Classifies degenerations of complex projective plane with rational singularities.
problem Classifying singularities of complex projective plane.
method Assuming Wahl's conjecture, classifies degenerations using rational homology disk smoothing.
result Classifies surfaces with rational singularities, including new degenerations with non-log canonical singularities.
Study on rational projective planes with small index singularities.
problem Existence and classification of rational homology projective planes with small index quotient singularities.
method Topological and smooth obstructions analysis, classification of singularities.
result Classification of quotient singularities for rational homology projective planes with indices up to three.
Compact RCD spaces derived from singular Kahler metrics on 3D projective varieties.
problem Understanding geometric structures of singular Kahler spaces.
method Proving RCD spaces homeomorphic to 3D projective varieties with bounded Nash entropy and Ricci curvature.
result Compact RCD spaces are equivalent to underlying projective varieties.
Paper relates curvature ellipse and parabola of surface projections.
problem Relating local geometry of surfaces in different dimensions.
method Analyzes curvature ellipses and parabolas in R4 and R3. result Relates curvature geometry of surfaces to their projections.
Projections of knotted spheres in high dimensions show complex singularities.
problem Understanding the singularities of projections of knotted spheres in high-dimensional spaces.
method Construction of specific knotted spheres and analysis of their projections.
result Projections of knotted spheres can have double points and connected embedded double point sets.
Formula for foliations' singularities in complex projective spaces.
problem Counting singularities of foliations on complex projective spaces.
method Global residue formula for logarithmic indices of foliations with isolated singularities.
result Formula for the number of singularities in the complement of the invariant divisor on complex projective spaces.
The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.
problem Extending the Poincaré-Hopf theorem to projective varieties with isolated singularities.
method Using generalized Poincaré-Hopf indices for a projective variety with isolated determinantal singularities.
result A Poincaré-Hopf type theorem is proven for projective varieties with isolated singularities.
New hyperbolic metrics with isolated singularities constructed.
problem Constructing hyperbolic metrics with specific singularities.
method Correspondence between metrics and projective functions.
result Explicit construction of new hyperbolic metrics.
The Poincaré-Hopf theorem is extended to projective varieties with isolated singularities.
problem Extending the Poincaré-Hopf theorem to varieties with isolated singularities.
method Using generalizations of the Poincaré-Hopf index.
result A Poincaré-Hopf type theorem for projective varieties with isolated singularities.
Simple construction for classifying spaces of projections of immersions.
problem Classifying spaces for projections of immersions with controlled singularities.
method Explicit simple construction for classifying spaces of maps obtained as hyperplane projections of immersions.
result Structure theorems for these classifying spaces.
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.
Characterizes monodromies of projective structures on finite-type surfaces.
problem Understanding monodromies of projective structures on finite-type surfaces.
method Geometrical/topological study of local conical projective structures.
result Any representation can be represented as the holonomy of a branched projective structure.
In this paper, we prove that any two birational projective varieties with finite quotient singularities can be realized as two geometric GIT quotients of a non-singular projective variety by a reductive algebraic group. Then, by applying the theory of Variation of Geometric Invariant Theory Quotients ([3]), we show tha…
Study projective KLT varieties with projectively flat cotangent sheaves.
problem Uniformisation problems on projective varieties with klt singularities.
method Generalising Jahnke-Radloff's work, study torus quotients and varieties with semistable cotangent sheaves and extremal Chern classes.
result Torus quotients are the only klt varieties with semistable cotangent sheaves and extremal Chern classes.
Paper proves conditions for rational homology complex projective planes with singularities.
problem Proving conditions for rational homology complex projective planes with singularities.
method Leveraging results from smooth 4-manifolds, including Donaldson diagonalization theorem and Heegaard Floer correction terms.
result Eliminates the possibility of a rational homology complex projective plane with four singularities and identifies families of singularities obstructed by smooth conditions.
Using standard methods for studying singularities of projections and of contacts, we classify the stable singularities of affine λ-equidistants of n-dimensional closed submanifolds of Rq, for q≤2n, whenever (2n,q) is a pair of nice dimensions.
Investigate the local geometry of smooth surfaces in 4-space via contact with 2-planes and apparent contours.
problem Local geometry of smooth surfaces in 4-space
method Contact with 2-planes and apparent contour
result Prove connections between singularities of parallel projections, orthogonal projections, and height functions.
