4132:
6013:: every smooth projective variety with ample canonical bundle has a Kähler–Einstein metric (with constant negative Ricci curvature), and every Calabi–Yau manifold has a Kähler–Einstein metric (with zero Ricci curvature). These results are important for the classification of algebraic varieties, with applications such as the
5886:
showed that a compact complex surface has a Kähler metric if and only if its first Betti number is even. An alternative proof of this result which does not require the hard case-by-case study using the classification of compact complex surfaces was provided independently by
Buchdahl and Lamari. Thus
5891:
shows, however, that this fails in dimensions at least 3. In more detail, the example is a 1-parameter family of smooth compact complex 3-folds such that most fibers are Kähler (and even projective), but one fiber is not Kähler. Thus a compact Kähler manifold can be diffeomorphic to a non-Kähler
2948:
3975:
3458:. The identities form the basis of the analytical toolkit on Kähler manifolds, and combined with Hodge theory are fundamental in proving many important properties of Kähler manifolds and their cohomology. In particular the Kähler identities are critical in proving the
3964:
6223:
For holomorphic maps between
Hermitian manifolds, the holomorphic sectional curvature is not strong enough to control the target curvature term appearing in the Schwarz lemma second-order estimate. This motivated the consideration of the
6211:
A remarkable feature of complex geometry is that holomorphic sectional curvature decreases on complex submanifolds. (The same goes for a more general concept, holomorphic bisectional curvature.) For example, every complex submanifold of
1134:
4813:
591:
4354:
5856:-lemma, and in particular agree when the manifold is Kähler. In general the kernel of the natural map from Bott–Chern cohomology to Dolbeault cohomology contains information about the failure of the manifold to be Kähler.
2780:
4231:
3385:
2128:
80:
refers to the study of Kähler manifolds, their geometry and topology, as well as the study of structures and constructions that can be performed on Kähler manifolds, such as the existence of special connections like
3136:
1777:
5114:
5745:
is an integral differential form) is also called the Hodge form, and the Kähler metric at this time is called the Hodge metric. The compact Kähler manifolds with Hodge metric are also called Hodge manifolds.
2846:
2530:
4127:{\displaystyle \Delta _{\bar {\partial }}={\bar {\partial }}{\bar {\partial }}^{*}+{\bar {\partial }}^{*}{\bar {\partial }},\ \ \ \ \Delta _{\partial }=\partial \partial ^{*}+\partial ^{*}\partial .}
5491:
3688:
816:
6640:
1211:
6133:
is a
Hermitian manifold with a Hermitian metric of negative holomorphic sectional curvature (bounded above by a negative constant), then it is Brody hyperbolic (i.e., every holomorphic map
5356:
2408:
6101:
The holomorphic sectional curvature is intimately related to the complex geometry of the underlying complex manifold. It is an elementary consequence of the
Ahlfors Schwarz lemma that if
3325:
1815:
5854:
5812:
5779:
5212:
3265:
As a consequence of the strong interaction between the smooth, complex, and
Riemannian structures on a Kähler manifold, there are natural identities between the various operators on the
3011:
2460:
2225:
1904:
2564:
3591:
2319:
5000:
4678:
1458:
1361:
4954:
4517:
274:
5420:
4388:
3420:
6159:
5863:
are not projective. One may ask whether every compact Kähler manifold can at least be deformed (by continuously varying the complex structure) to a smooth projective variety.
4480:
1048:
959:
5270:
1663:
1292:
3456:
2838:
2348:
681:
6202:
6131:
5297:
2266:
1938:
159:
4905:
2622:
3865:
7850:
2662:
1561:
4572:
4451:
3053:
class. In a sense, this means that the geometry of a complex subspace is bounded in terms of its topology. (This fails completely for real submanifolds.) Explicitly,
2483:
2368:
2172:
1863:
1314:
724:
496:
405:
209:
3708:
2428:
4604:
4550:
3857:
3739:
2642:
2148:
2038:
1839:
1719:
5859:
Every compact complex curve is projective, but in complex dimension at least 2, there are many compact Kähler manifolds that are not projective; for example, most
7452:
1053:
2694:
2596:
1525:
1419:
456:
5138:
5024:
4856:
4836:
4698:
4627:
4431:
4411:
4254:
4155:
3830:
3810:
3790:
3763:
3611:
3532:
3508:
3287:
2058:
2018:
1998:
1978:
1958:
1683:
1601:
1581:
1498:
1478:
1396:
1254:
1234:
1019:
999:
979:
930:
903:
883:
863:
839:
748:
701:
654:
634:
614:
476:
429:
379:
356:
325:
301:
182:
2977:
2814:
2723:
5547:
The "Kähler package" is a collection of further restrictions on the cohomology of compact Kähler manifolds, building on Hodge theory. The results include the
6036:
4714:
504:
6020:
By contrast, not every smooth Fano variety has a Kähler–Einstein metric (which would have constant positive Ricci curvature). However, Xiuxiong Chen,
5937:
Although Ricci curvature is defined for any
Riemannian manifold, it plays a special role in Kähler geometry: the Ricci curvature of a Kähler manifold
4262:
2728:
6003:), respectively. By the Kodaira embedding theorem, Fano manifolds and manifolds with ample canonical bundle are automatically projective varieties.
5516:
of a compact Kähler manifold is even, by Hodge symmetry. This is not true for compact complex manifolds in general, as shown by the example of the
4167:
3330:
2063:
7375:
7228:
7069:
7032:
6959:
6838:
6793:
5882:
One can also ask for a characterization of compact Kähler manifolds among all compact complex manifolds. In complex dimension 2, Kodaira and
5875:
found, however, that this fails in dimensions at least 4. She constructed a compact Kähler manifold of complex dimension 4 that is not even
3063:
1731:
82:
5029:
2943:{\displaystyle {\mathcal {K}}_{}:=\{\varphi :X\to \mathbb {R} {\text{ smooth}}\mid \omega +i\partial {\bar {\partial }}\varphi >0\}.}
1869:
408:
7445:
6922:
7817:
7352:
2953:
If two Kähler potentials differ by a constant, then they define the same Kähler metric, so the space of Kähler metrics in the class
6228:, introduced by Xiaokui Yang and Fangyang Zheng. This also appears in the work of Man-Chun Lee and Jeffrey Streets under the name
2488:
5556:
3475:
5991:, depending on whether the Einstein constant λ is positive, zero, or negative. Kähler manifolds of those three types are called
8047:
7782:
5868:
6281:. Conversely, every Riemann surface is Kähler since the Kähler form of any Hermitian metric is closed for dimensional reasons.
8409:
118:
Since Kähler manifolds are equipped with several compatible structures, they can be described from different points of view:
5425:
5901:
5607:
arises as the fundamental group of some compact complex manifold of dimension 3. (Conversely, the fundamental group of any
3021:
Kähler metrics in a given class simultaneously, and this perspective in the study of existence results for Kähler metrics.
86:
8635:
7989:
7495:
7438:
7410:
7367:
6777:
6423:
7759:
5871:
implies that every compact Kähler manifold of complex dimension 2 can indeed be deformed to a smooth projective variety.
5623:
characterizes smooth complex projective varieties among all compact Kähler manifolds. Namely, a compact complex manifold
5574:, is wide open. Hodge theory gives many restrictions on the possible Kähler groups. The simplest restriction is that the
3616:
8650:
8434:
7490:
6204:
is a compact Kähler manifold with a Kähler metric of positive holomorphic sectional curvature, Yang
Xiaokui showed that
6035:
In situations where there cannot exist a Kähler–Einstein metric, it is possible to study mild generalizations including
5548:
3467:
8014:
6821:, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol. 4,
3269:
of Kähler manifolds which do not hold for arbitrary complex manifolds. These identities relate the exterior derivative
756:
8640:
7405:
6615:
1142:
7640:
6014:
7086:
Kodaira, K. (1954). "On Kahler
Varieties of Restricted Type an Intrinsic Characterization of Algebraic Varieties)".
5714:
with a hermitian metric whose curvature form ω is positive (since ω is then a Kähler form that represents the first
8645:
7935:
7587:
7202:
6943:
5302:
2624:, and by the Poincaré lemma any Kähler form will locally be cohomologous to zero. Thus the local Kähler potential
6078:
means the sectional curvature restricted to complex lines in the tangent space. This behaves more simply, in that
5817:
3295:
1785:
8274:
8229:
6377:
6331:
6040:
5827:
5785:
5752:
5620:
5146:
4705:
4633:
gives an interpretation of the splitting above which does not depend on the choice of Kähler metric. Namely, the
4520:
3459:
3266:
2982:
2433:
2198:
1877:
1818:
1722:
727:
2535:
8454:
8374:
8189:
8123:
7485:
6285:
6241:
5604:
5560:
3537:
3463:
8334:
7956:
7930:
7671:
6292:
6017:
for varieties with ample canonical bundle and the
Beauville–Bogomolov decomposition for Calabi–Yau manifolds.
5996:
2373:
2271:
4959:
8594:
8404:
8118:
7961:
7802:
7560:
7502:
6418:
6413:
6381:
6376:, with holomorphic sectional curvature equal to −1. A natural generalization of the ball is provided by the
6043:. When a Kähler–Einstein metric can exist, these broader generalizations are automatically Kähler–Einstein.
5704:
304:
185:
162:
6094:
Kähler metric with holomorphic sectional curvature equal to −1. (With this metric, the ball is also called
4639:
1431:
1322:
8359:
8100:
7906:
7797:
7769:
7592:
6278:
6074:) varies between 1/4 and 1 at every point. For a Hermitian manifold (for example, a Kähler manifold), the
5593:
5552:
4910:
4485:
3471:
3042:
220:
8509:
7845:
7812:
7676:
7518:
5361:
8214:
8154:
8095:
8062:
8057:
7855:
7553:
7548:
7543:
7528:
6273:
SO(2), which is equal to the unitary group U(1).) In particular, an oriented
Riemannian 2-manifold is a
4362:
3393:
61:
37:
8384:
6338:. The natural Kähler metric on a Hermitian symmetric space of compact type has sectional curvature ≥ 0.
6136:
5888:
6310:
that preserve the standard Hermitian form. The Fubini–Study metric is the unique Riemannian metric on
4456:
2174:. There is no comparable way of describing a general Riemannian metric in terms of a single function.
1024:
935:
8599:
7597:
7582:
7538:
7400:
7267:
7061:
6951:
5821:
5223:
4158:
3260:
1606:
1259:
3428:
2819:
8514:
8399:
8052:
7951:
7577:
7049:
6898:
6428:
6342:
6091:
6056:
5927:
5876:
3711:
3479:
3290:
3244:
3240:
1376:
127:
57:
53:
5824:
of a compact complex manifolds, and they are isomorphic if and only if the manifold satisfies the
3959:{\displaystyle d=\partial +{\bar {\partial }},\ \ \ \ d^{*}=\partial ^{*}+{\bar {\partial }}^{*},}
659:
8559:
8479:
8379:
8339:
8219:
8184:
8019:
7896:
7792:
7533:
7329:
7311:
7291:
7257:
7206:
7185:
7167:
7103:
7005:
6175:
6104:
6083:
5923:
5275:
3014:
2239:
2191:
1911:
1686:
1317:
842:
382:
359:
132:
107:
99:
8269:
7302:
Yang, Xiaokui; Zheng, Fangyang (2018). "On real bisectional curvature for Hermitian manifolds".
5592:
cannot be the fundamental group of a compact Kähler manifold.) Extensions of the theory such as
4868:
2601:
6082:
has holomorphic sectional curvature equal to 1 everywhere. At the other extreme, the open unit
6028:–Donaldson conjecture: a smooth Fano variety has a Kähler–Einstein metric if and only if it is
5665:. (Because a positive multiple of a Kähler form is a Kähler form, it is equivalent to say that
8524:
8429:
8264:
8174:
8144:
7940:
7835:
7787:
7681:
7371:
7348:
7342:
7224:
7158:
Lee, Man-Chun; Streets, Jeffrey (2021). "Complex Manifolds with Negative Curvature Operator".
