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.

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 EXTENDED ABSOLUTE PARALLELISM GEOMETRY

2008 ◽  
Vol 05 (07) ◽  
pp. 1109-1135 ◽  
Author(s):  
NABIL. L. YOUSSEF ◽  
A. M. SID-AHMED
Keyword(s):  
Tangent Bundle ◽  
Special Form ◽  
Vector Fields ◽  
A Priori ◽  
Building Blocks ◽  
Absolute Parallelism ◽  
Linearly Independent ◽  
Geometric Objects ◽  
Directional Argument ◽  
Curvature Tensors

In this paper, we study Absolute Parallelism (AP-) geometry on the tangent bundle TM of a manifold M. Accordingly, all geometric objects defined in this geometry are not only functions of the positional argument x, but also depend on the directional argument y. Moreover, many new geometric objects, which have no counterpart in the classical AP-geometry, emerge in this different framework. We refer to such a geometry as an Extended Absolute Parallelism (EAP-) geometry. The building blocks of the EAP-geometry are a nonlinear connection (assumed given a priori) and 2n linearly independent vector fields (of special form) defined globally on TM defining the parallelization. Four different d-connections are used to explore the properties of this geometry. Simple and compact formulae for the curvature tensors and the W-tensors of the four defined d-connections are obtained, expressed in terms of the torsion and the contortion tensors of the EAP-space. Further conditions are imposed on the canonical d-connection assuming that it is of Cartan type (resp. Berwald type). Important consequences of these assumptions are investigated. Finally, a special form of the canonical d-connection is studied under which the classical AP-geometry is recovered naturally from the EAP-geometry. Physical aspects of some of the geometric objects investigated are pointed out and possible physical implications of the EAP-space are discussed, including an outline of a generalized field theory on the tangent bundle TM of M.

scholarly journals EXPLICIT HORIZONTAL OPEN BOOKS ON SOME PLUMBINGS

2006 ◽  
Vol 17 (09) ◽  
pp. 1013-1031 ◽  
Author(s):  
TOLGA ETGÜ ◽  
BURAK OZBAGCI
Keyword(s):  
Contact Structure ◽  
Complex Surface ◽  
Contact Structures ◽  
Open Book ◽  
Open Books ◽  
Tight Contact ◽  
Weinstein Conjecture ◽  
Circle Bundles ◽  
Surface Singularities

We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact structure is also horizontal and Stein fillable. In particular, on some Seifert fibered 3-manifolds we describe open books which are horizontal with respect to their plumbing description. As another application we describe horizontal open books isomorphic to Milnor open books for some complex surface singularities. Moreover we give examples of tight contact 3-manifolds supported by planar open books. As a consequence, the Weinstein conjecture holds for these tight contact structures [1].

scholarly journals The topological symmetric orbifold

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Songyuan Li ◽  
Jan Troost
Keyword(s):  
Complex Surface ◽  
Conformal Field Theories ◽  
Hurwitz Numbers ◽  
Conformal Field ◽  
Witten Theory ◽  
Higher Genus ◽  
Chiral Ring ◽  
Physics Literature ◽  
Structure Constants ◽  
Genus One

Abstract We analyze topological orbifold conformal field theories on the symmetric product of a complex surface M. By exploiting the mathematics literature we show that a canonical quotient of the operator ring has structure constants given by Hurwitz numbers. This proves a conjecture in the physics literature on extremal correlators. Moreover, it allows to leverage results on the combinatorics of the symmetric group to compute more structure constants explicitly. We recall that the full orbifold chiral ring is given by a symmetric orbifold Frobenius algebra. This construction enables the computation of topological genus zero and genus one correlators, and to prove the vanishing of higher genus contributions. The efficient description of all topological correlators sets the stage for a proof of a topological AdS/CFT correspondence. Indeed, we propose a concrete mathematical incarnation of the proof, relating Gromow-Witten theory in the bulk to the cohomology of the Hilbert scheme on the boundary.

scholarly journals Commuting planar polynomial vector fields for conservative Newton systems

2019 ◽  
Vol 22 (04) ◽  
pp. 1950025 ◽  
Author(s):  
Joel Nagloo ◽  
Alexey Ovchinnikov ◽  
Peter Thompson
Keyword(s):  
Differential Equation ◽  
Vector Field ◽  
Elliptic Equation ◽  
Characteristic Zero ◽  
Classical Result ◽  
Vector Fields ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Linearly Independent ◽  
Planar Vector

We study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field on a plane. It is a classical result that one can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. This problem is also central to the question of linearizability of vector fields. Let [Formula: see text], where [Formula: see text] is a field of characteristic zero, and [Formula: see text] the derivation that corresponds to the differential equation [Formula: see text] in a standard way. Let also [Formula: see text] be the Hamiltonian polynomial for [Formula: see text], that is [Formula: see text]. It is known that the set of all polynomial derivations that commute with [Formula: see text] forms a [Formula: see text]-module [Formula: see text]. In this paper, we show that, for every such [Formula: see text], the module [Formula: see text] is of rank [Formula: see text] if and only if [Formula: see text]. For example, the classical elliptic equation [Formula: see text], where [Formula: see text], falls into this category.

The Equality of a Manifold's Rank and Dimension

1969 ◽  
Vol 12 (2) ◽  
pp. 183-184
Author(s):  
R. S. D. Thomas
Keyword(s):  
Vector Fields ◽  
Present Note ◽  
Linearly Independent ◽  
Commuting Vector Fields

T. J. Willmore has shown that if a differentiable manifold's rank (the maximum number of everywhere linearly independent commuting vector fields definable on it) equals the manifold's dimension, then the manifold is a torus of the appropriate dimension [1]. This theorem is proved more simply and without any differentiability hypothesis in the present note.

A GENERALIZATION OF THE ATIYAH–DUPONT VECTOR FIELDS THEORY

2002 ◽  
Vol 04 (04) ◽  
pp. 777-796 ◽  
Author(s):  
ZIZHOU TANG ◽  
WEIPING ZHANG
Keyword(s):  
Vector Bundle ◽  
Cross Sections ◽  
Vector Fields ◽  
Complex Structure ◽  
Index Theory ◽  
Index Theorem ◽  
Obstruction Theory ◽  
Real Vector ◽  
Complete Answer ◽  
Linearly Independent

To generalize the Hopf index theorem and the Atiyah–Dupont vector fields theory, one is interested in the following problem: for a real vector bundle E over a closed manifold M with rank E = dim M, whether there exist two linearly independent cross sections of E? We provide, among others, a complete answer to this problem when both E and M are orientable. It extends the corresponding results for E = TM of Thomas, Atiyah, and Atiyah–Dupont. Moreover we prove a vanishing result of a certain mod 2 index when the bundle E admits a complex structure. This vanishing result implies many known famous results as consequences. Ideas and methods from obstruction theory, K-theory and index theory are used in getting our results.

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

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