This paper explores how local behavior of meromorphic connections on the projective line determines the global connection.
problem Determining the global meromorphic connection based on specified local behavior at singular points.
method Expository discussion of various problems related to meromorphic connections with specified local behavior, including Deligne-Simpson and rigidity problems.
result The existence and nonemptiness of moduli spaces of meromorphic connections with specified local behavior.
Curve Shortening Flow preserves circularity for convex projections.
problem Understanding the behavior of curves under Curve Shortening Flow.
method Contradiction argument and analysis of tangent flows.
result Smooth curves with convex projections become asymptotically circular under Curve Shortening Flow.
The paper proves a theorem for non-integrable projective distributions of degree one.
problem Classifying non-integrable projective distributions of degree one.
method Proving a singular Darboux type theorem for homogeneous polynomial closed 2-forms of degree one.
result Classification of non-integrable codimension one distributions of degree one.
By the quantization condition compact quantizable Kaehler manifolds can be embedded into projective space. In this way they become projective varieties. The quantum Hilbert space of the Berezin-Toeplitz quantization (and of the geometric quantization) is the projective coordinate ring of the embedded manifold. This all…
The paper updates SVD of evolving matrices using projection techniques.
problem Updating the rank-k truncated SVD of evolving matrices.
method Projection viewpoint, building subspaces to approximate singular vectors.
result The proposed algorithm leads to higher accuracy, especially for large singular values.
Study on 2D special Kaehler structures focusing on singularities.
problem Characterizing singularities in 2D special Kaehler structures.
method Analysis of isolated singularities and computation of monodromy.
result Construction of continuous families of special Kaehler structures with isolated singularities.
The paper relates curvature loci of different manifold types through projections and normal sections.
problem Understanding the geometry of manifolds and their curvature loci.
method Using normal sections and projections to relate curvature loci of different manifold types.
result A commutative diagram of projections and normal sections that relates the curvature loci of different types of manifolds.
Computes L∞-algebroid for linear foliations on vector spaces.
problem Invariants of singular foliations on vector spaces induced by Lie subalgebras.
method Explicitly constructs projective resolutions and computes L∞-algebroid structure. result Provides invariants and constant-rank replacements of singular foliations.
A new framework for dimension reduction using ensemble of random projections.
problem High-dimensional regression problems with limited data.
method Aggregating an ensemble of carefully chosen random projections, retaining based on empirical performance, and selecting singular vectors.
result The proposed method stabilizes error as the number of projection groups increases.
We study the geometry of the cuspidal edge M in R3 derived from its contact with planes and lines (referred to as flat geometry). The contact of M with planes is measured by the singularities of the height functions on M. We classify submersions on a model of M by diffeomorphisms and recover the cont…
Proves certain Calabi-Yau varieties are projective.
problem Compact Calabi-Yau varieties with isolated singularities are not always projective.
method Analysis and Ohsawa's degenerate spectral sequence in higher dimensions.
result Proves compact Calabi-Yau varieties with certain isolated singularities are projective.
Optimizes bounds for multiple T-singularities on surfaces.
problem Bounding T-singularities on non-rational projective surfaces with many singularities.
method Analyzes combinatorial configurations and classifies them to find optimal bounds.
result Classifies all combinatorial configurations leading to high bounds, proving their non-existence gives optimal bounds.
Singular Finsler metrics, such as Kropina metrics and m-Kropina metrics, have a lot of applications in the real world. In this paper, we study a class of singular Finsler metrics defined by a Riemann metric α and 1-form β and characterize those which are respectively Douglasian and locally projectively flat in di…
The paper classifies surfaces with pseudo-effective tangent bundles.
problem Understanding projective manifolds with pseudo-effective tangent bundles.
method Developed singular hermitian metrics on vector bundles and applied to projective manifolds.
result Projective manifolds with pseudo-effective tangent bundles admit a smooth fibration to a flat projective manifold.
Singular Finsler metrics, such as Kropina metrics and m-Kropina metrics, have a lot of applications in the real world. In this paper, we classify a class of singular (α,β)-metrics which are locally projectively flat with constant flag curvature in dimension n=2 and n≥3 respectively. Further, we determine t…
In 4-space, cross-ratios help classify surface singularities.
problem Classifying surfaces in 4-dimensional space.
method Establish cross-ratios for surfaces in 4-space, study singularities of projections.
result Recover two moduli from cross-ratios at specific points, show stable configurations of asymptotic curves.