7065:
7028:
7016:
6955:
6939:
6918:
6834:
6789:
6354:
6010:
5988:
5931:
5567:
3050:
2647:
2328:
1873:
1530:
385:
65:
4557:
4436:
3182:. These volumes are always positive, which expresses a strong positivity of the Kähler class
2465:
2353:
2157:
1848:
1299:
1129:{\textstyle h_{ab}=({\frac {\partial }{\partial z_{a}}},{\frac {\partial }{\partial z_{b}}})}
709:
481:
390:
194:
8534:
8469:
8439:
8319:
8259:
8224:
8169:
8159:
8139:
8072:
8024:
7982:
7887:
7880:
7873:
7866:
7859:
7777:
7567:
7475:
7321:
7275:
7216:
7177:
7146:
7128:
7095:
6997:
6989:
6910:
6886:
6868:
6826:
6781:
6166:
5976:
5864:
3742:
3693:
2413:
909:
730:. A Kähler manifold can also be viewed as a Riemannian manifold, with the Riemannian metric
336:
49:
7385:
7287:
7238:
7142:
7079:
7042:
6969:
6932:
6882:
6848:
6803:
4577:
4523:
3835:
3717:
2627:
2133:
2023:
1824:
1704:
8614:
8569:
8519:
8504:
8494:
8389:
8354:
8179:
7749:
7523:
7381:
7283:
7234:
7150:
7138:
7075:
7038:
7024:
7001:
6965:
6928:
6906:
6890:
6878:
6844:
6822:
6799:
6274:
6255:
6021:
5911:
5608:
5600:
1698:
189:
8459:
6977:
6770:
Amorós, Jaume; Burger, Marc; Corlette, Kevin; Kotschick, Dieter; Toledo, Domingo (1996),
69:
7271:
2667:
2569:
1507:
1401:
438:
8589:
8584:
8544:
8484:
8474:
8394:
8314:
8304:
8299:
8294:
8209:
8204:
8199:
8164:
8149:
8077:
7754:
7617:
7053:
6771:
6373:
6346:
6270:
6006:
5953:
5575:
5123:
5009:
4841:
4821:
4683:
4612:
4416:
4396:
4239:
4140:
3815:
3795:
3775:
3748:
3596:
3517:
3493:
3272:
2043:
2003:
1983:
1963:
1943:
1668:
1586:
1566:
1483:
1463:
1422:
1381:
1239:
1219:
1004:
984:
964:
915:
888:
868:
848:
824:
733:
686:
639:
619:
599:
461:
432:
414:
364:
341:
310:
286:
167:
96:
2956:
2793:
2702:
8629:
8579:
8564:
8539:
8529:
8499:
8444:
8419:
8364:
8349:
8344:
8309:
8284:
8244:
8034:
7686:
7602:
7480:
7461:
7333:
7245:
7009:
6296:
5915:
5872:
5860:
5521:
5494:
3030:
2783:
1426:
280:
212:
93:
73:
7295:
7248:(2004), "On the homotopy types of compact Kähler and complex projective manifolds",
7189:
8609:
8449:
8329:
8279:
8249:
8234:
8042:
8009:
7901:
7827:
7807:
7744:
7607:
6335:
6000:
5992:
5883:
5517:
5117:
4630:
2230:
103:
4808:{\displaystyle H^{r}(X,\mathbf {C} )\cong \bigoplus _{p+q=r}H^{q}(X,\Omega ^{p}).}
586:{\displaystyle \omega (u,v)=\operatorname {Re} h(iu,v)=\operatorname {Im} h(u,v)}
8604:
8574:
8554:
8414:
8369:
8324:
8289:
8239:
8004:
7973:
7734:
7691:
7419:
6814:
6059:, which is a real number associated to any real 2-plane in the tangent space of
5782:
5715:
4349:{\displaystyle {\mathcal {H}}^{r}(X)=\bigoplus _{p+q=r}{\mathcal {H}}^{p,q}(X),}
2195:
906:
33:
5749:
Many properties of Kähler manifolds hold in the slightly greater generality of
5217:
The Hodge numbers of a compact Kähler manifold satisfy several identities. The
2775:{\displaystyle \omega _{\varphi }=\omega +i\partial {\bar {\partial }}\varphi }
17:
8549:
8489:
8424:
8087:
8067:
7966:
7925:
7739:
7279:
6914:
6830:
6810:
6400:
5914:. Equivalently, the Ricci curvature tensor is equal to a constant λ times the
5615:
Characterizations of complex projective varieties and compact Kähler manifolds
4701:
4634:
3038:
1501:
7220:
4226:{\displaystyle \Delta _{d}=2\Delta _{\bar {\partial }}=2\Delta _{\partial }.}
8464:
8254:
8194:
7840:
7714:
7635:
7630:
7625:
6025:
3511:
6063:
at a point. For example, the sectional curvature of the standard metric on
7181:
3380:{\displaystyle \Delta _{d},\Delta _{\partial },\Delta _{\bar {\partial }}}
8110:
7999:
7994:
7724:
7719:
7656:
7572:
6266:
6029:
5926:, which asserts in the absence of mass that spacetime is a 4-dimensional
3766:
2123:{\displaystyle {\omega \vert }_{U}=(i/2)\partial {\bar {\partial }}\rho }
45:
7325:
5887:"Kähler" is a purely topological property for compact complex surfaces.
1908:, every Kähler metric can locally be described in this way. That is, if
7729:
7709:
7666:
7661:
7107:
6993:
6785:
5586:
6269:
2-manifold is Kähler. (Indeed, its holonomy group is contained in the
5422:
can be proved using that the Hodge star operator gives an isomorphism
3131:{\displaystyle \mathrm {vol} (Y)={\frac {1}{r!}}\int _{Y}\omega ^{r},}
1772:{\displaystyle \omega ={\frac {i}{2}}\partial {\bar {\partial }}\rho }
7262:
7211:
7133:
7116:
6873:
6856:
6165:
happens to be compact, then this is equivalent to the manifold being
4862:
connects topology and complex geometry for compact Kähler manifolds.
3251:
minimizes volume among all (real) cycles in the same homology class.
7099:
5505:
A simple consequence of Hodge theory is that every odd Betti number
7316:
7172:
7701:
6314:(up to a positive multiple) that is invariant under the action of
5109:{\displaystyle h^{p,q}(X)=\mathrm {dim} _{\mathbf {C} }H^{p,q}(X)}
2725:
is a Kähler class, then any other Kähler metric can be written as
2566:. In the local discussion above, one takes the local Kähler class
3017:. In this way the space of Kähler potentials allows one to study
7430:
5596:
give further restrictions on which groups can be Kähler groups.
4838:
as a topological space, while the groups on the right depend on
2186:
using a single Kähler potential, it is possible to describe the
7434:
2816:
is defined as those positive cases, and is commonly denoted by
5578:
of a Kähler group must have even rank, since the Betti number
6577:
6566:
6388:
of noncompact type is isomorphic to a bounded domain in some
5002:
of harmonic forms with respect to a given Kähler metric. The
2782:
for such a smooth function. This form is not automatically a
5781:-manifolds, that is compact complex manifolds for which the
5559:. A related result is that every compact Kähler manifold is
4966:
4492:
4369:
4317:
4269:
2988:
2853:
2825:
2525:{\displaystyle d\beta =i\partial {\bar {\partial }}\varphi }
2190:
of two Kähler forms this way, provided they are in the same
7201:, London Mathematical Society Student Texts, vol. 69,
6055:
from the standard metric on Euclidean space is measured by
2182:
Whilst it is not always possible to describe a Kähler form
6673:
5941:
can be viewed as a real closed (1,1)-form that represents
4161:
imply these Laplacians are all the same up to a constant:
29:
Manifold with Riemannian, complex and symplectic structure
6396:
is a complete Kähler metric with sectional curvature ≤ 0.
5116:. The Hodge decomposition implies a decomposition of the
6247:
with the standard Hermitian metric is a Kähler manifold.
5585:
of a compact Kähler manifold is even. (For example, the
3243:(outside its singular set). Even more: by the theory of
2268:
is a compact Kähler manifold, then the cohomology class
6980:(1933), "Ùber eine bemerkenswerte Hermitesche Metrik",
5599:
Without the Kähler condition, the situation is simple:
3792:-forms with compact support.) For a Hermitian manifold
932:
in which the metric agrees with the standard metric on
6277:
in a canonical way; this is known as the existence of
6258:) inherits a flat metric from the Euclidean metric on
5741:
that satisfies these conditions (that is, Kähler form
5486:{\displaystyle H^{p,q}\cong {\overline {H^{n-p,n-q}}}}
3157:
is closed, this integral depends only on the class of
1056:
6776:, Mathematical Surveys and Monographs, vol. 44,
6618:
6178:
6139:
6107:
5971:. It follows that a compact Kähler–Einstein manifold
5830:
5788:
5755:
5428:
5364:
5305:
5278:
5226:
5149:
5126:
5032:
5012:
4962:
4913:
4871:
4844:
4824:
4717:
4686:
4642:
4615:
4580:
4560:
4526:
4488:
4459:
4439:
4419:
4399:
4365:
4265:
4242:
4170:
4143:
3978:
3868:
3838:
3818:
3798:
3778:
3751:
3720:
3696:
3619:
3599:
3540:
3520:
3496:
3431:
3396:
3333:
3298:
3275:
3231:
A related fact is that every closed complex subspace
3066:
2985:
2959:
2849:
2822:
2796:
2731:
2705:
2670:
2650:
2630:
2604:
2572:
2538:
2491:
2468:
2436:
2416:
2376:
2356:
2331:
2274:
2242:
2201:
2160:
2136:
2066:
2046:
2026:
2006:
1986:
1966:
1946:
1914:
1880:
1851:
1827:
1788:
1734:
1707:
1671:
1609:
1589:
1569:
1533:
1510:
1486:
1466:
1434:
1404:
1384:
1325:
1302:
1262:
1242:
1222:
1145:
1027:
1007:
987:
967:
938:
918:
891:
871:
851:
827:
759:
736:
712:
689:
662:
642:
622:
602:
507:
484:
464:
441:
417:
393:
367:
344:
313:
289:
223:
197:
170:
135:
6817:; Peters, Chris A.M.; Van de Ven, Antonius (2004) ,
5627:
is projective if and only if there is a Kähler form
1050:, and the metric is written in these coordinates as
8132:
8109:
8086:
8033:
7918:
7826:
7768:
7700:
7649:
7616:
7511:
7468:
6345:of a Kähler manifold is Kähler. In particular, any
3683:{\displaystyle d^{*}=-(-1)^{n(r+1)}\star d\,\star }
6634:
6196:
6153:
6125:
5848:
5806:
5773:
5485:
5414:
5350:
5291:
5264:
5206:
5132:
5108:
5018:
4994:
4948:
4899:
4850:
4830:
4807:
4692:
4672:
4621:
4598:
4566:
4544:
4511:
4474:
4445:
4425:
4405:
4382:
4348:
4248:
4225:
4149:
4126:
3958:
3851:
3824:
3804:
3784:
3757:
3733:
3702:
3682:
3605:
3585:
3526:
3502:
3450:
3414:
3379:
3319:
3281:
3201:with respect to complex subspaces. In particular,
3130:
3005:
2971:
2942:
2832:
2808:
2774:
2717:
2688:
2656:
2636:
2616:
2590:
2558:
2524:
2477:
2454:
2422:
2402:
2362:
2342:
2313:
2260:
2219:
2166:
2142:
2122:
2052:
2032:
2012:
1992:
1972:
1952:
1932:
1898:
1857:
1833:
1809:
1771:
1713:
1677:
1657:
1595:
1575:
1555:
1519:
1492:
1472:
1452:
1413:
1390:
1355:
1308:
1286:
1248:
1228:
1205:
1128:
1042:
1013:
993:
973:
953:
924:
897:
877:
857:
833:
810:
742:
718:
695:
675:
648:
628:
608:
585:
490:
470:
450:
423:
399:
373:
350:
319:
295:
268:
203:
176:
153:
7304:Transactions of the American Mathematical Society
5650:is in the image of the integral cohomology group
4236:These identities imply that on a Kähler manifold
811:{\displaystyle g(u,v)=\operatorname {Re} h(u,v).}
6905:, Lecture Notes in Mathematics, vol. 1764,
6635:{\displaystyle \partial {\overline {\partial }}}
5999:, or with ample canonical bundle (which implies
1206:{\displaystyle h_{ab}=\delta _{ab}+O(\|z\|^{2})}
6731:
6470:
6446:
6361:) is Kähler. This is a large class of examples.
6284:There is a standard choice of Kähler metric on
6773:Fundamental Groups of Compact Kähler Manifolds
5930:with zero Ricci curvature. See the article on
7446:
6262:, and is therefore a compact Kähler manifold.
5987:either anti-ample, homologically trivial, or
5351:{\displaystyle H^{p,q}={\overline {H^{q,p}}}}
1460:. Equivalently, there is a complex structure
48:with three mutually compatible structures: a
8:
6612:Angella, D. and Tomassini, A., 2013. On the
3320:{\displaystyle \partial ,{\bar {\partial }}}
2934:
2873:
2072:
1810:{\displaystyle \partial ,{\bar {\partial }}}
1281:
1263:
1191:
1184:
7117:"Courants kählériens et surfaces compactes"
6220:) has holomorphic sectional curvature ≤ 0.
5849:{\displaystyle \partial {\bar {\partial }}}
5807:{\displaystyle \partial {\bar {\partial }}}
5774:{\displaystyle \partial {\bar {\partial }}}
5207:{\displaystyle b_{r}=\sum _{p+q=r}h^{p,q}.}
3006:{\displaystyle {\mathcal {K}}/\mathbb {R} }
2462:-lemma further states that this exact form
2455:{\displaystyle \partial {\bar {\partial }}}
2220:{\displaystyle \partial {\bar {\partial }}}
1940:is a Kähler manifold, then for every point
1899:{\displaystyle \partial {\bar {\partial }}}
72:in 1933. The terminology has been fixed by
7453:
7439:
7431:
7344:Differential Analysis on Complex Manifolds
7160:International Mathematics Research Notices
6754:
6554:
6542:
6530:
6518:
6506:
5879:to any smooth complex projective variety.
5563:in the sense of rational homotopy theory.
