793:
Essentially, different cohomology theories on singular target spaces yield different results thereby making it difficult to determine which theory physics may favor. Several important characteristics of the cohomology, which correspond to the massless fields, are based on general properties of field
142:
and has a very strong algebraic flavour, although the main examples arising from
Goresky–MacPherson theory are topological in nature because the simple objects in the category of perverse sheaves are the intersection cohomology complexes. This motivated MacPherson to recast the whole theory in
805:
package (T. Hubsch, 1992), should hold for singular and smooth target spaces. Paul Green and
Tristan Hubsch (P. Green & T. Hubsch, 1988) determined that the manner in which you move between singular CY target spaces require moving through either a
118:
comes through rough translation of the French "faisceaux pervers". The justification is that perverse sheaves are complexes of sheaves which have several features in common with sheaves: they form an abelian category, they have
517:
71:
in general represented by a complex of sheaves. The concept of perverse sheaves is already implicit in a 75's paper of
Kashiwara on the constructibility of solutions of holonomic D-modules.
817:
theory should be for singular target spaces. Tristan Hubsch and Abdul Rahman (T. Hubsch and A. Rahman, 2005) worked to solve the Hubsch conjecture by analyzing the non-transversal case of
581:
549:
86:
could be described using sheaf complexes that are actually perverse sheaves. It was clear from the outset that perverse sheaves are fundamental mathematical objects at the crossroads of
240:
297:
864:
required revisiting Hubsch's conjecture of a
Stringy Singular Cohomology (T. Hubsch, 1997). In the winter of 2002, T. Hubsch and A. Rahman met with R.M. Goresky to discuss this
1026:
Corollary 3.2. of A. Beilinson. How to glue perverse sheaves. In: K-theory, arithmetic and geometry (Moscow, 1984), Lecture Notes in Math. 1289, Springer-Verlag, 1987, 42 – 51.
398:
368:
475:
441:
681:
932:
884:
and T. Hubsch advised A. Rahman's Ph.D. dissertation on the construction of a self-dual perverse sheaf (A. Rahman, 2009) using the zig-zag construction of
907:
is still an open problem. Markus Banagl (M. Banagl, 2010; M. Banagl, et al., 2014) addressed the Hubsch conjecture through intersection spaces for higher
622:
Perverse sheaves are a fundamental tool for the geometry of singular spaces. Therefore, they are applied in a variety of mathematical areas. In the
67:. Perverse sheaves are the objects in the latter that correspond to individual D-modules (and not more general complexes thereof); a perverse sheaf
1903:
876:, R. MacPherson made the observation that there was such a perverse sheaf that could have the cohomology that satisfied Hubsch's conjecture and
147:. For many applications in representation theory, perverse sheaves can be treated as a 'black box', a category with certain formal properties.
606:
is equivalent to the derived category of constructible sheaves and similarly for sheaves on the complex analytic space associated to a scheme
900:, and aligned with some of the properties of the Kähler package. Satisfaction of all of the Kähler package by this Perverse sheaf for higher
591:
The category of perverse sheaves is an abelian subcategory of the (non-abelian) derived category of sheaves, equal to the core of a suitable
1063:
1347:, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics , vol. 42, Berlin, New York:
484:
914:
inspired by Hubsch's work (T. Hubsch, 1992, 1997; P. Green and T. Hubsch, 1988) and A. Rahman's original ansatz (A. Rahman, 2009) for
848:
point. T. Hubsch and A. Rahman determined the (co)-homology of this ground state variety in all dimensions, found it compatible with
748:. A singular target space means that only the CY manifold part is singular as the Minkowski space factor is smooth. Such a singular
1202:
1887:
1704:
1474:
1356:
1163:
1827:
1373:
885:
873:
83:
123:, and to construct one, it suffices to construct it locally everywhere. The adjective "perverse" originates in the
554:
522:
1716:
849:
623:
52:
33:
1714:
HĂĽbsch, Tristan; Rahman, Abdul (2005). "On the geometry and homology of certain simple stratified varieties".
