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Journal articles on the topic 'Donaldson's theorem'

Author: Grafiati

Published: 4 June 2021

Last updated: 11 February 2022

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1

Greene, Joshua Evan. "A note on applications of the d-invariant and Donaldson's theorem." Journal of Knot Theory and Its Ramifications 26, no. 02 (February 2017): 1740006. http://dx.doi.org/10.1142/s0218216517400065.

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This paper contains two remarks about the application of the [Formula: see text]-invariant in Heegaard Floer homology and Donaldson's diagonalization theorem to knot theory. The first is the equivalence of two obstructions they give to a 2-bridge knot being smoothly slice. The second carries out a suggestion by Stefan Friedl to replace the use of Heegaard Floer homology by Donaldson's theorem in the proof of the main result of [J. E. Greene, Lattices, graphs, and Conway mutation, Invent. Math. 192(3) (2013) 717–750] concerning Conway mutation of alternating links.

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2

Tosatti, Valentino, and Ben Weinkove. "The Calabi–Yau equation on the Kodaira–Thurston manifold." Journal of the Institute of Mathematics of Jussieu 10, no. 2 (September 24, 2010): 437–47. http://dx.doi.org/10.1017/s1474748010000289.

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AbstractWe prove that the Calabi–Yau equation can be solved on the Kodaira–Thurston manifold for all given T2-invariant volume forms. This provides support for Donaldson's conjecture that Yau's theorem has an extension to symplectic 4-manifolds with compatible but non-integrable almost complex structures.

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3

Greene, Joshua Evan. "Donaldson's theorem, Heegaard Floer homology, and knots with unknotting number one." Advances in Mathematics 255 (April 2014): 672–705. http://dx.doi.org/10.1016/j.aim.2014.01.018.

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4

Lisca, Paolo. "A vanishing theorem for Donaldson invariants." Proceedings of the American Mathematical Society 123, no. 2 (February 1, 1995): 607. http://dx.doi.org/10.1090/s0002-9939-1995-1233978-5.

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5

Owens, Brendan. "SMOOTH, NONSYMPLECTIC EMBEDDINGS OF RATIONAL BALLS IN THE COMPLEX PROJECTIVE PLANE." Quarterly Journal of Mathematics 71, no. 3 (June 30, 2020): 997–1007. http://dx.doi.org/10.1093/qmathj/haaa013.

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Abstract We exhibit an infinite family of rational homology balls, which embed smoothly but not symplectically in the complex projective plane. We also obtain a new lattice embedding obstruction from Donaldson’s diagonalization theorem and use this to show that no two of our examples may be embedded disjointly.

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6

Steer, Brian, and Andrew Wren. "The Donaldson-Hitchin-Kobayashi Correspondence for Parabolic Bundles over Orbifold Surfaces." Canadian Journal of Mathematics 53, no. 6 (December 1, 2001): 1309–39. http://dx.doi.org/10.4153/cjm-2001-047-x.

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AbstractA theorem of Donaldson on the existence of Hermitian-Einstein metrics on stable holomorphic bundles over a compact Kähler surface is extended to bundles which are parabolic along an effective divisor with normal crossings. Orbifold methods, together with a suitable approximation theorem, are used following an approach successful for the case of Riemann surfaces.

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7

Efimov, Alexander I. "Cohomological Hall algebra of a symmetric quiver." Compositio Mathematica 148, no. 4 (May 15, 2012): 1133–46. http://dx.doi.org/10.1112/s0010437x12000152.