2559:{\displaystyle \varphi :X\to \mathbb {C} }
2325:. Any other representative of this class,
1868:Conversely, by the complex version of the
1782:is positive, that is, a Kähler form. Here
7315:
7261:
7210:
7171:
7132:
6872:
6743:
6622:
6617:
6177:
6141:
6140:
6138:
6106:
5835:
5834:
5829:
5793:
5792:
5787:
5760:
5759:
5754:
5454:
5448:
5433:
5427:
5388:
5369:
5363:
5331:
5325:
5310:
5304:
5283:
5277:
5250:
5231:
5225:
5189:
5167:
5154:
5148:
5125:
5085:
5074:
5073:
5062:
5037:
5031:
5011:
4971:
4965:
4964:
4961:
4956:, which can be identified with the space
4937:
4918:
4912:
4876:
4870:
4843:
4823:
4793:
4774:
4752:
4737:
4722:
4716:
4685:
4662:
4647:
4641:
4614:
4579:
4559:
4525:
4497:
4491:
4490:
4487:
4458:
4438:
4418:
4398:
4374:
4368:
4367:
4364:
4322:
4316:
4315:
4296:
4274:
4268:
4267:
4264:
4241:
4214:
4192:
4191:
4175:
4169:
4142:
4112:
4099:
4083:
4053:
4052:
4046:
4035:
4034:
4024:
4013:
4012:
4000:
3999:
3984:
3983:
3977:
3947:
3936:
3935:
3925:
3912:
3882:
3881:
3867:
3843:
3837:
3817:
3797:
3777:
3750:
3725:
3719:
3695:
3676:
3649:
3624:
3618:
3598:
3586:{\displaystyle \Delta _{d}=dd^{*}+d^{*}d}
3574:
3561:
3545:
3539:
3519:
3495:
3442:
3430:
3395:
3365:
3364:
3351:
3338:
3332:
3306:
3305:
3297:
3274:
3149:-dimensional closed complex subspace and
3119:
3109:
3090:
3067:
3065:
2999:
2998:
2993:
2987:
2986:
2984:
2958:
2914:
2913:
2893:
2889:
2888:
2858:
2852:
2851:
2848:
2824:
2823:
2821:
2795:
2758:
2757:
2736:
2730:
2704:
2669:
2649:
2629:
2603:
2571:
2552:
2551:
2537:
2508:
2507:
2490:
2467:
2441:
2440:
2435:
2415:
2375:
2355:
2330:
2296:
2291:
2273:
2241:
2206:
2205:
2200:
2159:
2135:
2106:
2105:
2091:
2076:
2068:
2065:
2045:
2025:
2005:
1985:
1965:
1945:
1913:
1885:
1884:
1879:
1850:
1826:
1796:
1795:
1787:
1755:
1754:
1741:
1733:
1706:
1670:
1608:
1588:
1568:
1538:
1532:
1509:
1485:
1465:
1433:
1403:
1383:
1346:
1345:
1330:
1324:
1301:
1261:
1241:
1221:
1194:
1166:
1150:
1144:
1114:
1101:
1089:
1076:
1061:
1055:
1034:
1030:
1029:
1026:
1006:
986:
966:
945:
941:
940:
937:
917:
890:
870:
850:
826:
758:
735:
711:
688:
663:
661:
641:
621:
601:
506:
483:
463:
440:
416:
392:
366:
343:
312:
288:
222:
196:
169:
134:
6685:
6482:
6372:has a complete Kähler metric called the
6037:constant scalar curvature Kähler metrics
6032:, a purely algebro-geometric condition.
5703:is projective if and only if there is a
2403:{\displaystyle \omega '=\omega +d\beta }
2314:{\displaystyle \in H_{\text{dR}}^{2}(X)}
6669:
6667:
6601:
6439:
6306:, the group of linear automorphisms of
6051:The deviation of a Riemannian manifold
5922:. The reference to Einstein comes from
4995:{\displaystyle {\mathcal {H}}^{p,q}(X)}
4609:Further, for a compact Kähler manifold
4574:is harmonic if and only if each of its
1316:is closed, it determines an element in
6696:
6659:
4818:The group on the left depends only on
4700:with complex coefficients splits as a
3969:and two other Laplacians are defined:
3490:On a Riemannian manifold of dimension
3025:Kähler manifolds and volume minimizers
3013:. The space of Kähler potentials is a
6719:
6707:
6589:
6494:
6458:
4673:{\displaystyle H^{r}(X,\mathbf {C} )}
1453:{\displaystyle \operatorname {U} (n)}
1356:{\displaystyle H^{2}(X,\mathbb {R} )}
7:
7851:Bogomol'nyi–Prasad–Sommerfield bound
7058:Foundations of Differential Geometry
5570:of compact Kähler manifolds, called
5566:The question of which groups can be
5501:Topology of compact Kähler manifolds
4949:{\displaystyle H^{q}(X,\Omega ^{p})}
4512:{\displaystyle {\mathcal {H}}^{p,q}}
2979:can be identified with the quotient
2194:class. This is a consequence of the
269:{\displaystyle g(u,v)=\omega (u,Jv)}
5415:{\displaystyle h^{p,q}=h^{n-p,n-q}}
3327:and their adjoints, the Laplacians
163:integrable almost-complex structure
60:. The concept was first studied by
6642:-Lemma and Bott-Chern cohomology.
6624:
6619:
6384:. Every Hermitian symmetric space
5837:
5831:
5795:
5789:
5762:
5756:
5280:
5069:
5066:
5063:
4934:
4790:
4460:
4383:{\displaystyle {\mathcal {H}}^{r}}
4215:
4211:
4194:
4188:
4172:
4118:
4109:
4096:
4092:
4084:
4080:
4055:
4037:
4015:
4002:
3986:
3980:
3938:
3922:
3884:
3875:
3542:
3486:The Laplacian on a Kähler manifold
3432:
3415:{\displaystyle L:=\omega \wedge -}
3367:
3361:
3352:
3348:
3335:
3308:
3299:
3074:
3071:
3068:
2916:
2910:
2760:
2754:
2510:
2504:
2443:
2437:
2208:
2202:
2108:
2102:
2020:and a smooth real-valued function
1887:
1881:
1798:
1789:
1757:
1751:
1435:
1107:
1103:
1082:
1078:
307:(and hence a Riemannian metric on
25:
7021:Complex Geometry: An Introduction
6154:{\displaystyle \mathbb {C} \to X}
5669:has a Kähler form whose class in
6948:Principles of Algebraic Geometry
6380:of noncompact type, such as the
6326:. One natural generalization of
5952:) (the first Chern class of the
5557:Hodge-Riemann bilinear relations
5075:
4738:
4663:
4475:{\displaystyle \Delta \alpha =0}
3476:Hodge-Riemann bilinear relations
3220:, for a compact Kähler manifold
1721:on a complex manifold is called
1043:{\displaystyle \mathbb {C} ^{n}}
954:{\displaystyle \mathbb {C} ^{n}}
821:Equivalently, a Kähler manifold
83:Hermitian Yang–Mills connections
68:in 1930, and then introduced by
8048:Eleven-dimensional supergravity
6903:Lectures on Symplectic Geometry
6295:. One description involves the
6076:holomorphic sectional curvature
6047:Holomorphic sectional curvature
5265:{\displaystyle h^{p,q}=h^{q,p}}
5140:in terms of its Hodge numbers:
4858:as a complex manifold. So this
4554:. That is, a differential form
3613:is the exterior derivative and
1658:{\displaystyle g(Ju,Jv)=g(u,v)}
1500:at each point (that is, a real
1287:{\displaystyle \{1,\cdots ,n\}}
303:at each point is symmetric and
110:, proved using Kähler metrics.
6265:Every Riemannian metric on an
6216:(with the induced metric from
6191:
6179:
6145:
6120:
6108:
6024:, and Song Sun proved the Yau–
5840:
5798:
5765:
5103:
5097:
5055:
5049:
4989:
4983:
4943:
4924:
4894:
4888:
4799:
4780:
4742:
4728:
4667:
4653:
4593:
4581:
4539:
4527:
4340:
4334:
4286:
4280:
4197:
4058:
4040:
4018:
4005:
3989:
3941:
3887:
3665:
3653:
3646:
3636:
3451:{\displaystyle \Lambda =L^{*}}
3370:
3311:
3084:
3078:
2966:
2960:
2919:
2885:
2865:
2859:
2833:{\displaystyle {\mathcal {K}}}
2803:
2797:
2763:
2712:
2706:
2677:
2671:
2579:
2573:
2548:
2513:
2446:
2308:
2302:
2281:
2275:
2255:
2243:
2211:
2111:
2099:
2085:
1927:
1915:
1890:
1801:
1760:
1725:if the real closed (1,1)-form
1652:
1640:
1631:
1613:
1447:
1441:
1350:
1336:
1200:
1181:
1123:
1073:
981:. That is, if the chart takes
802:
790:
775:
763:
580:
568:
553:
538:
523:
511:
263:
248:
239:
227:
148:
136:
1:
7496:Second superstring revolution
7368:American Mathematical Society
7364:Complex Differential Geometry
7121:Annales de l'Institut Fourier
6982:Abh. Math. Sem. Univ. Hamburg
6861:Annales de l'Institut Fourier
6778:American Mathematical Society
6732:Kobayashi & Nomizu (1996)
6471:Kobayashi & Nomizu (1996)
5120:of a compact Kähler manifold
3235:of a compact Kähler manifold
85:, or special metrics such as
7990:Generalized complex manifold
7491:First superstring revolution
6857:"On compact Kähler surfaces"
6627:
6392:, and the Bergman metric of
5906:A Kähler manifold is called
5549:Lefschetz hyperplane theorem
5478:
5343:
5272:holds because the Laplacian
4907:be the complex vector space
3468:Lefschetz hyperplane theorem
676:{\displaystyle {\sqrt {-1}}}
7421:Lectures on Kähler Geometry
7406:Encyclopedia of Mathematics
7199:Lectures on Kähler Geometry
6855:Buchdahl, Nicholas (1999).
6734:, v. 2, Proposition IX.9.2.
6197:{\displaystyle (X,\omega )}
6126:{\displaystyle (X,\omega )}
5299:is a real operator, and so
5292:{\displaystyle \Delta _{d}}
4860:Hodge decomposition theorem
2261:{\displaystyle (X,\omega )}
1933:{\displaystyle (X,\omega )}
154:{\displaystyle (X,\omega )}
8667:
7588:Non-critical string theory
7341:Wells, Raymond O. (2007).
7203:Cambridge University Press
6424:Quaternion-Kähler manifold
6378:Hermitian symmetric spaces
6332:Hermitian symmetric spaces
6230:complex curvature operator
6226:real bisectional curvature
6208:is rationally connected.
5899:
5869:classification of surfaces
4900:{\displaystyle H^{p,q}(X)}
3267:complex differential forms
3258:
3153:is the Kähler form. Since
2617:{\displaystyle U\subset X}
2178:Space of Kähler potentials
865:such that for every point
431:gives a positive definite
7418:Moroianu, Andrei (2004),
7280:10.1007/s00222-003-0352-1
7197:Moroianu, Andrei (2007),
6915:10.1007/978-3-540-45330-7
6831:10.1007/978-3-642-57739-0
6334:of compact type, such as
5896:Kähler–Einstein manifolds
5820:is an alternative to the
5816:holds. In particular the
5621:Kodaira embedding theorem
4706:coherent sheaf cohomology
4606:-components is harmonic.
4519:is the space of harmonic
3464:Nakano vanishing theorems
1723:strictly plurisubharmonic
683:). For a Kähler manifold
8124:Introduction to M-theory
7818:Wess–Zumino–Witten model
7760:Hanany–Witten transition
7486:History of string theory
7362:Zheng, Fangyang (2000),
7250:Inventiones Mathematicae
7221:10.1017/CBO9780511618666
6819:Compact Complex Surfaces
6755:Lee & Streets (2021)
6644:Inventiones mathematicae
6341:The induced metric on a
6286:complex projective space
6250:A compact complex torus
6096:complex hyperbolic space
5611:is finitely presented.)
5605:finitely presented group
5594:non-abelian Hodge theory
2657:{\displaystyle \varphi }
2343:{\displaystyle \omega '}
1980:there is a neighborhood
1556:{\displaystyle J^{2}=-1}
1480:on the tangent space of
7803:Vertex operator algebra
7503:String theory landscape
7115:Lamari, Ahcène (1999).
6744:Yang & Zheng (2018)
6545:, sections 3.3 and 5.2,
6414:Almost complex manifold
6382:Siegel upper half space
6353:) or smooth projective
6041:extremal Kähler metrics
5705:holomorphic line bundle
5493:. It also follows from
4567:{\displaystyle \alpha }
4446:{\displaystyle \alpha }
2478:{\displaystyle d\beta }
2363:{\displaystyle \omega }
2167:{\displaystyle \omega }
1858:{\displaystyle \omega }
1375:A Kähler manifold is a
1309:{\displaystyle \omega }
719:{\displaystyle \omega }
491:{\displaystyle \omega }
400:{\displaystyle \omega }
335:A Kähler manifold is a
204:{\displaystyle \omega }
126:A Kähler manifold is a
87:Kähler–Einstein metrics
8101:AdS/CFT correspondence
7856:Exceptional Lie groups
7798:Superconformal algebra
7770:Conformal field theory
7641:Montonen–Olive duality
7593:Non-linear sigma model
6636:
6447:Cannas da Silva (2001)
6279:isothermal coordinates
6198:
6172:On the other hand, if
6155:
6127:
6015:Miyaoka–Yau inequality
5902:Kähler–Einstein metric
5850:
5808:
5775:
5553:hard Lefschetz theorem
5487:
5416:
5352:
5293:
5266:
5208:
5134:
5110:
5020:
4996:
4950:
4901:
4852:
4832:
4809:
4694:
4674:
4623:
4600:
4568:
4546:
4513:
4476:
4447:
4427:
4407:
4384:
4350:
4250:
4227:
4151:
4128:
3960:
3853:
3826:
3806:
3786:
3759:
3735:
3704:
3703:{\displaystyle \star }
3684:
3607:
3587:
3528:
3504:
3472:Hard Lefschetz theorem
3452:
3416:
3381:
3321:
3283:
3132:
3007:
2973:
2944:
2834:
2810:
2776:
2719:
2690:
2658:
2638:
2618:
2592:
2560:
2532:for a smooth function
2526:
2479:
2456:
2424:
2423:{\displaystyle \beta }
2404:
2364:
2344:
2315:
2262:
2221:
2168:
2152:local Kähler potential
2144:
2124:
2054:
2034:
2014:
1994:
1974:
1954:
1934:
1900:
1859:
1835:
1811:
1773:
1715:
1679:
1659:
1597:
1577:
1557:
1521:
1494:
1474:
1454:
1415:
1392:
1357:
1310:
1288:
1250:
1230:
1207:
1130:
1044:
1015:
995:
975:
955:
926:
899:
879:
859:
835:
812:
744:
720:
697:
677:
656:is the complex number
650:
630:
610:
587:
492:
472:
452:
425:
401:
375:
352:
321:
297:
270:
205:
178:
155:
102:is a Kähler manifold.
8096:Holographic principle
8063:Type IIB supergravity
8058:Type IIA supergravity
7910:-form electrodynamics
7529:Bosonic string theory
7088:Annals of Mathematics
7062:John Wiley & Sons
6952:John Wiley & Sons
6637:
6557:, Proposition 3.A.28.
6521:, Proposition 3.1.12.