1238:
Bremer, Christopher L.; Sage, Daniel S. (2013), "Generalized Serre conditions and perverse coherent sheaves",
185:
245:
892:(R. MacPherson & K. Vilonen, 1986). This perverse sheaf proved the HĂĽbsch conjecture for isolated conic
1972:
911:
904:
822:
647:
124:
1088:
1956:
718:
477:
stratified so that the origin is the unique singular stratum. Then the category of perverse sheaves on
1392:(2007), "Geometric Langlands duality and representations of algebraic groups over commutative rings",
782:
can change. These singular target spaces, i.e. conifolds, correspond to certain mild degenerations of
976:
Les faisceaux pervers n'etant ni des faisceaux, ni pervers, la terminologie requiert une explication.
379:
340:
1781:
458:
749:
730:
408:
135:
134:
The
Beilinson–Bernstein–Deligne definition of a perverse sheaf proceeds through the machinery of
1837:
Rahman, Abdul (2009). "A perverse sheaf approach toward a cohomology theory for string theory".
733:). The determination of the matter and interaction content requires a detailed analysis of the
635:
95:
414:
1977:
1394:
937:
631:
103:
75:
656:
1934:
1790:
1763:
1735:
1665:
1618:
1577:
1503:
1433:
1366:
1324:
1271:
1229:
1210:
1173:
1127:
1107:
1076:
798:
775:
1779:
MacPherson, Robert; Vilonen, Kari (1986). "Elementary constructions of perverse sheaves".
630:. This application establishes the notion of perverse sheaf as occurring 'in nature'. The
8:
1182:
1143:
651:
168:
139:
64:
44:
1794:
1739:
1669:
1622:
1581:
1507:
1111:
897:
1938:
1912:
1864:
1846:
1806:
1767:
1725:
1681:
1655:
1634:
1593:
1567:
1519:
1493:
1449:
1437:
1403:
1328:
1275:
1249:
1240:
1131:
1097:
1055:
927:
915:
893:
845:
844:
over a certain base with a 1-dimensional exocurve (termed exo-strata) attached at each
841:
830:
757:
745:
715:
87:
1464:
1119:
1883:
1823:
1810:
1771:
1751:
1700:
1638:
1589:
1515:
1470:
1421:
1352:
1332:
1159:
826:
783:
763:
25:
1942:
1868:
1685:
1597:
1441:
1279:
1135:
1922:
1898:
1856:
1798:
1747:
1743:
1673:
1626:
1585:
1544:
1523:
1511:
1413:
1312:
1263:
1259:
1186:
1151:
1115:
699:
164:
56:
40:
21:
1484:
Strominger, Andrew (1995). "Massless black holes and conifolds in string theory".
1150:, Lecture Notes in Mathematics, vol. 1289, Berlin: Springer, pp. 27–41,
942:
583:
are invertible. More generally, quivers can be used to describe perverse sheaves.
1930:
1877:
1759:
1694:
1606:
1429:
1362:
1348:
1340:
1320:
1267:
1225:
1206:
1169:
1123:
1072:
1051:
726:
688:
596:
1926:
1860:
1377:
836:
Under certain conditions it was determined that this ground state variety was a
802:
1463:
Beilinson, Alexander; Bernstein, Joseph; Deligne, Pierre; Gabber, Ofer (2018).
1417:
1190:
877:
865:
861:
857:
807:
684:
405:
48:
1677:
1316:
404:
is a flat, locally complete intersection (for example, regular) scheme over a
1966:
1755:
1549:
1532:
1425:
957:
952:
853:
818:
787:
99:
1389:
1288:
1086:
Arinkin, Dmitry; Bezrukavnikov, Roman (2010). "Perverse coherent sheaves".
991:
889:
881:
869:
771:
643:
639:
374:
144:
79:
1300:
908:
901:
592:
1730:
1454:
1802:
1660:
1630:
1572:
1498:
1155:
814:
795:
767:
734:
722:
120:
1558:
Witten, Edward (1993). "Phases of n = 2 theories in two dimensions".
1408:
840:(P. Green & T.Hubsch, 1988; T. Hubsch, 1992) with isolated conic
766:
observed (A. Strominger, 1995) that conifolds correspond to massless
602:
The bounded derived category of perverse l-adic sheaves on a scheme
1879:
Intersection Spaces, Spatial
Homology Truncation, and String Theory
1292:
987:
947:
837:
779:
778:—including the fact that the space can tear near the cone, and its
753:
627:
91:
60:
29:
1917:
1899:"Intersection spaces, perverse sheaves and type IIB string theory"
1851:
1448:
Rietsch, Konstanze (2003). "An introduction to perverse sheaves".