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AbstractIn [M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, Preprint (2011), arXiv:1006.2706v2[math.AG]], the authors, in particular, associate to each finite quiver Q with a set of vertices I the so-called cohomological Hall algebra ℋ, which is ℤI≥0-graded. Its graded component ℋγ is defined as cohomology of the Artin moduli stack of representations with dimension vector γ. The product comes from natural correspondences which parameterize extensions of representations. In the case of a symmetric quiver, one can refine the grading to ℤI≥0×ℤ, and modify the product by a sign to get a super-commutative algebra (ℋ,⋆) (with parity induced by the ℤ-grading). It is conjectured in [M. Kontsevich and Y. Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants, Preprint (2011), arXiv:1006.2706v2[math.AG]] that in this case the algebra (ℋ⊗ℚ,⋆) is free super-commutative generated by a ℤI≥0×ℤ-graded vector space of the form V =Vprim ⊗ℚ[x] , where x is a variable of bidegree (0,2)∈ℤI≥0×ℤ, and all the spaces ⨁ k∈ℤVprimγ,k, γ∈ℤI≥0. are finite-dimensional. In this paper we prove this conjecture (Theorem 1.1). We also prove some explicit bounds on pairs (γ,k) for which Vprimγ,k≠0 (Theorem 1.2). Passing to generating functions, we obtain the positivity result for quantum Donaldson–Thomas invariants, which was used by Mozgovoy to prove Kac’s conjecture for quivers with sufficiently many loops [S. Mozgovoy, Motivic Donaldson–Thomas invariants and Kac conjecture, Preprint (2011), arXiv:1103.2100v2[math.AG]]. Finally, we mention a connection with the paper of Reineke [M. Reineke, Degenerate cohomological Hall algebra and quantized Donaldson–Thomas invariants for m-loop quivers, Preprint (2011), arXiv:1102.3978v1[math.RT]].

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8

Anderson, Judith H. "Commenting on Donaldson's Commentaries." Chaucer Review 41, no. 3 (2007): 271–78. http://dx.doi.org/10.1353/cr.2007.0000.

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9

Lê, Quy Thuong. "The motivic Thom–Sebastiani theorem for regular and formal functions." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 735 (February 1, 2018): 175–98. http://dx.doi.org/10.1515/crelle-2015-0022.

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AbstractThanks to the work of Hrushovski and Loeser on motivic Milnor fibers, we give a model-theoretic proof for the motivic Thom–Sebastiani theorem in the case of regular functions. Moreover, slightly extending Hrushovski–Loeser’s construction adjusted to Sebag, Loeser and Nicaise’s motivic integration for formal schemes and rigid varieties, we formulate and prove an analogous result for formal functions. The latter is meaningful as it has been a crucial element of constructing Kontsevich–Soibelman’s theory of motivic Donaldson–Thomas invariants.

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10

Farrell, Thomas J. "The Persistence of Donaldson's Memory." Chaucer Review 41, no. 3 (2007): 289–98. http://dx.doi.org/10.1353/cr.2007.0003.

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11

IGA, KEVIN. "WHAT DO TOPOLOGISTS WANT FROM SEIBERG–WITTEN THEORY?" International Journal of Modern Physics A 17, no. 30 (December 10, 2002): 4463–514. http://dx.doi.org/10.1142/s0217751x0201217x.

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In 1983, Donaldson shocked the topology world by using instantons from physics to prove new theorems about four-dimensional manifolds, and he developed new topological invariants. In 1988, Witten showed how these invariants could be obtained by correlation functions for a twisted N = 2 SUSY gauge theory. In 1994, Seiberg and Witten discovered dualities for such theories, and in particular, developed a new way of looking at four-dimensional manifolds that turns out to be easier, and is conjectured to be equivalent to, Donaldson theory. This review describes the development of this mathematical subject, and shows how the physics played a pivotal role in the current understanding of this area of topology.

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12

Nicaise, Johannes, and Sam Payne. "A tropical motivic Fubini theorem with applications to Donaldson–Thomas theory." Duke Mathematical Journal 168, no. 10 (July 2019): 1843–86. http://dx.doi.org/10.1215/00127094-2019-0003.

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13

Hurtubise, Jacques. "Monopoles and rational maps: A note on a theorem of Donaldson." Communications in Mathematical Physics 100, no. 2 (June 1985): 191–96. http://dx.doi.org/10.1007/bf01212447.

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14

Cochran, Tim D., and Robert E. Gompf. "Applications of Donaldson's theorems to classical knot concordance, homology 3-spheres and Property P." Topology 27, no. 4 (1988): 495–512. http://dx.doi.org/10.1016/0040-9383(88)90028-6.