6199:
6156:
6128:
5851:
5818:Bott–Chern cohomology
5809:
5776:
5488:
5417:
5353:
5294:
5267:
5209:
5135:
5111:
5021:
4997:
4951:
4902:
4853:
4833:
4810:
4695:
4675:
4624:
4601:
4599:{\displaystyle (p,q)}
4569:
4547:
4545:{\displaystyle (p,q)}
4514:
4477:
4448:
4428:
4408:
4385:
4351:
4251:
4228:
4152:
4129:
3961:
3854:
3852:{\displaystyle d^{*}}
3827:
3807:
3787:
3760:
3736:
3734:{\displaystyle d^{*}}
3705:
3685:
3608:
3588:
3534:-forms is defined by
3529:
3505:
3453:
3422:and its adjoint, the
3417:
3382:
3322:
3284:
3224:of complex dimension
3133:
3049:is determined by its
3008:
2974:
2945:
2835:
2811:
2777:
2720:
2691:
2659:
2639:
2637:{\displaystyle \rho }
2619:
2593:
2561:
2527:
2480:
2457:
2425:
2405:
2365:
2345:
2316:
2263:
2222:
2169:
2145:
2143:{\displaystyle \rho }
2125:
2055:
2035:
2033:{\displaystyle \rho }
2015:
1995:
1975:
1955:
1935:
1901:
1860:
1836:
1834:{\displaystyle \rho }
1812:
1774:
1716:
1714:{\displaystyle \rho }
1701:real-valued function
1680:
1660:
1598:
1583:preserves the metric
1578:
1558:
1522:
1495:
1475:
1455:
1416:
1393:
1358:
1311:
1289:
1251:
1231:
1208:
1131:
1045:
1016:
996:
976:
956:
927:
900:
880:
860:
845:of complex dimension
836:
813:
745:
721:
698:
678:
651:
631:
611:
588:
493:
473:
453:
435:on the tangent space
426:
402:
376:
353:
322:
298:
271:
206:
179:
156:
106:is a central part of
62:Jan Arnoldus Schouten
38:differential geometry
8636:Riemannian manifolds
8015:Hořava–Witten theory
7962:Hyperkähler manifold
7650:Particles and fields
7598:Tachyon condensation
7583:Matrix string theory
7050:Kobayashi, Shoshichi
6899:Cannas da Silva, Ana
6616:
6592:p.217 Definition 1.1
6578:Amorós et al. (1996)
6567:Amorós et al. (1996)
6419:Hyperkähler manifold
6176:
6167:Kobayashi hyperbolic
6137:
6105:
5861:compact complex tori
5828:
5822:Dolbeault cohomology
5786:
5753:
5426:
5362:
5303:
5276:
5224:
5147:
5124:
5030:
5010:
4960:
4911:
4869:
4842:
4822:
4715:
4684:
4640:
4613:
4578:
4558:
4524:
4486:
4457:
4437:
4417:
4397:
4363:
4263:
4240:
4168:
4141:
3976:
3866:
3836:
3816:
3796:
3776:
3765:with respect to the
3749:
3718:
3694:
3617:
3597:
3538:
3518:
3494:
3429:
3424:contraction operator
3394:
3331:
3296:
3273:
3064:
2983:
2957:
2847:
2820:
2794:
2729:
2703:
2668:
2648:
2628:
2602:
2570:
2536:
2489:
2466:
2434:
2414:
2374:
2354:
2329:
2272:
2240:
2199:
2158:
2134:
2064:
2044:
2024:
2004:
1984:
1964:
1944:
1912:
1878:
1849:
1825:
1786:
1732:
1705:
1669:
1607:
1587:
1567:
1531:
1508:
1484:
1464:
1432:
1425:is contained in the
1402:
1382:
1371:Riemannian viewpoint
1323:
1300:
1260:
1240:
1220:
1143:
1054:
1025:
1005:
985:
965:
936:
916:
889:
869:
849:
825:
757:
734:
710:
687:
660:
640:
620:
600:
596:for tangent vectors
505:
482:
462:
439:
415:
391:
365:
342:
311:
287:
221:
195:
168:
133:
122:Symplectic viewpoint
58:symplectic structure
54:Riemannian structure
8651:Symplectic geometry
8053:Type I supergravity
7957:Calabi–Yau manifold
7952:Ricci-flat manifold
7931:Kaluza–Klein theory
7672:Ramond–Ramond field
7578:String field theory
7272:2004InMat.157..329V
7182:10.1093/imrn/rnz331
7166:(24): 18520–18528.
6674:Barth et al. (2004)
6533:, Corollary 3.2.12.
6461:, Proposition 7.14.
6429:K-energy functional
6403:is Kähler (by Siu).
6364:The open unit ball
6343:complex submanifold
6330:is provided by the
6293:Fubini–Study metric
6057:sectional curvature
5928:Lorentzian manifold
5910:if it has constant
5877:homotopy equivalent
5737:). The Kähler form
3712:Hodge star operator
3480:Hodge index theorem
3291:Dolbeault operators
3245:calibrated geometry
3241:minimal submanifold
3055:Wirtinger's formula
2301:
1819:Dolbeault operators
1377:Riemannian manifold
211:, meaning that the
128:symplectic manifold
8641:Algebraic geometry
8020:K-theory (physics)
7897:ADE classification
7534:Superstring theory
7017:Huybrechts, Daniel
6994:10.1007/BF02940642
6940:Griffiths, Phillip
6632:
6485:, Proposition 8.8.
6449:, Definition 16.1.
6194:
6151:
6123:
5934:for more details.
5932:Einstein manifolds
5924:general relativity
5892:complex manifold.
5889:Hironaka's example
5846:
5804:
5771:
5603:showed that every
5568:fundamental groups
5483:
5412:
5348:
5289:
5262:
5204:
5184:
5130:
5106:
5016:
4992:
4946:
4897:
4848:
4828:
4805:
4769:
4690:
4670:
4619:
4596:
4564:
4542:
4509:
4472:
4443:
4423:
4403:
4380:
4346:
4313:
4246:
4223:
4147:
4124:
3956:
3859:are decomposed as
3849:
3822:
3802:
3782:
3755:
3731:
3700:
3680:
3603:
3583:
3524:
3500:
3448:
3412:
3389:Lefschetz operator
3377:
3317:
3279:
3128:
3037:, the volume of a
3015:contractible space
3003:
2969:
2940:
2830:
2806:
2786:, so the space of
2772:
2715:
2689:{\displaystyle =0}
2686:
2654:
2634:
2614:
2598:on an open subset
2591:{\displaystyle =0}
2588:
2556:
2522:
2485:may be written as
2475:
2452:
2420:
2410:for some one-form
2400:
2360:
2350:say, differs from
2340:
2311:
2287:
2258:
2217:
2192:de Rham cohomology
2164:
2140:
2120:
2050:
2030:
2010:
1990:
1970:
1950:
1930:
1896:
1855:
1831:
1807:
1769:
1711:
1687:parallel transport
1675:
1655:
1593:
1573:
1553:
1520:{\displaystyle TX}
1517:
1490:
1470:
1450:
1414:{\displaystyle 2n}
1411:
1398:of even dimension
1388:
1353:
1318:de Rham cohomology
1306:
1284:
1246:
1226:
1203:
1126:
1040:
1011:
991:
971:
951:
922:
895:
875:
855:
843:Hermitian manifold
831:
808:
740:
716:
693:
673:
646:
626:
606:
583:
488:
468:
451:{\displaystyle TX}
448:
421:
411:. In more detail,
397:
371:
348:
317:
293:
266:
201:
174:
151:
108:algebraic geometry
100:projective variety
8646:Complex manifolds
8623:
8622:
8405:van Nieuwenhuizen
7941:Why 10 dimensions
7846:Chern–Simons form
7813:Kac–Moody algebra
7793:Conformal algebra
7788:Conformal anomaly
7682:Magnetic monopole
7677:Kalb–Ramond field
7519:Nambu–Goto action
7401:"Kähler manifold"
7377:978-0-8218-2163-3
7326:10.1090/tran/7445
7230:978-0-521-68897-0
7071:978-0-471-15732-8
7034:978-3-540-21290-4
6961:978-0-471-05059-9
6840:978-3-540-00832-3
6795:978-0-8218-0498-8
6630:
6580:, Corollary 1.66.
6555:Huybrechts (2005)
6543:Huybrechts (2005)
6531:Huybrechts (2005)
6519:Huybrechts (2005)
6507:Huybrechts (2005)
6355:algebraic variety
6161:is constant). If
6011:Calabi conjecture
5843:
5801:
5768:
5699:.) Equivalently,
5481:
5346:
5163:
5133:{\displaystyle X}
5019:{\displaystyle X}
4851:{\displaystyle X}
4831:{\displaystyle X}
4748:
4693:{\displaystyle X}
4622:{\displaystyle X}
4426:{\displaystyle X}
4406:{\displaystyle r}
4292:
4249:{\displaystyle X}
4200:
4159:Kähler identities
4150:{\displaystyle X}
4078:
4075:
4072:
4069:
4061:
4043:
4021:
4008:
3992:
3944:
3907:
3904:
3901:
3898:
3890:
3825:{\displaystyle d}
3805:{\displaystyle X}
3785:{\displaystyle r}
3758:{\displaystyle d}
3714:. (Equivalently,
3606:{\displaystyle d}
3527:{\displaystyle r}
3503:{\displaystyle n}
3373:
3314:
3282:{\displaystyle d}
3261:Kähler identities
3255:Kähler identities
3103:
2922:
2896:
2788:Kähler potentials
2766:
2516:
2449:
2294:
2214:
2114:
2053:{\displaystyle U}
2013:{\displaystyle p}
1993:{\displaystyle U}
1973:{\displaystyle X}
1953:{\displaystyle p}
1893:
1804:
1763:
1749:
1678:{\displaystyle J}
1596:{\displaystyle g}
1576:{\displaystyle J}
1493:{\displaystyle X}
1473:{\displaystyle J}
1391:{\displaystyle X}
1296:Since the 2-form
1249:{\displaystyle b}
1229:{\displaystyle a}
1121:
1096:
1014:{\displaystyle 0}
994:{\displaystyle p}
974:{\displaystyle p}
925:{\displaystyle p}
898:{\displaystyle X}
878:{\displaystyle p}
858:{\displaystyle n}
834:{\displaystyle X}
743:{\displaystyle g}
726:is a real closed
696:{\displaystyle X}
671:
649:{\displaystyle i}
629:{\displaystyle v}
609:{\displaystyle u}
478:, and the 2-form
471:{\displaystyle X}
458:at each point of
424:{\displaystyle h}
374:{\displaystyle h}
351:{\displaystyle X}
331:Complex viewpoint
320:{\displaystyle X}
305:positive definite
296:{\displaystyle X}
177:{\displaystyle J}
161:equipped with an
66:David van Dantzig
50:complex structure
16:(Redirected from
8658:
8133:String theorists
8073:Lie superalgebra
8025:Twisted K-theory
7983:Spin(7)-manifold
7936:Compactification
7778:Virasoro algebra
7561:Heterotic string
7455:
7448:
7441:
7432:
7427:
7426:
7414:
7388:
7358:
7337:
7319:
7310:(4): 2703–2718.
7298:
7265:
7241:
7214:
7193:
7175:
7154:
7136:
7134:10.5802/aif.1673
7111:
7082:
7045:
7012:
6973:
6935:
6894:
6876:
6874:10.5802/aif.1674
6851:
6806:
6786:10.1090/surv/044
6757:
6752:
6746:
6741:
6735:
6729:
6723:
6717:
6711:
6710:, Corollary 9.8.