1254:
1102:
829:(termed the ground state variety) in the case of isolated conical
512:{\displaystyle V{\overset {u}{\underset {v}{\rightleftarrows }}}W}
330:
is a smooth complex algebraic variety and everywhere of dimension
744:
However, a troubling consequence occurs when the target space is
738:
1462:
1345:
Weil conjectures, perverse sheaves and l'adic
Fourier transform
794:
theories, specifically, the (2,2)-supersymmetric 2-dimensional
737:
of these spaces: nearly all massless fields in the effective
1646:
HĂĽbsch, Tristan (1997). "On a stringy singular cohomology".
821:
gauged linear sigma model (E. Witten, 1993) which induces a
774:
explains the physics of conifolds in
Chapter 13 of his book
1224:, Instituto Nacional de Matemática Pura e Aplicada (IMPA),
810:(T. Hubsch, 1992) and called it the 'conifold transition'.
481:
is equivalent to the category of diagrams of vector spaces
1882:. Lecture Notes in Mathematics. Vol. 1997. Springer.
1222:
Introduction to intersection homology and perverse sheaves
790:
theories, including superstring theory (E. Witten, 1982).
1181:
1002:
741:
model are represented by certain (co)homology elements.
1146:(1987), "On the derived category of perverse sheaves",
813:
Tristan Hubsch (T. Hubsch, 1997) conjectured what this
638:
decomposition, requires the usage of perverse sheaves.
1897:
Banagl, Markus; Budur, Nero; Maxim, Laurențiu (2014).
1293:"What is the etymology of the term "perverse sheaf"?"
1148:
K-theory, arithmetic and geometry (Moscow, 1984–1986)
659:
557:
525:
487:
461:
417:
382:
343:
248:
188:
1085:
1050:
770:. Conifolds are important objects in string theory:
1607:"Connecting moduli spaces of Calabi-Yau threefolds"
988:
What is the etymology of the term "perverse sheaf"?
725:classes on the target space (i.e. four-dimensional
626:, perverse sheaves correspond to regular holonomic
808:small resolution or deformation of the singularity
675:
575:
543:
511:
469:
435:
392:
362:
291:
234:
1778:
1964:
1904:Advances in Theoretical and Mathematical Physics
1896:
1839:Advances in Theoretical and Mathematical Physics
1696:Calabi-Yau Manifolds: A Bestiary for Physicists
1387:
695:
650:identifies equivariant perverse sheaves on the
1339:
703:
55:, which establishes a connection between the
1713:
1469:. Astérisque. Vol. 100 (2nd ed.).
1378:"Intersection Homology and Perverse Sheaves"
1064:Notices of the American Mathematical Society
576:{\displaystyle \operatorname {id} -v\circ u}
544:{\displaystyle \operatorname {id} -u\circ v}
1604:
756:as it is a CY manifold that admits conical
1957:Intersection homology and perverse sheaves
1483:
1372:
1237:
150:
39:The concept was introduced in the work of
1916:
1850:
1729:
1659:
1571:
1548:
1497:
1453:
1407:
1253:
1219:
1142:
1101:
1035:
463:
1605:Green, Paul S.; HĂĽbsch, Tristan (1988).
1003:Beilinson, Bernstein & Deligne (1982
235:{\displaystyle H^{-i}(j_{x}^{*}C)\neq 0}
127:theory, and its origin was explained by
1447:
1299:
1287:
1014:
292:{\displaystyle H^{i}(j_{x}^{!}C)\neq 0}
128:
1965:
1875:
1836:
1817:
1692:
1645:
1611:Communications in Mathematical Physics
1557:
1530:
860:(T. Hubsch and A. Rahman, 2005). This
109:
98:. They also play an important role in
411:, then the constant sheaf shifted by
646:refinement of perverse sheaves. The
1303:(2003). "Perversité et variation".
