By Robert Friedman
This publication covers the speculation of algebraic surfaces and holomorphic vector bundles in an built-in demeanour. it really is aimed toward graduate scholars who've had an intensive first-year direction in algebraic geometry (at the extent of Hartshorne's Algebraic Geometry), in addition to extra complex graduate scholars and researchers within the parts of algebraic geometry, gauge conception, or 4-manifold topology. a number of the effects on vector bundles must also be of curiosity to physicists learning string idea. a unique function of the booklet is its built-in method of algebraic floor concept and the examine of vector package deal concept on either curves and surfaces. whereas the 2 matters stay separate during the first few chapters, and are studied in exchange chapters, they develop into even more tightly interconnected because the publication progresses. therefore vector bundles over curves are studied to appreciate governed surfaces, after which reappear within the evidence of Bogomolov's inequality for good bundles, that is itself utilized to check canonical embeddings of surfaces through Reider's technique. equally, governed and elliptic surfaces are mentioned intimately, after which the geometry of vector bundles over such surfaces is analyzed. a number of the effects on vector bundles look for the 1st time in e-book shape, compatible for graduate scholars. The ebook additionally has a powerful emphasis on examples, either one of surfaces and vector bundles. There are over a hundred workouts which shape an essential component of the textual content.
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Extra info for Algebraic Surfaces and Holomorphic Vector Bundles
Here R is also a variable. (V1) The value f ((E, φN , φp )/R ) ∈ R depends only on the isomorphism class of (E, φN , φp )/R . (V2) If ϕ : R → R is an B-algebra homomorphism continuous under the p-adic topology, we have f ((E, φN , φp )R ⊗ R ) = ϕ(f ((E, φN , φp )/R )). (V3) f (q) = f ((Tate(q), zφp,can , φN )/R[[q1/d ]][q−1/d ] ) ∈ R[[q 1/d ]] for any level N -structures φN on Tate(q) over R[[q 1/d ]][q −1/d ] for 0 < d|N and z ∈ Z× p . Again, we note f (q) = f ((Tate(q), φp,can , φN,can )/R[[q]][q−1 ] ) ∈ R[[q]].
In this sense, μN is “small” (see [CRT] Sect. 5 for Krull dimension). Return to a general base ring B. Since [l] is essentially surjective with small “ﬁnite” kernel, we call [l] an isogeny. We will see later that [l] is an ´etale isogeny if l is invertible in B (see Sect. 4 for the exact notion of ´etale map, but here “´etale” basically means that for any maximal ideal m of B, B/m[μl ] is a separable extension of B/m; see [CRT] Sect. 26 for separability). Then taking t to be the variable on the target Gm , we have t1/l in the source Gm .
R) and all elements in μl (Fp ) over Fp . We write lN0 = k ∩ μl∞ (Fp ) . 15 applied to K = k, we have Trk(ζ)/k (ζ) = 0 if ζ ∈ μl∞ (Fp ) − k. Put M = N0 + N . If f (ζ) = 0 for a root of unity ζ ∈ μlM (Fp ), 0 = Trk(ζ)/k (ζ −bi f (ζ)) = [k(ζ) : k]ci , because ζ bj −bi ∈ k (i = j). Since [k(ζ) : k] is an l-power, we ﬁnd that ci = 0 for all i, a contradiction; hence, the zero set is contained in μlM (Fp ). 14 by Sinnott: Since a1 , . . , ar are linearly independent over Z, for all monomials y k = y1k1 .
Algebraic Surfaces and Holomorphic Vector Bundles by Robert Friedman