Complex Geometry : Introduction to holomorphic line bundles

1. Complex Geometry :Introduction to holomorphic line bundles 20110158 Sungkyung Kang

2. Preliminaries • Basic homological algebra • Sheaf cohomology • Cechcohomology

3. Contents • Complex manifolds: definitions • Holomorphic vector bundles • Divisors and line bundles • Calculation of line bundles on .

4. Complex Manifolds

5. Complex Manifolds • A holomorphic atlas on a differentiable manifold is an atlas of the form , such that the transition functions are holomorphic. • A complex manifold of dimension is a real -manifold with an equivalence class of holomorphic atlases.

6. Complex Manifolds • A holomorphic function on a complex manifold is a function , such that is holomorphic for each local coordinate . • By we denote the sheaf of holomorphic functions on . • A continuous map is (bi)holomorphic if for any holomorphic charts of and , the induced map is (bi)holomorphic.

7. Complex Manifolds • A meromorphic function on a complex manifold is a map which associates to any an element in such that for any there exists an open nbdand two holomorphic functions with for all . • The field of meromorphic functions called “the function field of ”.

8. Complex Manifolds • Prop. (Siegel) Let be a compact connected complex manifold of dimension . Then . • The algebraic dimension of a compact connected manifold is .

9. Complex Manifolds • Let be a complex manifold of complex dimension and be a real submanifold in of real dimension is a complex submanifold if there exists a holomorphic atlas of such that . • A complex manifold is projective if it can be embedded in some complex projective space.

10. Complex Manifolds • Let be a complex manifold. An analytic subvariety of is a closed subset such that for any point in there exists an open nbdcontaining such that is a zero set of a finitely generated ideal of . A point in is a smooth point of if a generating set of can be chosen so that is a regular point of the holomorphic map is holomorphic. • A point is singular if it is not regular.

11. Complex Manifolds • An analytic subvariety in a neighbourhood of a regular point is a complex submanifold. • The set of regular points is also a complex submanifold. • An analytic subvarietyis irreducible if it cannot be written as a union of two proper analytic subvarieties. • A dimension of an irreducible analytic subvarietyis defined by . • A hypersurface is an analytic subvariety of codimension one.

12. Holomorphic Vector Bundles

13. Holomorphic Vector Bundles • Let be a complex manifold. A holomorphic vector bundle of rank on is a complex manifold together with a holomorphic surjection and a structure of an -dimensional -vector space on each fiber of satisfying: • There exists an open covering of and biholomorphic maps commuting with the projections to such that the induced map is -linear.

14. Holomorphic Vector Bundles • Note that the induced transition functions is -linear for each in . • A vector bundle homomorphism is a biholomorphism between two vector bundles over the same complex manifold such that the induced maps over the attached vector spaces have the same rank.

15. Holomorphic Vector Bundles • A holomorphic rank vector bundle is determined by the holomorphic cocycle, thus by an element in the 1stCechcohomology. • Note that the 1stCechcohomology is canonically isomorphic to the 1st sheaf cohomology.

16. Holomorphic Vector Bundles • Meta-theorem. Any canonical construction in linear algebra gives rise to a geometric version for holomorphic vector bundles. • Direct sum • Tensor product • Exterior product, Symmetric product • Dual bundle • Determinant line bundle; , where is a holomorphic vector bundle of rank precisely • Projectivization (explained later)

17. Holomorphic Vector Bundles • Meta-theorem. Any canonical construction in linear algebra gives rise to a geometric version for holomorphic vector bundles. • Kernel • Cokernel • Short exact sequence • Pullback (explained later)

18. Holomorphic Vector Bundles • The map that maps to is the zero section. • The quotient is a complex manifold that admits a holomorphic projection such that is isomorphic to . • : the projectivization of .

19. Holomorphic Vector Bundles • Let be a holomorphic map between complex mfds and let be a holo vector bundle on given by a cocycle. • Then the pullback is the holo vector bundle on that is given by the cocycle. For any there is a canonical isomorphism . • If is a complex submfd of and is the inclusion, then is the restriction of to .

20. Holomorphic Vector Bundles • Prop. The set that consists of all with forms a holomorphic vector bundle over . • The line bundle is the dual of . • For let be the tensor product of copies of . • Similar for . • We denote the trivial line bundle by .

21. Holomorphic Vector Bundles • Prop. The isomclasses of holo line bundles on a cplxmfdwith tensor product and dualization operation forms an abelian group. • This group is the Picard group of . • Proof: We only need to show that is trivial. This can be easily seen by the cocycle description of and the induced one for .

22. Holomorphic Vector Bundles • Cor. The Picard group is naturally isomorphic to . • Cor. Let be holomorphic. Then is a group homfrom to .This map is induced by the sheaf hom.

23. Holomorphic Vector Bundles • The exponential sequence on a cplxmfdis the short exact seq. Here, is the locally constant sheaf and is the natural inclusion. The map is given by the exponential . • By snake lemma, we get an induced long exact cohomology sequence . • Note that if is compact, the first map is injective.(since any holo map on a compact cplxmfd is const)

24. Holomorphic Vector Bundles • Thus, can be computed by the 1st and 2ndcohomology groups of and , and the induced maps between them. • The first Chern class of a holomorphic line bundle is the image of under the boundary map .

25. Holomorphic Vector Bundles • The holo tangent bundle of a complex manifold is the holo vector bundle on of rank which is given by the transition functions ,where is a local coordinate and is a transition ftn. • The holo cotangent bundle is the dual of the tangent bundle . • The bundle of holo-forms is the exterior product bundle , and is the canonical bundle of .