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15

Tanaka, Yuuji. "A weak compactness theorem of the Donaldson–Thomas instantons on compact Kähler threefolds." Journal of Mathematical Analysis and Applications 408, no. 1 (December 2013): 27–34. http://dx.doi.org/10.1016/j.jmaa.2013.05.059.

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16

Tanaka, Yuuji. "A removal singularity theorem of the Donaldson–Thomas instanton on compact Kähler threefolds." Journal of Mathematical Analysis and Applications 411, no. 1 (March 2014): 422–28. http://dx.doi.org/10.1016/j.jmaa.2013.09.053.

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17

DAVISON, BEN, JARED ONGARO, and BALÁZS SZENDRŐI. "Enumerating coloured partitions in 2 and 3 dimensions." Mathematical Proceedings of the Cambridge Philosophical Society 169, no. 3 (July 19, 2019): 479–505. http://dx.doi.org/10.1017/s0305004119000252.

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AbstractWe study generating functions of ordinary and plane partitions coloured by the action of a finite subgroup of the corresponding special linear group. After reviewing known results for the case of ordinary partitions, we formulate a conjecture concerning a basic factorisation property of the generating function of coloured plane partitions that can be thought of as an orbifold analogue of a conjecture of Maulik et al., now a theorem, in three-dimensional Donaldson–Thomas theory. We study natural quantisations of the generating functions arising from geometry, discuss a quantised version of our conjecture, and prove a positivity result for the quantised coloured plane partition function under a geometric assumption.

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18

Lawton, David. "Donaldson and Irony." Chaucer Review 41, no. 3 (2007): 231–39. http://dx.doi.org/10.1353/cr.2007.0008.

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19

Harvey, Melanee C. "Jeff Donaldson." Callaloo 40, no. 5 (2017): 19–28. http://dx.doi.org/10.1353/cal.2017.0151.

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20

Gursky, Matthew J., and Jeffrey Streets. "A formal Riemannian structure on conformal classes and the inverse Gauss curvature flow." Geometric Flows 4, no. 1 (January 1, 2019): 30–50. http://dx.doi.org/10.1515/geofl-2019-0003.

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Abstract We define a formal Riemannian metric on a given conformal class of metrics with signed curvature on a closed Riemann surface. As it turns out this metric is the well-known Mabuchi-Semmes-Donaldson metric of Kähler geometry in a different guise. The metric has many interesting properties, and in particular we show that the classical Liouville energy is geodesically convex. This suggests a different approach to the uniformization theorem by studying the negative gradient flow of the normalized Liouville energy with respect to this metric, a new geometric flow whose principal term is the inverse of the Gauss curvature. We prove long time existence of solutions with arbitrary initial data and weak convergence to constant scalar curvature metrics by exploiting the metric space structure.

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21

Kirk, Elizabeth D. "Donaldson Teaching and Learning." Chaucer Review 41, no. 3 (2007): 279–88. http://dx.doi.org/10.1353/cr.2007.0007.

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22

Gust, Geoffrey W. "Revaluating "Chaucer the Pilgrim" and Donaldson's Enduring Persona." Chaucer Review 41, no. 3 (2007): 311–23. http://dx.doi.org/10.1353/cr.2007.0004.

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23

Kronheimer, P. B., and T. S. Mrowka. "Embedded surfaces and the structure of Donaldson's polynomial invariants." Journal of Differential Geometry 41, no. 3 (1995): 573–734. http://dx.doi.org/10.4310/jdg/1214456482.

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24

Borroff, Marie. "Donaldson and the Romantic Poets." Chaucer Review 41, no. 3 (2007): 225–30. http://dx.doi.org/10.1353/cr.2007.0001.

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25

TELEMAN, ANDREI. "FAMILIES OF HOLOMORPHIC BUNDLES." Communications in Contemporary Mathematics 10, no. 04 (August 2008): 523–51. http://dx.doi.org/10.1142/s0219199708002892.