Solved a conjecture about rational homology projective planes with quotient singularities.
problem A conjecture about rational homology projective planes with quotient singularities.
method Combining Donaldson's diagonalization theorem with a distinguished spin^c structure on the smooth locus.
result Proved that rational homology projective planes with quotient singularities have at most three singular points.
Uniform bounds prove connection between Kähler metrics and RCD spaces.
problem Bounding Nash entropy and Calabi energy for Kähler metrics.
method Proving uniform Sobolev bounds for Kähler manifolds.
result Establishes connection to RCD spaces and provides examples.
It is given the diffeomorphism classification on generic singularities of tangent varieties to curves with arbitrary codimension in a projective space. The generic classifications are performed in terms of certain geometric structures and differential systems on flag manifolds, via several techniques in differentiable …
We study fundamental groups of projective varieties with normal crossing singularities and of germs of complex singularities. We prove that for every finitely-presented group G there is a complex projective surface S with simple normal crossing singularities only, so that the fundamental group of S is isomorphic to G. …
Study of curves and surfaces from single-direction projections.
problem Obtaining complete shape information from a single view.
method Theoretical study of differential geometric information from multiple orthogonal projections.
result Formulae for recovering certain information on curves or surfaces from their projections.
Extends Serre-Swan theorem to all finitely generated modules over smooth functions.
problem Classical Serre-Swan theorem limitations.
method Introduces tepui fibrations and singular vector bundles.
result Realizes all finitely generated modules over smooth functions.
We show that on any compact Riemann surface with variable negative curvature there exists a measure which is invariant and ergodic under the geodesic flow and whose projection to the base manifold is 2-dimensional and singular with respect to the 2-dimensional Lebesgue measure.
Characterizes and projects convolutional layers' singular values for improved deep learning performance.
problem Improving deep learning models' performance using convolutional layers.
method Characterizes and projects the singular values of convolutional layers, providing an effective regularizer.
result Improves test error of a deep residual network on CIFAR-10 from 6.2% to 5.3%.
Establishes K"ahler-Ricci flow on log canonical varieties.
problem Existence and convergence of K"ahler-Ricci flow on varieties with log canonical singularities.
method Generalizes previous results for klt singularities, proves convergence, and constructs solutions with flips.
result Existence and convergence of K"ahler-Ricci flow on semi-log canonical models.
The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C^1 of double numbers, and the algebra C^0 of dual numbers. In the space RP^5, to these smooth lines there correspond families of straight lines describing point three-dimensional tangentially degenerate submanifolds …
Study singularities of Weinstein skeleta and arboreal singularities.
problem Understanding singularities in Weinstein skeleta and their relation to arboreal singularities.
method Deforming the skeleton via homotopies, Morse-Bott representative, generic perturbation, localization procedure.
result Singularities of Weinstein skeleta can be classified into arboreal singularities or tangency singularities.
Complex projective varieties are quotients of polydiscs under specific group actions.
problem Characterizing complex projective varieties as quotients of polydiscs.
method Proving varieties are quotients by groups acting properly discontinuously and freely in codimension one.
result Complex projective varieties with klt singularities and ample canonical divisors are quotients of the polydisc.
Decomposes singular Kähler spaces with trivial first Chern class into simpler components.
problem Understanding the structure of singular Kähler spaces with specific properties.
method Beauville-Bogomolov decomposition and small projective deformations.
result Compact Kähler fourfolds with trivial first Chern class decompose into simpler components.
The study of projective varieties with nef anticanonical divisors and log terminal singularities.
problem Understanding the structure and properties of projective varieties with specific divisor conditions.
method Analyzing the Albanese map and MRC fibration for klt projective varieties, showing locally constant fibrations and product decompositions.
result Generalization of results for smooth projective varieties to the klt case, including decomposition into rationally connected and projective varieties with trivial canonical divisor.
The paper sharpens the analysis of sketch-and-project methods using randomized singular value decomposition.
problem Improving convergence rates of sketch-and-project methods for solving linear systems and non-linear optimization problems.
method Developing a theoretical framework and new spectral bounds for the expected sketched projection matrix.
result The convergence rate improves linearly with sketch size and even faster with certain spectral decays.