6705:
6699:
6694:
6688:
6683:
6677:
6671:
6662:
6657:
6651:
6641:
6639:
6638:
6633:
6631:
6623:
6610:
6604:
6599:
6593:
6587:
6581:
6575:
6569:
6564:
6558:
6552:
6546:
6540:
6534:
6528:
6522:
6516:
6510:
6504:
6498:
6492:
6486:
6480:
6474:
6468:
6462:
6456:
6450:
6444:
6321:
6305:
6203:
6201:
6200:
6195:
6160:
6158:
6157:
6152:
6144:
6132:
6130:
6129:
6124:
6073:
5977:canonical bundle
5970:
5865:Kunihiko Kodaira
5855:
5853:
5852:
5847:
5845:
5844:
5836:
5813:
5811:
5810:
5805:
5803:
5802:
5794:
5780:
5778:
5777:
5772:
5770:
5769:
5761:
5736:
5698:
5683:
5664:
5649:
5543:
5533:
5492:
5490:
5489:
5484:
5482:
5477:
5476:
5449:
5444:
5443:
5421:
5419:
5418:
5413:
5411:
5410:
5380:
5379:
5357:
5355:
5354:
5349:
5347:
5342:
5341:
5326:
5321:
5320:
5298:
5296:
5295:
5290:
5288:
5287:
5271:
5269:
5268:
5263:
5261:
5260:
5242:
5241:
5213:
5211:
5210:
5205:
5200:
5199:
5183:
5159:
5158:
5139:
5137:
5136:
5131:
5115:
5113:
5112:
5107:
5096:
5095:
5080:
5079:
5078:
5072:
5048:
5047:
5025:
5023:
5022:
5017:
5001:
4999:
4998:
4993:
4982:
4981:
4970:
4969:
4955:
4953:
4952:
4947:
4942:
4941:
4923:
4922:
4906:
4904:
4903:
4898:
4887:
4886:
4857:
4855:
4854:
4849:
4837:
4835:
4834:
4829:
4814:
4812:
4811:
4806:
4798:
4797:
4779:
4778:
4768:
4741:
4727:
4726:
4699:
4697:
4696:
4691:
4679:
4677:
4676:
4671:
4666:
4652:
4651:
4628:
4626:
4625:
4620:
4605:
4603:
4602:
4597:
4573:
4571:
4570:
4565:
4551:
4549:
4548:
4543:
4518:
4516:
4515:
4510:
4508:
4507:
4496:
4495:
4481:
4479:
4478:
4473:
4452:
4450:
4449:
4444:
4432:
4430:
4429:
4424:
4412:
4410:
4409:
4404:
4390:is the space of
4389:
4387:
4386:
4381:
4379:
4378:
4373:
4372:
4355:
4353:
4352:
4347:
4333:
4332:
4321:
4320:
4312:
4279:
4278:
4273:
4272:
4255:
4253:
4252:
4247:
4232:
4230:
4229:
4224:
4219:
4218:
4203:
4202:
4201:
4193:
4180:
4179:
4156:
4154:
4153:
4148:
4133:
4131:
4130:
4125:
4117:
4116:
4104:
4103:
4088:
4087:
4076:
4073:
4070:
4067:
4063:
4062:
4054:
4051:
4050:
4045:
4044:
4036:
4029:
4028:
4023:
4022:
4014:
4010:
4009:
4001:
3995:
3994:
3993:
3985:
3965:
3963:
3962:
3957:
3952:
3951:
3946:
3945:
3937:
3930:
3929:
3917:
3916:
3905:
3902:
3899:
3896:
3892:
3891:
3883:
3858:
3856:
3855:
3850:
3848:
3847:
3831:
3829:
3828:
3823:
3811:
3809:
3808:
3803:
3791:
3789:
3788:
3783:
3764:
3762:
3761:
3756:
3740:
3738:
3737:
3732:
3730:
3729:
3709:
3707:
3706:
3701:
3689:
3687:
3686:
3681:
3669:
3668:
3629:
3628:
3612:
3610:
3609:
3604:
3592:
3590:
3589:
3584:
3579:
3578:
3566:
3565:
3550:
3549:
3533:
3531:
3530:
3525:
3509:
3507:
3506:
3501:
3457:
3455:
3454:
3449:
3447:
3446:
3421:
3419:
3418:
3413:
3386:
3384:
3383:
3378:
3376:
3375:
3374:
3366:
3356:
3355:
3343:
3342:
3326:
3324:
3323:
3318:
3316:
3315:
3307:
3288:
3286:
3285:
3280:
3219:
3200:
3181:
3137:
3135:
3134:
3129:
3124:
3123:
3114:
3113:
3104:
3102:
3091:
3077:
3033:Kähler manifold
3012:
3010:
3009:
3004:
3002:
2997:
2992:
2991:
2978:
2976:
2975:
2972:{\displaystyle }
2970:
2949:
2947:
2946:
2941:
2924:
2923:
2915:
2897:
2894:
2892:
2869:
2868:
2857:
2856:
2839:
2837:
2836:
2831:
2829:
2828:
2815:
2813:
2812:
2809:{\displaystyle }
2807:
2781:
2779:
2778:
2773:
2768:
2767:
2759:
2741:
2740:
2724:
2722:
2721:
2718:{\displaystyle }
2716:
2695:
2693:
2692:
2687:
2663:
2661:
2660:
2655:
2643:
2641:
2640:
2635:
2623:
2621:
2620:
2615:
2597:
2595:
2594:
2589:
2565:
2563:
2562:
2557:
2555:
2531:
2529:
2528:
2523:
2518:
2517:
2509:
2484:
2482:
2481:
2476:
2461:
2459:
2458:
2453:
2451:
2450:
2442:
2429:
2427:
2426:
2421:
2409:
2407:
2406:
2401:
2384:
2369:
2367:
2366:
2361:
2349:
2347:
2346:
2341:
2339:
2320:
2318:
2317:
2312:
2300:
2295:
2292:
2267:
2265:
2264:
2259:
2226:
2224:
2223:
2218:
2216:
2215:
2207:
2173:
2171:
2170:
2165:
2149:
2147:
2146:
2141:
2129:
2127:
2126:
2121:
2116:
2115:
2107:
2095:
2081:
2080:
2075:
2059:
2057:
2056:
2051:
2039:
2037:
2036:
2031:
2019:
2017:
2016:
2011:
1999:
1997:
1996:
1991:
1979:
1977:
1976:
1971:
1959:
1957:
1956:
1951:
1939:
1937:
1936:
1931:
1905:
1903:
1902:
1897:
1895:
1894:
1886:
1864:
1862:
1861:
1856:
1843:Kähler potential
1840:
1838:
1837:
1832:
1816:
1814:
1813:
1808:
1806:
1805:
1797:
1778:
1776:
1775:
1770:
1765:
1764:
1756:
1750:
1742:
1720:
1718:
1717:
1712:
1693:Kähler potential
1685:is preserved by
1684:
1682:
1681:
1676:
1664:
1662:
1661:
1656:
1602:
1600:
1599:
1594:
1582:
1580:
1579:
1574:
1562:
1560:
1559:
1554:
1543:
1542:
1526:
1524:
1523:
1518:
1499:
1497:
1496:
1491:
1479:
1477:
1476:
1471:
1459:
1457:
1456:
1451:
1420:
1418:
1417:
1412:
1397:
1395:
1394:
1389:
1362:
1360:
1359:
1354:
1349:
1335:
1334:
1315:
1313:
1312:
1307:
1293:
1291:
1290:
1285:
1255:
1253:
1252:
1247:
1235:
1233:
1232:
1227:
1212:
1210:
1209:
1204:
1199:
1198:
1174:
1173:
1158:
1157:
1135:
1133:
1132:
1127:
1122:
1120:
1119:
1118:
1102:
1097:
1095:
1094:
1093:
1077:
1069:
1068:
1049:
1047:
1046:
1041:
1039:
1038:
1033:
1020:
1018:
1017:
1012:
1000:
998:
997:
992:
980:
978:
977:
972:
961:to order 2 near
960:
958:
957:
952:
950:
949:
944:
931:
929:
928:
923:
910:coordinate chart
904:
902:
901:
896:
884:
882:
881:
876:
864:
862:
861:
856:
840:
838:
837:
832:
817:
815:
814:
809:
749:
747:
746:
741:
725:
723:
722:
717:
702:
700:
699:
694:
682:
680:
679:
674:
672:
664:
655:
653:
652:
647:
635:
633:
632:
627:
615:
613:
612:
607:
592:
590:
589:
584:
497:
495:
494:
489:
477:
475:
474:
469:
457:
455:
454:
449:
430:
428:
427:
422:
406:
404:
403:
398:
380:
378:
377:
372:
360:Hermitian metric
357:
355:
354:
349:
337:complex manifold
326:
324:
323:
318:
302:
300:
299:
294:
275:
273:
272:
267:
210:
208:
207:
202:
183:
181:
180:
175:
160:
158:
157:
152:
21:
8666:
8665:
8661:
8660:
8659:
8657:
8656:
8655:
8626:
8625:
8624:
8619:
8128:
8105:
8082:
8029:
7977:
7947:Kähler manifold
7914:
7891:
7884:
7877:
7870:
7863:
7822:
7783:Mirror symmetry
7764:
7750:Brane cosmology
7696:
7645:
7612:
7568:N=2 superstring
7554:Type IIB string
7549:Type IIA string
7524:Polyakov action
7507:
7464:
7459:
7424:
7417:
7399:
7396:
7391:
7378:
7361:
7355:
7340:
7301:
7244:
7231:
7196:
7157:
7114:
7100:10.2307/1969701
7085:
7072:
7060:, vol. 2,
7054:Nomizu, Katsumi
7048:
7035:
7015:
6976:
6962:
6938:
6925:
6897:
6854:
6841:
6809:
6796:
6769:
6765:
6760:
6753:
6749:
6742:
6738:
6730:
6726:
6718:
6714:
6706:
6702:
6695:
6691:
6686:Buchdahl (1999)
6684:
6680:
6676:, section IV.3.
6672:
6665:
6658:
6654:
6614:
6613:
6611:
6607:
6600:
6596:
6588:
6584:
6576:
6572:
6565:
6561:
6553:
6549:
6541:
6537:
6529:
6525:
6517:
6513:
6505:
6501:
6493:
6489:
6483:Moroianu (2007)
6481:
6477:
6473:, v. 2, p. 149.
6469:
6465:
6457:
6453:
6445:
6441:
6437:
6410:
6315:
6299:
6275:Riemann surface
6238:
6174:
6173:
6135:
6134:
6103:
6102:
6068:
6049:
6022:Simon Donaldson
5986:
5957:
5947:
5912:Ricci curvature
5908:Kähler–Einstein
5904:
5898:
5867:'s work on the
5826:
5825:
5784:
5783:
5751:
5750:
5723:
5685:
5670:
5651:
5636:
5635:whose class in
5617:
5609:closed manifold
5601:Clifford Taubes
5584:
5541:
5535:
5525:
5515:
5503:
5450:
5429:
5424:
5423:
5384:
5365:
5360:
5359:
5358:. The identity
5327:
5306:
5301:
5300:
5279:
5274:
5273:
5246:
5227:
5222:
5221:
5185:
5150:
5145:
5144:
5122:
5121:
5081:
5061:
5033:
5028:
5027:
5026:are defined by
5008:
5007:
4963:
4958:
4957:
4933:
4914:
4909:
4908:
4872:
4867:
4866:
4840:
4839:
4820:
4819:
4789:
4770:
4718:
4713:
4712:
4682:
4681:
4643:
4638:
4637:
4611:
4610:
4576:
4575:
4556:
4555:
4522:
4521:
4489:
4484:
4483:
4455:
4454:
4435:
4434:
4415:
4414:
4395:
4394:
4366:
4361:
4360:
4314:
4266:
4261:
4260:
4238:
4237:
4210:
4187:
4171:
4166:
4165:
4157:is Kähler, the
4139:
4138:
4108:
4095:
4079:
4033:
4011:
3979:
3974:
3973:
3934:
3921:
3908:
3864:
3863:
3839:
3834:
3833:
3814:
3813:
3794:
3793:
3774:
3773:
3747:
3746:
3721:
3716:
3715:
3692:
3691:
3645:
3620:
3615:
3614:
3595:
3594:
3570:
3557:
3541:
3536:
3535:
3516:
3515:
3492:
3491:
3488:
3438:
3427:
3426:
3392:
3391:
3360:
3347:
3334:
3329:
3328:
3294:
3293:
3271:
3270:
3263:
3257:
3206:
3205:is not zero in
3187:
3171:
3162:
3115:
3105:
3095:
3062:
3061:
3027:
2981:
2980:
2955:
2954:
2850:
2845:
2844:
2818:
2817:
2792:
2791:
2732:
2727:
2726:
2701:
2700:
2666:
2665:
2646:
2645:
2626:
2625:
2600:
2599:
2568:
2567:
2534:
2533:
2487:
2486:
2464:
2463:
2432:
2431:
2412:
2411:
2377:
2372:
2371:
2352:
2351:
2332:
2327:
2326:
2270:
2269:
2238:
2237:
2197:
2196:
2180:
2156:
2155:
2132:
2131:
2067:
2062:
2061:
2042:
2041:
2022:
2021:
2002:
2001:
1982:
1981:
1962:
1961:
1942:
1941:
1910:
1909:
1876:
1875:
1872:, known as the
1847:
1846:
1823:
1822:
1821:. The function
1784:
1783:
1730:
1729:
1703:
1702:
1695:
1667:
1666:
1605:
1604:
1585:
1584:
1565:
1564:
1534:
1529:
1528:
1527:to itself with
1506:
1505:
1482:
1481:
1462:
1461:
1430:
1429:
1400:
1399:
1380:
1379:
1373:
1363:, known as the
1326:
1321:
1320:
1298:
1297:
1258:
1257:
1238:
1237:
1218:
1217:
1190:
1162:
1146:
1141:
1140:
1110:
1106:
1085:
1081:
1057:
1052:
1051:
1028:
1023:
1022:
1003:
1002:
983:
982:
963:
962:
939:
934:
933:
914:
913:
887:
886:
867:
866:
847:
846:
823:
822:
755:
754:
732:
731:
708:
707:
685:
684:
658:
657:
638:
637:
618:
617:
598:
597:
503:
502:
480:
479:
460:
459:
437:
436:
413:
412:
389:
388:
363:
362:
340:
339:
333:
309:
308:
285:
284:
219:
218:
193:
192:
190:symplectic form
166:
165:
131:
130:
124:
116:
78:Kähler geometry
42:Kähler manifold
36:and especially
30:
23:
22:
18:Kähler geometry
15:
12:
11:
5:
8664:
8662:
8654:
8653:
8648:
8643:
8638:
8628:
8627:
8621:
8620:
8618:
8617:
8612:
8607:
8602:
8597:
8592:
8587:
8582:
8577:
8572:
8567:
8562:
8557:
8552:
8547:
8542:
8537:
8532:
8527:
8522:
8517:
8512:
8507:
8502:
8497:
8492:
8487:
8482:
8477:
8472:
8467:
8462:
8457:
8455:Randjbar-Daemi
8452:
8447:
8442:
8437:
8432:
8427:
8422:
8417:
8412:
8407:
8402:
8397:
8392:
8387:
8382:
8377:
8372:
8367:
8362:
8357:
8352:
8347:
8342:
8337:
8332:
8327:
8322:
8317:
8312:
8307:
8302:
8297:
8292:
8287:
8282:
8277:
8272:
8267:
8262:
8257:
8252:
8247:
8242:
8237:
8232:
8227:
8222:
8217:
8212:
8207:
8202:
8197:
8192:
8187:
8182:
8177:
8172:
8167:
8162:
8157:
8152:
8147:
8142:
8136:
8134:
8130:
8129:
8127:
8126:
8121:
8115:
8113:
8107:
8106:
8104:
8103:
8098:
8092:
8090:
8084:
8083:
8081:
8080:
8078:Lie supergroup
8075:
8070:
8065:
8060:
8055:
8050:
8045:
8039:
8037:
8031:
8030:
8028:
8027:
8022:
8017:
8012:
8007:
8002:
7997:
7992:
7987:
7986:
7985:
7980:
7975:
7971:
7970:
7969:
7959:
7949:
7944:
7938:
7933:
7928:
7922:
7920:
7916:
7915:
7913:
7912:
7904:
7899:
7894:
7889:
7882:
7875:
7868:
7861:
7853:
7848:
7843:
7838:
7832:
7830:
7824:
7823:
7821:
7820:
7815:
7810:
7805:
7800:
7795:
7790:
7785:
7780:
7774:
7772:
7766:
7765:
7763:
7762:
7757:
7755:Quiver diagram
7752:
7747:
7742:
7737:
7732:
7727:
7722:
7717:
7712:
7706:
7704:
7698:
7697:
7695:
7694:
7689:
7684:
7679:
7674:
7669:
7664:
7659:
7653:
7651:
7647:
7646:
7644:
7643:
7638:
7633:
7628:
7622:
7620:
7618:String duality
7614:
7613:
7611:
7610:
7605:
7600:
7595:
7590:
7585:
7580:
7575:
7570:
7565:
7564:
7563:
7558:
7557:
7556:
7551:
7544:Type II string
7541:
7531:
7526:
7521:
7515:
7513:
7509:
7508:
7506:
7505:
7500:
7499:
7498:
7493:
7483:
7481:Cosmic strings
7478:
7472:
7470:
7466:
7465:
7460:
7458:
7457:
7450:
7443:
7435:
7429:
7428:
7415:
7395:
7394:External links
7392:
7390:
7389:
7376:
7359:
7353:
7338:
7299:
7256:(2): 329–343,
7246:Voisin, Claire
7242:
7229:
7194:
7155:
7127:(1): 263–285.
