New rational homology balls found in specific knot 2-handlebodies.
problem Finding rational homology balls in 2-handlebodies.
method Proving existence and properties of rational homology balls through smooth embeddings and rational blow-ups.
result Rationally blown-up 4-manifolds cannot be obtained by ordinary blow-ups under certain conditions.
Study symplectic forms on manifolds to find Lagrangian pinwheels that can be separated.
problem Determine conditions for symplectic forms to carry disjoint Lagrangian pinwheels.
method Use rational blow-up to analyze Lagrangian pinwheels in symplectic manifolds.
result Conditions for disjunction of Lagrangian pinwheels in specific manifolds.
Constructs inequivalent Lefschetz fibrations on various surfaces.
problem Finding inequivalent Lefschetz fibrations on surfaces.
method Explicit construction of fibrations on rational and ruled surfaces.
result Explicit construction of inequivalent fibrations on all rational and ruled surfaces.
Fintushel and Stern defined the rational blow-down construction [FS] for smooth 4-manifolds, where a linear plumbing configuration of spheres Cn is replaced with a rational homology ball Bn, n≥2. Subsequently, Symington [Sy] defined this procedure in the symplectic category, where a symplectic Cn (given…
We prove a gluing formula for Seiberg--Witten invariants which describes in particular the behaviour of the invariant under blow-up and rational blow-down.
Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.
problem Investigate the homotopy of blow ups in algebraic and symplectic geometry.
method Develops fibrewise surgery theory and a purely homotopy theoretic approach.
result Obtained homotopy decompositions of the based loop space on blow ups.
The study explores pinwheels in symplectic surfaces and non-squeezing of rational homology balls.
problem Understanding when Lagrangian pinwheels embed in symplectic rational and ruled surfaces.
method Almost toric fibrations and symplectic rational blow-up.
result A rational homology ball embeds into a rational homology cylinder if and only if the parameter is greater than or equal to 1.
Let X be an oriented 4-manifold which does not have simple SW-type, for example a blow-up of a rational or ruled surface. We show that any two cohomologous and deformation equivalent symplectic forms on X are isotopic. This implies that blow-ups of these manifolds are unique, thus extending work of Biran. We also e…
Positive holomorphic sectional curvature on rational surfaces is characterized by Kähler metrics.
problem Characterizing rational surfaces by the existence of a Kähler metric with positive holomorphic sectional curvature.
method Constructing Kähler metrics on projective manifolds obtained from toric manifolds.
result Every projective manifold obtained from a projective toric manifold by a finite sequence of blow-ups at points admits a Kähler metric with positive holomorphic sectional curvature.
A complex ruled surface admits an iterated blow-up encoded by a parabolic structure with rational weights. Under a condition of parabolic stability, one can construct a Kaehler metric of constant scalar curvature on the blow-up according to math.DG/0412405. We present a generalization of this construction to the case o…
Simple embeddings of rational homology balls are shown via antiflips under certain conditions.
problem Finding simple embeddings of rational homology balls in specific geometric contexts.
method Using the semi-stable minimal model program and antiflips.
result Simple embeddings of rational homology balls are possible via a sequence of antiflips.
New exotic 4-manifolds created from lines and quadrics in CP^2.
problem Creating new exotic 4-manifolds homeomorphic but not diffeomorphic to CP^2 # 8 \overline{CP^2} and CP^2 # 9 \overline{CP^2}.
method Rational blowdown surgery along 4-valent plumbing graphs formed by complex lines and quadrics in CP^2.
result Graph classes from \cite{weighted} have representatives admitting rational blowdown leading to exotic manifolds.
New forms generalize Whitney forms with rational coefficients for numerical analysis.
problem Numerical problems with singularities near simplex faces.
method Introduce shadow forms and degrees of freedom for integration over faces of blow-up simplices.
result Obtain isomorphism between shadow forms cohomology and cellular cohomology of blow-up simplices.
In this note, we investigate pluri-half-anticanonical systems on the so called LeBrun twistor spaces. We determine its dimension, the base locus, structure of the associated rational map, and also structure of general members, in precise form. In particular, we show that if n>2 and m>1, the base locus of the system |mK…
Let X be any rational ruled symplectic four-manifold. Given a symplectic embedding $ι:B_{c}\into X$ of the standard ball of capacity c into X, consider the corresponding symplectic blow-up $\tX_ι$. In this paper, we study the homotopy type of the symplectomorphism group $\Symp(\tX_ι)$, simplifying and extending t…
Study Cremona transformations in weighted projective planes to find rational cuspidal curves and Zariski pairs.
problem Finding rational cuspidal curves and Zariski pairs in weighted projective planes.
method Construct families of curves using Cremona transformations, compute fundamental groups, and use blow-up-down decompositions.
result Discover new examples of rational cuspidal curves and Zariski pairs in weighted projective planes.
