Algebraic Geometry Proc. conf. Chicago, 1989 by Spencer Bloch, Igor V. Dolgachev, William Fulton

By Spencer Bloch, Igor V. Dolgachev, William Fulton

This quantity comprises the court cases of a joint USA-USSR symposium on algebraic geometry, held in Chicago, united states, in June-July 1989.

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A coarse equivalence is a coarse map f : X → Y such that there exists a coarse map g : Y → X and there is a constant K > 0 with dX (x, g ◦ f (x)) ≤ K and dY (y, f ◦ g(y)) ≤ K for all x ∈ X and y ∈ Y . 17. The inclusion map Z → R (when Z and R are equipped with their standard metrics) is a coarse equivalence, with coarse inverse the “rounding down” map x → x . More generally, if M is a connected compact manifold with fundamental group π, and if M is the universal cover of M , then M is coarsely equivalent to |π|, the group π viewed as a metric space with respect to a word-length metric (defined by a choice of a finite generating set).

Identical boundary components Figure 4. A Z/3-manifold One should really think of a Z/k-manifold M as the singular space MΣ = M/ ∼ obtained by identifying all k of the boundary components with one another. This space is not a manifold (if k > 2), and does not satisfy Poincar´e duality. The neighborhood of a point on βM is a cone on k copies of B n−1 joined along S n−2 , as illustrated in Figure 5. If M is an oriented Z/2-manifold, then MΣ is a manifold, but is not orientable, because of the way the two copies of βM have been glued together.

Since this foliation is linear, it has a transverse measure given by Haar measure on a 2-torus transverse to the leaves of F . Let D be the ∂ operator along the leaves, acting on E. Then on a leaf L = im{(z, w + z) : z ∈ C}, a holomorphic section of E can be identified with a meromorphic function on C with all its poles simple and contained in z + Λ1 , for some z, and with zeros at points of Λ2 . Apply the foliation index theorem to deduce an existence result about such meromorphic functions in L2 .

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