7112:
7083:
7070:
7046:
7033:
7013:
6974:
6960:
6944:Harris, Joseph
6936:
6924:978-3540421955
6923:
6895:
6867:(1): 287–302.
6852:
6839:
6811:Barth, Wolf P.
6807:
6794:
6766:
6764:
6761:
6759:
6758:
6747:
6736:
6724:
6712:
6700:
6689:
6678:
6663:
6652:
6650:(1), pp.71-81.
6629:
6626:
6621:
6605:
6602:Kodaira (1954)
6594:
6582:
6570:
6559:
6547:
6535:
6523:
6511:
6509:, Section 3.1.
6499:
6497:, section 7.4.
6487:
6475:
6463:
6451:
6438:
6436:
6433:
6432:
6431:
6426:
6421:
6416:
6409:
6406:
6405:
6404:
6397:
6374:Bergman metric
6362:
6347:Stein manifold
6339:
6282:
6271:rotation group
6263:
6254:/Λ (Λ a full
6248:
6237:
6234:
6193:
6190:
6187:
6184:
6181:
6150:
6147:
6143:
6122:
6119:
6116:
6113:
6110:
6048:
6045:
6007:Shing-Tung Yau
5982:
5954:tangent bundle
5945:
5900:Main article:
5897:
5894:
5842:
5839:
5833:
5800:
5797:
5791:
5767:
5764:
5758:
5616:
5613:
5582:
5576:abelianization
5539:
5534:and hence has
5509:
5502:
5499:
5480:
5475:
5472:
5469:
5466:
5463:
5460:
5457:
5453:
5447:
5442:
5439:
5436:
5432:
5409:
5406:
5403:
5400:
5397:
5394:
5391:
5387:
5383:
5378:
5375:
5372:
5368:
5345:
5340:
5337:
5334:
5330:
5324:
5319:
5316:
5313:
5309:
5286:
5282:
5259:
5256:
5253:
5249:
5245:
5240:
5237:
5234:
5230:
5219:Hodge symmetry
5215:
5214:
5203:
5198:
5195:
5192:
5188:
5182:
5179:
5176:
5173:
5170:
5166:
5162:
5157:
5153:
5129:
5105:
5102:
5099:
5094:
5091:
5088:
5084:
5077:
5071:
5068:
5065:
5060:
5057:
5054:
5051:
5046:
5043:
5040:
5036:
5015:
4991:
4988:
4985:
4980:
4977:
4974:
4968:
4945:
4940:
4936:
4932:
4929:
4926:
4921:
4917:
4896:
4893:
4890:
4885:
4882:
4879:
4875:
4847:
4827:
4816:
4815:
4804:
4801:
4796:
4792:
4788:
4785:
4782:
4777:
4773:
4767:
4764:
4761:
4758:
4755:
4751:
4747:
4744:
4740:
4736:
4733:
4730:
4725:
4721:
4689:
4669:
4665:
4661:
4658:
4655:
4650:
4646:
4618:
4595:
4592:
4589:
4586:
4583:
4563:
4541:
4538:
4535:
4532:
4529:
4506:
4503:
4500:
4494:
4471:
4468:
4465:
4462:
4442:
4422:
4402:
4377:
4371:
4357:
4356:
4345:
4342:
4339:
4336:
4331:
4328:
4325:
4319:
4311:
4308:
4305:
4302:
4299:
4295:
4291:
4288:
4285:
4282:
4277:
4271:
4245:
4234:
4233:
4222:
4217:
4213:
4209:
4206:
4199:
4196:
4190:
4186:
4183:
4178:
4174:
4146:
4135:
4134:
4123:
4120:
4115:
4111:
4107:
4102:
4098:
4094:
4091:
4086:
4082:
4066:
4060:
4057:
4049:
4042:
4039:
4032:
4027:
4020:
4017:
4007:
4004:
3998:
3991:
3988:
3982:
3967:
3966:
3955:
3950:
3943:
3940:
3933:
3928:
3924:
3920:
3915:
3911:
3895:
3889:
3886:
3880:
3877:
3874:
3871:
3846:
3842:
3821:
3801:
3781:
3754:
3728:
3724:
3699:
3679:
3675:
3672:
3667:
3664:
3661:
3658:
3655:
3652:
3648:
3644:
3641:
3638:
3635:
3632:
3627:
3623:
3602:
3582:
3577:
3573:
3569:
3564:
3560:
3556:
3553:
3548:
3544:
3523:
3499:
3487:
3484:
3445:
3441:
3437:
3434:
3411:
3408:
3405:
3402:
3399:
3372:
3369:
3363:
3359:
3354:
3350:
3346:
3341:
3337:
3313:
3310:
3304:
3301:
3278:
3259:Main article:
3256:
3253:
3166:
3139:
3138:
3127:
3122:
3118:
3112:
3108:
3101:
3098:
3094:
3089:
3086:
3083:
3080:
3076:
3073:
3070:
3026:
3023:
3001:
2996:
2990:
2968:
2965:
2962:
2951:
2950:
2939:
2936:
2933:
2930:
2927:
2921:
2918:
2912:
2909:
2906:
2903:
2900:
2891:
2887:
2884:
2881:
2878:
2875:
2872:
2867:
2864:
2861:
2855:
2827:
2805:
2802:
2799:
2790:for the class
2771:
2765:
2762:
2756:
2753:
2750:
2747:
2744:
2739:
2735:
2714:
2711:
2708:
2699:In general if
2685:
2682:
2679:
2676:
2673:
2653:
2633:
2613:
2610:
2607:
2587:
2584:
2581:
2578:
2575:
2554:
2550:
2547:
2544:
2541:
2521:
2515:
2512:
2506:
2503:
2500:
2497:
2494:
2474:
2471:
2448:
2445:
2439:
2419:
2399:
2396:
2393:
2390:
2387:
2383:
2380:
2359:
2338:
2335:
2310:
2307:
2304:
2299:
2290:
2286:
2283:
2280:
2277:
2257:
2254:
2251:
2248:
2245:
2213:
2210:
2204:
2179:
2176:
2163:
2139:
2119:
2113:
2110:
2104:
2101:
2098:
2094:
2090:
2087:
2084:
2079:
2074:
2071:
2049:
2029:
2009:
1989:
1969:
1949:
1929:
1926:
1923:
1920:
1917:
1892:
1889:
1883:
1870:Poincaré lemma
1854:
1830:
1803:
1800:
1794:
1791:
1780:
1779:
1768:
1762:
1759:
1753:
1748:
1745:
1740:
1737:
1710:
1694:
1691:
1674:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1603:(meaning that
1592:
1572:
1552:
1549:
1546:
1541:
1537:
1516:
1513:
1489:
1469:
1449:
1446:
1443:
1440:
1437:
1423:holonomy group
1410:
1407:
1387:
1372:
1369:
1352:
1348:
1344:
1341:
1338:
1333:
1329:
1305:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1245:
1225:
1214:
1213:
1202:
1197:
1193:
1189:
1186:
1183:
1180:
1177:
1172:
1169:
1165:
1161:
1156:
1153:
1149:
1125:
1117:
1113:
1109:
1105:
1100:
1092:
1088:
1084:
1080:
1075:
1072:
1067:
1064:
1060:
1037:
1032:
1010:
990:
970:
948:
943:
921:
894:
874:
854:
830:
819:
818:
807:
804:
801:
798:
795:
792:
789:
786:
783:
780:
777:
774:
771:
768:
765:
762:
739:
715:
692:
670:
667:
645:
625:
605:
594:
593:
582:
579:
576:
573:
570:
567:
564:
561:
558:
555:
552:
549:
546:
543:
540:
537:
534:
531:
528:
525:
522:
519:
516:
513:
510:
498:is defined by
487:
467:
447:
444:
433:Hermitian form
420:
396:
370:
347:
332:
329:
316:
292:
277:
276:
265:
262:
259:
256:
253:
250:
247:
244:
241:
238:
235:
232:
229:
226:
200:
173:
150:
147:
144:
141:
138:
123:
120:
115:
112:
28:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8663:
8652:
8649:
8647:
8644:
8642:
8639:
8637:
8634:
8633:
8631:
8616:
8613:
8611:
8608:
8606:
8603:
8601:
8600:Zamolodchikov
8598:
8596:
8595:Zamolodchikov
8593:
8591:
8588:
8586:
8583:
8581:
8578:
8576:
8573:
8571:
8568:
8566:
8563:
8561:
8558:
8556:
8553:
8551:
8548:
8546:
8543:
8541:
8538:
8536:
8533:
8531:
8528:
8526:
8523:
8521:
8518:
8516:
8513:
8511:
8508:
8506:
8503:
8501:
8498:
8496:
8493:
8491:
8488:
8486:
8483:
8481:
8478:
8476:
8473:
8471:
8468:
8466:
8463:
8461:
8458:
8456:
8453:
8451:
8448:
8446:
8443:
8441:
8438:
8436:
8433:
8431:
8428:
8426:
8423:
8421:
8418:
8416:
8413:
8411:
8408:
8406:
8403:
8401:
8398:
8396:
8393:
8391:
8388:
8386:
8383:
8381:
8378:
8376:
8373:
8371:
8368:
8366:
8363:
8361:
8358:
8356:
8353:
8351:
8348:
8346:
8343:
8341:
8338:
8336:
8333:
8331:
8328:
8326:
8323:
8321:
8318:
8316:
8313:
8311:
8308:
8306:
8303:
8301:
8298:
8296:
8293:
8291:
8288:
8286:
8283:
8281:
8278:
8276:
8273:
8271:
8268:
8266:
8263:
8261:
8258:
8256:
8253:
8251:
8248:
8246:
8243:
8241:
8238:
8236:
8233:
8231:
8228:
8226:
8223:
8221:
8218:
8216:
8213:
8211:
8208:
8206:
8203:
8201:
8198:
8196:
8193:
8191:
8188:
8186:
8183:
8181:
8178:
8176:
8173:
8171:
8168:
8166:
8163:
8161:
8158:
8156:
8153:
8151:
8148:
8146:
8143:
8141:
8138:
8137:
8135:
8131:
8125:
8122:
8120:
8119:Matrix theory
8117:
8116:
8114:
8112:
8108:
8102:
8099:
8097:
8094:
8093:
8091:
8089:
8085:
8079:
8076:
8074:
8071:
8069:
8066:
8064:
8061:
8059:
8056:
8054:
8051:
8049:
8046:
8044:
8041:
8040:
8038:
8036:
8035:Supersymmetry
8032:
8026:
8023:
8021:
8018:
8016:
8013:
8011:
8008:
8006:
8003:
8001:
7998:
7996:
7993:
7991:
7988:
7984:
7981:
7979:
7972:
7968:
7965:
7964:
7963:
7960:
7958:
7955:
7954:
7953:
7950:
7948:
7945:
7942:
7939:
7937:
7934:
7932:
7929:
7927:
7924:
7923:
7921:
7917:
7911:
7909:
7905:
7903:
7900:
7898:
7895:
7892:
7885:
7878:
7871:
7864:
7857:
7854:
7852:
7849:
7847:
7844:
7842:
7839:
7837:
7834:
7833:
7831:
7829:
7825:
7819:
7816:
7814:
7811:
7809:
7806:
7804:
7801:
7799:
7796:
7794:
7791:
7789:
7786:
7784:
7781:
7779:
7776:
7775:
7773:
7771:
7767:
7761:
7758:
7756:
7753:
7751:
7748:
7746:
7743:
7741:
7738:
7736:
7733:
7731:
7728:
7726:
7723:
7721:
7718:
7716:
7713:
7711:
7708:
7707:
7705:
7703:
7699:
7693:
7690:
7688:
7687:Dual graviton
7685:
7683:
7680:
7678:
7675:
7673:
7670:
7668:
7665:
7663:
7660:
7658:
7655:
7654:
7652:
7648:
7642:
7639:
7637:
7634:
7632:
7629:
7627:
7624:
7623:
7621:
7619:
7615:
7609:
7606:
7604:
7603:RNS formalism
7601:
7599:
7596:
7594:
7591:
7589:
7586:
7584:
7581:
7579:
7576:
7574:
7571:
7569:
7566:
7562:
7559:
7555:
7552:
7550:
7547:
7546:
7545:
7542:
7540:
7539:Type I string
7537:
7536:
7535:
7532:
7530:
7527:
7525:
7522:
7520:
7517:
7516:
7514:
7510:
7504:
7501:
7497:
7494:
7492:
7489:
7488:
7487:
7484:
7482:
7479:
7477:
7474:
7473:
7471:
7467:
7463:
7462:String theory
7456:
7451:
7449:
7444:
7442:
7437:
7436:
7433:
7423:
7422:
7416:
7412:
7408:
7407:
7402:
7398:
7397:
7393:
7387:
7383:
7379:
7373:
7369:
7365:
7360:
7356:
7354:9780387738925
7350:
7346:
7345:
7339:
7335:
7331:
7327:
7323:
7318:
7313:
7309:
7305:
7300:
7297:
7293:
7289:
7285:
7281:
7277:
7273:
7269:
7264:
7259:
7255:
7251:
7247:
7243:
7240:
7236:
7232:
7226:
7222:
7218:
7213:
7208:
7204:
7200:
7195:
7191:
7187:
7183:
7179:
7174:
7169:
7165:
7161:
7156:
7152:
7148:
7144:
7140:
7135:
7130:
7126:
7122:
7118:
7113:
7109:
7105:
7101:
7097:
7093:
7089:
7084:
7081:
7077:
7073:
7067:
7063:
7059:
7055:
7051:
7047:
7044:
7040:
7036:
7030:
7026:
7022:
7018:
7014:
7011:
7007:
7003:
6999:
6995:
6991:
6987:
6983:
6979:
6978:Kähler, Erich
6975:
6971:
6967:
6963:
6957:
6953:
6949:
6945:
6941:
6937:
6934:
6930:
6926:
6920:
6916:
6912:
6908:
6904:
6900:
6896:
6892:
6888:
6884:
6880:
6875:
6870:
6866:
6862:
6858:
6853:
6850:
6846:
6842:
6836:
6832:
6828:
6824:
6820:
6816:
6812:
6808:
6805:
6801:
6797:
6791:
6787:
6783:
6779:
6775:
6774:
6768:
6767:
6762:
6756:
6751:
6748:
6745:
6740:
6737:
6733:
6728:
6725:
6722:, Lemma 9.14.
