A regularity theorem for deformations of a compact complex surface

1995 ◽  
Vol 5 (4) ◽  
pp. 533-550 ◽  
Author(s):  
Adam Harris
Keyword(s):  
Complex Surface ◽  
Regularity Theorem ◽  
Compact Complex Surface

scholarly journals Non Kählerian surfaces with a cycle of rational curves

2021 ◽  
Vol 8 (1) ◽  
pp. 208-222
Author(s):  
Georges Dloussky
Keyword(s):  
Deformation Theory ◽  
Complex Surface ◽  
Rational Curves ◽  
Intersection Form ◽  
Connected Component ◽  
Hirzebruch Surface ◽  
Compact Complex Surface ◽  
A Chain

Abstract Let S be a compact complex surface in class VII0 + containing a cycle of rational curves C = ∑Dj . Let D = C + A be the maximal connected divisor containing C. If there is another connected component of curves C ′ then C ′ is a cycle of rational curves, A = 0 and S is a Inoue-Hirzebruch surface. If there is only one connected component D then each connected component Ai of A is a chain of rational curves which intersects a curve Dj of the cycle and for each curve Dj of the cycle there at most one chain which meets Dj . In other words, we do not prove the existence of curves other those of the cycle C, but if some other curves exist the maximal divisor looks like the maximal divisor of a Kato surface with perhaps missing curves. The proof of this topological result is an application of Donaldson theorem on trivialization of the intersection form and of deformation theory. We apply this result to show that a twisted logarithmic 1-form has a trivial vanishing divisor.

scholarly journals From non-Kählerian surfaces to Cremona group of P2(C)

2014 ◽  
Vol 1 (1) ◽  
Author(s):  
Georges Dloussky
Keyword(s):  
Complex Surface ◽  
Cremona Group ◽  
Compact Complex Surface

AbstractFor any minimal compact complex surface S with n = b

scholarly journals A rigid, not infinitesimally rigid surface with K ample

2021 ◽  
Author(s):  
Christian Böhning ◽  
Hans-Christian Graf von Bothmer ◽  
Roberto Pignatelli
Keyword(s):  
Complex Surface ◽  
Canonical Bundle ◽  
Rigid Surface ◽  
Compact Complex Surface ◽  
Arrangements Of Lines ◽  
Rigid Smooth

AbstractWe produce an example of a rigid, but not infinitesimally rigid smooth compact complex surface with ample canonical bundle using results about arrangements of lines inspired by work of Hirzebruch, Kapovich & Millson, Manetti and Vakil.

Line Arrangements in P2(C) and Their Finite Covers

2017 ◽  
Author(s):  
Paula Tretkoff
Keyword(s):  
Projective Plane ◽  
Complex Surface ◽  
Linear Arrangement ◽  
Line Arrangements ◽  
Finite Covers ◽  
Compact Complex Surface ◽  
Blowing Up ◽  
Ball Quotients ◽  
Intersection Points ◽  
Complex Projective

This chapter discusses the free 2-ball quotients arising as finite covers of the projective plane branched along line arrangements. It first considers a surface X obtained by blowing up the singular intersection points of a linear arrangement in the complex projective plane, as well as a smooth compact complex surface Y that is a finite covering of X. If Y is of general type with vanishing proportionality deviation, then it is a free 2-ball quotient. The chapter then looks at line arrangements that have equal ramification indices along each of the proper transforms of the original lines, along with cases of blowing down rational curves and removing elliptic curves. It also enumerates all possibilities for the assigned weights of the arrangements, under the assumption that divisors of negative or infinite weight on the blown-up line arrangements do not intersect.

scholarly journals Branched Holomorphic Cartan Geometries and Calabi–Yau Manifolds

2018 ◽  
Vol 2019 (23) ◽  
pp. 7428-7458 ◽  
Author(s):  
Indranil Biswas ◽  
Sorin Dumitrescu
Keyword(s):  
Vector Bundle ◽  
Holomorphic Vector Bundle ◽  
Complex Surface ◽  
Projective Structure ◽  
Compact Complex Surface ◽  
Cartan Geometries ◽  
Projective Manifolds ◽  
Simply Connected ◽  
Complex Projective ◽  
Calabi Yau Manifolds

Abstract We introduce the concept of a branched holomorphic Cartan geometry. It generalizes to higher dimension the definition of branched (flat) complex projective structure on a Riemann surface introduced by Mandelbaum [25]. This new framework is much more flexible than that of the usual holomorphic Cartan geometries. We show that all compact complex projective manifolds admit a branched flat holomorphic projective structure. We also give an example of a non-flat branched holomorphic normal projective structure on a compact complex surface. It is known that no compact complex surface admits such a structure with empty branching locus. We prove that non-projective compact simply connected Kähler Calabi–Yau manifolds do not admit any branched holomorphic projective structure. The key ingredient of its proof is the following result of independent interest: if E is a holomorphic vector bundle over a compact simply connected Kähler Calabi–Yau manifold and E admits a holomorphic connection, then E is a trivial holomorphic vector bundle and any holomorphic connection on E is trivial.

scholarly journals DESINGULARIZATION OF QUASIPLURISUBHARMONIC FUNCTIONS

2005 ◽  
Vol 16 (05) ◽  
pp. 555-560 ◽  
Author(s):  
VINCENT GUEDJ
Keyword(s):  
Complex Surface ◽  
Lelong Numbers ◽  
Compact Complex Surface ◽  
Closed Current ◽  
Positive Closed Current