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The first goal of the article is to solve several fundamental problems in the theory of holomorphic bundles over non-algebraic manifolds. For instance, we prove that stability and semi-stability are Zariski open properties in families when the Gauduchon degree map is a topological invariant, or when the parameter manifold is compact. Second, we show that, for a generically stable family of bundles over a Kähler manifold, the Petersson–Weil form extends as a closed positive current on the whole parameter space of the family. This extension theorem uses classical tools from Yang–Mills theory (e.g., the Donaldson functional on the space of Hermitian metrics and its properties). We apply these results to study families of bundles over a Kählerian manifold Y parametrized by a non-Kählerian surface X, proving that such families must satisfy very restrictive conditions. These results play an important role in our program to prove existence of curves on class VII surfaces [22–24].

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26

Hanna, Ralph. "Donaldson and Robertson: An Obligatory Conjunction." Chaucer Review 41, no. 3 (2007): 240–49. http://dx.doi.org/10.1353/cr.2007.0005.

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27

Maulik, Davesh, and Alexei Oblomkov. "Donaldson–Thomas theory of 𝒜n×P1." Compositio Mathematica 145, no. 5 (August 18, 2009): 1249–76. http://dx.doi.org/10.1112/s0010437x09003972.

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AbstractWe study the relative Donaldson–Thomas theory of 𝒜n×P1, where 𝒜n is the surface resolution of type An singularity. The action of divisor operators in the theory is expressed in terms of operators of the affine algebra $\glh $ on Fock space. Assuming a nondegeneracy conjecture, this gives a complete solution for the theory. The results complete the comparison of this theory with the Gromov–Witten theory of 𝒜n×P1 and the quantum cohomology of the Hilbert scheme of points on 𝒜n.

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28

Göttsche, Lothar, Hiraku Nakajima, and Kōta Yoshioka. "Instanton counting and Donaldson invariants." Journal of Differential Geometry 80, no. 3 (November 2008): 343–90. http://dx.doi.org/10.4310/jdg/1226090481.

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29

Dloussky, Georges. "Non Kählerian surfaces with a cycle of rational curves." Complex Manifolds 8, no. 1 (January 1, 2021): 208–22. http://dx.doi.org/10.1515/coma-2020-0114.

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Abstract Let S be a compact complex surface in class VII0 + containing a cycle of rational curves C = ∑Dj . Let D = C + A be the maximal connected divisor containing C. If there is another connected component of curves C ′ then C ′ is a cycle of rational curves, A = 0 and S is a Inoue-Hirzebruch surface. If there is only one connected component D then each connected component Ai of A is a chain of rational curves which intersects a curve Dj of the cycle and for each curve Dj of the cycle there at most one chain which meets Dj . In other words, we do not prove the existence of curves other those of the cycle C, but if some other curves exist the maximal divisor looks like the maximal divisor of a Kato surface with perhaps missing curves. The proof of this topological result is an application of Donaldson theorem on trivialization of the intersection form and of deformation theory. We apply this result to show that a twisted logarithmic 1-form has a trivial vanishing divisor.

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30

Mozgovoy, Sergey. "Motivic Donaldson–Thomas invariants and the Kac conjecture." Compositio Mathematica 149, no. 3 (February 28, 2013): 495–504. http://dx.doi.org/10.1112/s0010437x13007148.

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AbstractWe derive some combinatorial consequences from the positivity of Donaldson–Thomas invariants for symmetric quivers conjectured by Kontsevich and Soibelman and proved recently by Efimov. These results are used to prove the Kac conjecture for quivers having at least one loop at every vertex.

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31

Wofford, Tobias. "Jeff Donaldson’s Howard: A Center of Trans-African Art." Callaloo 40, no. 5 (2017): 29–35. http://dx.doi.org/10.1353/cal.2017.0152.

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32

Van Dyke, Carolynn. "Amorous Behavior: Sexism, Sin, and the Donaldson Persona." Chaucer Review 41, no. 3 (2007): 250–60. http://dx.doi.org/10.1353/cr.2007.0011.