858:obstruction in the middle dimension
702:using perverse sheaves is given in
446:
13:
1950:
634:, a far-reaching extension of the
385:
346:
319:is the inclusion map of the point
14:
1989:
1120:10.17323/1609-4514-2010-10-1-3-29
801:. These properties, known as the
786:which appear in a large class of
28:, which may be a real or complex
20:refers to the objects of certain
1537:Journal of Differential Geometry
1533:"Supersymmetry and Morse theory"
709:
455:be a disk around the origin in
1717:Journal of Geometry and Physics
617:
74:A key observation was that the
51:(1982) as a consequence of the
34:topologically stratified spaces
1748:10.1016/j.geomphys.2004.04.010
1264:10.1016/j.jalgebra.2013.06.018
1203:Société Mathématique de France
1029:
1020:
1008:
996:
981:
969:
624:Riemann-Hilbert correspondence
494:
393:{\displaystyle {\mathcal {F}}}
363:{\displaystyle {\mathcal {F}}}
357:
351:
280:
259:
223:
202:
143:geometric terms on a basis of
53:Riemann-Hilbert correspondence
1:
1220:Brasselet, Jean-Paul (2009),
1193:(1982). "Faisceaux pervers".
1044:
696:Mirković & Vilonen (2007)
586:
1590:10.1016/0550-3213(93)90033-L
1516:10.1016/0550-3213(95)00287-3
1343:; Weissauer, Rainer (2001),
704:Kiehl & Weissauer (2001)
683:with representations of the
648:geometric Satake equivalence
470:{\displaystyle \mathbb {C} }
443:is an Ă©tale perverse sheaf.
373:is a perverse sheaf for any
302:has real dimension at most 2
175:such that the set of points
7:
1927:10.4310/ATMP.2014.v18.n2.a3
1861:10.4310/ATMP.2009.v13.n3.a3
1089:Moscow Mathematical Journal
1056:"What is a perverse sheaf?"
1054:; Migliorini, Luca (2010).
921:
868:and in discussions between
10:
1994:
1959:, notes by Bruno Klingler.
1418:10.4007/annals.2007.166.95
1005:, Proposition 2.2.2, §4.0)
721:have been identified with
1678:10.1142/S0217732397000546
1383:(unpublished manuscript).
1317:10.1007/s00229-003-0407-z
642:are, roughly speaking, a
1782:Inventiones Mathematicae
1693:HĂĽbsch, Tristan (1994).
1648:Modern Physics Letters A
963:
878:resolved the obstruction
731:Calabi-Yau (CY) manifold
436:{\displaystyle \dim X+1}
1876:Banagl, Markus (2010).
1531:Witten, Edward (1982).
1305:Manuscripta Mathematica
1183:Beilinson, Alexander A.
1144:Beilinson, Alexander A.
1052:de Cataldo, Mark Andrea
729:with a six-dimensional
595:, and is preserved by
409:discrete valuation ring
151:Definition and examples
136:triangulated categories
1818:Greene, Brian (2003).
1550:10.4310/jdg/1214437492
916:isolated singularities
677:
676:{\displaystyle Gr_{G}}
636:hard Lefschetz theorem
577:
545:
513:
471:
437:
394:
364:
293:
236:
171:cohomology on a space
96:differential equations
36:, possibly singular.
16:The mathematical term
1395:Annals of Mathematics
1376:(December 15, 1990).
938:Intersection homology
678:
632:decomposition theorem
578:
546:
514:
472:
438:
395:
365:
294:
237:
125:intersection homology
104:representation theory
76:intersection homology
65:constructible sheaves
1820:The Elegant Universe
1699:. World Scientific.
