Elliptic Singularities Research Articles

Smooth invariants of focus-focus singularities and obstructions to product decomposition

We study focus-focus singularities (also known as nodal singularities, or pinched tori) of Lagrangian fibrations on symplectic 4-manifolds. We show that, in contrast to elliptic and hyperbolic singularities, there exist homeomorphic focus-focus singularities which are not diffeomorphic. Furthermore, we obtain an algebraic description of the moduli space of focus-focus singularities up to smooth equivalence, and show that for double pinched tori this space is one-dimensional. Finally, we apply our construction to disprove Zung’s conjecture which says that any non-degenerate singularity can be smoothly decomposed into an almost direct product of standard singularities.

Random walk algorithms for elliptic equations and boundary singularities

Abstract The influence of boundary singularities on the behavior of the random walk algorithms is studied and illustrated by the results of several computational experiments of solving boundary-value problems for elliptic equations.

On the GIT quotient space of quintic surfaces

We describe the GIT compactification for the moduli space of smooth quintic surfaces in ${\mathbb {P}^{3}}$. In particular, we show that a normal quintic surface with at worst isolated double points or minimal elliptic singularities is stable. We also describe the boundary of the GIT quotient, and we discuss the stability of the nonnormal surfaces.

Singularity analysis for single gimbal control moment gyroscope system using space expansion method

Control Moment Gyroscope (CMG) is an effective candidate for agile satellites and large spacecraft attitude control because of its powerful torque amplification capability. The most serious situation, however, in using CMG is the inherent geometric singularity problem, where there’s no torque output along a particular direction. Space expansion method has been proposed in this work for the singularity analysis. Based on inverse mapping transformation, an expanded Jacobian matrix which is a full rank square matrix is obtained. The singular angle sets of the 3-parallel cluster and pyramid cluster are distinguished using space expansion method. An effective hybrid steering strategy, able to deal with the elliptic singularity, is further proposed. Simulation results demonstrate the excellent performance of the proposed steering logic compared to the generalized singular robust logic and pseudo inverse logic in terms of energy consumption and torque error.

Attitude control for on-orbit servicing spacecraft using hybrid actuator

On-orbit servicing is one of the research hotspots of space missions. A small satellite equipped with multiple robotic manipulators is expected to carry out device replacement task for target large spacecraft. Attitude hyperstable control of a small satellite platform under rotations of the manipulators is a challenging problem. A hybrid momentum exchanging actuator consists of Control Moment Gyro (CMG) and Reaction Wheel (RW) is proposed to tackle the above issue, due to its huge amount of momentum storage capacity of the CMG and high control accuracy of the RW, in which the CMG produces large command torque while the RW offers additional control degrees. The constructed dynamic model of the servicing satellite advises that it’s feasible for attitude hyperstable control of the platform with arbitrary manipulators through compensating the disturbance generated by rapid rotation of the manipulators. Then, null motion between the CMG and RW is exploited to drive the system to the expected target with favorable performance, and to overcome the CMG inherent geometric singularity and RW saturation. Simulations with different initial situations, including CMG hyperbolic and elliptic singularities and RW saturation, are executed. Compared to the scenarios where the CMG or RW fails stabilizing the platform, large control torque, precise control effect and escape of singularity are guaranteed by the introduced hybrid actuator, CMGRW (CMGRW refers to the hybrid momentum exchanging devices in this paper, consisting of 4 CMGs in classical pyramid cluster and 3 RWs in an orthogonal group (specific description can been found in Section 4)). The feasible performance of the satellite, CMG and RW under large disturbance demonstrates that the control architecture proposed is capable of attitude control for on-orbit servicing satellite with multiple robotic manipulators.

A note on distinguished bases of singularities

For an isolated hypersurface singularity which is neither simple nor simple elliptic, it is shown that there exists a distinguished basis of vanishing cycles which contains two basis elements with an arbitrary intersection number. This implies that the set of Coxeter–Dynkin diagrams of such a singularity is infinite, whereas it is finite for the simple and simple elliptic singularities. For the simple elliptic singularities, it is shown that the set of distinguished bases of vanishing cycles is also infinite. We also show that some of the hyperbolic unimodal singularities have Coxeter–Dynkin diagrams like the exceptional unimodal singularities.

$$\mathrm{SL}(2, \mathbb {C})$$ SL ( 2 , C ) group action on cohomological field theories

We introduce the $\mathrm{SL}(2,\mathbb{C})$ group action on a partition function of a Cohomological field theory via the certain Givental's action. Restricted to the small phase space we describe the action via the explicit formulae on a CohFT genus $g$ potential. We prove that applied to the total ancestor potential of a simple elliptic singularity the action introduced coincides with the transformation of Milanov--Ruan changing the primitive form (cf. arXiv:1106.2321).

