Real line arrangements with 3n lines intersect each other in n+1 points are related to finite complex reflection groups.
problem Identifying all real line arrangements in CP^2 with the Hirzebruch property.
method Analyzing the intersections of lines and confirming the relationship with finite complex reflection groups.
result There exist exactly four real line arrangements in CP^2 satisfying the Hirzebruch property.
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…
The article studies cohomology on complex manifolds and proves vanishing theorems.
problem Investigating topological properties of complex manifolds.
method Using Dolbeault-Morse-Novikov cohomology and integral inequalities.
result The Hirzebruch χ_y-genus vanishes on certain complex manifolds.
New metrics found on Hirzebruch surfaces solve complex geometry questions.
problem Finding new conformally Kähler, Einstein-Maxwell metrics on Hirzebruch surfaces.
method Used special Killing potentials found by Futaki and Ono.
result Proved existence of new conformally Kähler, Einstein-Maxwell metrics.
Proves non-positivity of Hirzebruch form on stable weights and connects to flat logarithmic connections.
problem Non-positivity of Hirzebruch form on stable weights
method Kempf--Ness and frame-potential inequality
result Zero locus of Hirzebruch form on stable weights corresponds to flat logarithmic connections
Study homological mirror symmetry for Hirzebruch surfaces using Morse homotopy.
problem Homological mirror symmetry for Hirzebruch surfaces Fk. method Using Strominger-Yau-Zaslow construction and Morse homotopy.
result Homological mirror symmetry holds for Hirzebruch surfaces Fk. Study K-R flow on Hirzebruch surfaces, showing tangent flows are K-R flows with orbifold singularities.
problem Finite time singularities in Kähler-Ricci flow on Hirzebruch surfaces.
method Analyze tangent flows based at singular points.
result Tangent flows are K-R flows with orbifold singularities.
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 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.
problem Verifying a conjecture about Kähler-Einstein edge metrics on Hirzebruch surfaces.
method Using the Calabi ansatz, constructing a family of metrics and studying their angle deformation.
result Verification of a conjecture and finding a rigid singularity.
This paper proves multiplicativity of Hirzebruch χ_y genera modulo 8 for odd y.
problem Multiplicativity of Hirzebruch χ_y genera modulo 8 for odd integers y.
method Analyzing smooth fiber bundles of compact complex algebraic manifolds.
result Hirzebruch χ_y genera are multiplicative modulo 8 for y ≡ 3 (mod 4), and for y ≡ 1 (mod 4), modulo 8 non-multiplicativity is identified with an Arf-Kervaire invariant.
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.
The paper studies almost complex torus manifolds using graphs and Hirzebruch genera, proving properties of their fixed points and cohomology.
problem Understanding the fixed points and cohomology of almost complex torus manifolds.
method Using directed labeled multigraphs and Hirzebruch genera to encode and analyze the manifolds.
result Almost complex torus manifolds have positive Todd genus and at least n+1 fixed points.
New method calculates signatures of biquotients.
problem Signature calculation for homogeneous spaces.
method Generalization of Hirzebruch's computation to biquotients.
result Signature of biquotients computed for equal rank cases.
Cohomological and homological spectral sequences are shown to be isomorphic.
problem Cohomological and homological Atiyah-Hirzebruch spectral sequences are not always isomorphic.
method Spanier-Whitehead duality is used to establish an isomorphism between the two spectral sequences.
result Cohomological and homological Atiyah-Hirzebruch spectral sequences are isomorphic for finite spectra.
This note finds metrics with specific curvature pinching on Hirzebruch surfaces.
problem Finding metrics with controlled curvature pinching on Hirzebruch surfaces.
method Proving a general pinching result for product metrics and applying it to Hirzebruch surfaces.
result Existence of Hodge metrics with controlled curvature pinching on 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 P1 or contracts an exc…
The paper constructs metrics on Hirzebruch surfaces and ruled surfaces.
problem Existence of Hermitian metrics with constant Chern scalar curvature.
method Using Page--Bérard-Bergery's ansatz to construct metrics on Hirzebruch surfaces.
result Construction of Hermitian metrics of positive constant Chern scalar curvature on Hirzebruch surfaces.
Study of trigonal curves in abelian differentials with specific divisor properties.
problem Characterizing locally closed subspaces of abelian differentials.
method Using linear systems on Segre-Hirzebruch surfaces to describe orbifold structure and orbifold fundamental groups.
result Identified the orbifold fundamental group of a specific subspace as a quotient of the Artin group of type E8. 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.
problem Understanding circle actions on 4-dimensional manifolds.
method Derive the Atiyah-Hirzebruch formula and associate graphs to fixed point data.
result Show existence of 4D oriented S^1-manifolds from satisfying graphs.
The paper constructs new Kähler manifolds with positive holomorphic sectional curvature.
problem Finding Kähler manifolds with positive holomorphic sectional curvature.
method Generalizing Hitchin's construction to Hirzebruch manifolds.
result Hirzebruch manifolds admit Kähler metrics with positive holomorphic sectional curvature.
