By Kichoon Yang

ISBN-10: 904815149X

ISBN-13: 9789048151493

ISBN-10: 9401591512

ISBN-13: 9789401591515

This booklet comprises an exposition of the speculation of meromorphic capabilities and linear sequence on a compact Riemann floor. hence the most subject material contains holomorphic maps from a compact Riemann floor to complicated projective area. Our emphasis is on households of meromorphic capabilities and holomorphic curves. Our strategy is extra geometric than algebraic alongside the strains of [Griffiths-Harrisl]. AIso, we have now depended on the books [Namba] and [Arbarello-Cornalba-Griffiths-Harris] to agreat exten- approximately each bring about Chapters 1 via four are available within the union of those books. Our fundamental motivation used to be to appreciate the totality of meromorphic features on an algebraic curve. even though this can be a classical topic and masses is understood approximately meromorphic capabilities, we felt that an obtainable exposition used to be missing within the present literature. hence our e-book will be regarded as a modest attempt to show elements of the identified concept of meromorphic features and holomorphic curves with a geometrical bent. now we have attempted to make the publication self-contained and concise which intended that a number of significant proofs now not necessary to extra improvement of the speculation needed to be passed over. The booklet is focused on the non-expert who needs to leam sufficient approximately meromorphic services and holomorphic curves in order that helshe might be capable of follow the consequences in hislher personal learn. for instance, a differential geometer operating in minimum floor idea will want to tind out extra concerning the distribution trend of poles and zeros of a meromorphic function.

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It is necessary to generalize the notion of a family somewhat in order to study the space of holomorphic maps. Definition 40 Let X and S be complex spaces. ) A surjective proper holomorphic map 1r:X-S is said to define an analytic family if every fiber is connectedj if there are an open cover {Xi} of X, open subsets {Ui } ofCd (d is fixed), open subsets {Si} of S and biholomorphisms"'i : Xi Ui X Si such that where 1r2 : Ui x Si Si is the obvious projection. Note that each fiber M t , t E S, is a compact complex manifold of dimension d.

3) Conversely, for any X E Sp(g,Z) ifÎi is defined (I II) is the period matrix of as in the above, then g , canonica! homology basis e. The Siegel upper half space is given by Hg = {Z E GL(g,q: zt = Mrelative to some Z, Im(Z) > O}. It is a g(gtLdimensional complex manifold. )) is well-defined. We have the famous Torelli theorem concerning this map. Theorem 28 Compact Riemann surfaces M, M' both of genus 9 are biholo- morphic to each other if and only if j (M) = j (M'). 6. THE JACOBIAN VARIETY Let M d = M x ...

By a variety we shall always mean at least a quasi-projective variety. The Zariski topology is to be used unless otherwise specified. When we speak of a subvariety of a given variety we will usually mean a closed subvariety so that a subvariety of a projective (respectively, affine) variety is also projective (respectively, affine). Definition 44 For any variety B we dejine a family of projective varieties in ]pn with base B to be a closed subvariety V of the product B x ]pn. The idea is that if B c pm is projective, the family V c pm x ]pn will be described by a collection of polynomials Fa (Z, W) bihomogeneous in the coordinates Z on pm and W on ]pn, which we may think of as a collection of polynomials in W whose coefficients are polynomials on B; likewise, if B is affine, then we may describe V C Cm X ]pn by a collection of polynomials Fa (z, W), which we may think of as homogeneous polynomials in the variables W whose coefficients are regular functions on B.

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