arXiv research

Study homological mirror symmetry for Hirzebruch surfaces using Morse homotopy.

Study K-R flow on Hirzebruch surfaces, showing tangent flows are K-R flows with orbifold singularities.

We construct smooth Riemannian metrics with constant scalar curvature on each Hirzebruch surface. These metrics respect the complex structures, fiber bundle structures, and Lie group actions of cohomogeneity one on these manifolds. Our construction is reduced to an ordinary differential equation called Duffing equation…

Study Kähler-Einstein edge metrics on Hirzebruch surfaces, verifying a conjecture and finding a rigid singularity.

Using different forms of the arithmetic Riemann-Roch theorem and the computations of Bott-Chern secondary classes, we compute the analytic torsion and the height of Hirzebruch surfaces.

We study rational cuspidal curves in Hirzebruch surfaces. We provide two obstructions for the existence of rational cuspidal curves in Hirzebruch surfaces with prescribed types of singular points. The first result comes from Heegaard--Floer theory and is a generalization of a result by Livingston and the first author. …

We investigate the metric behavior of the Kahler-Ricci flow on the Hirzebruch surfaces, assuming the initial metric is invariant under a maximal compact subgroup of the automorphism group. We show that, in the sense of Gromov-Hausdorff, the flow either shrinks to a point, collapses to $\mathbb{P}^1$ or contracts an exc…

Study on curvature blow-up and convergence of continuity method on Hirzebruch surface.

In a recent preprint Yael Karshon showed that there exist non-conjugate tori in a group of symplectomorphisms of a Hirzebruch surface. She counted them in terms of the cohomology class of the symplectic structure. We show that a similar phenomenon exists in the contactomorphism groups of pre-quantum circle bundles over…

In this note we prove that a special family of Killing potentials on certain Hirzebruch complex surfaces, found by Futaki and Ono, gives rise to new conformally Kähler, Einstein-Maxwell metrics. The correspondent Kähler metrics are ambitoric but they are not given by the Calabi ansatz. This answers in positive question…

In this paper a study of $G$-minimality, i.e., minimality of four-manifolds equipped with an action of a finite group $G$, is initiated. We focus on cyclic actions on $CP^2\# \overline{CP^2}$, and our work shows that even in this simple setting, the comparison of $G$-minimality in the various categories, i.e., locally …

It is known that Hirzebruch surfaces of non zero degree do not admit any constant scalar curvature Kähler metric \cite{ACGT,G,M17}. In this note, we describe how to construct Hermitian metrics of positive constant Chern scalar curvature on Hirzebruch surfaces using Page--Bérard-Bergery's ansatz \cite{P78,B82}. We also …

The paper introduces twins in Kähler and Sasaki geometry, generalizing known concepts.

This paper is the first step in a systematic project to study examples of Kähler manifolds with positive holomorphic sectional curvature ($H > 0$). Previously Hitchin proved that any compact Kähler surface with $H>0$ must be rational and he constructed such examples on Hirzebruch surfaces $M_{2, k}=\mathbb{P}(H^{k}\opl…

Newly discovered Eguchi-Hanson metric arises from edge metrics.

The main result of this note is that, for each $n\in \{1,2,3,\ldots\}$, there exists a Hodge metric on the $n$-th Hirzebruch surface whose positive holomorphic sectional curvature is $\frac{1}{(1+2n)^2}$-pinched. The type of metric under consideration was first studied by Hitchin in this context. In order to address th…

We prove that all minimal symplectic four-manifolds are essentially irreducible. We also clarify the relationship between holomorphic and symplectic minimality of Kähler surfaces. This leads to a new proof of the deformation-invariance of holomorphic minimality for complex surfaces with even first Betti number which ar…

New symplectic 4-manifolds with non-negative signatures are constructed using complex surfaces and quotients.

We count the conjugacy classes of maximal tori in the groups of symplectomorphisms of S^2 \times S^2 and of the blow-up of CP^2 at a point.

The study classifies Kähler threefolds with special fiber bundles.

The paper proves the existence of a special Kähler metric on a minimal ruled surface.

Paper studies Frobenius manifolds and quantum differential equations, proving Dubrovin Conjecture for Hirzebruch surfaces.

We list up all the candidates for the real isotopy types of real anti-bicanonical curves with one real nondegenerate double point on the 4-th real Hirzebruch surface RF_4 by enumerating the connected components of the moduli space of real 2-elementary K3 surfaces of type (S,θ)=((3,1,1), -id). We also list up all the ca…

We show that any hyperbolic Inoue surface (or Inoue-Hirzebruch surface of even type) admits anti-self-dual bihermitian structures. The same result also holds for any of its small deformations as far as its anti-canonical system is non-empty. Similar results are obtained for parabolic Inoue surfaces. Our method also yie…

Study of trigonal curves in abelian differentials with specific divisor properties.