Homological mirror symmetry for hypersurface cusp singularities

We study versions of homological mirror symmetry for hypersurface cusp singularities and the three hypersurface simple elliptic singularities. We show that the Milnor fibres of each of these carries a distinguished Lefschetz fibration; its derived directed Fukaya category is equivalent to the derived category of coherent sheaves on a smooth rational surface Y_{p,q,r}. By using localization techniques on both sides, we get an isomorphism between the derived wrapped Fukaya category of the Milnor fibre and the derived category of coherent sheaves on a quasi-projective surface given by deleting an anti-canonical divisor D from Y_{p,q,r}. In the cusp case, the pair (Y_{p,q,r}, D) is naturally associated to the dual cusp singularity, tying into Gross, Hacking and Keel’s proof of Looijenga’s conjecture.

Elliptic singularities on log symplectic manifolds and Feigin–Odesskii Poisson brackets

A log symplectic manifold is a complex manifold equipped with a complex symplectic form that has simple poles on a hypersurface. The possible singularities of such a hypersurface are heavily constrained. We introduce the notion of an elliptic point of a log symplectic structure, which is a singular point at which a natural transversality condition involving the modular vector field is satisfied, and we prove a local normal form for such points that involves the simple elliptic surface singularities$\widetilde{E}_{6},\widetilde{E}_{7}$and$\widetilde{E}_{8}$. Our main application is to the classification of Poisson brackets on Fano fourfolds. For example, we show that Feigin and Odesskii’s Poisson structures of type$q_{5,1}$are the only log symplectic structures on projective four-space whose singular points are all elliptic.

The Craighero–Gattazzo surface is simply connected

We show that the Craighero–Gattazzo surface, the minimal resolution of an explicit complex quintic surface with four elliptic singularities, is simply connected. This was conjectured by Dolgachev and Werner, who proved that its fundamental group has a trivial profinite completion. The Craighero–Gattazzo surface is the only explicit example of a smooth simply connected complex surface of geometric genus zero with ample canonical class. We hope that our method will find other applications: to prove a topological fact about a complex surface we use an algebraic reduction mod$p$technique and deformation theory.

Cohomology of ideals in elliptic surface singularities

We introduce the the normal reduction number of two-dimensional normal singularities and prove that elliptic singularity has normal reduction number two. We also prove that for a two-dimensional normal singularity which is not rational, it is Gorenstein and its maximal ideal is a $p_{g}$-ideal if and only if it is a maximally elliptic singularity of degree $1$.

Extension of CR functions from boundaries in Cn x R

Let $\Omega \subset {\mathbb C}^n \times {\mathbb R}$ be a bounded domain with smooth boundary such that $\partial \Omega$ has only nondegenerate elliptic CR singularities, and let $f \colon \partial \Omega \to {\mathbb C}$ be a smooth function that is CR at CR points of $\partial \Omega$ (when $n=1$ we require separate holomorphic extensions for each real parameter). Then $f$ extends to a smooth CR function on $\bar{\Omega}$, that is, an analogue of Hartogs-Bochner holds. In addition, if $f$ and $\partial \Omega$ are real-analytic, then $f$ is the restriction of a function that is holomorphic on a neighborhood of $\bar{\Omega}$ in ${\mathbb C}^{n+1}$. An immediate application is a (possibly singular) solution of the Levi-flat Plateau problem for codimension 2 submanifolds that are CR images of $\partial \Omega$ as above. The extension also holds locally near nondegenerate, holomorphically flat, elliptic CR singularities.

Algebraicity of normal analytic compactifications of ℂ2 with one irreducible curve at infinity

We present an effective criterion to determine if a normal analytic compactification of C^2 with one irreducible curve at infinity is algebraic or not. As a by product we establish a correspondence between normal algebraic compactifications of C^2 with one irreducible curve at infinity and algebraic curves contained in C^2 with one place at infinity. Using our criterion we construct pairs of homeomorphic normal analytic surfaces with minimally elliptic singularities such that one of the surfaces is algebraic and the other is not. Our main technical tool is the sequence of "key forms" - a 'global' variant of the sequence of "key polynomials" introduced by MacLane to study valuations in the 'local' setting - which also extends the notion of "approximate roots" of polynomials considered by Abhyankar and Moh.

KSBA surfaces with elliptic quotient singularities, π1 = 1, p g = 0, and K 2 = 1, 2

Among log canonical surface singularities, those which have a rational homology disk smoothing are the cyclic quotient singularities \(\frac{1}{{{n^2}}}\left( {1,na - 1} \right)\) with gcd(a, n) = 1, and three distinguished elliptic quotient singularities. We show the existence of smoothable KSBA normal surfaces with π1 = 1, p g = 0, and K 2 = 1, 2 for each of these three singularities. We also give a list of new (and old) normal surface singularities in smoothable KSBA surfaces for invariants π1 = 1, p g = 0, and K 2 = 1, 2, 3, 4.