26. Holomorphic Vector Bundles • Let be a cplxsubmfd. Then there is a canonical injection . • The normal bundle of in is the holo vector bundle is the cokernel of . • Prop. (Adjunction Formula) The canonical bundle of is naturally isomorphic to the line bundle .

27. Holomorphic Vector Bundles • Let be a holovec bundle. Its sheaf of sections is in a natural way a sheaf of -modules. • Prop. The association is a canonical bijection between the set of holovec bundles of rank and the set of locally free -modules of rank .

28. Holomorphic Vector Bundles • If is a holo vector bundle on a cplxmfd, then denotes the thcohomology of its sheaf of sections. • The Hodge numbers of a compact cplxmfd are the numbers • Related to the cohomology of as real manifolds and the Betti numbers of

29. Holomorphic Vector Bundles • Note that for two holovec bundles and on a cplxmfd, we have a natural map . • Thus, for every line bundle on a cplxmfd, the space has a natural ring structure. • The canonical ring of a cplxmfd is the ring . • If is connected, is an ID. Thus we may form the quotient field .

30. Holomorphic Vector Bundles • The Kodaira dimension of a connected complex manifold is defined as . • When , . • One has . • In general, is not f.g., but is expected to be f.g. at least if is projective. (Abundance Conjecture)

31. Divisors and Line Bundles

32. Divisors and Line Bundles • Recal: An analytic hypersurface of is an analytic subvariety of codimension one. • The divisor group is the free abelian group generated by the irreducible hypersurfaces. • Its elements are called “divisors”. • A divisor is effective if its coefficients are all nonnegative. In this case, one writes .

33. Divisors and Line Bundles • Let be a hypersurface and let . Suppose that is locally a zero set of an irreducible . • Let be a meromorphic function in a nbd of . Then the order is given by the equality with . • Note that the definition of order does not depend on the function , as long as we choose to be irreducible.

34. Divisors and Line Bundles • One defines the order along an irreducible hypersurfaceas where is irreducible in . • Such always exists! • Note that the order function satisfies . • For a meroftn, we define the divisor associated to by . • Such divisor is called principal.

35. Divisors and Line Bundles • The divisor can be written as , where is the summation along all zeroes of and is the summation along all poles of . • Next, we shall globalize this construction to local meromorphic functions.

36. Divisors and Line Bundles • Prop. There exists a natural isomorphism . • Note that the order is invariant under the multiplication by nowhere vanishing holoftns. • Cor. There exists a natural group hom, , where the definition of will be given in the proof.

37. Divisors and Line Bundles • Proof. Let corresponds to , which is given by functions for an open covering of . Then we define as the line bundle with transition ftns. • By the definition and thus we obtain an element in . • The rest is easy.

38. Divisors and Line Bundles • Suppose that is a smooth hypersurface of compact cplxmfdof dimension . • By Poincare duality, the linear map on defines an element in , called “the fundamental class of ”. • We have .

39. Divisors and Line Bundles • The pull-back of a divisor under a morphismis not always well-defined, so one has to assume that is dominant, i.e. has dense image. • Let be holo and let be an irrhypersurface such that no component of is contained in . Then becomes a hypersurface.

40. Divisors and Line Bundles • Let be the irr components of . Then , where is determined as follows. • Consider a smooth point and its image . Then locally near the hypersurfaceis defined by a holoftn, and its pullback can be decomposed by irr factors . • We extend this definition linearly to all divisors in .

41. Divisors and Line Bundles • Prop. Let be dominant holo map of conn cplxmfds. Then is a group homomorphism. • Cor. Let be holo. If is a divisor on such that is defined, • Thus, if is connected and is dominant then one obtains a commutative diagram of groups between .

42. Divisors and Line Bundles • Two divisors are linearly equivalent if the difference is principal. • Lemma. is pricipaliffis trivial. • This lemma shows that the group homrestricts to . • The inclusion is strict in general, but if a line bundle admits a global section, it is contained in the image.

43. Divisors and Line Bundles • Prop.(i) Let . Then the line bundle is isomorphic to .(ii) For any effective divisor , there exists a nontrivial section with . • Cor. Non-trivial sections of and of define linearly equivalent divisors iff.

44. Divisors and Line Bundles • Cor. The image of the natural map is generated by those line bundles with . • Proof. Prop above shows that any with is contained in the image. • Conversely, any divisor can be written as . • Hence, and the line bundles are both associated to effective divisors, thus admit nontrivial global sections.

45. Calculation in

46. Calculation in • Recall: is the holo line bundle which is naturally embedded into the trivial vector bundle such that the fibre of over is . • Recall: If is a holosubbundle of , then the inclusion naturally induces an inclusion of all tensor products. In particular, for the line bundle is a subbundle of .

47. Calculation in • Any homogeneous poly of degree defines a linear map which is linear on every fibre of the projection to . The restriction to thus provides a holo section of . • This way we associate to any homogeneous poly of degree a global holo section of , which will also be called .

48. Calculation in • Prop. For the space is canonically isomorphic to . • Proof. Note that the section associated to a non-trivial polynomial is nontrivial. Since that map is clearly linear, it suffices to show surjectivity.

49. Calculation in • Let . Choose a nontrivial induced by a homogeneous polynomial of degree and consider the meroftn. • Composing with the projection yields a mero map . • Moreover, is a holoftn. This function can be extended to the whole space by Hartogs’ theorem.

50. Calculation in • By definition of and , this function is homogeneous of degree , i.e. . • Using the power series expansion of at this proves that is a homogeneous polynomial of degree . Clearly, the section of induced by equals .