Complex Geometry : Introduction to holomorphic line bundles

Complex Geometry :Introduction to holomorphic line

bundles20110158

Sungkyung Kang

Basic homological algebra Sheaf cohomology Cech cohomology

Preliminaries

Complex manifolds: definitions

Holomorphic vector bundles Divisors and line bundles Calculation of line bundles on .

Contents

Complex Manifolds

A holomorphic atlas on a differentiable man-ifold 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 holo-morphic atlases.

Complex Manifolds

A holomorphic function on a complex mani-fold 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 in-duced map is (bi)holomorphic.

Complex Manifolds

A meromorphic function on a complex mani-fold is a map which associates to any an element in such that for any there exists an open nbd and two holomorphic func-tions with for all .

The field of meromorphic functions called “the function field of ”.

Complex Manifolds

Prop. (Siegel) Let be a compact connected complex manifold of dimension . Then .

The algebraic dimension of a compact con-nected manifold is .

Complex Manifolds

Let be a complex manifold of complex di-mension 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.

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 nbd containing such that is a zero set of a fi-nitely 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.

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 subvariety is irreducible if it cannot be written as a union of two proper analytic subvarieties.

A dimension of an irreducible analytic sub-variety is defined by .

A hypersurface is an analytic subvariety of codimension one.

Complex Manifolds

Holomorphic Vector Bundles

Let be a complex manifold. A holomorphic vector bundle of rank on is a complex manifold together with a holomorphic sur-jection and a structure of an -dimensional -vector space on each fiber of satisfying:◦ There exists an open covering of and biholomor-

phic maps commuting with the projections to such that the induced map is -linear.

Holomorphic Vector Bun-dles

Note that the induced transition functions is -linear for each in .

A vector bundle homomorphism is a bi-holomorphism between two vector bundles over the same complex manifold such that the induced maps over the attached vector spaces have the same rank.


A holomorphic rank vector bundle is de-termined by the holomorphic cocycle , thus by an element in the 1st Cech cohomology .

Note that the 1st Cech cohomology is canon-ically isomorphic to the 1st sheaf cohomol-ogy.


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 holomor-

phic vector bundle of rank precisely ◦ Projectivization (explained later)


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)


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 .


Let be a holomorphic map between com-plex 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 inclu-sion, then is the restriction of to .


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 .


Prop. The isom classes of holo line bundles on a cplx mfd with tensor product and dual-ization 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 de-scription of and the induced one for .


Cor. The Picard group is naturally isomor-phic to .

Cor. Let be holomorphic. Then is a group hom from to . This map is induced by the sheaf hom .


The exponential sequence on a cplx mfd is 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 cplx mfd is const)


Thus, can be computed by the 1st and 2nd cohomology groups of and , and the in-duced maps between them.

The first Chern class of a holomorphic line bundle is the image of under the boundary map .


The holo tangent bundle of a complex mani-fold 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 bun-dle of .


Let be a cplx submfd. 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 .


Let be a holo vec bundle. Its sheaf of sec-tions is in a natural way a sheaf of -mod-ules.

Prop. The association is a canonical bijec-tion between the set of holo vec bundles of rank and the set of locally free -modules of rank .


If is a holo vector bundle on a cplx mfd , then denotes the th cohomology of its sheaf of sections.

The Hodge numbers of a compact cplx mfd are the numbers ◦ Related to the cohomology of as real manifolds

and the Betti numbers of


Note that for two holo vec bundles and on a cplx mfd , we have a natural map .

Thus, for every line bundle on a cplx mfd , the space has a natural ring structure.

The canonical ring of a cplx mfd is the ring .

If is connected, is an ID. Thus we may form the quotient field .


The Kodaira dimension of a connected com-plex 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)


Divisors and Line Bundles

Recal: An analytic hypersurface of is an an-alytic 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 .


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.


One defines the order along an irreducible hypersurface as where is irreducible in .

Such always exists! Note that the order function satisfies . For a mero ftn , we define the divisor asso-

ciated to by . Such divisor is called principal.


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.


Prop. There exists a natural isomorphism .◦ Note that the order is invariant under the multipli-

cation by nowhere vanishing holo ftns. Cor. There exists a natural group hom , ,

where the definition of will be given in the proof.


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 el-ement in .

The rest is easy.


Suppose that is a smooth hypersurface of compact cplx mfd of dimension .

By Poincare duality, the linear map on de-fines an element in , called “the fundamen-tal class of ”.

We have .


The pull-back of a divisor under a mor-phism is not always well-defined, so one has to assume that is dominant, i.e. has dense image.

Let be holo and let be an irr hypersurface such that no component of is contained in . Then becomes a hypersurface.


Let be the irr components of . Then , where is determined as follows.

Consider a smooth point and its image . Then locally near the hypersurface is de-fined by a holo ftn , and its pullback can be decomposed by irr factors .

We extend this definition linearly to all divi-sors in .


Prop. Let be dominant holo map of conn cplx mfds. 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 .


Two divisors are linearly equivalent if the difference is principal.

Lemma. is pricipal iff is trivial.

This lemma shows that the group hom re-stricts to .

The inclusion is strict in general, but if a line bundle admits a global section, it is con-tained in the image.


Prop.(i) Let . Then the line bundle is iso-morphic to .(ii) For any effective divisor , there exists a nontrivial section with .

Cor. Non-trivial sections of and of define linearly equivalent divisors iff .


Cor. The image of the natural map is gen-erated 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 asso-

ciated to effective divisors, thus admit non-trivial global sections.


Calculation in

Recall: is the holo line bundle which is nat-urally embedded into the trivial vector bun-dle such that the fibre of over is .

Recall: If is a holo subbundle of , then the inclusion naturally induces an inclusion of all tensor products. In particular, for the line bundle is a subbundle of .

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 .

Calculation in

Prop. For the space is canonically isomor-phic 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.

Calculation in

Let . Choose a nontrivial induced by a ho-mogeneous polynomial of degree and con-sider the mero ftn .

Composing with the projection yields a mero map .

Moreover, is a holo ftn . This function can be extended to the whole space by Hartogs’ theorem.

Calculation in

By definition of and , this function is homo-geneous 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 .

Calculation in

By the proposition, we obtain a ring isom

Lemma. Let be a line bundle on a compact cplx mfd Then both and admit nontrivial global sections iff is trivial.

Cor. For the line bundle admits no global sections, i.e. .

Calculation in

Prop. (Euler sequence) On there exists a natural short exact sequence of holo vector bundles .

Recall that the canonical bundle is defined by .

Recall that any SES of holo vector bundles induces a canonical isomorphism .

Cor. The canonical bundle is isomorphic to .

Calculation in

Cor. One has . Proof. Since the line bundles have global

sections for , none of the powers for ad-mits non-trivial global sections. Thus the canonical ring is isomorphic to .

Calculation in

We need one more result to compute the canonical bundle of hypersurfaces in .

Prop. Let be a smooth hypersurface of a cplx mfd defined by a section of some holo line bundle on . Then and thus .

Calculation in

Cor. If is a smooth hypersurface of degree , i.e. defined by a section , then .

So, everything in can be expressed in poly-nomials. This turns complex geometry on into algebraic geometry!!!

WoW! Incredible!! Interesting!!! Unbeliev-able!!!!

Calculation in

Thank you!I hope you enjoyed the presentation.

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Complex Geometry : Introduction to holomorphic line bundles