We prove that the group of Hamiltonian automorphisms of a symplectic 4-manifold contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group. We also extend to rational and ruled manifolds a result of Kedra which asserts that, if M is a simply co…
Let BlP1Pn be a Kähler manifold obtained by blowing up a complex projective space Pn along a line P1. We prove that BlP1Pn does not admit constant scalar curvature Kähler metrics in any rational Kähler class, but admits extremal m…
Let M be either S2×S2 or the one point blow-up $\cp# \bcp$ of $\cp$. In both cases M carries a family of symplectic forms $\om_\la$, where $\la > -1$ determines the cohomology class $[\om_\la]$. This paper calculates the rational (co)homology of the group $G_\la$ of symplectomorphisms of $(M,\om_\la)$ as …
We introduce hyperelliptic simplified (more generally, directed) broken Lefschetz fibrations, which is a generalization of hyperelliptic Lefschetz fibrations. We construct involutions on the total spaces of such fibrations of genus g≥3 and extend these involutions to the four-manifolds obtained by blowing up the …
In this article, using combinatorial techniques of mapping class groups, we show that a Stein fillable integral homology 3-sphere supported by an open book decomposition with page a 4-holed sphere admits a unique Stein filling up to diffeomorphism. Furthermore, according to a property of deforming symplectic fillin…
New proof shows unique symplectic fillings for certain surface singularity links.
problem Uniqueness of symplectic fillings for specific rational surface singularity links.
method Analysis of positive monodromy factorizations for planar open books.
result Unique symplectic fillings proven for specified contact structures.
We prove that any symplectic 4-manifold which is not a rational or ruled surface, after sufficiently many blow-ups, admits an arbitrary number of nonisomorphic Lefschetz fibrations of the same genus which cannot be obtained from one another via Luttinger surgeries. This generalizes results of Park and Yun who construct…
In this paper we compute the homotopy groups of the symplectomorphism groups of the 3-, 4- and 5-point blow-ups of the projective plane (considered as monotone symplectic Del Pezzo surfaces). Along the way, we need to compute the homotopy groups of the compactly supported symplectomorphism groups of the cotangent bundl…
Let X be a pseudomanifold. In this text, we use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson. We do it simplicially in the setting of a filtered version of face sets, also cal…
Nearby pinwheels are isotopic, solving Arnold's conjecture.
problem Proving isotopy of Lagrangian pinwheels in rational homology balls.
method Combining neck-stretching, symplectic blow-up, and computation of isotopy groups.
result Two pinwheels are isotopic, confirming Arnold's conjecture.
The study constructs symplectic 4-manifolds with exotic structures.
problem Creating symplectic 4-manifolds with exotic smooth structures.
method Using star surgeries and complex singularities, the study constructs these manifolds.
result Symplectic 4-manifolds with one Seiberg-Witten basic class are constructed.
Study on half spheres solves Nirenberg problem with complex blow-up analysis.
problem Prescribing scalar curvature on half spheres.
method Refined blow-up analysis of finite energy approximated solutions.
result Complex blow-up points and vortex problems reveal new connections.
The paper defines and constructs almost complex blow-ups on 4D almost complex manifolds.
problem Existence and uniqueness of almost complex blow-ups on almost complex manifolds.
method Definition and construction of almost complex blow-ups, proving their existence and uniqueness.
result Existence and uniqueness of almost complex blow-ups on 4D almost complex manifolds.
Formula derived for holomorphic Poisson blow-ups.
problem Invariance of Koszul-Brylinski homology under Poisson blow-ups.
method Blow-up formula derivation for holomorphic Koszul-Brylinski homologies.
result Invariance of E1-degeneracy of Dolbeault-Koszul-Brylinski spectral sequence. Study identifies numerical signs of blow-up in hydrodynamic equations.
problem Determining if numerical results of blow-up are genuine or artifacts.
method Geometrically consistent spatiotemporal discretization of complexified Euler equations.
result Identification of a signature based on supremum norm growth rates of vorticity.
Formula derived for Bott-Chern classes in complex blow-ups.
problem Calculating Bott-Chern classes in blow-ups of complex manifolds.
method Proved blow-up formula for Bott-Chern classes, established Riemann-Roch without denominators.
result Formula for Bott-Chern classes in blow-ups.
