Singular Fibers Research Articles

Local curvature estimates of long-time solutions to the Kähler-Ricci flow

We study the local curvature estimates of long-time solutions to the normalized Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle. Using these estimates, we prove that on such a manifold, the set of singular fibers of the semi-ample fibration on which the Riemann curvature blows up at time-infinity is independent of the choice of the initial Kähler metric. Moreover, when a regular fiber of the semi-ample fibration is not a finite quotient of a torus, we determine the exact curvature blow-up rate of the Kähler-Ricci flow near the regular fiber.

The number of singular fibers in hyperelliptic Lefschetz fibrations

We consider complex surfaces, viewed as smooth 4-dimensional manifolds, that admit hyperelliptic Lefschetz fibrations over the 2-sphere. In this paper, we show that the minimal number of singular fibers of such fibrations is equal to $2g+4$ for even $g\geq4$. For odd $g\geq7$, we show that the number is greater than or equal to $2g+6$. Moreover, we discuss the minimal number of singular fibers in all hyperelliptic Lefschetz fibrations over the 2-sphere as well.

Hilbert curves of conic fibrations over smooth surfaces

Let (X, L) be a complex polarized threefold which is a conic fibration over a smooth surface. The complex affine cubic Γ representing the Hilbert curve of (X, L) is studied, paying special attention to its reducibility. In particular, Γ contains a specific line ℓ 0 if and only if X has no singular fibers. This leads to characterize the existence of a triple point simply in terms of numerical invariants of X. Other lines may cause the reducibility of Γ, which in this case depends also on the polarization. This situation is analyzed for a special class of conic fibrations.

Fibred algebraic surfaces and commutators in the Symplectic group

We describe the minimal number of critical points and the minimal number s of singular fibres for a non isotrivial fibration of a surface S over a curve B of genus 1, exhibiting several examples and in particular constructing a fibration with s=1 and irreducible singular fibre with 4 nodes.Then we consider the associated factorizations in the mapping class group and in the symplectic group. We describe explicitly which products of transvections on homologically independent and disjoint circles are a commutator in the Symplectic group Sp(2g,Z).

McShane-Type Identities for Quasifuchsian Representations of Nonorientable Surfaces

Abstract We show that Norbury’s McShane identity for nonorientable cusped hyperbolic surfaces $N$ generalizes to quasifuchsian representations of $\pi _1(N)$ as well as pseudo-Anosov mapping Klein bottles with singular fibers given by $N$.

Threefolds fibred by mirror sextic double planes

AbstractWe present a systematic study of threefolds fibred by K3 surfaces that are mirror to sextic double planes. There are many parallels between this theory and the theory of elliptic surfaces. We show that the geometry of such threefolds is controlled by a pair of invariants, called the generalized functional and generalized homological invariants, and we derive an explicit birational model for them, which we call the Weierstrass form. We then describe how to resolve the singularities of the Weierstrass form to obtain the “minimal form”, which has mild singularities and is unique up to birational maps in codimension 2. Finally, we describe some of the geometric properties of threefolds in minimal form, including their singular fibres, canonical divisor, and Betti numbers.

Global Higher-Order Estimates for Collapsing Calabi–Yau Metrics on Elliptic K3 Surfaces

We improve Gross-Wilson's local estimates to global ones. As an application, we study the blow-up limits of the degenerating Calabi-Yau metrics on singular fibers.

UNWEIGHTED DONALDSON–THOMAS THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSES

AbstractWe further the study of the Donaldson–Thomas theory of the banana 3-folds which were recently discovered and studied by Bryan [3]. These are smooth proper Calabi–Yau 3-folds which are fibred by Abelian surfaces such that the singular locus of a singular fibre is a non-normal toric curve known as a ‘banana configuration’. In [3], the Donaldson–Thomas partition function for the rank 3 sub-lattice generated by the banana configurations is calculated. In this article, we provide calculations with a view towards the rank 4 sub-lattice generated by a section and the banana configurations. We relate the findings to the Pandharipande–Thomas theory for a rational elliptic surface and present new Gopakumar–Vafa invariants for the banana 3-fold.

Algorithmic Construction of Two-Dimensional Singular Fibers of Atoms of Billiards in Non-Convex Domains

Algorithmic Construction of Two-Dimensional Singular Fibers of Atoms of Billiards in Non-Convex Domains

Smoothly embedding Seifert fibered spaces in 𝑆⁴

Using an obstruction based on Donaldson’s theorem, we derive strong restrictions on when a Seifert fibered space Y = F ( e ; p 1 q 1 , … , p k q k ) Y = F(e; \frac {p_1}{q_1}, \ldots , \frac {p_k}{q_k}) over an orientable base surface F F can smoothly embed in S 4 S^4 . This allows us to classify precisely when Y Y smoothly embeds provided e > k / 2 e > k/2 , where e e is the normalized central weight and k k is the number of singular fibers. Based on these results and an analysis of the Neumann-Siebenmann invariant μ ¯ \overline {\mu } , we make some conjectures concerning Seifert fibered spaces which embed in S 4 S^4 . Finally, we also provide some applications to doubly slice Montesinos links, including a classification of the smoothly doubly slice odd pretzel knots up to mutation.

