Publications
68 results found
Donaldson S, 2021, ATIYAH'S WORK ON HOLOMORPHIC VECTOR BUNDLES AND GAUGE THEORIES, BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 58, Pages: 567-610, ISSN: 0273-0979
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- Citations: 2
Donaldson S, 2021, INTRODUCTION TO THE ATIYAH ISSUE, BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 58, Pages: 479-480, ISSN: 0273-0979
Donaldson S, 2021, DEFORMATIONS OF MULTIVALUED HARMONIC FUNCTIONS, QUARTERLY JOURNAL OF MATHEMATICS, Vol: 72, Pages: 199-235, ISSN: 0033-5606
Donaldson S, 2019, An elliptic boundary value problem for G2-structures, Annales de l'Institut Fourier, Vol: 68, Pages: 2783-2809, ISSN: 0373-0956
We show that the G2 holonomy equation on a manifold with boundary, with prescribed 3-form on the boundary, is elliptic. The main point is to set up a suitable linear elliptic boundary value problem. This result leads to a deformation theory. In particular we establish the existence of certain G2 cobordisms between two small deformations of a Calabi-Yau 3-fold.
Donaldson SK, 2018, Remarks on G2 structures with boundary, Journal of Differential Geometry Conference
Donaldson SK, 2018, Stability of algebraic varieties and Kahler geometry, American-Mathematical-Society Summer Research Institute on Algebraic Geometry, Publisher: AMER MATHEMATICAL SOC, Pages: 199-221, ISSN: 2324-707X
Donaldson S, Sun S, 2017, Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry, II, Journal of Differential Geometry, Vol: 107, Pages: 327-371, ISSN: 0022-040X
We study Gromov–Hausdorff limits of Kähler–Einstein manifolds, in particular, their singularities, and connections with algebraic geometry. This is a continuation of our previous work.
Donaldson S, 2016, Kahler-Einstein metrics and algebraic structures on limit spaces, Conference on Geometry and Topology, Publisher: INTERNATIONAL PRESS INC BOSTON, Pages: 85-94, ISSN: 1052-9233
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- Citations: 2
Chen X, Donaldson S, Sun S, 2015, Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2 pi and completion of the main proof, Journal of the American Mathematical Society, Vol: 28, Pages: 235-278, ISSN: 0894-0347
This is the third and final article in a series which prove the fact that a K-stable Fano manifold admits a Kähler-Einstein metric. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle approaches 2$ \pi $. We also put all our technical results together to complete the proof of the main theorem.
Chen X, Donaldson S, Sun S, 2015, Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2 pi, Journal of the American Mathematical Society, Vol: 28, Pages: 199-234, ISSN: 0894-0347
This is the second of a series of three papers which prove the fact that a K-stable Fano manifold admits a Kähler-Einstein metric. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle is less than 2$ {\pi }$. We show that these are in a natrual way projective algebraic varieties. In the case when the limiting variety and the limiting divisor are smooth we show that the limiting metric also has standard cone singularities.
