Download e-book for iPad: A survey of the Hodge conjecture by James D. Lewis

By James D. Lewis

ISBN-10: 0821805681

ISBN-13: 9780821805688

This e-book offers an advent to an issue of relevant curiosity in transcendental algebraic geometry: the Hodge conjecture. such as 15 lectures plus addenda and appendices, the amount relies on a sequence of lectures introduced by way of Professor Lewis on the Centre de Recherches Mathematiques (CRM). The ebook is a self-contained presentation, thoroughly dedicated to the Hodge conjecture and similar issues. It comprises many examples, and such a lot effects are thoroughly confirmed or sketched. the incentive at the back of the various effects and history fabric is equipped. This finished method of the ebook offers it a ``user-friendly'' type. Readers don't need to seek in different places for numerous effects. The publication is appropriate to be used as a textual content for a themes path in algebraic geometry; contains an appendix via B. Brent Gordon.

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CHAPTER 2 Framed moduli space of torsion free sheaves on P2 Throughout this chapter, we assume k = C. e. ⎧ ⎫ E is a torsion free sheaf of rank E =⎪ ⎪ ⎪ ⎪ ⎨ r, c2 (E) = n which is locally free in a⎬ isomorphism, M(r, n) = (E, Φ) neighbourhood of ∞, ⎪ ⎪ ⎪ ⎪ ∼ ⎩ ⎭ ⊕r Φ : E| → O : framing at infinity ∞ ∞ where ∞ = {[0 : z1 : z2 ] ∈ P } ⊂ P is the line at infinity. Notice that the existence of a framing Φ implies c1 (E) = 0. In the first section we shall give the following description of M(r, n) which is essentially due to Barth [6].

G · x) = Ad∗g−1 μ(x). 2) dμi (v), ξ = ωi (ξ ∗ , v) for any v ∈ T X, any ξ ∈ g and i = 1, 2, 3, where ξ ∗ is a vector field generated by ξ. Take ζ = (ζ1 , ζ2 , ζ3 ) ∈ R3 ⊗ g∗ which satisfies Ad∗g (ζi ) = ζi for any g ∈ G, (i = 1, 2, 3). Then μ−1 (ζ) is invariant under the G-action. So we can consider the quotient space μ−1 (ζ)/G. [57]). Suppose G-action on μ−1 (ζ) is free. Then the quotient space μ−1 (ζ)/G is a smooth manifold and has a Riemannian metric and a hyper-K¨ ahler structure induced from those on X.

Hence there exists a set of generators λ1 , . . 4) σ ∨ = R≥0 λ1 + R≥0 λ2 + · · · + R≥0 λs . 5. If x = σ = M ⊗ R. t (1, . . , 1) ∈ V , then T C · x is closed if and only if Proof. Suppose σ = M ⊗ R, and hence σ ∨ = {0}. Then we can take a set of generators λ1 , . . 4). The one-parameter subgroup λj : C∗ → T C satisfies χi , λj ≥ 0 for all i = 1, . . , n. Then, as t → 0, λj (t) · x converges to a point which is not contained in the orbit of x. Hence T C · x is not closed. Suppose T C · x is not closed.

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A survey of the Hodge conjecture by James D. Lewis

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