The 9-vertex complex projective plane

Kühnel, Wolfgang; Banchoff, Thomas

doi:10.1007/bf03026567

Cited by 70 publications

(71 citation statements)

References 4 publications

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“…This statement can be deduced from Mochizuki [13; 14] (see also Li [12]). We explain this very briefly skipping all details.…”

Section: Definition 63mentioning

confidence: 70%

“…Known examples An explicit example of a nonnegatively curved polyhedral CP 2 is provided by Kühnel's 9 vertices triangulation [10]. This polyhedral CP 2 can be obtained as a finite isometric quotient of a flat complex 2-torus, and the holonomy of the metric on CP 2 is finite.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…The following inequality (called Bogomolov-Gieseker inequality) is proven in [13] and was also proven in different terms in [12].…”

Section: Definition 63mentioning

confidence: 97%

“…We prove that constructed parabolic bundle is stable and calculate its parabolic Chern characters. Inequality (1-5) is a consequence of Bogomolov-Gieseker inequality [13] (see also [12]). The existence of a PK metric in the case of equality follows form the parabolic version of Kobayashi-Hitchin correspondence from [13] and additional statements about logarithmic connection the we prove in Section 4.…”

Section: Theorem 112mentioning

confidence: 99%

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Polyhedral Kähler manifolds

Panov

2009

Geom. Topol.

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In this article we introduce the notion of polyhedral Kähler manifolds, even dimensional polyhedral manifolds with unitary holonomy. We concentrate on the 4-dimensional case, prove that such manifolds are smooth complex surfaces and classify the singularities of the metric. The singularities form a divisor and the residues of the flat connection on the complement of the divisor give us a system of cohomological equations. A parabolic version of the Kobayshi-Hitchin correspondence of T Mochizuki permits us to characterize polyhedral Kähler metrics of nonnegative curvature on CP 2 with singularities at complex line arrangements.53C56; 32Q15, 53C55

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“…This statement can be deduced from Mochizuki [13; 14] (see also Li [12]). We explain this very briefly skipping all details.…”

Section: Definition 63mentioning

confidence: 70%

Section: Introduction and Resultsmentioning

confidence: 99%

“…The following inequality (called Bogomolov-Gieseker inequality) is proven in [13] and was also proven in different terms in [12].…”

Section: Definition 63mentioning

confidence: 97%

Section: Theorem 112mentioning

confidence: 99%

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Polyhedral Kähler manifolds

Panov

2009

Geom. Topol.

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“…The existence of 3-neighborly combinatorial 4-manifolds and n ≤ 13 vertices was studied in [15]. Besides the trivial cases n = 5 (the dihedron which is not simplicial) and n = 6 (the boundary of a 5-simplex) there is exactly one combinatorial type, the unique 9-vertex triangulation of the complex projective plane [2,3,14,16]. This can be interpreted as a design S 10 (2, 5, 9).…”

Section: Definition ([4]) a Block Design S λ (T K N) Is A Collectimentioning

confidence: 98%

Block DesignsS2n−8(2,5,n) and Triangulated Eulerian 4-manifolds

Kühnel

Laßmann

1998

European Journal of Combinatorics

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Three-neighborly triangulations of eulerian 4-manifolds with n vertices can be interpreted as block designs S 2n−8 (2, 5, n). We discuss this correspondence and present a new cyclic example with 14 vertices. c 1998 Academic Press LimitedIt is well known that any 2-neighborly triangulation of a compact 2-manifold M with n vertices can be regarded as a block design S 2 (2, 3, n) or twofold triple system if one interprets the triangles as abstract triples (or blocks). This is possible only if n(7 − n) = 6χ(M) or, equivalent,, the so-called regular cases in Heawood's map color problem [18, 20]. As usual, χ denotes the Euler characteristic. In general, k-neighborliness means that the number of (k − 1)-dimensional simplices is n k where n is the number of vertices. Vice versa, any twofold triple system S 2 (2, 3, n) can be regarded as a triangular (and 2-neighborly) decomposition of a pseudo-surface. A pseudo-surface is a surface except at isolated points, see [1,10, 21]. In the literature, these regular cases are also known as triangular embeddings of the complete graph K n into surfaces or pseudo-surfaces.For compact triangulated 4-manifolds M with n vertices there are similar regular cases, characterized by the equation. This coincides with the case of 3-neighborly triangulations, meaning that any triple of vertices determines a triangle of the triangulation [12,13]. The 3-neighborliness implies that the link of every edge contains all the remaining n −2 vertices. If it is a 2-sphere then the number of triangles is 2(n −4), the number of edges is 3(n −4), in accordance with the Euler equation χ = n −2−3(n −4)+2(n −4) = 2. The same holds for any 4-dimensional simplicial complex such that the link of each simplex has the same Euler characteristic as one would expect in a 4-manifold. Such complexes are called eulerian manifolds in [12]. In this paper we study the induced block designs S 2n−8 (2, 5, n).

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Computer Graphics Applications in Geometry: “Because the Light is Better Over Here”

Banchoff

1986

The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics

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The 9-vertex complex projective plane

Cited by 70 publications

References 4 publications

Polyhedral Kähler manifolds

Polyhedral Kähler manifolds

Block DesignsS2n−8(2,5,n) and Triangulated Eulerian 4-manifolds

Computer Graphics Applications in Geometry: “Because the Light is Better Over Here”

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