Proves non-positivity of Hirzebruch form on stable weights and connects to flat logarithmic connections.

The paper classifies fiber structures of discontinuity domains for Anosov representations.

The paper classifies circle actions on 6D manifolds with isolated fixed points.

We consider projective rational strong Calabi dream surfaces: projective smooth rational surfaces which admit a constant scalar curvature Kähler metric for every Kähler class. We show that there are only two such rational surfaces, namely the projective plane and the quadric surface. In particular, we show that all rat…

We define the notion of a moment map and reduction in both generalized complex geometry and generalized Kähler geometry. As an application, we give very simple explicit constructions of bi-Hermitian structures on $\C¶^n$, Hirzebruch surfaces, the blow up of $\CP^2$ at arbitrarily many points, and other toric varieties,…

A line arrangement of $3n$ lines in $\mathbb CP^2$ satisfies Hirzebruch property if each line intersect others in $n+1$ points. Hirzebruch asked if all such arrangements are related to finite complex reflection groups. We give a positive answer to this question in the case when the line arrangement in $\mathbb CP^2$ is…

We prove that any two irreducible cuspidal Hurwitz curves $C_0$ and $C_1$ (or more generally, curves with A-type singularities) in the Hirzebruch surface $F_N$ with coinciding homology classes and sets of singularities are regular homotopic; and symplectically regular homotopic if $C_0$ and $C_1$ are symplectic with re…

We classify the minimum volume smooth complex hyperbolic surfaces that admit smooth toroidal compactifications, and we explicitly construct their compactifications. There are five such surfaces and they are all arithmetic, i.e., they are associated with quotients of the ball by an arithmetic lattice. Moreover, the asso…

New method calculates signatures of biquotients.

Cohomological and homological spectral sequences are shown to be isomorphic.

Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that e…

In the paper we describe obstructions for the existence of symplectic and Hamiltonian symplectic circle actions on closed compact manifolds in terms of Hirzebruch genera and relations between differential and homotopic invariants of such manifolds.

The article studies cohomology on complex manifolds and proves vanishing theorems.

Study circle actions on 4-manifolds, deriving formulas and graphs.

We study the number of distinct ways in which a smooth projective surface $X$ can be realized as a smooth toroidal compactification of a ball quotient. It follows from work of Hirzebruch that there are infinitely many distinct ball quotients with birational smooth toroidal compactifications. We take this to its natural…

The Hirzebruch $χ_y$-genus and Poincare polynomial share some similar features. In this article we investigate two of their similar features simultaneously. Through this process we shall derive several new results as well as reprove and improve some known results.

We survey the use of continued fraction expansions in the algebraical and topological study of complex analytic singularities. We also prove new results, firstly concerning a geometric duality with respect to a lattice between plane supplementary cones and secondly concerning the existence of a canonical plumbing struc…

Legendrian surgery describes canonical contact structures and calculates Gompf's θ-invariant.

We consider a family of Kähler structures on products of 2-spheres, arising from complex Bott manifolds. These are obtained via iterated $\mathbb P^1$-bundle constructions, generalizing the classical Hirzebruch surfaces. We show that the resulting Kähler structures all have identical Chern classes. We construct Bott di…

We express the coefficients of the Hirzebruch L-polynomials in terms of certain alternating multiple zeta values. In particular, we show that every monomial in the Pontryagin classes appears with a non-zero coefficient, with the expected sign. Similar results hold for the polynomials associated to the A-hat genus.

We complete the topological classification of real algebraic non-singular curves of bidegree $(5, 5)$ on the quadric ellipsoid. We show in particular that previously known restrictions form a complete system for this bidegree. Therefore, the main part of the paper concerns the construction of real algebraic curves. Our…

We calculate the Hirzebruch $χ_y$ and $\hatχ_y$-genera of symmetric products of closed complex manifolds by the holomorphic Lefschetz formula of Atiyah and Singer \cite{Ati-Sin}. Such calculation rederive some formulas proved in an earlier paper \cite{Zho} by a different method.

I. Hambleton, A. Korzeniewski and A. Ranicki proved that the signature of a fibre bundle of closed, connected, compatibly oriented PL manifolds is always multiplicative modulo 4. In this paper, we consider the Hirzebruch $χ_y$-genera for odd integers $y$ for a smooth fiber bundle such that the base, fibre, and total sp…