Analysis and Geometry in Several Complex Variables (Trends - download pdf or read online
By Gen Komatsu, Masatake Kuranishi
This quantity is an outgrowth of the fortieth Taniguchi Symposium research and Geometry in numerous complicated Variables held in Katata, Japan. Highlighted are the latest advancements on the interface of complicated research and genuine research, together with the Bergman kernel/projection and the CR constitution. the gathering additionally contains articles exploring mathematical interactions with different fields resembling algebraic geometry and theoretical physics. This paintings will function a superb source for either researchers and graduate scholars attracted to new tendencies in a few diverse branches of study and geometry.
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Extra resources for Analysis and Geometry in Several Complex Variables (Trends in Mathematics)
A very simple example is provided by the product of two curves, one elliptic and the other of genus > 2. Of course, the surface in this example has the algebraic dimension 2. In order to give some other examples of properly elliptic surfaces, consider again principal elliptic fibre bundles X over a curve B of genus > 2, given by ~ ~ H I ( B , EB) with c(~) ~ 0. 34 that b~(X) is odd, so X is non-k/ihlerian, hence a(X) = 1. Clearly, we have kod(X) = 1. (10) A surface of general type is a surface with kod(X) = 2.
In particular, it follows a(X) > 1. 36 2. 15 Let X be a nonalgebraic surface with a(X) = O. Then: (i) h~ h~ (2) hl,~ <_ 1 for any line bundle L E Pic(X); in particular pg(X) 02x) <_ 1; := dime H~ g2}) _< 2. Proof. (1) If 81 and s2 are two linearly independent (over r sections of the line bundle L, then st~s2 is a (global) meromorphic function on X which is not constant. It follows a(X) > 1, contradiction. (2) Let cot, w2 and Caa be three linearly independent holomorphic 1-forms on X. Then cot Acoz and wt Awa are not identically zero on X, otherwise it would follow a(X) > 1 (see the previous remark).
Then any irreducible curve on X is contained in some fibre and thus the fibration is unique. Proof. Let D be an irreducible curve contained in no fibre and let x0 E D be a point on D. D > 0. 10. Let X be a compact surface. IV , Prop. 1). In particular, it follows a(X) > 1. 36 2. 15 Let X be a nonalgebraic surface with a(X) = O. Then: (i) h~ h~ (2) hl,~ <_ 1 for any line bundle L E Pic(X); in particular pg(X) 02x) <_ 1; := dime H~ g2}) _< 2. Proof. (1) If 81 and s2 are two linearly independent (over r sections of the line bundle L, then st~s2 is a (global) meromorphic function on X which is not constant.
Analysis and Geometry in Several Complex Variables (Trends in Mathematics) by Gen Komatsu, Masatake Kuranishi