Volume 17, issue 4 (2013)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 26
Issue 2, 477–936
Issue 1, 1–476

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
Asymptotically cylindrical Calabi–Yau $3$–folds from weak Fano $3$–folds

Alessio Corti, Mark Haskins, Johannes Nordström and Tommaso Pacini

Geometry & Topology 17 (2013) 1955–2059
1 Y Akizuki, S Nakano, Note on Kodaira–Spencer's proof of Lefschetz theorems, Proc. Japan Acad. 30 (1954) 266 MR0066694
2 M Arap, J W Cutrone, N A Marshburn, On the existence of certain weak Fano threefolds of Picard number two arXiv:1112.2611
3 E Arbarello, M Cornalba, P A Griffiths, J Harris, Geometry of algebraic curves, Vol I, Grundl. Math. Wissen. 267, Springer (1985) MR770932
4 M Artin, D Mumford, Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. 25 (1972) 75 MR0321934
5 V V Batyrev, Toric Fano threefolds, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981) 704, 927 MR631434
6 A Beauville, Fano threefolds and $K3$ surfaces, from: "The Fano Conference", Univ. Torino, Turin (2004) 175 MR2112574
7 C Birkar, P Cascini, C D Hacon, J McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010) 405 MR2601039
8 J Blanc, S Lamy, Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links, Proc. Lond. Math. Soc. 105 (2012) 1047 MR2997046
9 G Brown, A Kasprzyk, The graded rings database homepage
10 F Campana, On twistor spaces of the class $\mathcal{C}$, J. Differential Geom. 33 (1991) 541 MR1094468
11 C Casagrande, P Jahnke, I Radloff, On the Picard number of almost Fano threefolds with pseudo-index $\gt 1$, Internat. J. Math. 19 (2008) 173 MR2384898
12 K Chan, S C Lau, N C Leung, H H Tseng, Open Gromov–Witten invariants and mirror maps for semi-Fano toric manifolds arXiv:1112.0388
13 I Cheltsov, Nonrational nodal quartic threefolds, Pacific J. Math. 226 (2006) 65 MR2247856
14 C H Clemens, P A Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. 95 (1972) 281 MR0302652
15 T Coates, M Haskins, A M Kasprzyk, J Nordström, in preparation
16 A Conte, J P Murre, On quartic threefolds with a double line, I, Nederl. Akad. Wetensch. Proc. Ser. A 39 (1977) 145 MR0498594
17 A Corti, M Haskins, J Nordström, T Pacini, $\mathrm{G}_2$–manifolds and associative submanifolds via semi-Fano $3$–folds arXiv:1207.4470
18 J W Cutrone, N A Marshburn, Towards the classification of weak Fano threefolds with $\rho=2$ arXiv:1009.5036
19 O Debarre, Higher-dimensional algebraic geometry, Springer (2001) MR1841091
20 I Dolgachev, Integral quadratic forms: applications to algebraic geometry (after V Nikulin), from: "Bourbaki seminar, Vol. 1982/83", Astérisque 105, Soc. Math. France (1983) 251 MR728992
21 I V Dolgachev, Mirror symmetry for lattice polarized $K3$ surfaces, J. Math. Sci. 81 (1996) 2599 MR1420220
22 S Endraß, On the divisor class group of double solids, Manuscripta Math. 99 (1999) 341 MR1702593
23 H Esnault, E Viehweg, Lectures on vanishing theorems, DMV Seminar 20, Birkhäuser (1992) MR1193913
24 H Finkelnberg, Small resolutions of the Segre cubic, Nederl. Akad. Wetensch. Indag. Math. 49 (1987) 261 MR914085
25 H Finkelnberg, On the Geometry of the Burkhardt Quartic, PhD thesis, Leiden (1989)
26 H Finkelnberg, J Werner, Small resolutions of nodal cubic threefolds, Nederl. Akad. Wetensch. Indag. Math. 51 (1989) 185 MR1005050
27 R Friedman, Simultaneous resolution of threefold double points, Math. Ann. 274 (1986) 671 MR848512
28 A Fujiki, S Nakano, Supplement to “On the inverse of monoidal transformation”, Publ. Res. Inst. Math. Sci. 7 (1971/72) 637 MR0294712
29 O Fujino, Y Gongyo, On images of weak Fano manifolds, Math. Z. 270 (2012) 531 MR2875847
30 W Fulton, Intersection theory, Ergeb. Math. Grenzgeb. 2, Springer (1984) MR732620
31 I M Gel’fand, A V Zelevinskiĭ, M M Kapranov, Hypergeometric functions and toric varieties, Funktsional. Anal. i Prilozhen. 23 (1989) 12 MR1011353
32 M Goresky, R MacPherson, Stratified Morse theory, Ergeb. Math. Grenzgeb. 14, Springer (1988) MR932724
33 M Haskins, H J Hein, J Nordström, Asymptotically cylindrical Calabi–Yau manifolds arXiv:1212.6929
34 A Iliev, L Katzarkov, V Przyjalkowski, Double solids, categories and non-rationality arXiv:1102.2130
