality in the setting of cr manifolds we prove approximation theorems and we study the hartogsbochner phenomenon in 1concave cr generic manifolds. Abstract let f be a polarisation, in the sense of kostant, of a 2 ndimensional symplectic manifold m, and let l be a. Lecture 25 proof of serre duality mit opencourseware. Carleman estimates for the laplacebeltrami equation on complex manifolds,inst. The serre duality theorem for holomorphic vector bundles over a strongly pseudoconvex manifold. Hermitian metrics, connections, curvature and chern classes for complex vector bundles.
Serre duality is a special case of the duality called the coherent duality in a much general setting. I am wondering if there is some relationship between serre duality and pontryagin duality for compact complex manifolds. In fact, you do not need the hodge theorem, since the hodge theorem itself is a corollary of the theorem. The serre duality for holomorphic vector bundles over strongly pseudo convex cr manifolds 19 we call a s. The riemannroch theorem and serre duality 3 locally around x. In fact the serre duality is a kind of corollary of thefundamental theorem as i know it. Then on an arbitrary projective scheme x, we show that there is a coherent sheaf. We will cover fundamental theorems such as the uniformization theorem, the riemannroch theorem, serre duality, the torelli theorem and the brillnoether theorem. Finiteness and duality on complex symplectic manifolds. Complex manifolds example sheet 3 the two questions marked can be handed in to be marked. Special cases of serreduality using l received by the editors june 15, 2010. Citeseerx document details isaac councill, lee giles, pradeep teregowda.
Ext groups of sheaves and sheaf cohomology in the context of duality theorems. Foundations of grothendieck duality for diagrams of schemes, 1259, lecture notes in math. This establishes a duality between the cohomology of a complex manifold. Introduction letm beastronglypseudoconvexcrmanifold,ans.
Namely, i believe although, math history is not my strong suit that this treatment of serre duality is a bit anachronistic. Joseph lipman, notes on derived functors and grothendieck duality, in. In this case serre duality reduces to the commutativity of hodgestar operator with the laplace operator. The serre duality for holomorphic vector bundles over. Let x x be a hermitian manifold of complex dimension dim. Complex manifold duality theorem these keywords were added by machine and not by the authors. Brian conrad, grothendieck duality and base change, springer lec. The statements of poincare duality for manifolds and serre duality for coherent sheaves on algebraic varieties or analytic spaces look tantalizingly similar. We will call this set of data the generalized serre duality on c. Complex manifolds l24 ruadha dervan complex manifolds are the holomorphic analogue of smooth manifolds. This process is experimental and the keywords may be updated as the learning algorithm improves. They roughly thought about serre duality as giving a sort of pairing. A gentle introduction to homology, cohomology, and sheaf.
Cpn cannot be expressed nontrivially as a product of complex manifolds. Very often you want to compute some cohomology group, and then serre duality tells you its the same as another cohomology group thats perhaps easier to compute. Observe that the category chas serre duality in the sense of 7 if and only if r c l. Finiteness and duality on complex symplectic manifolds pierre schapira abstract for a complex compact manifold x, denote by tthe category db cohox. We introduce a notion of generalized serre duality on a hom. If the variety is defined over the complex numbers, this yields different information. It relates sheaf cohomology groups on nonsingular projective algebraic varieties. A separation theorem and serre duality for the dolbeault. A gentle introduction to homology, cohomology, and sheaf cohomology jean gallier and jocelyn quaintance department of computer and information science. Ams transactions of the american mathematical society. A duality theorem for complex manifolds springerlink. In this setting, the serre duality theorem is an application of hodge theory for dolbeault cohomology, and may be seen as a result in the theory of elliptic operators. Aaschool of mathematics, trinity college, dublin, ireland. Citeseerx some applications of serre duality in cr.
A separation theorem and serre duality for the dolbeault cohomology christine laurentthi6baut and jfirgen leiterer abstract. The main point is that one has to be particularly careful and always to differentiate cohomology with. The usual proof for compact complex manifolds uses hodge theory. A separation theorem and serre duality for the dolbeault cohomology. The serre duality theorem is also true in complex geometry more generally, for compact complex manifolds that are not necessarily projective complex algebraic varieties. This category is a ctriangulated category, this category is ext. In this paper we study serre duality in cr manifolds to get some approximation theorem and a better understanding of the hartogsbochner. Because of the lack of the dolbeault isomorphism for the. Serre duality and applications 3 furthermore, if g is a sheaf of o ymodules, then f 1g is a f 1o ymodule. To prove the result we first have to study serre duality for the case of more general families of supports instead of the family of all closed sets and the family of all compact sets which is the main part of the paper.
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