933:Mixed perverse sheaf
776:The Elegant Universe
657:
555:
523:
485:
459:
415:
380:
341:
246:
186:
1973:Homological algebra
1795:1986InMat..84..403M
1740:2005JGP....53...31H
1670:1997MPLA...12..521H
1623:1988CMaPh.119..431G
1582:1993NuPhB.403..159W
1508:1995NuPhB.451...96S
1112:2009arXiv0902.0349A
827:algebraic varieties
784:algebraic varieties
714:Massless fields in
652:affine Grassmannian
276:
219:
140:homological algebra
110:Preliminary remarks
45:Alexander Beilinson
1803:10.1007/BF01388812
1631:10.1007/BF01218081
1374:MacPherson, Robert
1241:Journal of Algebra
1156:10.1007/BFb0078365
928:Mixed Hodge module
673:
573:
541:
509:
500:
467:
433:
390:
360:
289:
262:
232:
205:
88:algebraic geometry
59:regular holonomic
57:derived categories
32:, or more general
26:topological spaces
22:abelian categories
1889:978-3-642-12588-1
1706:978-981-02-1927-7
1560:Nuclear Physics B
1486:Nuclear Physics B
1476:978-2-85629-878-7
1466:Faisceaux Pervers
1398:, Second Series,
1358:978-3-540-41457-5
1187:Bernstein, Joseph
1165:978-3-540-18571-0
1017:, Corollaire 2.7)
764:Andrew Strominger
719:compactifications
698:. A proof of the
504:
493:
84:Robert MacPherson
1985:
1946:
1920:
1893:
1872:
1854:
1833:
1814:
1775:
1733:
1710:
1689:
1663:
1642:
1601:
1575:
1566:(1–2): 159–222.
1554:
1552:
1527:
1501:
1480:
1459:
1457:
1444:
1411:
1388:Mirković, Ivan;
1384:
1382:
1369:
1341:Kiehl, Reinhardt
1336:
1296:
1282:
1257:
1232:
1214:
1176:
1139:
1105:
1080:
1060:
1039:
1033:
1027:
1024:
1018:
1012:
1006:
1000:
994:
985:
979:
973:
898:Poincaré duality
700:Weil conjectures
682:
680:
679:
674:
672:
671:
582:
580:
579:
574:
550:
548:
547:
542:
518:
516:
515:
510:
505:
492:
476:
474:
473:
468:
466:
447:A simple example
442:
440:
439:
434:
399:
397:
396:
391:
389:
388:
369:
367:
366:
361:
350:
349:
298:
296:
295:
290:
275:
270:
258:
257:
241:
239:
238:
233:
218:
213:
201:
200:
167:of sheaves with
165:derived category
41:Joseph Bernstein
18:perverse sheaves
1993:
1992:
1988:
1987:
1986:
1984:
1983:
1982:
1963:
1962:
1953:
1951:Further reading
1890:
1830:
1731:math.AG/0210394
1707:
1492:(1–2): 96–108.
1477:
1455:math.RT/0307349
1380:
1359:
1349:Springer-Verlag
1191:Deligne, Pierre
1166:
1058:
1047:
1042:
1036:Beilinson (1987
1034:
1030:
1025:
1021:
1013:
1009:
1001:
997:
986:
982:
974:
970:
966:
924:
850:Mirror symmetry
727:Minkowski space
712:
689:reductive group
667:
663:
658:
655:
654:
644:Hodge-theoretic
620:
597:Verdier duality
589:
556:
553:
552:
524:
521:
520:
491:
486:
483:
482:
462:
460:
457:
456:
449:
416:
413:
412:
384:
383:
381:
378:
377:
345:
344:
342:
339:
338:
318:
271:
266:
253:
249:
247:
244:
243:
214:
209:
193:
189:
187:
184:
183:
163:of the bounded
153:
112:
102:, algebra, and
94:, analysis and
12:
11:
5:
1991:
1981:
1980:
1975:
1961:
1960:
1952:
1949:
1948:
1947:
1911:(2): 363–399.
1894:
1888:
1873:
1845:(3): 667–693.
1834:
1828:
1815:
1789:(2): 403–435.
1776:
1711:
1705:
1690:
1661:hep-th/9612075
1654:(8): 521–533.
1643:
1617:(3): 431–441.
1602:
1573:hep-th/9301042
1555:
1543:(4): 661–692.
1528:
1499:hep-th/9504090
1481:
1475:
1460:
1445:
1385:
1370:
1357:
1337:
1311:(3): 271–295.
1297:
1284:
1283:
1234:
1233:
1216:
1215:
1178:
1177:
1164:
1140:
1082:
1081:
1071:(5): 632–634.