The Hirzebruch χy-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.
problem Understanding limits of compact singular Einstein spaces.
method Constructing Kahler-Einstein edge metrics on Calabi-Hirzebruch manifolds.
result Eguchi-Hanson metric emerges as a Gromov-Hausdorff limit.
Study on curvature blow-up and convergence of continuity method on Hirzebruch surface.
problem Curvature blow-up and convergence of continuity method on Hirzebruch surface.
method Continuity method applied to generalised Hirzebruch surface, focusing on Gromov-Hausdorff convergence and scalar curvature estimates.
result A general solution to the continuity method either exists or all times, or the scalar curvature blows up.
Proves uniqueness of complex projective space's Kahler structure.
problem Uniqueness of complex projective space's Kahler structure.
method Exposition of Hirzebruch, Kodaira, Yau's theorem and Yau's resolution.
result Proves uniqueness of complex projective space's Kahler structure.
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…
This paper shows that, away from 6, the kernel of the Witten genus is precisely the ideal consisting of (bordism classes of) Cayley plane bundles with connected structure group, but only after restricting the Witten genus to string bordism. It does so by showing that the divisibility properties of Cayley plane bundle c…
The paper classifies fiber structures of discontinuity domains for Anosov representations.
problem Understanding the topology of discontinuity domains for Anosov representations.
method Explicitly working out a smooth version of Fintushel's classification theorem for S1-actions on 4-manifolds. result The action on the fiber is equivalent to a circle action on a Hirzebruch surface.
We calculate the Hirzebruch χy and χ^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.
Study proves Hirzebruch genus inequality for almost Kähler manifolds with negative curvature.
problem Proving Hirzebruch genus inequality for almost Kähler manifolds with negative sectional curvature.
method Combining \(L^2\)-estimates for harmonic forms, refined vanishing theorem, and Atiyah's \(L^2\)-index theorem.
result Components of Hirzebruch genus satisfy inequality \((-1)^{n-p}χ_{p}(X) \geq 1\) for all \(p\).
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 c^1 …
We prove properties of the Schweitzer complex and its cohomologies.
problem Properties of the Schweitzer complex and its cohomologies.
method Elliptic complex and cohomological functors.
result Finite dimensionality and equivalence to Hirzebruch-Riemann-Roch relations.
Study shows circle actions on complex manifolds with specific fixed points.
problem Understanding circle actions on complex manifolds with two fixed points.
method Used rigid Hirzebruch genera to prove conditions on n.
result Proved n=1 or 3 for circle actions with two fixed points on stably complex manifolds.
Expressing L-polynomials coefficients in terms of multiple zeta values.
problem Expressing L-polynomials coefficients in terms of multiple zeta values.
method Expressing L-polynomials coefficients in terms of multiple zeta values.
result Every monomial in the Pontryagin classes appears with a non-zero coefficient, with the expected sign.
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.
problem Computing signatures of torus links and their cores.
method Use Neumann's equivariant signatures and rewrite Hirzebruch's formula for torus links (without cores) in terms of integral points in a parallelogram.
result Rewritten Hirzebruch's formula for torus links with cores using integral points in a parallelogram.
The paper introduces twins in Kähler and Sasaki geometry, generalizing known concepts.
problem Understanding twins in Kähler and Sasaki geometry.
method Introducing weighted extremal Kähler twins and extremal Sasaki twins, studying their properties.
result Many twins appear in the Kähler setting and more than one extremal ray in the Sasaki cone.
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 CP2#CP2, and our work shows that even in this simple setting, the comparison of G-minimality in the various categories, i.e., locally …
Kähler structures on products of 2-spheres have identical Chern classes.
problem Classifying Kähler structures on products of 2-spheres.
method Using complex Bott manifolds and iterated P1-bundle constructions, we classify Kähler structures up to biholomorphism via Bott diagrams. result Kähler structures on products of 2-spheres have identical Chern classes.
Study shows lower central subgroups of a subgroup don't contain those of the free group.
problem Whether lower central subgroups of a subgroup contain those of the free group.
method Analyzes the relationship between lower central subgroups of a free group and its subgroup.
result Lower central subgroups of a subgroup do not contain those of the free group if the subgroup does not normally generate the free group.
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…
New solutions found for Einstein-Maxwell equations on complex manifolds.
problem Finding solutions to Einstein-Maxwell equations on complex manifolds.
method Constructing a family of conformally Kähler solutions that deform the Page metric.
result Displaying infinitely many geometrically distinct families of solutions.
Classifies real algebraic curves on a quadric ellipsoid of specific degree.
problem Classifying real algebraic curves of bidegree (5,5) on the quadric ellipsoid.
method Reduction to curves in the second Hirzebruch surface, combining classical construction methods on toric surfaces.
result Previously known restrictions form a complete system for this bidegree.
The paper proves that a specific manifold is unitary cobordant to S^2 × S^6.
problem Characterizing 8D almost complex manifolds with 4 fixed points.
method Analyzing Chern numbers and Hirzebruch χy-genus. result An 8D compact almost complex manifold with 4 fixed points is unitary cobordant to S^2 × S^6.
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.