Explains blow-ups for Lie groupoids and algebroids, comparing different methods.
problem Blow-ups of Lie groupoids and algebroids.
method Detailed explanation of various blow-up constructions.
result Different blow-up constructions for Lie groupoids and algebroids are shown to fit into a general geometric framework.
Researchers study eigenvalues on singular Riemannian manifolds, showing how curvature affects Weyl's law.
problem Analyzing eigenvalues of Laplace-Beltrami operator on singular Riemannian manifolds with unbounded geometrical invariants.
method Developed a new quantitative estimate for the remainder of the heat trace and Weyl's function on Riemannian manifolds.
result Constructed singular Riemannian metrics with prescribed non-classical Weyl's law for various slowly varying functions.
The paper derives a formula for Chow weights of toric blow-ups.
problem Chow weights of toric blow-ups.
method Combinatorial formula derived from toric manifold and Delzant polytope.
result Explicit formula for Chow weights of blow-ups.
New proof of blow-up formula for Morse-Novikov cohomology.
problem Blow-up formula for Morse-Novikov cohomology.
method Introducing relative Morse-Novikov cohomology and using sheaf cohomology.
result Explicit isomorphism in relative Morse-Novikov cohomology.
Study on curvature blow-up rates in black hole interiors from gravitational collapse.
problem Understanding curvature blow-up rates in black hole interiors during gravitational collapse.
method Investigation of spherically symmetric Einstein-scalar field spacetimes, focusing on blow-up rates of curvature and mass.
result Kretschmann scalar blows up faster than in Schwarzschild setting, indicating a new blow-up phenomenon.
The paper examines the blow-up of Ricci curvatures in conformal metrics.
problem Characterizing the blow-up set of Ricci curvatures in conformal metrics.
method Analyzing the blow-up phenomena of Ricci curvatures on domains close to a limit set of lower dimension.
result Characterization of the blow-up set according to the Yamabe invariant of the manifold.
We show that the blow-up of a generalized Kahler 4-manifold in a nondegenerate complex point admits a generalized Kahler metric. As with the blow-up of complex surfaces, this metric may be chosen to coincide with the original outside a tubular neighbourhood of the exceptional divisor. To accomplish this, we develop a b…
This paper studies a specific blow-up algorithm for sop polynomials and their RLCT.
problem Determining the RLCT of sum-of-products polynomials through blow-up.
method Investigates a specific blow-up algorithm for sop polynomials to resolve their singularities.
result It is possible to resolve the singularities of sop polynomials using a specific blow-up algorithm.
We study blow-ups in generalized complex geometry. To that end we introduce the concept of holomorphic ideal, which allows one to define a blow-up in the category of smooth manifolds. We then investigate which generalized complex submanifolds are suitable for blowing up. Two classes naturally appear; generalized Poisso…
Computes cohomologies of blow-ups and projective bundles.
problem Cohomologies of blow-ups and projective bundles.
method Computes double complex of differential forms on projective bundles and blow-ups.
result Formulas for all cohomologies associated with the complex.
The abstract proves a Leray-Hirsch theorem and blow-up formula for Dolbeault cohomology.
problem Cohomology on complex manifolds, especially non-compact ones.
method Proves a Leray-Hirsch theorem and an explicit blow-up formula.
result Explicit blow-up formula for Dolbeault cohomology on complex manifolds.
Solutions blow up for Yamabe problem on umbilic manifolds with nonzero Weyl tensor.
problem Yamabe problem on manifolds with umbilic boundary
method Building blowing-up solutions
result Solutions blow up for linear perturbation of Yamabe problem
New non-Kähler 3-folds constructed via log conifold transitions.
problem Constructing new non-Kähler 3-folds from Fano threefold pairs.
method Defining log conifold transitions and studying their deformation theory.
result Local smoothings of nodes can be lifted to global first-order deformations.
The paper analyzes high-dimensional sphere solutions to the Nirenberg problem with residual mass.
problem The Nirenberg problem on high-dimensional spheres with residual mass.
method Analysis of subcritical approximations and blowing up solutions.
result Comprehensive description of blowing up solutions, including blow-up points and rates.
The Yamabe flow can blow up in infinite time with small perturbations.
problem Understanding the behavior of the Yamabe flow under small perturbations.
method Constructive proof using solutions of the Yamabe problem on the unit sphere as blow-up profiles.
result The Yamabe flow can blow up at multiple points on a Riemannian manifold in infinite time with small perturbations.
The generalized Jang equation was introduced in an attempt to prove the Penrose inequality in the setting of general initial data for the Einstein equations. In this paper we give an extensive study of this equation, proving existence, regularity, and blow-up results. In particular, precise asymptotics for the blow-up …