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33

Bayer, Arend, Martí Lahoz, Emanuele Macrì, Howard Nuer, Alexander Perry, and Paolo Stellari. "Stability conditions in families." Publications mathématiques de l'IHÉS 133, no. 1 (May 17, 2021): 157–325. http://dx.doi.org/10.1007/s10240-021-00124-6.

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AbstractWe develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers.Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type.Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

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34

Mieszkowski, Gretchen. ""The Least Innocent of All Innocent-Sounding Lines": The Legacy of Donaldson's Troilus Criticism." Chaucer Review 41, no. 3 (2007): 299–310. http://dx.doi.org/10.1353/cr.2007.0009.

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35

Teleman, Andrei. "Analytic cycles in flip passages and in instanton moduli spaces over non-Kählerian surfaces." International Journal of Mathematics 27, no. 07 (June 2016): 1640009. http://dx.doi.org/10.1142/s0129167x16400097.

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Let [Formula: see text] ([Formula: see text]) be a moduli space of stable (polystable) bundles with fixed determinant on a complex surface with [Formula: see text] and let [Formula: see text] be a pure [Formula: see text]-dimensional analytic set. We prove a general formula for the homological boundary [Formula: see text] of the Borel–Moore fundamental class of [Formula: see text] in the boundary of the blown up moduli space [Formula: see text]. The proof is based on the holomorphic model theorem of [A. Teleman, Instanton moduli spaces on non-Kählerian surfaces, Holomorphic models around the reduction loci, J. Geom. Phys. 91 (2015) 66–87] which identifies a neighborhood of a boundary component of [Formula: see text] with a neighborhood of the boundary of a “blown up flip passage”. We then focus on a particular instanton moduli space which intervenes in our program for proving the existence of curves on class VII surfaces. Using our result, combined with general properties of the Donaldson cohomology classes, we prove incidence relations between the Zariski closures (in the considered moduli space) of certain families of extensions. These incidence relations are crucial for understanding the geometry of the moduli space, and cannot be obtained using classical complex geometric deformation theory.

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36

Cazzaniga, Alberto, Andrew Morrison, Brent Pym, and Balázs Szendrői. "Motivic Donaldson–Thomas invariants of some quantized threefolds." Journal of Noncommutative Geometry 11, no. 3 (2017): 1115–39. http://dx.doi.org/10.4171/jncg/11-3-10.

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37

Wheeler, Bonnie. "The Legacy of New Criticism: Revisiting the Work of Donaldson." Chaucer Review 41, no. 3 (2007): 216–24. http://dx.doi.org/10.1353/cr.2007.0012.

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38

Butzer, Karl W., and Pascal James Imperato. "Arthur Donaldson Smith and the Exploration of Lake Rudolf." African Arts 22, no. 1 (November 1988): 96. http://dx.doi.org/10.2307/3336703.

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39

EICHHORN, JÜRGEN. "GAUGE THEORY ON OPEN MANIFOLDS OF BOUNDED GEOMETRY." International Journal of Modern Physics A 07, no. 17 (July 10, 1992): 3927–77. http://dx.doi.org/10.1142/s0217751x92001769.

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On compact manifolds (Mn, g) and for r>n/2+1 the configuration space [Formula: see text] is a well-defined object. [Formula: see text] is an affine space with a Sobolev space as vector space, and [Formula: see text] a Hilbert Lie group which acts smoothly and properly on [Formula: see text]. [Formula: see text] is a stratified space with Hilbert manifolds as strata. The existence problem has been solved for many interesting cases by Cliff Taubes and the description of the moduli space of instantons has been given by Donaldson. On noncompact manifolds none of the approaches of the compact case is further valid. We present here an intrinsic, self-consistent approach for gauge theory on open manifolds of bounded geometry up to order n/2+2. The main idea is to endow the space CP of gauge potentials and the gauge group with an intrinsic Sobolev topology. Bounded geometry of the underlying manifold and the considered connections provides all the Sobolev theorems which are needed to prove the existence of instantons if G=SU(2). We prove the existence of instantons if (M4, g) satisfies a certain spectral condition and has a positive definite L2 intersection form.