6721:
6716:
6713:
6709:
6704:
6701:
6698:
6697:Lamari (1999)
6693:
6690:
6687:
6682:
6679:
6675:
6670:
6668:
6664:
6661:
6660:Voisin (2004)
6656:
6653:
6649:
6645:
6609:
6606:
6603:
6598:
6595:
6591:
6586:
6583:
6579:
6574:
6571:
6568:
6563:
6560:
6556:
6551:
6548:
6544:
6539:
6536:
6532:
6527:
6524:
6520:
6515:
6512:
6508:
6503:
6500:
6496:
6491:
6488:
6484:
6479:
6476:
6472:
6467:
6464:
6460:
6455:
6452:
6448:
6443:
6440:
6434:
6430:
6427:
6425:
6422:
6420:
6417:
6415:
6412:
6411:
6407:
6402:
6398:
6395:
6391:
6387:
6383:
6379:
6375:
6371:
6367:
6363:
6360:
6357:(embedded in
6356:
6352:
6349:(embedded in
6348:
6344:
6340:
6337:
6336:Grassmannians
6333:
6329:
6325:
6319:
6313:
6309:
6303:
6298:
6297:unitary group
6294:
6290:
6287:
6283:
6280:
6276:
6272:
6268:
6264:
6261:
6257:
6253:
6249:
6246:
6243:
6242:Complex space
6240:
6239:
6235:
6233:
6231:
6227:
6221:
6219:
6215:
6209:
6207:
6188:
6185:
6182:
6170:
6168:
6164:
6148:
6117:
6114:
6111:
6099:
6097:
6093:
6089:
6085:
6081:
6077:
6071:
6066:
6062:
6058:
6054:
6046:
6044:
6042:
6038:
6033:
6031:
6027:
6023:
6018:
6016:
6012:
6008:
6004:
6002:
5998:
5994:
5990:
5985:
5981:
5978:
5974:
5968:
5964:
5960:
5955:
5951:
5944:
5940:
5935:
5933:
5929:
5925:
5921:
5917:
5916:metric tensor
5913:
5909:
5903:
5895:
5893:
5890:
5885:
5880:
5878:
5874:
5873:Claire Voisin
5870:
5866:
5862:
5857:
5823:
5819:
5815:
5747:
5744:
5740:
5734:
5730:
5726:
5721:
5717:
5713:
5709:
5706:
5702:
5696:
5692:
5688:
5681:
5677:
5673:
5668:
5662:
5658:
5654:
5647:
5643:
5639:
5634:
5630:
5626:
5622:
5614:
5612:
5610:
5606:
5602:
5597:
5595:
5591:
5588:
5581:
5577:
5573:
5572:Kähler groups
5569:
5564:
5562:
5558:
5554:
5550:
5545:
5538:
5532:
5528:
5523:
5522:diffeomorphic
5519:
5513:
5508:
5500:
5498:
5496:
5495:Serre duality
5473:
5470:
5467:
5464:
5461:
5458:
5455:
5451:
5445:
5440:
5437:
5434:
5430:
5407:
5404:
5401:
5398:
5395:
5392:
5389:
5385:
5381:
5376:
5373:
5370:
5366:
5338:
5335:
5332:
5328:
5322:
5317:
5314:
5311:
5307:
5284:
5257:
5254:
5251:
5247:
5243:
5238:
5235:
5232:
5228:
5220:
5201:
5196:
5193:
5190:
5186:
5180:
5177:
5174:
5171:
5168:
5164:
5160:
5155:
5151:
5143:
5142:
5141:
5127:
5119:
5118:Betti numbers
5100:
5092:
5089:
5086:
5082:
5058:
5052:
5044:
5041:
5038:
5034:
5013:
5005:
5004:Hodge numbers
4986:
4978:
4975:
4972:
4938:
4930:
4927:
4919:
4915:
4891:
4883:
4880:
4877:
4873:
4863:
4861:
4845:
4825:
4802:
4794:
4786:
4783:
4775:
4771:
4765:
4762:
4759:
4756:
4753:
4749:
4745:
4734:
4731:
4723:
4719:
4711:
4710:
4709:
4707:
4703:
4687:
4659:
4656:
4648:
4644:
4636:
4632:
4616:
4607:
4590:
4587:
4584:
4561:
4553:
4536:
4533:
4530:
4504:
4501:
4498:
4469:
4466:
4463:
4440:
4420:
4400:
4393:
4375:
4343:
4337:
4329:
4326:
4323:
4309:
4306:
4303:
4300:
4297:
4293:
4289:
4283:
4275:
4259:
4258:
4257:
4243:
4220:
4207:
4204:
4184:
4181:
4176:
4164:
4163:
4162:
4160:
4144:
4121:
4113:
4105:
4100:
4089:
4064:
4047:
4030:
4025:
3996:
3972:
3971:
3970:
3953:
3948:
3931:
3926:
3918:
3913:
3909:
3893:
3878:
3872:
3869:
3862:
3861:
3860:
3844:
3840:
3819:
3799:
3779:
3771:
3770:inner product
3769:
3752:
3744:
3726:
3722:
3713:
3697:
3677:
3673:
3670:
3662:
3659:
3656:
3650:
3642:
3639:
3633:
3630:
3625:
3621:
3600:
3580:
3575:
3571:
3567:
3562:
3558:
3554:
3551:
3546:
3521:
3513:
3497:
3485:
3483:
3481:
3477:
3473:
3469:
3465:
3461:
3443:
3439:
3435:
3425:
3409:
3406:
3403:
3400:
3397:
3390:
3357:
3344:
3339:
3302:
3292:
3276:
3268:
3262:
3254:
3252:
3250:
3246:
3242:
3238:
3234:
3229:
3227:
3223:
3217:
3213:
3209:
3204:
3198:
3194:
3190:
3185:
3179:
3175:
3170:
3165:
3160:
3156:
3152:
3148:
3144:
3125:
3120:
3116:
3110:
3106:
3099:
3096:
3092:
3087:
3081:
3060:
3059:
3058:
3056:
3052:
3048:
3044:
3040:
3036:
3032:
3024:
3022:
3020:
3016:
2994:
2963:
2937:
2931:
2928:
2925:
2907:
2904:
2901:
2898:
2882:
2879:
2876:
2870:
2862:
2843:
2842:
2841:
2800:
2789:
2785:
2784:positive form
2769:
2751:
2748:
2745:
2742:
2737:
2733:
2709:
2697:
2683:
2680:
2674:
2651:
2631:
2611:
2608:
2605:
2585:
2582:
2576:
2545:
2542:
2539:
2519:
2501:
2498:
2495:
2492:
2472:
2469:
2417:
2397:
2394:
2391:
2388:
2385:
2381:
2378:
2357:
2336:
2333:
2324:
2305:
2297:
2288:
2284:
2278:
2252:
2249:
2246:
2234:
2232:
2228:
2193:
2189:
2185:
2177:
2175:
2161:
2153:
2137:
2117:
2096:
2092:
2088:
2082:
2077:
2069:
2047:
2027:
2007:
1987:
1967:
1947:
1924:
1921:
1918:
1907:
1871:
1866:
1852:
1844:
1828:
1820:
1792:
1766:
1746:
1743:
1738:
1735:
1728:
1727:
1726:
1724:
1708:
1700:
1692:
1690:
1688:
1672:
1649:
1646:
1643:
1637:
1634:
1628:
1625:
1622:
1619:
1616:
1610:
1590:
1570:
1550:
1547:
1544:
1539:
1535:
1514:
1511:
1503:
1487:
1467:
1444:
1438:
1428:
1427:unitary group
1424:
1408:
1405:
1385:
1378:
1370:
1368:
1366:
1342:
1339:
1331:
1327:
1319:
1303:
1294:
1278:
1275:
1272:
1269:
1266:
1243:
1223:
1195:
1187:
1178:
1175:
1170:
1167:
1163:
1159:
1154:
1151:
1147:
1139:
1138:
1137:
1115:
1111:
1098:
1090:
1086:
1070:
1065:
1062:
1058:
1035:
1008:
988:
968:
946:
919:
911:
908:
905:, there is a
892:
872:
852:
844:
828:
805:
799:
796:
793:
787:
784:
781:
778:
772:
769:
766:
760:
753:
752:
751:
737:
729:
713:
706:
690:
668:
665:
643:
623:
603:
577:
574:
571:
565:
562:
559:
556:
550:
547:
544:
541:
535:
532:
529:
526:
520:
517:
514:
508:
501:
500:
499:
485:
465:
445:
442:
434:
418:
410:
394:
387:
384:
368:
361:
345:
338:
330:
328:
314:
306:
290:
282:
281:tangent space
260:
257:
254:
251:
245:
242:
236:
233:
230:
224:
217:
216:
215:
214:
213:bilinear form
198:
191:
187:
171:
164:
145:
142:
139:
129:
121:
119:
113:
111:
109:
105:
101:
98:
95:
90:
88:
84:
79:
75:
71:
67:
63:
59:
55:
51:
47:
43:
39:
35:
27:
19:
8145:Arkani-Hamed
8043:Supergravity
8010:Moduli space
7946:
7907:
7902:Dirac string
7828:Gauge theory
7808:Loop algebra
7745:Black string
7608:GS formalism
7420:
7404:
7363:
7347:. Springer.
7343:
7307:
7303:
7263:math/0312032
7253:
7249:
7212:math/0402223
7198:
7163:
7159:
7124:
7120:
7094:(1): 28–48.
7091:
7087:
7057:
7020:
6985:
6981:
6947:
6902:
6864:
6860:
6818:
6815:Hulek, Klaus
6772:
6750:
6739:
6727:
6720:Zheng (2000)
6715:
6708:Zheng (2000)
6703:
6692:
6681:
6655:
6647:
6643:
6608:
6597:
6590:Wells (2007)
6585:
6573:
6562:
6550:
6538:
6526:
6514:
6502:
6495:Zheng (2000)
6490:
6478:
6466:
6459:Zheng (2000)
6454:
6442:
6393:
6389:
6385:
6369:
6365:
6358:
6350:
6327:
6323:
6317:
6311:
6307:
6301:
6288:
6259:
6251:
6244:
6229:
6225:
6222:
6217:
6213:
6210:
6205:
6171:
6162:
6100:
6095:
6087:
6079:
6075:
6069:
6064:
6060:
6052:
6050:
6034:
6019:
6005:
6001:general type
5983:
5979:
5972:
5966:
5962:
5958:
5949:
5942:
5938:
5936:
5919:
5907:
5905:
5884:Yum-Tong Siu
5881:
5858:
5748:
5742:
5738:
5732:
5728:
5724:
5719:
5711:
5707:
5700:
5694:
5690:
5686:
5679:
5675:
5671:
5666:
5660:
5656:
5652:
5645:
5641:
5637:
5632:
5628:
5624:
5618:
5598:
5589:
5579:
5571:
5565:
5546:
5536:
5530:
5526:
5518:Hopf surface
5511:
5506:
5504:
5218:
5216:
5003:
4864:
4859:
4817:
4631:Hodge theory
4608:
4391:
4358:
4235:
4136:
3968:
3767:
3489:
3423:
3388:
3264:
3248:
3236:
3232:
3230:
3225:
3221:
3215:
3211:
3207:
3202:
3196:
3192:
3188:
3183:
3177:
3173:
3168:
3163:
3158:
3154:
3150:
3146:
3142:
3140:
3054:
3046:
3034:
3028:
3018:
2952:
2895: smooth
2787:
2698:
2644:is the same
2323:Kähler class
2322:
2321:is called a
2235:
2231:Hodge theory
2187:
2183:
2181:
2151:
2150:is called a
1867:
1842:
1841:is called a
1781:
1696:
1563:) such that
1374:
1365:Kähler class
1364:
1295:
1215:
820:
704:
595:
334:
278:
125:
117:
104:Hodge theory
91:
77:
70:Erich Kähler
41:
31:
26:
8505:Silverstein
8005:Orientifold
7740:Black holes
7735:Black brane
7692:Dual photon
6988:: 173–186,
6009:proved the
5716:Chern class
5684:comes from
5520:, which is
4704:of certain
2236:Namely, if
907:holomorphic
750:defined by
705:Kähler form
114:Definitions
34:mathematics
8630:Categories
8525:Strominger
8520:Steinhardt
8515:Staudacher
8430:Polchinski
8380:Nanopoulos
8340:Mandelstam
8320:Kontsevich
8160:Berenstein
8088:Holography
8068:Superspace
7967:K3 surface
7926:Worldsheet
7841:Instantons
7469:Background
7317:1610.07165
7173:1903.12645
7151:0926.32026
7002:58.0780.02
6891:0926.32025
6763:References
6401:K3 surface
5997:Calabi–Yau
5975:must have
5555:, and the
4702:direct sum
4635:cohomology
4413:-forms on
3514:on smooth
3387:, and the
3057:says that
2188:difference
2060:such that
1502:linear map
728:(1,1)-form
383:associated
186:compatible
74:André Weil
8560:Veneziano
8440:Rajaraman
8335:Maldacena
8225:Gopakumar
8175:Dijkgraaf
8170:Curtright
7836:Anomalies
7715:NS5-brane
7636:U-duality
7631:S-duality
7626:T-duality
7411:EMS Press
7334:119669591
7056:(1996) ,
7010:122246578
6946:(1994) .
