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 Characterization algorithms for shift radix systems with finiteness property

2014 ◽  
Vol 11 (01) ◽  
pp. 211-232 ◽  
Author(s):  
Mario Weitzer
Keyword(s):  
Convex Hull ◽  
Convex Polyhedron ◽  
The Other ◽  
Closed Convex Hull ◽  
Connected Component ◽  
Complicated Structure ◽  
Finiteness Property ◽  
A Chain ◽  
Interior Points

For d ∈ ℕ and r ∈ ℝd, let τr : ℤd → ℤd, where τr(a) = (a2, …, ad, -⌊ra⌋) for a = (a1, …, ad), denote the (d-dimensional) shift radix system associated with r. τr is said to have the finiteness property if and only if all orbits of τr end up in (0, …, 0); the set of all corresponding r ∈ ℝd is denoted by [Formula: see text], whereas 𝒟d consists of those r ∈ ℝd for which all orbits are eventually periodic. [Formula: see text] has a very complicated structure even for d = 2. In the present paper, two algorithms are presented which allow the characterization of the intersection of [Formula: see text] and any closed convex hull of finitely many interior points of 𝒟d which is completely contained in the interior of 𝒟d. One of the algorithms is used to determine the structure of [Formula: see text] in a region considerably larger than previously possible, and to settle two questions on its topology: It is shown that [Formula: see text] is disconnected and that the largest connected component has non-trivial fundamental group. The other is the first algorithm characterizing [Formula: see text] in a given convex polyhedron which terminates for all inputs. Furthermore, several infinite families of "cutout polygons" are deduced settling the finiteness property for a chain of regions touching the boundary of 𝒟2.

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.

Configurations of 2-spheres in the K3 surface and other 4-manifolds

1996 ◽  
Vol 120 (2) ◽  
pp. 247-253 ◽  
Author(s):  
Daniel Ruberman
Keyword(s):  
Algebraic Geometry ◽  
Gauge Theory ◽  
Homology Class ◽  
Classical Case ◽  
Complex Surface ◽  
Rational Curves ◽  
K3 Surface ◽  
Complex Curve ◽  
Adjunction Formula ◽  
A Current

A current theme in the theory of 4-manifolds is the study of which properties of complex surface are determined the underlying smooth 4-manifold. For instance, the genus of a complex curve in a complex surface is determined by its homology class, via the adjunction formula. Recent work in gauge theory [10–12] has shown that, to a great degree, a similar principal holds for an arbitrary (i.e. not necessarily complex) smooth representative of a 2-dimensional homology class. Another question, still unsolved even in the context of algebraic geometry, is to find the number of disjoint rational curves on a complex surface. The classical case, namely that of hypersurfaces in CP3, has only been settled for degrees d ≤ 6. The papers [1, 2, 4, 8, 14, 15] contain bounds on the number of such curves and constructions of surfaces with many ( — 2)-curves; the last two together establish that 65 is the correct bound in degree 6.

scholarly journals A differential-geometric approach to deformations of pairs (X, E)

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Kwokwai Chan ◽  
Yat-Hin Suen
Keyword(s):  
Vector Bundle ◽  
Deformation Theory ◽  
Holomorphic Vector Bundle ◽  
Geometric Approach ◽  
Compact Complex Manifold ◽  
Deformation Problem ◽  
Graded Lie Algebra ◽  
A Chain ◽  
Differential Graded Lie Algebra ◽  
Chain Level

AbstractThis article gives an exposition of the deformation theory for pairs (X, E), where X is a compact complex manifold and E is a holomorphic vector bundle over X, adapting an analytic viewpoint `a la Kodaira- Spencer. By introducing and exploiting an auxiliary differential operator, we derive the Maurer–Cartan equation and differential graded Lie algebra (DGLA) governing the deformation problem, and express them in terms of differential-geometric notions such as the connection and curvature of E, obtaining a chain level refinement of the classical results that the tangent space and obstruction space of the moduli problem are respectively given by the first and second cohomology groups of the Atiyah extension of E over X. As an application, we give examples where deformations of pairs are unobstructed.

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.

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