Holomorphic Families of Strongly Pseudoconvex Domains in a Kähler Manifold

Let $p:X\rightarrow Y$ be a surjective holomorphic mapping between K\"ahler manifolds. Let $D$ be a bounded smooth domain in $X$ such that every generic fiber $D_y:=D\cap p^{-1}(y)$ for $y\in Y$ is a strongly pseudoconvex domain in $X_y:=p^{-1}(y)$, which admits the complete K\"ahler-Einstein metric. This family of K\"ahler-Einstein metrics induces a smooth $(1,1)$-form $\rho$ on $D$. In this paper, we prove that $\rho$ is positive-definite on $D$ if $D$ is strongly pseudoconvex. We also discuss the extensioin of $\rho$ as a positive current across singular fibers.

Change in phase singularities of a partially coherent Gaussian vortex beam propagating in a GRIN fiber.

In this paper, we have derived the analytical formulae for the cross-spectral densities of partially coherent Gaussian vortex beams propagating in a gradient-index (GRIN) fiber. In numerical analysis, the variations of the intensity and the phase distributions are demonstrated to illustrate the change in singularities within a GRIN fiber. It turns out that the beam intensity and phase distribution change periodically in the propagation process. The partially coherent Gaussian vortex beams do not typically possess the center intensity zero in the focal plane, which usually called 'hidden' singularities in intensities detection. We demonstrated the phase singularities more clearly by the phase distribution, one finds that the phase vortex of a partially coherent beam will crack near the focus, and opposite topological charge will be generated, we attribute to the wave-front decomposition and reconstruction of the vortex beams by the GRIN fiber. Our results show that the change in phase singularities not only affected by the GRIN fiber, but also by the initial coherence of the beam source, and high initial coherence will be more conducive to maintaining the phase singularities in the propagation. Our results may find applications in singular optics, wave-front reconstruction and optical fiber communications.

Bott vanishing for algebraic surfaces

Bott proved a strong vanishing theorem for sheaf cohomology on projective space. It holds for toric varieties, but not for most other varieties. We prove Bott vanishing for the quintic del Pezzo surface, also known as the moduli space M_{0,5}^bar of 5-pointed stable curves of genus zero. This is the first non-toric Fano variety for which Bott vanishing has been shown, answering a question by Achinger, Witaszek, and Zdanowicz. In another direction, we prove sharp results on which K3 surfaces satisfy Bott vanishing. For example, a K3 surface with Picard number 1 satisfies Bott vanishing if and only if the degree is 20 or at least 24. For K3 surfaces of any Picard number, we have complete results when the degree is big enough. We build on Beauville, Mori, and Mukai's work on moduli spaces of K3 surfaces, as well as recent advances by Arbarello-Bruno-Sernesi, Ciliberto-Dedieu-Sernesi, and Feyzbakhsh. The most novel aspect of the paper is our analysis of what happens when there is an elliptic curve of low degree. (In other terminology, this concerns K3 surfaces that are unigonal, hyperelliptic, trigonal, or tetragonal.) It turns out that the crucial issue is whether an elliptic fibration has a certain special type of singular fiber.

On the Fibres of Mishchenko-Fomenko Systems

This work is concerned with Mishchenko and Fomenko's celebrated theory of completely integrable systems on a complex semisimple Lie algebra \mathfrak{g} . Their theory associates a maximal Poisson-commutative subalgebra of \mathbb{C}[\mathfrak{g}] to each regular element a\in\mathfrak{g} , and one can assemble free generators of this subalgebra into a moment map F_a:\mathfrak{g}\rightarrow\mathbb{C}^b . This leads one to pose basic structural questions about F_a and its fibres, e.g. questions concerning the singular points and irreducible components of such fibres. We examine the structure of fibres in Mishchenko-Fomenko systems, building on the foundation laid by Bolsinov, Charbonnel-Moreau, Moreau, and others. This includes proving that the critical values of F_a have codimension 1 or 2 in \mathbb{C}^b , and that each codimension is achievable in examples. Our results on singularities make use of a subalgebra \mathfrak{b}^a\subseteq\mathfrak{g} , defined to be the intersection of all Borel subalgebras of \mathfrak{g} containing a . In the case of a non-nilpotent a\in\mathfrak{g}_{\mathrm{reg}} and an element x\in\mathfrak{b}^a , we prove the following: x+[\mathfrak{b}^a,\mathfrak{b}^a] lies in the singular locus of F_a^{-1}(F_a(x)) , and the fibres through points in \mathfrak{b}^a form a \text{rank}(\mathfrak{g}) -dimensional family of singular fibres. We next consider the irreducible components of our fibres, giving a systematic way to construct many components via Mishchenko-Fomenko systems on Levi subalgebras \mathfrak{l}\subseteq\mathfrak{g} . In addition, we obtain concrete results on irreducible components that do not arise from the aforementioned construction. Our final main result is a recursive formula for the number of irreducible components in F_a^{-1}(0) , and it generalizes a result of Charbonnel-Moreau. Illustrative examples are included at the end of this paper.