35 N O Ilten, Deformations of Rational Varieties with Codimension-One Torus Action, PhD thesis, Freie Universität Berlin (2010)
36 V A Iskovskih, Fano threefolds, I, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977) 516, 717 MR463151
37 V A Iskovskih, Fano threefolds, II, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978) 506 MR503430
38 V A Iskovskikh, Y G Prokhorov, Fano varieties, from: "Algebraic geometry, V", Encyclopaedia Math. Sci. 47, Springer (1999) 1 MR1668579
39 P Jahnke, T Peternell, Almost del Pezzo manifolds, Adv. Geom. 8 (2008) 387 MR2427467
40 P Jahnke, T Peternell, I Radloff, Threefolds with big and nef anticanonical bundles, I, Math. Ann. 333 (2005) 569 MR2198800
41 P Jahnke, T Peternell, I Radloff, Threefolds with big and nef anticanonical bundles, II, Cent. Eur. J. Math. 9 (2011) 449 MR2784025
42 P Jahnke, I Radloff, Gorenstein Fano threefolds with base points in the anticanonical system, Compos. Math. 142 (2006) 422 MR2218903
43 A J de Jong, N I Shepherd-Barron, A Van de Ven, On the Burkhardt quartic, Math. Ann. 286 (1990) 309 MR1032936
44 D D Joyce, Compact manifolds with special holonomy, Oxford University Press (2000) MR1787733
45 A S Kaloghiros, The topology of terminal quartic $3$–folds arXiv:0707.1852
46 A S Kaloghiros, The defect of Fano $3$–folds, J. Algebraic Geom. 20 (2011) 127 MR2729277
47 A S Kaloghiros, A classification of terminal quartic $3$–folds and applications to rationality questions, Math. Ann. 354 (2012) 263 MR2957627
48 S Katz, D R Morrison, Gorenstein threefold singularities with small resolutions via invariant theory for Weyl groups, J. Algebraic Geom. 1 (1992) 449 MR1158626
49 Y Kawamata, Crepant blowing-up of $3$–dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math. 127 (1988) 93 MR924674
50 Y Kawamata, General hyperplane sections of nonsingular flops in dimension $3$, Math. Res. Lett. 1 (1994) 49 MR1258489
51 Y Kawamata, K Matsuki, The number of the minimal models for a $3$–fold of general type is finite, Math. Ann. 276 (1987) 595 MR879538
52 K Kodaira, On a differential-geometric method in the theory of analytic stacks, Proc. Nat. Acad. Sci. U. S. A. 39 (1953) 1268 MR0066693
53 J Kollár, Flips, flops, minimal models, etc, from: "Surveys in differential geometry (Cambridge, MA, 1990)", Lehigh Univ. (1991) 113 MR1144527
54 J Kollár, Y Miyaoka, S Mori, Rationally connected varieties, J. Algebraic Geom. 1 (1992) 429 MR1158625
55 J Kollár, Y Miyaoka, S Mori, H Takagi, Boundedness of canonical $\mathbb{Q}$–Fano $3$–folds, Proc. Japan Acad. Ser. A Math. Sci. 76 (2000) 73 MR1771144
56 J Kollár, S Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, Cambridge Univ. Press (1998) MR1658959
57 A Kovalev, Twisted connected sums and special Riemannian holonomy, J. Reine Angew. Math. 565 (2003) 125 MR2024648
58 A Kovalev, N H Lee, $K3$ surfaces with non-symplectic involution and compact irreducible $G_2$–manifolds, Math. Proc. Cambridge Philos. Soc. 151 (2011) 193 MR2823130
59 M Kreuzer, H Skarke, Classification of reflexive polyhedra in three dimensions, Adv. Theor. Math. Phys. 2 (1998) 853 MR1663339
60 P B Kronheimer, A Torelli-type theorem for gravitational instantons, J. Differential Geom. 29 (1989) 685 MR992335
61 H B Laufer, On $\mathbb{C}\!P^{1}$ as an exceptional set, from: "Recent developments in several complex variables", Ann. of Math. Stud. 100, Princeton Univ. Press (1981) 261 MR627762
62 R Lazarsfeld, Positivity in algebraic geometry, I, Ergeb. Math. Grenzgeb. 48, Springer (2004) MR2095471
63 K Matsuki, Introduction to the Mori program, Springer (2002) MR1875410
64 J W Milnor, J D Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton Univ. Press (1974) MR0440554
65 B G Moĭšezon, On $n$–dimensional compact complex manifolds having $n$ algebraically independent meromorphic functions I, Izv. Akad. Nauk SSSR Ser. Mat. 30 (1966) 133 MR0216522
66 S Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. 116 (1982) 133 MR662120
67 S Mori, S Mukai, Classification of Fano $3$–folds with $B_{2}\geq 2$, Manuscripta Math. 36 (1981/82) 147 MR641971
