arXiv research

New metrics found on Hirzebruch surfaces solve complex geometry questions.

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

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.

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…

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.

New method calculates signatures of biquotients.

Cohomological and homological spectral sequences are shown to be isomorphic.

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…

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

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.

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

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.

Newly discovered Eguchi-Hanson metric arises from edge metrics.

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…

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 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…

We present updates to the problems on Hirzebruch's 1954 problem list focussing on open problems, and on those where substantial progress has been made in recent years. We discuss some purely topological problems, as well as geometric problems about (almost) complex structures, both algebraic and non-algebraic, about co…

We show that a conjectural extension of a fixed point formula in Arakelov geometry implies results about a tautological subring in the arithmetic Chow ring of bases of abelian schemes. Among the results are an Arakelov version of the Hirzebruch proportionality principle and a formula for a critical power of $\hat c_1$ …

Study proves Hirzebruch genus inequality for almost Kähler manifolds with negative curvature.

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…

We use K-area homology to summarize some results about the Novikov conjecture and the Hirzebruch L-class. In fact, we provide necessary and sufficient conditions for closed manifolds to have a homotopy invariant L-class. In order to obtain additional properties we also introduce the cohomology of infinite K--area which…

We obtain general formulae expressing Hirzebruch genera of a manifold with Z/p-action in terms of invariants of this action (the sets of weights of fixed points). As an illustration, we consider numerous particular cases of well-known genera, in particular, the elliptic genus. We also describe the connection with the s…

Study signatures of torus links and their cores using Neumann's equivariant signatures and Hirzebruch's formula.

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

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 …

The paper studies almost complex torus manifolds using graphs and Hirzebruch genera, proving properties of their fixed points and cohomology.

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 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…

In the present paper we discuss an independent on the Grothendieck-Sato isomorphism approach to the Riemann-Roch-Hirzebruch formula for an arbitrary differential operator. Instead of the Grothendieck-Sato isomorphism, we use the Topological Quantum Mechanics (more or less equivalent to the well-known constructions with…

This article continues a line of research aimed at solving an important problem of T. Kobayashi of the existence of compact Clifford-Klein forms of reductive homogeneous spaces. We contribute to this topic by showing that almost all symmetric spaces and 3-symmetric spaces do not admit solvable compact CliffordfKlein fo…

The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and polyhedral metrics on the complex projective plane with singularities at a colle…

We give an exposition of a theorem of Hirzebruch, Kodaira and Yau which proves the uniqueness of the Kahler structure of complex projective space, and of Yau's resolution of the Severi Conjecture.

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 paper proves that a specific manifold is unitary cobordant to S^2 × S^6.

The study classifies Kähler threefolds with special fiber bundles.

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

The paper shows conditions for vanishing of certain topological invariants on specific types of manifolds.

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

We employ Hirzebruch-type invariants obtained from iterated p-covers to investigate concordance of links and string links. We show that the invariants naturally give various group homomorphisms of the string link concordance group into L-groups over number fields. We also obtain homomorphisms of successive quotients of…

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

We establish an S^1-equivariant index theorem for Dirac operators on Z/k-manifolds. As an application, we generalize the Atiyah-Hirzebruch vanishing theorem for S^1-actions on closed spin manifolds to the case of Z/k-manifolds.

We prove that if the circle group acts smooth and unitary on 2n-dimensional stably complex manifold with two isolated fixed points and it is not bound equivariantly, then n=1 or 3. Our proof relies on the rigid Hirzebruch genera.