Let T be a positive closed current of bidegree (1,1) on a compact complex surface. We show that for all ε > 0, one can find a finite composition of blow-ups π such that π*T decomposes as the sum of a divisorial part and a positive closed current whose Lelong numbers are all less than ε.

scholarly journals 3-MANIFOLD GROUPS, KÄHLER GROUPS AND COMPLEX SURFACES

2012 ◽  
Vol 14 (06) ◽  
pp. 1250038 ◽  
Author(s):  
INDRANIL BISWAS ◽  
MAHAN MJ ◽  
HARISH SESHADRI
Keyword(s):  
Fundamental Group ◽  
Finite Index ◽  
Cartesian Product ◽  
Surface Group ◽  
Complex Surface ◽  
Index Subgroup ◽  
Minimal Action ◽  
Compact Complex Surface ◽  
Kähler Groups ◽  
Finite Index Subgroup

Let G be a Kähler group admitting a short exact sequence [Formula: see text] where N is finitely generated. (i) Then Q cannot be non-nilpotent solvable. (ii) Suppose in addition that Q satisfies one of the following: (a) Q admits a discrete faithful non-elementary action on ℍn for some n ≥ 2. (b) Q admits a discrete faithful non-elementary minimal action on a simplicial tree with more than two ends. (c) Q admits a (strong-stable) cut R such that the intersection of all conjugates of R is trivial. Then G is virtually a surface group. It follows that if Q is infinite, not virtually cyclic, and is the fundamental group of some closed 3-manifold, then Q contains as a finite index subgroup either a finite index subgroup of the three-dimensional Heisenberg group or the fundamental group of the Cartesian product of a closed oriented surface of positive genus and the circle. As a corollary, we obtain a new proof of a theorem of Dimca and Suciu in [Which 3-manifold groups are Kähler groups? J. Eur. Math. Soc.11 (2009) 521–528] by taking N to be the trivial group. If instead, G is the fundamental group of a compact complex surface, and N is finitely presented, then we show that Q must contain the fundamental group of a Seifert-fibered 3-manifold as a finite index subgroup, and G contains as a finite index subgroup the fundamental group of an elliptic fibration. We also give an example showing that the relation of quasi-isometry does not preserve Kähler groups. This gives a negative answer to a question of Gromov which asks whether Kähler groups can be characterized by their asymptotic geometry.

Holomorphic curvatures of twistor spaces

2014 ◽  
Vol 11 (03) ◽  
pp. 1450022
Author(s):  
Danish Ali ◽  
Johann Davidov ◽  
Oleg Mushkarov
Keyword(s):  
Complex Surface ◽  
Hermitian Manifold ◽  
Negative Answer ◽  
Einstein Metric ◽  
Holomorphic Sectional Curvature ◽  
Almost Hermitian Manifold ◽  
Twistor Spaces ◽  
Bisectional Curvature ◽  
Holomorphic Bisectional Curvature ◽  
Compact Complex Surface

We study the twistor spaces of oriented Riemannian 4-manifolds as a source of almost Hermitian 6-manifolds of constant or strictly positive holomorphic, Hermitian and orthogonal bisectional curvatures. In particular, we obtain explicit formulas for these curvatures in the case when the base manifold is Einstein and self-dual, and observe that the "squashed" metric on ℂℙ3 is a non-Kähler Hermitian–Einstein metric of positive holomorphic bisectional curvature. This shows that a recent result of Kalafat and Koca [M. Kalafat and C. Koca, Einstein–Hermitian 4-manifolds of positive bisectional curvature, preprint (2012), arXiv: 1206.3941v1 [math.DG]] in dimension four cannot be extended to higher dimensions. We prove that the Hermitian bisectional curvature of a non-Kähler Hermitian manifold is never a nonzero constant which gives a partial negative answer to a question of Balas and Gauduchon [A. Balas and P. Gauduchon, Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler, Math. Z.190 (1985) 39–43]. Finally, motivated by an integrability result of Vezzoni [L. Vezzoni, On the Hermitian curvature of symplectic manifolds, Adv. Geom.7 (2007) 207–214] for almost Kähler manifolds, we study the problem when the holomorphic and the Hermitian bisectional curvatures of an almost Hermitian manifold coincide. We extend the result of Vezzoni to a more general class of almost Hermitian manifolds and describe the twistor spaces having this curvature property.

scholarly journals THE LIPMAN–ZARISKI CONJECTURE IN GENUS ONE HIGHER

2020 ◽  
Vol 8 ◽  
Author(s):  
HANNAH BERGNER ◽  
PATRICK GRAF
Keyword(s):  
Vector Fields ◽  
Dual Graph ◽  
Complex Surface ◽  
Geometric Genus ◽  
Canonical Sheaf ◽  
Linearly Independent ◽  
Compact Complex Surface ◽  
Surface Singularities ◽  
Tangent Sheaf ◽  
Genus One

We prove the Lipman–Zariski conjecture for complex surface singularities with $p_{g}-g-b\leqslant 2$ . Here $p_{g}$ is the geometric genus, $g$ is the sum of the genera of exceptional curves and $b$ is the first Betti number of the dual graph. This improves on a previous result of the second author. As an application, we show that a compact complex surface with a locally free tangent sheaf is smooth as soon as it admits two generically linearly independent twisted vector fields and its canonical sheaf has at most two global sections.

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