68 S Mori, S Mukai, Erratum: Classification of Fano $3$–folds with $B_{2}\geq 2$, Manuscripta Math. 110 (2003) 407 MR641971
69 S Mori, S Mukai, Extremal rays and Fano $3$–folds, from: "The Fano Conference", Univ. Torino, Turin (2004) 37 MR2112566
70 S Mukai, New developments in Fano manifold theory related to the vector bundle method and moduli problems, Sūgaku 47 (1995) 125 MR1364825
71 S Mukai, Plane quartics and Fano threefolds of genus twelve, from: "The Fano Conference", Univ. Torino, Turin (2004) 563 MR2112592
72 S Mukai, H Umemura, Minimal rational threefolds, from: "Algebraic geometry", Lecture Notes in Math. 1016, Springer (1983) 490 MR726439
73 S Nakano, On the inverse of monoidal transformation, Publ. Res. Inst. Math. Sci. 6 (1970/71) 483 MR0294710
74 Y Namikawa, Smoothing Fano $3$–folds, J. Algebraic Geom. 6 (1997) 307 MR1489117
75 Y Namikawa, J H M Steenbrink, Global smoothing of Calabi–Yau threefolds, Invent. Math. 122 (1995) 403 MR1358982
76 B Nill, Gorenstein toric Fano varieties, PhD thesis, Universität Tübingen (2005)
77 T Oda, Convex bodies and algebraic geometry, Ergeb. Math. Grenzgeb. 15, Springer (1988) MR922894
78 K Oguiso, Birational automorphism groups and the movable cone theorem for Calabi–Yau manifolds of Wehler type via universal Coxeter groups arXiv:1107.5862
79 R Paoletti, The Kähler cone in families of quasi–Fano threefolds, Math. Z. 227 (1998) 45 MR1605373
80 K Pettersen, On Nodal Determinantal Quartic Hypersurfaces in $\mathbb{P}^{4}$, PhD thesis, University of Oslo (1998)
81 H C Pinkham, Factorization of birational maps in dimension $3$, from: "Singularities, Part 2", Proc. Sympos. Pure Math. 40, Amer. Math. Soc. (1983) 343 MR713260
82 J Rambau, TOPCOM: triangulations of point configurations and oriented matroids, from: "Mathematical software", World Sci. Publ. (2002) 330 MR1932619
83 Z Ran, Deformations of manifolds with torsion or negative canonical bundle, J. Algebraic Geom. 1 (1992) 279 MR1144440
84 M Reid, Minimal models of canonical $3$–folds, from: "Algebraic varieties and analytic varieties", Adv. Stud. Pure Math. 1, North-Holland (1983) 131 MR715649
85 M Reid, Projective morphisms according to Kawamata, Warwick preprint (1983)
86 M Reid, Young person's guide to canonical singularities, from: "Algebraic geometry", Proc. Sympos. Pure Math. 46, Amer. Math. Soc. (1987) 345 MR927963
87 A N Rudakov, I R Shafarevich, Surfaces of type $K3$ over fields of finite characteristic, from: "Current problems in mathematics", Akad. Nauk SSSR (1981) 115 MR633161
88 T Sano, Unobstructedness of deformations of weak Fano manifolds, Int. Math. Res. Not. (to appear) arXiv:1302.0705
89 K H Shin, $3$–dimensional Fano varieties with canonical singularities, Tokyo J. Math. 12 (1989) 375 MR1030501
90 K A Shramov, $\mathbb{Q}$–factorial quartic threefolds, Mat. Sb. 198 (2007) 103 MR2355438
91 V V Šokurov, Smoothness of a general anticanonical divisor on a Fano variety, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979) 430 MR534602
92 N E Steenrod, Cohomology operations, Annals of Mathematics Studies 50, Princeton Univ. Press (1962) MR0145525
93 K Takeuchi, Weak Fano threefolds with del Pezzo fibration arXiv:0910.2188
94 K Takeuchi, Some birational maps of Fano $3$–folds, Compositio Math. 71 (1989) 265 MR1022045
95 G Tian, S T Yau, Complete Kähler manifolds with zero Ricci curvature, I, J. Amer. Math. Soc. 3 (1990) 579 MR1040196
96 G Tian, S T Yau, Complete Kähler manifolds with zero Ricci curvature, II, Invent. Math. 106 (1991) 27 MR1123371
97 J A Todd, On a quartic primal with forty-five nodes, in space of four dimensions, Q. J. Math. 7 (1936) 168
98 A N Varchenko, Semicontinuity of the spectrum and an upper bound for the number of singular points of the projective hypersurface, Dokl. Akad. Nauk SSSR 270 (1983) 1294 MR712934
99 J Werner, Kleine Auflösungen spezieller dreidimensionaler Varietäten, Bonner Mathematische Schriften 186, Universität Bonn Mathematisches Institut (1987) MR930270
100 J A Wiśniewski, On deformation of nef values, Duke Math. J. 64 (1991) 325 MR1136378
101 S T Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I, Comm. Pure Appl. Math. 31 (1978) 339 MR480350