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40

Toda, Yukinobu. "On a computation of rank two Donaldson–Thomas invariants." Communications in Number Theory and Physics 4, no. 1 (2010): 49–102. http://dx.doi.org/10.4310/cntp.2010.v4.n1.a2.

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41

Diaconescu, Duiliu-Emanuel, Zheng Hua, and Yan Soibelman. "HOMFLY polynomials, stable pairs and motivic Donaldson–Thomas invariants." Communications in Number Theory and Physics 6, no. 3 (2012): 517–600. http://dx.doi.org/10.4310/cntp.2012.v6.n3.a1.

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42

Mozgovoy, Sergey, and Olivier Schiffmann. "Counting Higgs bundles and type quiver bundles." Compositio Mathematica 156, no. 4 (February 27, 2020): 744–69. http://dx.doi.org/10.1112/s0010437x20007010.

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We prove a closed formula counting semistable twisted (or meromorphic) Higgs bundles of fixed rank and degree over a smooth projective curve of genus $g$ defined over a finite field, when the twisting line bundle degree is at least $2g-2$ (this includes the case of usual Higgs bundles). This yields a closed expression for the Donaldson–Thomas invariants of the moduli spaces of twisted Higgs bundles. We similarly deal with twisted quiver sheaves of type $A$ (finite or affine), obtaining in particular a Harder–Narasimhan-type formula counting semistable $U(p,q)$-Higgs bundles over a smooth projective curve defined over a finite field.

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43

Hanning, Robert W. "No [One] Way to Treat a Text: Donaldson and the Criticism of Engagement." Chaucer Review 41, no. 3 (2007): 261–70. http://dx.doi.org/10.1353/cr.2007.0006.

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44

Kontsevich, Maxim, and Yan Soibelman. "Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson–Thomas invariants." Communications in Number Theory and Physics 5, no. 2 (2011): 231–52. http://dx.doi.org/10.4310/cntp.2011.v5.n2.a1.

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45

Göttsche, Lothar, and Yao Yuan. "Generating functions for $K$-theoretic Donaldson invariants and Le Potier’s strange duality." Journal of Algebraic Geometry 28, no. 1 (September 4, 2018): 43–98. http://dx.doi.org/10.1090/jag/703.

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46

Malmendier, Andreas, and Ken Ono. "Moonshine for $M_{24}$ and Donaldson invariants of $\mathbb{C} \mathrm{P}^2$." Communications in Number Theory and Physics 6, no. 4 (2012): 759–70. http://dx.doi.org/10.4310/cntp.2012.v6.n4.a1.

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47

Chiwome, Emmanuel. "BOOK REVIEW: Zimbabwe.THE ZIMBABWE REVIEW, 3 Donaldson Lane, P/Bag A 6177, Avondale, Harare, E-mail: zimreview@mango.zw." Research in African Literatures 32, no. 1 (March 2001): 164–65. http://dx.doi.org/10.2979/ral.2001.32.1.164.

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48

Elliott, J. E. "Conjecturing the Common in English Common Law: Donaldson v. Beckett and the Rhetoric of Ancient Right." Forum for Modern Language Studies 42, no. 4 (October 1, 2006): 431–46. http://dx.doi.org/10.1093/fmls/cql074.

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49

LETHBRIDGE, R. "Review. A Woman's Revenge: The Chronology of Dispossession in Maupassant's Fiction. Donaldson-Evans, Mary." French Studies 42, no. 3 (July 1, 1988): 358. http://dx.doi.org/10.1093/fs/42.3.358.

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50

Hassan, Hussein. "Review: Sharecropping in the Yemen: A Study in Islamic Theory, Custom and Pragmatism William J. Donaldson." Journal of Islamic Studies 14, no. 1 (January 1, 2003): 76–79. http://dx.doi.org/10.1093/jis/14.1.76.

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