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By Leonard Lovering Barrett
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Additional info for An introduction to tensor analysis
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.
An introduction to tensor analysis by Leonard Lovering Barrett