6628:¯
6625:∂
6620:∂
6189:ω
6146:→
6118:ω
5841:¯
5838:∂
5832:∂
5799:¯
5796:∂
5790:∂
5766:¯
5763:∂
5757:∂
5479:¯
5471:−
5459:−
5446:≅
5405:−
5393:−
5344:¯
5281:Δ
5165:∑
4935:Ω
4791:Ω
4750:⨁
4746:≅
4562:α
4464:α
4461:Δ
4441:α
4294:⨁
4216:∂
4212:Δ
4198:¯
4195:∂
4189:Δ
4173:Δ
4119:∂
4114:∗
4110:∂
4101:∗
4097:∂
4093:∂
4085:∂
4081:Δ
4059:¯
4056:∂
4048:∗
4041:¯
4038:∂
4026:∗
4019:¯
4016:∂
4006:¯
4003:∂
3990:¯
3987:∂
3981:Δ
3949:∗
3942:¯
3939:∂
3927:∗
3923:∂
3914:∗
3888:¯
3885:∂
3876:∂
3845:∗
3727:∗
3698:⋆
3678:⋆
3671:⋆
3640:−
3634:−
3626:∗
3576:∗
3563:∗
3543:Δ
3512:Laplacian
3444:∗
3433:Λ
3410:−
3407:∧
3404:ω
3371:¯
3368:∂
3362:Δ
3353:∂
3349:Δ
3336:Δ
3312:¯
3309:∂
3300:∂
3117:ω
3107:∫
2964:ω
2926:φ
2920:¯
2917:∂
2911:∂
2902:ω
2899:∣
2886:→
2877:φ
2863:ω
2801:ω
2770:φ
2764:¯
2761:∂
2755:∂
2746:ω
2738:φ
2734:ω
2710:ω
2696:locally.
2675:ω
2652:φ
2632:ρ
2609:⊂
2577:ω
2549:→
2540:φ
2520:φ
2514:¯
2511:∂
2505:∂
2496:β
2473:β
2447:¯
2444:∂
2438:∂
2418:β
2398:β
2389:ω
2379:ω
2358:ω
2334:ω
2285:∈
2279:ω
2253:ω
2212:¯
2209:∂
2203:∂
2162:ω
2138:ρ
2118:ρ
2112:¯
2109:∂
2103:∂
2070:ω
2028:ρ
1925:ω
1891:¯
1888:∂
1882:∂
1853:ω
1829:ρ
1802:¯
1799:∂
1790:∂
1767:ρ
1761:¯
1758:∂
1752:∂
1736:ω
1709:ρ
1548:−
1439:
1304:ω
1273:⋯
1192:‖
1185:‖
1164:δ
1108:∂
1104:∂
1083:∂
1079:∂
785:
714:ω
666:−
563:
533:
509:ω
486:ω
395:ω
246:ω
199:ω
188:with the
184:which is
146:ω
8615:Zwiebach
8570:Verlinde
8565:Verlinde
8540:Townsend
8535:Susskind
8470:Sagnotti
8435:Polyakov
8390:Nekrasov
8355:Minwalla
8350:Martinec
8315:Knizhnik
8310:Klebanov
8305:Kapustin
8270:'t Hooft
8205:Fischler
8140:Aganagić
8111:M-theory
8000:Conifold
7995:Orbifold
7978:manifold
7919:Geometry
7725:M5-brane
7720:M2-brane
7657:Graviton
7573:F-theory
7296:11984149
7190:88524040
7025:Springer
7019:(2005),
6907:Springer
6901:(2001),
6823:Springer
6408:See also
6267:oriented
6236:Examples
6092:complete
6030:K-stable
5918:, Ric =
5587:integers
4708:groups:
4392:harmonic
3690:, where
3051:homology
3043:subspace
3041:complex
2382:′
2337:′
2184:globally
1817:are the
1216:for all
56:, and a
46:manifold
8545:Trivedi
8530:Sundrum
8495:Shenker
8485:Seiberg
8480:Schwarz
8450:Randall
8410:Novikov
8400:Nielsen
8385:Năstase
8295:Kallosh
8280:Gibbons
8220:Gliozzi
8210:Friedan
8200:Ferrara
8185:Douglas
8180:Distler
7730:S-brane
7710:D-brane
7667:Tachyon
7662:Dilaton
7476:Strings
7413:, 2001
7386:1777835
7288:2076925
7268:Bibcode
7239:2325093
7143:1688140
7108:1969701
7080:1393941
7043:2093043
6970:0507725
6933:1853077
6883:1688136
6849:2030225
6804:1379330
6256:lattice
4433:(forms
3743:adjoint
3741:is the
3710:is the
3460:Kodaira
3031:compact
2130:. Here
1136:, then
912:around
636:(where
358:with a
279:on the
97:complex
8610:Zumino
8605:Zaslow
8590:Yoneya
8580:Witten
8500:Siegel
8475:Scherk
8445:Ramond
8420:Ooguri
8345:Marolf
8300:Kaluza
8285:Kachru
8275:Hořava
8265:Harvey
8260:Hanson
8245:Gubser
8235:Greene
8165:Bousso
8150:Atiyah
7702:Branes
7512:Theory
7384:
7374:
7351:
7332:
7294:
7286:
7237:
7227:
7188:
7149:
7141:
7106:
7078:
7068:
7041:
7031:
7008:
7000:
6968:
6958:
6931:
6921:
6889:
6881:
6847:
6837:
6802:
6792:
6399:Every
6291:, the
6090:has a
5814:-lemma
5561:formal
5551:, the
4552:-forms
4482:) and
4359:where
4077:
4074:
4071:
4068:
3906:
3903:
3900:
3897:
3593:where
3510:, the
3478:, and
3466:, the
3289:, the
3145:is an
3141:where
3039:closed
3029:For a
2430:. The
2227:-lemma
1906:-lemma
1874:local
1699:smooth
1665:) and
1421:whose
703:, the
409:closed
386:2-form
381:whose
94:smooth
92:Every
8550:Turok
8460:Roček
8425:Ovrut
8415:Olive
8395:Neveu
8375:Myers
8370:Mukhi
8360:Moore
8330:Linde
8325:Klein
8250:Gukov
8240:Gross
8230:Green
8215:Gates
8195:Dvali
8155:Banks
7425:(PDF)
7330:S2CID
7312:arXiv
7292:S2CID
7258:arXiv
7207:arXiv
7186:S2CID
7168:arXiv
7104:JSTOR
7006:S2CID
6435:Notes
6067:(for
5989:ample
5956:) in
4453:with
3239:is a
2229:from
1504:from
841:is a
44:is a
8575:Wess
8555:Vafa
8465:Rohm
8365:Motl
8290:Kaku
8255:Guth
8190:Duff
7372:ISBN
7349:ISBN
7225:ISBN
7164:2021
7066:ISBN
7029:ISBN
6956:ISBN
6919:ISBN
6835:ISBN
6790:ISBN
6320:+ 1)
6304:+ 1)
6084:ball
6039:and
6026:Tian
5993:Fano
5619:The
4865:Let
3832:and
3462:and
2929:>
2664:for
2154:for
1845:for
616:and
64:and
52:, a
40:, a
8585:Yau
8510:Sơn
8490:Sen
7322:doi
7308:371
7276:doi
7254:157
7217:doi
7178:doi
7147:Zbl
7129:doi
7096:doi
6998:JFM
6990:doi
6911:doi
6887:Zbl
6869:doi
6827:doi
6782:doi
6648:192
6368:in
6322:on
6098:.)
6086:in
6072:≥ 2
5722:in
5718:of
5710:on
5631:on
5542:= 1
5524:to
5006:of
4680:of
4256:,
4137:If
3772:on
3745:of
3186:in
3161:in
3045:of
3019:all
2370:by
2040:on
2000:of
1960:in
1256:in
1021:in
1001:to
885:of
407:is
327:).
283:of
32:In
8632::
7886:,
7879:,
7872:,
7865:,
7409:,
7403:,
7382:MR
7380:,
7370:,
7366:,
7328:.
7320:.
7306:.
7290:,
7284:MR
7282:,
7274:,
7266:,
7252:,
7235:MR
7233:,
7223:,
7215:,
7205:,
7184:.
7176:.
7162:.
7145:.
7139:MR
7137:.
7125:49
7123:.
7119:.
7102:.
7092:60
7090:.
7076:MR
7074:,
7064:,
7052:;
7039:MR
7037:,
7027:,
7023:,
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6996:,
6984:,
6966:MR
6964:.
6954:.
6950:.
6942:;
6929:MR
6927:,
6917:,
6909:,
6885:.
6879:MR
6877:.
6865:49
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6845:MR
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6833:,
6825:,
6813:;
6800:MR
6798:,
6788:,
6780:,
6666:^
6646:,
6359:CP
6328:CP
6324:CP
6316:U(
6312:CP
6300:U(
6289:CP
6232:.
6169:.
6080:CP
6065:CP
5995:,
5965:,
5920:λg
5731:,
5693:,
5678:,
5659:,
5644:,
5544:.
5529:×
5514:+1
5497:.
4629:,
3812:,
3482:.
3474:,
3470:,
3401::=
3247:,
3228:.
3214:,
3195:,
3176:,
2871::=
2840::
2293:dR
2233:.
1865:.
1697:A
1689:.
1367:.
1236:,
782:Re
560:Im
530:Re
89:.
76:.
7976:2
7974:G
7943:?
7908:p
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7890:8
7888:E
7883:7
7881:E
7876:6
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7869:4
7867:F
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7860:G
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7447:t
7440:v
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7314::
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7260::
7219::
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7110:.
7098::
6992::
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6972:.
6913::
6893:.
6871::
6829::
6784::
6394:X
6390:C
6386:X
6370:C
6366:B
6351:C
6318:n
6308:C
6302:n
6260:C
6252:C
6245:C
6218:C
6214:C
6206:X
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6186:,
6183:X
6180:(
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6149:X
6142:C
6121:)
6115:,
6112:X
6109:(
6088:C
6070:n
6061:X
6053:X
5984:X
5980:K
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5969:)
5967:R
5963:X
5961:(
5959:H
5950:X
5948:(
5946:1
5943:c
5939:X
5743:ω
5739:ω
5735:)
5733:Z
5729:X
5727:(
5725:H
5720:L
5712:X
5708:L
5701:X
5697:)
5695:Q
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5689:(
5687:H
5682:)
5680:R
5676:X
5674:(
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5667:X
5663:)
5661:Z
5657:X
5655:(
5653:H
5648:)
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5640:(
5638:H
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5629:ω
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5583:1
5580:b
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5537:b
5531:S
5527:S
5512:a
5510:2
5507:b
5474:q
5468:n
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5441:q
5438:,
5435:p
5431:H
5408:q
5402:n
5399:,
5396:p
5390:n
5386:h
5382:=
5377:q
5374:,
5371:p
5367:h
5339:p
5336:,
5333:q
5329:H
5323:=
5318:q
5315:,
5312:p
5308:H
5285:d
5258:p
5255:,
5252:q
5248:h
5244:=
5239:q
5236:,
5233:p
5229:h
5202:.
5197:q
5194:,
5191:p
5187:h
5181:r
5178:=
5175:q
5172:+
5169:p
5161:=
5156:r
5152:b
5128:X
5104:)
5101:X
5098:(
5093:q
5090:,
5087:p
5083:H
5076:C
5070:m
5067:i
5064:d
5059:=
5056:)
5053:X
5050:(
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5042:,
5039:p
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4803:.
4800:)
4795:p
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4781:(
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4766:r
4763:=
4760:q
4757:+
4754:p
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4505:q
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4470:0
4467:=
4421:X
4401:r
4376:r
4370:H
4344:,
4341:)
4338:X
4335:(
4330:q
4327:,
4324:p
4318:H
4310:r
4307:=
4304:q
4301:+
4298:p
4290:=
4287:)
4284:X
4281:(
4276:r
4270:H
4244:X
4221:.
4208:2
4205:=
4185:2
4182:=
4177:d
4145:X
4122:.
4106:+
4090:=
4065:,
4031:+
3997:=
3954:,
3932:+
3919:=
3910:d
3894:,
3879:+
3873:=
3870:d
3841:d
3820:d
3800:X
3780:r
3768:L
3753:d
3723:d
3674:d
3666:)
3663:1
3660:+
3657:r
3654:(
3651:n
3647:)
3643:1
3637:(
3631:=
3622:d
3601:d
3581:d
3572:d
3568:+
3559:d
3555:d
3552:=
3547:d
3522:r
3498:n
3440:L
3436:=
3398:L
3358:,
3345:,
3340:d
3303:,
3277:d
3249:Y
3237:X
3233:Y
3226:n
3222:X
3218:)
3216:R
3212:X
3210:(
3208:H
3203:ω
3199:)
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