Geometry & Topology Volume 25, issue 4 (2021)

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Complex algebraic compactifications of the moduli space of Hermitian Yang–Mills connections on a projective manifold

Daniel Greb, Benjamin Sibley, Matei Toma and Richard Wentworth

Geometry & Topology 25 (2021) 1719–1818

DOI: 10.2140/gt.2021.25.1719

Bibliography

1	V Ancona, G Tomassini, Modifications analytiques, 943, Springer (1982) MR673560
2	M Artin, Algebraization of formal moduli, II : Existence of modifications, Ann. of Math. 91 (1970) 88 MR260747
3	S Bando, Y T Siu, Stable sheaves and Einstein–Hermitian metrics, from: "Geometry and analysis on complex manifolds" (editors T Mabuchi, J Noguchi, T Ochiai), World Sci. (1994) 39 MR1463962
4	D Barlet, Convexité de l’espace des cycles, Bull. Soc. Math. France 106 (1978) 373 MR518045
5	D Barlet, J Magnússon, Cycles analytiques complexes, II : L’espace des cycles, 27, Soc. Math. France (2020) MR4242816
6	P Belmans, A J de Jong, others, The Stacks project, electronic reference (2005–)
7	E Bishop, Conditions for the analyticity of certain sets, Michigan Math. J. 11 (1964) 289 MR168801
8	N Buchdahl, A Teleman, M Toma, A continuity theorem for families of sheaves on complex surfaces, J. Topol. 10 (2017) 995 MR3743066
9	H Cartan, Quotients of complex analytic spaces, from: "Contributions to function theory" (editor K Chandrasekharan), Tata Inst. Fund. Res. (1960) 1 MR0139769
10	D Chakrabarti, M C Shaw, The Cauchy–Riemann equations on product domains, Math. Ann. 349 (2011) 977 MR2777041
11	X Chen, S Sun, Algebraic tangent cones of reflexive sheaves, Int. Math. Res. Not. 2020 (2020) 10042 MR4190396
12	X Chen, S Sun, Analytic tangent cones of admissible Hermitian–Yang–Mills connections, Geom. Topol. (2021) 2061
13	X Chen, S Sun, Reflexive sheaves, Hermitian–Yang–Mills connections, and tangent cones, Invent. Math. 225 (2021) 73 MR4270664
14	G D Daskalopoulos, R A Wentworth, Convergence properties of the Yang–Mills flow on Kähler surfaces, J. Reine Angew. Math. 575 (2004) 69 MR2097548
15	I Dolgachev, Lectures on invariant theory, 296, Cambridge Univ. Press (2003) MR2004511
16	S K Donaldson, Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. 50 (1985) 1 MR765366
17	S K Donaldson, Connections, cohomology and the intersection forms of 4–manifolds, J. Differential Geom. 24 (1986) 275 MR868974
18	S K Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J. 54 (1987) 231 MR885784
19	S K Donaldson, Compactification and completion of Yang–Mills moduli spaces, from: "Differential geometry" (editors F J Carreras, O Gil-Medrano, A M Naveira), Lecture Notes in Math. 1410, Springer (1989) 145 MR1034277
20	S K Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990) 257 MR1066174
21	S K Donaldson, P B Kronheimer, The geometry of four-manifolds, Oxford Univ. Press (1990) MR1079726
22	S Donaldson, E Segal, Gauge theory in higher dimensions, II, from: "Geometry of special holonomy and related topics" (editors N C Leung, S T Yau), Surv. Differential Geom. 16, International (2011) 1 MR2893675
23	G Ellingsrud, M Lehn, Irreducibility of the punctual quotient scheme of a surface, Ark. Mat. 37 (1999) 245 MR1714770
24	G B Folland, J J Kohn, The Neumann problem for the Cauchy–Riemann complex, 75, Princeton Univ. Press (1972) MR0461588
25	A Fujiki, G Schumacher, The moduli space of Hermite–Einstein bundles on a compact Kähler manifold, Proc. Japan Acad. Ser. A Math. Sci. 63 (1987) 69 MR894698
26	W Fulton, Intersection theory, 2, Springer (1984) MR732620
27	D Gieseker, On the moduli of vector bundles on an algebraic surface, Ann. of Math. 106 (1977) 45 MR466475
28	D Gilbarg, N S Trudinger, Elliptic partial differential equations of second order, 224, Springer (1983) MR737190
29	D Greb, S Kebekus, T Peternell, Étale fundamental groups of Kawamata log terminal spaces, flat sheaves, and quotients of abelian varieties, Duke Math. J. 165 (2016) 1965 MR3522654
30	D Greb, J Ross, M Toma, Moduli of vector bundles on higher-dimensional base manifolds: construction and variation, Int. J. Math. 27 (2016) MR3605660
31	D Greb, M Toma, Compact moduli spaces for slope-semistable sheaves, Algebr. Geom. 4 (2017) 40 MR3592465
32	A Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique, IV : Les schémas de Hilbert, from: "Séminaire Bourbaki, 1960/1961", Benjamin (1966) MR1611822