Hitchin Fibrations on Two-Dimensional Moduli Spaces of Irregular Higgs Bundles with One Singular Fiber

We analyze and completely describe the four cases when the Hitchin fibration on a $2$-dimensional moduli space of irregular Higgs bundles over $\mathbb{C}P^{1}$ has a single singular fiber. The case when the fiber at infinity is of type $I_0^*$ is further analyzed, and we give constructions of all the possible configurations of singular curves inelliptic fibrations having this type of singular fiber at infinity.

A Topological Approach to Indices of Geometric Operators on Manifolds with Fibered Boundaries

In this paper, we investigate topological aspects of indices of twisted geometric operators on manifolds equipped with fibered boundaries. We define K-groups relative to the pushforward for boundary fibration, and show that indices of twisted geometric operators, defined by complete Phi or edge metrics, can be regarded as the index pairing over these K-groups. We also prove various properties of these indices using groupoid deformation techniques. Using these properties, we give an application to the localization problem of signature operators for singular fiber bundles.

Topological strings on genus one fibered Calabi-Yau 3-folds and string dualities

We calculate the generating functions of BPS indices using their modular properties in Type II and M-theory compactifications on compact genus one fibered CY 3-folds with singular fibers and additional rational sections or just N -sections, in order to study string dualities in four and five dimensions as well as rigid limits in which gravity decouples. The generating functions are Jacobi-forms of Γ1(N) with the complexified fiber volume as modular parameter. The string coupling λ, or the ϵ± parameters in the rigid limit, as well as the masses of charged hypermultiplets and non-Abelian gauge bosons are elliptic parameters. To understand this structure, we show that specific auto-equivalences act on the category of topological B-branes on these geometries and generate an action of Γ1(N) on the stringy Kähler moduli space. We argue that these actions can always be expressed in terms of the generic Seidel-Thomas twist with respect to the 6-brane together with shifts of the B-field and are thus monodromies. This implies the elliptic transformation law that is satisfied by the generating functions. We use Higgs transitions in F-theory to extend the ansatz for the modular bootstrap to genus one fibrations with N -sections and boundary conditions fix the all genus generating functions for small base degrees completely. This allows us to study in depth a wide range of new, non-perturbative theories, which are Type II theory duals to the CHL ℤN orbifolds of the heterotic string on K3 × T2. In particular, we compare the BPS degeneracies in the large base limit to the perturbative heterotic one-loop amplitude with {R}_{+}^2{F}_{+}^{2g-2} insertions for many new Type II geometries. In the rigid limit we can refine the ansatz and obtain the elliptic genus of superconformal theories in 5d.

On the tropical discs counting on elliptic K3 surfaces with general singular fibres

Using Lagrangian Floer theory, we study the tropical geometry of K3 surfaces with more general singular fibres other than simple nodal curves. In particular, we give the local models for the types I n I_n , I I II , I I I III and I V IV singular fibres in the Kodaira classification. This generalizes the correspondence theorem in “Correspondence theorem between holomorphic discs and tropical discs on K3 surfaces”, arXiv:1703.00411 between open Gromov-Witten invariants/tropical discs counting to these cases.

Moderate Degeneration of Kähler–Einstein Manifolds with Negative Ricci Curvature

We study a family of Kähler–Einstein manifolds with negative Ricci curvature over a curve with a possibly singular special fiber. We show that if the diameters of the smooth fibers are uniformly bounded, then the special fiber has log-terminal singularities at worst after a finite base change and a birational modification.

Fibers of cyclic covering fibrations of a ruled surface

We give an algorithm to classify singular fibers of finite cyclic covering fibrations of a ruled surface by using singularity diagrams. As the first application, we classify all fibers of 3-cyclic covering fibrations of genus 4 of a ruled surface and show that the signature of a complex surface with this fibration is non-positive by computing the local signature for any fiber. As the second application, we classify all fibers of hyperelliptic fibrations of genus 3 into 12 types according to the Horikawa index. We also prove that finite cyclic covering fibrations of a ruled surface have no multiple fibers if the degree of the covering is greater than 3.