By S. Coen

ISBN-10: 0585367582

ISBN-13: 9780585367583

ISBN-10: 0824784456

ISBN-13: 9780824784454

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Japan 50 (1998) 615-621. [6] M. Chen, Kawamata-Viehweg vanishing and quint-canonical maps for threefolds of general type, Comm. in Algebra 27 (1999) 5471-5486. [7] M. Chen, Complex varieties of general type whose canonical systems are composed with pencils, J. Math. Soc. Japan 51 (1999) 331-335. [8] C. Ciliberto, The bicanonical map for surfaces of general type, Proc. Symposia in Pure Math. 62 (1997) 57-83. [9] L. Ein and R. Lazarsfeld, Global generation of pluricanonical and adjoint linear systems on smooth projective threefolds, J.

E. the kernel of the augmentation map Z[ni(X(Z),x)] -> Z. There is an analogue (which predates this theorem) of this result in Galois theory: the profinite completion of the fundamental group of a variety defined over any field (a notion which itself needs to be defined) carries an action of the absolute Galois group of the field. There is a philosophy, going back to Grothendieck, that these various homology and cohomology theories (Hodge and Galois are the examples we use here) are reflections of an underlying motivic theory which would subsume them both.

We can write L ~Hn S i | s ! + J ~ n u m CL2C + J , where J is an effective divisor on Si and C is contained in Si. So L-C J ~num I 1 )L The

### Geometry and Complex Variables by S. Coen

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