33	L Hörmander, L² estimates and existence theorems for the ∂ operator, Acta Math. 113 (1965) 89 MR179443
34	D Huybrechts, M Lehn, The geometry of moduli spaces of sheaves, Cambridge Univ. Press (2010) MR2665168
35	S Kobayashi, Differential geometry of complex vector bundles, 15, Princeton Univ. Press (1987) MR909698
36	J Kollár, Quotients by finite equivalence relations, from: "Current developments in algebraic geometry" (editors L Caporaso, J McKernan, M Mustaţă, M Popa), Math. Sci. Res. Inst. Publ. 59, Cambridge Univ. Press (2012) 227 MR2931872
37	A Langer, Semistable sheaves in positive characteristic, Ann. of Math. 159 (2004) 251 MR2051393
38	J Le Potier, Fibré déterminant et courbes de saut sur les surfaces algébriques, from: "Complex projective geometry" (editors G Ellingsrud, C Peskine, G Sacchiero, S A Strømme), Lond. Math. Soc. Lect. Note Ser. 179, Cambridge Univ. Press (1992) 213 MR1201385
39	J Li, Algebraic geometric interpretation of Donaldson’s polynomial invariants, J. Differential Geom. 37 (1993) 417 MR1205451
40	M Lübke, C Okonek, Moduli spaces of simple bundles and Hermitian–Einstein connections, Math. Ann. 276 (1987) 663 MR879544
41	M Lübke, A Teleman, The Kobayashi–Hitchin correspondence, World Sci. (1995) MR1370660
42	M Maruyama, Openness of a family of torsion free sheaves, J. Math. Kyoto Univ. 16 (1976) 627 MR429899
43	M Maruyama, On boundedness of families of torsion free sheaves, J. Math. Kyoto Univ. 21 (1981) 673 MR637512
44	V B Mehta, A Ramanathan, Semistable sheaves on projective varieties and their restriction to curves, Math. Ann. 258 (1982) 213 MR649194
45	K Miyajima, Kuranishi family of vector bundles and algebraic description of the moduli space of Einstein–Hermitian connections, Publ. Res. Inst. Math. Sci. 25 (1989) 301 MR1003790
46	J W Morgan, Comparison of the Donaldson polynomial invariants with their algebro-geometric analogues, Topology 32 (1993) 449 MR1231956
47	A Naber, D Valtorta, Energy identity for stationary Yang Mills, Invent. Math. 216 (2019) 847 MR3955711
48	H Nakajima, Compactness of the moduli space of Yang–Mills connections in higher dimensions, J. Math. Soc. Japan 40 (1988) 383 MR945342
49	C Okonek, M Schneider, H Spindler, Vector bundles on complex projective spaces, 3, Birkhäuser (1980) MR561910
50	M Pavel, Moduli spaces of slope-semistable pure sheaves, preprint (2021) arXiv:2105.09395
51	S Sedlacek, A direct method for minimizing the Yang–Mills functional over 4–manifolds, Comm. Math. Phys. 86 (1982) 515 MR679200
52	M C Shaw, Global solvability and regularity for ∂ on an annulus between two weakly pseudoconvex domains, Trans. Amer. Math. Soc. 291 (1985) 255 MR797058
53	M C Shaw, The closed range property for ∂ on domains with pseudoconcave boundary, from: "Complex analysis" (editors P Ebenfelt, N Hungerbühler, J J Kohn, N Mok, E J Straube), Birkhäuser (2010) 307 MR2885124
54	B Sibley, Asymptotics of the Yang–Mills flow for holomorphic vector bundles over Kähler manifolds : the canonical structure of the limit, J. Reine Angew. Math. 706 (2015) 123 MR3393366
55	B Sibley, R A Wentworth, Analytic cycles, Bott–Chern forms, and singular sets for the Yang–Mills flow on Kähler manifolds, Adv. Math. 279 (2015) 501 MR3345190
56	T Tao, G Tian, A singularity removal theorem for Yang–Mills fields in higher dimensions, J. Amer. Math. Soc. 17 (2004) 557 MR2053951
57	G Tian, Gauge theory and calibrated geometry, I, Ann. of Math. 151 (2000) 193 MR1745014
58	G Tian, B Yang, Compactification of the moduli spaces of vortices and coupled vortices, J. Reine Angew. Math. 553 (2002) 17 MR1944806
59	K Uhlenbeck, A priori estimates for Yang–Mills fields, unpublished manuscript
60	K K Uhlenbeck, Connections with L^p bounds on curvature, Comm. Math. Phys. 83 (1982) 31 MR648356
61	K K Uhlenbeck, Removable singularities in Yang–Mills fields, Comm. Math. Phys. 83 (1982) 11 MR648355
62	K Uhlenbeck, S T Yau, On the existence of Hermitian–Yang–Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39 (1986) MR861491
63	K Wehrheim, Uhlenbeck compactness, Eur. Math. Soc. (2004) MR2030823
64	B Yang, The uniqueness of tangent cones for Yang–Mills connections with isolated singularities, Adv. Math. 180 (2003) 648 MR2020554