Good ideals and $${p_{g}}$$ p g -ideals in two-dimensional normal singularities

In this paper, we introduce the notion of $${p_{g}}$$ -ideals and $${p_{g}}$$ -cycles, which inherits nice properties of integrally closed ideals on rational singularities. As an application, we prove an existence of good ideals for two-dimensional Gorenstein normal local rings. Moreover, we classify all Ulrich ideals for two-dimensional simple elliptic singularities.

Global mirror symmetry for invertible simple elliptic singularities

A simple elliptic singularity of type $E_N^{(1,1)}$ ($N=6,7,8$) can be described in terms of a marginal deformation of an invertible polynomial $W$. In the papers \cite{KS} and \cite{MR} the authors proved a mirror symmetry statement for some particular choices of $W$ and used it to prove quasi-modularity of Gromov-Witten invariants for certain elliptic orbifold $\mathbb{P}^1$s. However, the choice of the polynomial $W$ and its marginal deformation $\phi_{\mu}$ are not unique. In this paper, we investigate the global mirror symmetry phenomenon for the one-parameter family $W+\sigma\phi_{\mu}$. In each case the mirror symmetry is governed by a certain system of hypergeometric equations. We conjecture that the Saito-Givental theory of $W+\sigma\phi_{\mu}$ at any special limit $\sigma$ is mirror to either the Gromov-Witten theory of an elliptic orbifold $\mathbb{P}^1$ or the Fan-Jarvis-Ruan-Witten theory of an invertible simple elliptic singularity with diagonal symmetries, and the limits are classified by the Milnor number of the singularity and the $j$-invariant at the special limit. We prove the conjecture when $W$ is a Fermat polynomial. We also prove that the conjecture is true at the Gepner point $\sigma=0$ in all other cases.

Gromov–Witten theory and cycle-valued modular forms

Abstract In this paper, we review Teleman’s work on lifting Givental’s quantization of ℒ + ( 2 ) ⁢ GL ⁢ ( H ) {\mathcal{L}^{(2)}_{+}{\rm GL}(H)} action for semisimple formal Gromov–Witten potential into cohomological field theory level. We apply this to obtain a global cohomological field theory for simple elliptic singularities. The extension of those cohomological field theories over large complex structure limit are mirror to cohomological field theories from elliptic orbifold projective lines of weight ( 3 , 3 , 3 ) (3,3,3) , ( 2 , 4 , 4 ) (2,4,4) , ( 2 , 3 , 6 ) (2,3,6) . Via mirror symmetry, we prove generating functions of Gromov–Witten cycles for those orbifolds are cycle-valued (quasi)-modular forms.

On the Links of Simple Singularities, Simple Elliptic Singularities and Cusp Singularities

Abstract This is a survey article about the study of the links of some complex hypersurface singularities in ℂ3 . We study the links of simple singularities, simple elliptic singularities and cusp singularities, and the canonical contact structures on them. It is known that each singularity link is diffeomorphic to a compact quotient of a 3-dimensional Lie group SU (2), Nil3 or Sol3 , respectively. Moreover, the canonical contact structure is equivalent to the contact structure invariant under the action of each Lie group. We show a new proof of this fact using the moment polytope of S5 . Our proof gives a new aspect to the relation between simple elliptic singularities and cusp singularities, and visualizes how the singularity links are embedded in S5 as codimension two contact submanifolds.

Castelnuovo–Mumford regularity bounds for singular surfaces

We prove the regularity conjecture, namely Eisenbud–Goto conjecture, for a normal surface with rational, Gorenstein elliptic and log canonical singularities. Along the way, we bound the regularity for a dimension zero scheme by its Loewy length and for a curve allowing embedded or isolated point components by its arithmetic degree.

Calculation and fitting of boundaries between elliptic and hyperbolic singularities of pyramid-type control moment gyros

There exists a singularity problem in control moment gyros (CMGs). CMG singularities are classified into two types: hyperbolic and elliptic. Several gimbal steering control methods have been presented to avoid CMG singularities. Hyperbolic singularities can be avoided by null motion, but elliptic singularities cannot. The existing steering control methods are rarely designed by explicitly taking the singularity type into account. In order to effectively avoid elliptic singularities by perturbing gimbal angles, it is desirable to calculate and record the boundaries between elliptic and hyperbolic singularities in advance so that the determined boundaries can be utilized for developing model predictive steering control. To this end, the boundaries between elliptic and hyperbolic singularities of CMGs are calculated and represented in the form of fitted curves. Several numerical examples are presented to determine the perturbation gimbal angles for avoiding elliptic singularities without using singular value decomposition.