Chen X, Donaldson S, Sun S, 2015, KAHLER-EINSTEIN METRICS ON FANO MANIFOLDS. I: APPROXIMATION OF METRICS WITH CONE SINGULARITIES, JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 28, Pages: 183-197, ISSN: 0894-0347
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- Citations: 76
Donaldson S, 2015, Algebraic families of constant scalar curvature Kahler metrics, REGULARITY AND EVOLUTION OF NONLINEAR EQUATIONS: ESSAYS DEDICATED TO RICHARD HAMILTON, LEON SIMON, AND KAREN UHLENBECK, Vol: 19, Pages: 111-137, ISSN: 1052-9233
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- Citations: 14
Donaldson S, Sun S, 2014, Gromov-Hausdorff limits of Kahler manifolds and algebraic geometry, ACTA MATHEMATICA, Vol: 213, Pages: 63-106, ISSN: 0001-5962
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- Citations: 95
Chen X-X, Donaldson SK, 2014, Integral bounds on curvature and Gromov-Hausdorff limits, JOURNAL OF TOPOLOGY, Vol: 7, Pages: 543-556, ISSN: 1753-8416
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- Citations: 2
Donaldson SK, 2014, b-STABILITY AND BLOW-UPS, PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY, Vol: 57, Pages: 125-137, ISSN: 0013-0915
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- Citations: 3
Chen X, Donaldson S, Sun S, 2014, Kahler-Einstein Metrics and Stability, INTERNATIONAL MATHEMATICS RESEARCH NOTICES, Vol: 2014, Pages: 2119-2125, ISSN: 1073-7928
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- Citations: 48
Chen X-X, Donaldson SK, 2013, VOLUME ESTIMATES FOR KAHLER-EINSTEIN METRICS AND RIGIDITY OF COMPLEX STRUCTURES, JOURNAL OF DIFFERENTIAL GEOMETRY, Vol: 93, Pages: 191-201, ISSN: 0022-040X
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- Citations: 5
Chen X-X, Donaldson SK, 2013, VOLUME ESTIMATES FOR KAHLER-EINSTEIN METRICS: THE THREE-DIMENSIONAL CASE, JOURNAL OF DIFFERENTIAL GEOMETRY, Vol: 93, Pages: 175-189, ISSN: 0022-040X
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- Citations: 1
Donaldson S, 2012, Kahler metrics with cone singularities along a divisor, Essays in mathematics and its applications : In honor of Stephen Smale's 80th birthday., Editors: Pardalos, Rassias, Publisher: Springer, Pages: 49-79, ISBN: 978-3-642-28820-3
We develop some foundations for the study of Kahler-Einstein metrics withcone singularities transverse to a divisor. The main goal is a treatment of thedeformation of the cone angle.
Donaldson SK, 2009, Constant Scalar Curvature Metrics on Toric Surfaces, GEOMETRIC AND FUNCTIONAL ANALYSIS, Vol: 19, Pages: 83-136, ISSN: 1016-443X
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- Citations: 100
Donaldson SK, 2009, Lie Algebra Theory without Algebra, ALGEBRA, ARITHMETIC, AND GEOMETRY: IN HONOR OF YU I MANIN, VOL I, Vol: 269, Pages: 549-566, ISSN: 0743-1643
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- Citations: 1
Donaldson SK, 2009, Some numerical results in complex differential geometry, Pure and Applied Mathematics Quarterly, Vol: 5, Pages: 571-618, ISSN: 1558-8599
Donaldson SK, 2008, Some problems in differential geometry and topology, NONLINEARITY, Vol: 21, Pages: T157-T164, ISSN: 0951-7715
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- Citations: 5
Donaldson SK, 2008, Extremal metrics on toric surfaces: A continuity method, JOURNAL OF DIFFERENTIAL GEOMETRY, Vol: 79, Pages: 389-432, ISSN: 0022-040X
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- Citations: 66
Donaldson SK, 2008, A generalised Joyce construction for a family of nonlinear partial differential equations
We explain a simple construction of solutions to a family of PDE's in twodimensions which includes that defining zero scalar curvature Kahler metrics,with two Killing fields, and the affine maximal equation.
Donaldson S, Fine J, 2006, Tonic anti-self-dual 4-manifolds via complex geometry, MATHEMATISCHE ANNALEN, Vol: 336, Pages: 281-309, ISSN: 0025-5831
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- Citations: 4
Donaldson SK, 2005, Lower bounds on the Calabi functional, JOURNAL OF DIFFERENTIAL GEOMETRY, Vol: 70, Pages: 453-472, ISSN: 0022-040X
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- Citations: 139
Donaldson SK, 2005, Scalar curvature and projective embeddings, II, Quarterly Journal of Mathematics, Vol: 56, Pages: 345-356, ISSN: 0033-5606
Donaldson SK, 2005, Interior estimates for solutions of Abreu's equation, COLLECTANEA MATHEMATICA, Vol: 56, Pages: 103-142, ISSN: 0010-0757
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- Citations: 85
Auroux D, Donaldson SK, Katzarkov L, 2005, Singular Lefschetz pencils, GEOMETRY & TOPOLOGY, Vol: 9, Pages: 1043-1114, ISSN: 1465-3060
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- Citations: 53
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