1046:
1043:
1041:
1040:
1038:, Theorem 1.3)
1028:
1019:
1007:
995:
980:
967:
965:
962:
961:
960:
955:
950:
945:
940:
935:
930:
923:
920:
823:stratification
799:field theories
788:supersymmetric
711:
708:
685:Langlands dual
670:
666:
662:
619:
616:
588:
585:
572:
569:
566:
563:
560:
540:
537:
534:
531:
528:
508:
503:
499:
496:
490:
465:
448:
445:
432:
429:
426:
423:
420:
387:
371:
370:
359:
356:
353:
348:
314:
300:
299:
288:
285:
282:
279:
274:
269:
265:
261:
256:
252:
231:
228:
225:
222:
217:
212:
208:
204:
199:
196:
192:
157:perverse sheaf
152:
149:
129:Goresky (2010)
116:perverse sheaf
111:
108:
49:Pierre Deligne
24:associated to
9:
6:
4:
3:
2:
1990:
1979:
1976:
1974:
1971:
1970:
1968:
1958:
1955:
1954:
1944:
1940:
1936:
1932:
1928:
1924:
1919:
1914:
1910:
1906:
1905:
1900:
1895:
1891:
1885:
1881:
1880:
1874:
1870:
1866:
1862:
1858:
1853:
1848:
1844:
1840:
1835:
1831:
1829:0-393-05858-1
1825:
1821:
1816:
1812:
1808:
1804:
1800:
1796:
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1446:
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1410:
1405:
1402:(1): 95–143,
1401:
1397:
1396:
1391:
1390:Vilonen, Kari
1386:
1379:
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1368:
1364:
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1197:(in French).
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891:
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874:R. MacPherson
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867:
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856:but found an
855:
854:String Theory
851:
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842:singularities
839:
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831:singularities
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100:number theory
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66:
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58:
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50:
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37:
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27:
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1978:Morse theory
1908:
1902:
1878:
1842:
1838:
1819:
1786:
1780:
1724:(1): 31–48.
1721:
1715:
1695:
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1614:
1610:
1563:
1559:
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1536:
1489:
1485:
1465:
1409:math/0401222
1399:
1393:
1344:
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1301:Illusie, Luc
1245:
1239:
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1087:
1068:
1062:
1031:
1022:
1010:
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992:MathOverflow
983:
975:
971:
896:, satisfied
882:R.M. Goresky
870:R.M. Goresky
835:
812:
792:
772:Brian Greene
762:
752:is called a
743:
735:(co)homology
713:
691:
621:
618:Applications
611:
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478:
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176:
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156:
154:
145:Morse theory
133:
115:
113:
80:Mark Goresky
73:
68:
38:
17:
15:
1096:(1): 3–29.
909:codimension
902:codimension
866:obstruction
862:obstruction
796:world-sheet
750:CY manifold
716:superstring
687:group of a
593:t-structure
1967:Categories
1822:. Norton.
1195:Astérisque
1045:References
978:BBD, p. 10
886:MacPherson
815:cohomology
768:blackholes
723:cohomology
587:Properties
306:, for all
121:cohomology
1918:1212.2196
1852:0704.3298
1811:120183452
1772:119584805
1756:0393-0440
1639:119452483
1426:0003-486X
1333:122652995
1255:1106.2616
1248:: 85–96,
1201:. Paris:
1103:0902.0349
825:of these
628:D-modules
568:∘
562:−
536:∘
530:−
495:⇄
422:
406:henselian
284:≠
227:≠
216:∗
195:−
114:The name
61:D-modules
1943:62773026
1869:14787272
1686:11779832
1598:16122549
1442:14127684
1291:(2010).
1280:14754630
1136:14409918
948:Conifold
922:See also
846:singular
838:conifold
819:Witten's
780:topology
754:conifold
746:singular
92:topology
30:manifold
1935:3273317
1791:Bibcode
1764:2102048
1736:Bibcode
1666:Bibcode
1619:Bibcode
1578:Bibcode
1524:6035714
1504:Bibcode
1434:2342692
1367:1855066
1325:2067039
1272:3085024
1230:2533465
1211:0751966
1174:0923133
1128:2668828
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1077:2664042
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1939:S2CID
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1865:S2CID
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1807:S2CID
1768:S2CID
1726:arXiv
1682:S2CID
1656:arXiv
1635:S2CID
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1520:S2CID
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1450:arXiv
1438:S2CID
1404:arXiv
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1098:arXiv
1059:(PDF)
964:Notes
400:. If
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