Dr. Katz DEq Homework Solutions 8

Dr. Katz DEq Homework Solutions 8 - 8 N .M. Katz: w hich...

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8 N.M. Katz: which gives rise to an element p e Ext,,, (f21x/s (log D), f * (t2~/r)) (1.1.2) ~- H t (X, Dero (X/S) @r f* (I2~/r)). This element is called the Kodaira-Spencer class of the situation 1.0. It's image in U ~ (S, R'f, (Der o (X/S) | f* (f2~/r))) (1.1.3) ~-- U ~ (S, n I f. (Der o (X/S)) | (2~/r) Home~(Der(S/T)), R~ f. (Dero(X/S)) is the Kodaira-Spencer mapping (but still denoted p !), which may be explicated as follows. Let D be a section of Der(S/T) over an affine open set ~ c S. Then p (D) is an element of n I (f- 1 (q[), Dero (f - 1 (q/)/q/)) which may be given explicitly as follows. Let {Vii} be an affine open cover of f-l(q/). By the exactness of the dual of (1.1.1) over each Vii (or, more "directly", by the explicit description (1.0.3) via local coordinates), we may choose, for each i, a derivation Di~H~ Dero(X/T)) which extends the given derivation D ~H ~ (ql, Der(S/T)). Because D i and D~ extend the same derivation D of q/, the difference D i - Dj lies in H ~ (Vii n Vj, Dero(X/S)).
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