## Geometry & Topology

### Polyhedral Kähler manifolds

Dmitri Panov

#### Abstract

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 $ℂP2$ with singularities at complex line arrangements.

#### Article information

Source
Geom. Topol., Volume 13, Number 4 (2009), 2205-2252.

Dates
Revised: 5 May 2009
Accepted: 26 April 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800291

Digital Object Identifier
doi:10.2140/gt.2009.13.2205

Mathematical Reviews number (MathSciNet)
MR2507118

Zentralblatt MATH identifier
1175.53082

#### Citation

Panov, Dmitri. Polyhedral Kähler manifolds. Geom. Topol. 13 (2009), no. 4, 2205--2252. doi:10.2140/gt.2009.13.2205. https://projecteuclid.org/euclid.gt/1513800291

#### References

• A Borel, P-P Grivel, B Kaup, A Haefliger, B Malgrange, F Ehlers, Algebraic $D$–modules, Perspectives in Math. 2, Academic Press, Boston (1987)
• J Cheeger, A vanishing theorem for piecewise constant curvature spaces, from: “Curvature and topology of Riemannian manifolds (Katata, 1985)”, (K Shiohama, T Sakai, T Sunada, editors), Lecture Notes in Math. 1201, Springer, Berlin (1986) 33–40
• W Couwenberg, G Heckman, E Looijenga, Geometric structures on the complement of a projective arrangement, Publ. Math. Inst. Hautes Études Sci. (2005) 69–161
• R Friedman, Algebraic surfaces and holomorphic vector bundles, Universitext, Springer, New York (1998)
• B Grünbaum, Arrangements of hyperplanes, from: “Proceedings of the Second Louisiana Conference on Combinatorics, Graph Theory and Computing (Louisiana State Univ., Baton Rouge, La., 1971)”, (R C Mullin, K B Reid, D P Roselle, R S D Thomas, editors), Louisiana State Univ., Baton Rouge, La. (1971) 41–106
• B Grünbaum, Arrangements and spreads, Conf. Board of Math. Sci. Reg. Conf. Ser. in Math. 10, Amer. Math. Soc. (1972)
• F Hirzebruch, Algebraic surfaces with extremal Chern numbers (based on a dissertation by T Höfer, Bonn, 1984), Uspekhi Mat. Nauk 40 (1985) 121–129 Translated from the English by I A Skornyakov, International conference on current problems in algebra and analysis (Moscow-Leningrad, 1984)
• J N N Iyer, C Simpson, The Chern character of a parabolic bundle, and a parabolic Reznikov theorem in the case of finite order at infinity
• J Kaneko, S Tokunaga, M Yoshida, Complex crystallographic groups. II, J. Math. Soc. Japan 34 (1982) 595–605
• W Kühnel, T F Banchoff, The $9$–vertex complex projective plane, Math. Intelligencer 5 (1983) 11–22
• A Langer, Logarithmic orbifold Euler numbers of surfaces with applications, Proc. London Math. Soc. $(3)$ 86 (2003) 358–396
• J Li, Hermitian–Einstein metrics and Chern number inequalities on parabolic stable bundles over Kähler manifolds, Comm. Anal. Geom. 8 (2000) 445–475
• T Mochizuki, Kobayashi–Hitchin correspondence for tame harmonic bundles and an application, Astérisque (2006) viii+117
• T Mochizuki, Asymptotic behaviour of tame harmonic bundles and an application to pure twistor $D$–modules. I, Mem. Amer. Math. Soc. 185 (2007) xii+324
• M Ohtsuki, A residue formula for Chern classes associated with logarithmic connections, Tokyo J. Math. 5 (1982) 13–21
• S Orshanskiy, A PL–manifold of nonnegative curvature homeomorphic to $S^2 \times S^2$ is a direct metric product
• A Petrunin, Polyhedral approximations of Riemannian manifolds, Turkish J. Math. 27 (2003) 173–187
• C T Simpson, Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988) 867–918
• W P Thurston, Shapes of polyhedra and triangulations of the sphere, from: “The Epstein birthday schrift”, (I Rivin, C Rourke, C Series, editors), Geom. Topol. Monogr. 1, Geom. Topol. Publ., Coventry (1998) 511–549
• M Troyanov, Les surfaces euclidiennes à singularités coniques, Enseign. Math. $(2)$ 32 (1986) 79–94