The fundamental group at infinity of affine surface

Gurjar, R. V.; Shastri, Anant R.

doi:10.1007/bf02566361

Cited by 17 publications

(12 citation statements)

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“…Our construction avoids the deformation theory in order to get such examples. In [20] it is given an example of a smooth affine surface V which is topologically contractible and such that the fundamental group at infinity π ∞ 1 (V ) is finite and non-trivial (note that if V is topologically contractible and π ∞ 1 (V ) = 0, by Ramanujam's result [29], V is isomorphic to C 2 ). Therefore, as in the proof of Theorem 1.3., one gets examples of (1, 1)-convex-concave 2-dimensional manifolds whose holes cannot be filled.…”

Section: Remarkmentioning

confidence: 99%

On the disk theorem

Colţoiu

Tibăr

2009

Math. Ann.

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We construct an example of a 2-dimensional Stein normal space X with one singular point x 0 such that X \{x 0 } is simply connected and it satisfies the disk condition. This answers a question raised by Fornaess and Narasimhan. We also prove that any increasing union of Stein open sets contained in a Stein space of dimension 2 satisfies the disk condition. Starting from the above example we exhibit, without using deformation theory, a new type of 2-dimensional holes which cannot be filled.

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Section: Remarkmentioning

confidence: 99%

On the disk theorem

Colţoiu

Tibăr

2009

Math. Ann.

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“…Since conjugation preserves the weights of T , T must have weights ≤ −2 outside the central vertex (using that T is minimal, so there are no −1 weights). Now we are done by Gurjar-Shastri's Lemma 7, p. 467, which says that there is no minimal linear tree T with π 1 (T ) = Z/2 and with all weights ≤ −2 except for a single vertex which is not an endpoint of the tree [13]. So T must be a single vertex of weight ±2.…”

Section: Theorem 61 For M = S 1 There Is Exactly One Isomorphism mentioning

confidence: 99%

“…By Gurjar and Shastri [13], Theorem 1, p. 467, any minimal weighted tree satisfying certain conditions (E) and (H) such that the associated group π(T ) is isomorphic to Z/2 is linear. The conditions (E) and (H) hold for a tree T as above.…”

Section: Theorem 61 For M = S 1 There Is Exactly One Isomorphism mentioning

confidence: 99%

“…Namely, (E) says that every linear subspace of H 2 (X(C), R) which is positive definite for the intersection form has dimension at most 1, which follows from the Hodge index theorem on X C . And (H) holds for any normal crossing arrangement of rational curves on a complex surface with b 1 (X(C)) = 0 ( [13], p. 463). As a result, the weighted tree T associated to a compactification X of U as in the previous paragraph is linear.…”

Section: Theorem 61 For M = S 1 There Is Exactly One Isomorphism mentioning

confidence: 99%

“…More interestingly, we show in Theorem 6.1 that S 2 and RP 2 have only one good complexification if we strengthen the assumption that U(R) → U(C) is a homotopy equivalence to say that U(C) is diffeomorphic to the total space of the tangent bundle of M = U(R), as is true in all known examples of good complexifications. The proof for S 2 and RP 2 uses the methods of Ramanujam's topological characterization of C 2 [21] and related results by Gurjar and Shastri [13]. Aguilar and Burns found the same result independently, with a similar proof [2].…”

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confidence: 99%

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Complexifications of nonnegatively curved manifolds

Totaro¹

2003

J. Eur. Math. Soc.

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Define a good complexification of a closed smooth manifold M to be a smooth affine algebraic variety U over the real numbers such that M is diffeomorphic to U(R) and the inclusion U(R) → U(C) is a homotopy equivalence. Kulkarni showed that every manifold which has a good complexification has nonnegative Euler characteristic [16]. We strengthen his theorem to say that if the Euler characteristic is positive, then all the odd Betti numbers are zero. Also, if the Euler characteristic is zero, then all the Pontrjagin numbers are zero (see Theorem 1.1 and, for a stronger statement, Theorem 2.1). We also construct a new class of manifolds with good complexifications. As a result, all known closed manifolds which have Riemannian metrics of nonnegative sectional curvature, including those found by Cheeger [5] and Grove and Ziller [11], have good complexifications.We can in fact ask whether a closed manifold has a good complexification if and only if it has a Riemannian metric of nonnegative sectional curvature. The question is suggested by the work of Lempert and Szőke [17]. Lempert and Szőke, and independently Guillemin and Stenzel [12], constructed a canonical complex analytic structure on an open subset of the tangent bundle of M, given a real analytic Riemannian metric on M.

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Algebraic Characterization of the Affine Plane and the Affine 3-Space

Sugie

1989

Progress in Mathematics

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The fundamental group at infinity of affine surface

Cited by 17 publications

References 6 publications

On the disk theorem

On the disk theorem

Complexifications of nonnegatively curved manifolds

Algebraic Characterization of the Affine Plane and the Affine 3-Space

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