574:. He further conjectured that each cohomology class should have a distinguished representative with the property that both it and its dual vanish under the exterior derivative operator; these are now called harmonic forms. Hodge devoted most of the 1930s to this problem. His earliest published attempt at a proof appeared in 1933, but he considered it "crude in the extreme".
2797:
5574:
This example suggests several different roles played by Hodge theory in complex algebraic geometry. First, Hodge theory gives restrictions on which topological spaces can have the structure of a smooth complex projective variety. Second, Hodge theory gives information about the moduli space of smooth
818:
578:, one of the most brilliant mathematicians of the era, found himself unable to determine whether Hodge's proof was correct or not. In 1936, Hodge published a new proof. While Hodge considered the new proof much superior, a serious flaw was discovered by Bohnenblust. Independently, Hermann Weyl and
586:
In retrospect it is clear that the technical difficulties in the existence theorem did not really require any significant new ideas, but merely a careful extension of classical methods. The real novelty, which was Hodge’s major contribution, was in the conception of harmonic integrals and their
2359:
2659:
1510:
270:, and that therefore each of the terms on the left-hand side are vector space duals of one another. In contemporary language, de Rham's theorem is more often phrased as the statement that singular cohomology with real coefficients is isomorphic to de Rham cohomology:
565:
Hodge felt that these techniques should be applicable to higher dimensional varieties as well. His colleague Peter Fraser recommended de Rham's thesis to him. In reading de Rham's thesis, Hodge realized that the real and imaginary parts of a holomorphic 1-form on a
3308:
3640:
458:
is a positive volume form, from which
Lefschetz was able to rederive Riemann's inequalities. In 1929, W. V. D. Hodge learned of Lefschetz's paper. He immediately observed that similar principles applied to algebraic surfaces. More precisely, if
4959:
1924:
4833:
A crucial point is that the Hodge decomposition is a decomposition of cohomology with complex coefficients that usually does not come from a decomposition of cohomology with integral (or rational) coefficients. As a result, the intersection
261:
3465:
This decomposition is in fact independent of the choice of Kähler metric (but there is no analogous decomposition for a general compact complex manifold). On the other hand, the Hodge decomposition genuinely depends on the structure of
976:
358:
2648:
660:
1079:
2896:
2054:
5623:
showed that the intersection homology of any complex projective variety (not necessarily smooth) has a pure Hodge structure, just as in the smooth case. In fact, the whole Kähler package extends to intersection homology.
1580:
154:, in which he suggested — but did not prove — that differential forms and topology should be linked. Upon reading it, Georges de Rham, then a student, was inspired. In his 1931 thesis, he proved a result now called
3460:
2244:
2792:{\displaystyle {\begin{aligned}{\mathcal {E}}^{\bullet }&=\bigoplus \nolimits _{i}\Gamma (E_{i})\\L&=\bigoplus \nolimits _{i}L_{i}:{\mathcal {E}}^{\bullet }\to {\mathcal {E}}^{\bullet }\end{aligned}}}
1671:
2946:
507:
456:
1758:
2536:
5019:
2213:
1282:
1143:
3742:
1406:
5606:, extends Hodge theory to all complex algebraic varieties, not necessarily smooth or compact. Namely, the cohomology of any complex algebraic variety has a more general type of decomposition, a
3073:
2664:
4376:
The "Kähler package" is a powerful set of restrictions on the cohomology of smooth complex projective varieties (or compact Kähler manifolds), building on Hodge theory. The results include the
3964:
3205:
1398:
4345:
of a smooth complex projective variety (or compact Kähler manifold) are even, by Hodge symmetry. This is not true for compact complex manifolds in general, as shown by the example of the
4423:
1809:
2444:
4720:
4627:
4580:
5505:
5283:
2107:
of an elliptic operator on a closed manifold is always a finite-dimensional vector space. Another consequence of the Hodge theorem is that a
Riemannian metric on a closed manifold
5383:
5312:
556:
4840:
3889:
4018:
1318:
5569:
4784:
1192:
5547:
5354:
4762:
4669:
1832:
5794:
2078:
587:
relevance to algebraic geometry. This triumph of concept over technique is reminiscent of a similar episode in the work of Hodge’s great predecessor
Bernhard Riemann.
5052:
3824:
1346:
527:
3834:
is dual to a cohomology class which we will call , and the cap product can be computed by taking the cup product of and α and capping with the fundamental class of
1222:
5627:
A fundamental aspect of complex geometry is that there are continuous families of non-isomorphic complex manifolds (which are all diffeomorphic as real manifolds).
176:
5461:
5129:
885:
276:
5723:
5703:
5675:
5655:
5435:
5407:
5239:
5196:
5172:
5152:
5096:
5072:
4828:
4804:
4533:
4510:
4490:
4470:
3506:
813:{\displaystyle 0\to \Omega ^{0}(M)\xrightarrow {d_{0}} \Omega ^{1}(M)\xrightarrow {d_{1}} \cdots \xrightarrow {d_{n-1}} \Omega ^{n}(M)\xrightarrow {d_{n}} 0,}
2555:
108:, is encompassed by the latter case. Hodge theory has become an important tool in algebraic geometry, particularly through its connection to the study of
1000:
2830:
1978:
3117:
4388:. Many of these results follow from fundamental technical tools which may be proven for compact Kähler manifolds using Hodge theory, including the
570:
were in some sense dual to each other. He suspected that there should be a similar duality in higher dimensions; this duality is now known as the
1525:
6156:
6075:
6001:
3841:
Because is a cohomology class, it has a Hodge decomposition. By the computation we did above, if we cup this class with any class of type
2354:{\displaystyle \Omega ^{k}(M)\cong \operatorname {im} d_{k-1}\oplus \operatorname {im} \delta _{k+1}\oplus {\mathcal {H}}_{\Delta }^{k}(M).}
582:
modified Hodge's proof to repair the error. This established Hodge's sought-for isomorphism between harmonic forms and cohomology classes.
5916:
Huybrechts (2005), sections 3.3 and 5.2; Griffiths & Harris (1994), sections 0.7 and 1.2; Voisin (2007), v. 1, ch. 6, and v. 2, ch. 1.
5587:
from Chow groups to ordinary cohomology, but Hodge theory also gives information about the kernel of the cycle map, for example using the
5579:
holds, meaning that the variety is determined up to isomorphism by its Hodge structure. Finally, Hodge theory gives information about the
363:
De Rham's original statement is then a consequence of the fact that over the reals, singular cohomology is the dual of singular homology.
3368:
1965:
1595:
5102:
6105:
6034:
5967:
2904:
6130:
6097:
6026:
466:
415:
4385:
1690:
2464:
1505:{\displaystyle (\omega ,\tau )\mapsto \langle \omega ,\tau \rangle :=\int _{M}\langle \omega (p),\tau (p)\rangle _{p}\sigma .}
4967:
2171:
39:
1230:
1091:
4042:). These are important invariants of a smooth complex projective variety; they do not change when the complex structure of
3667:
562:
itself must represent a non-zero cohomology class, so its periods cannot all be zero. This resolved a question of Severi.
5749:
4046:
is varied continuously, and yet they are in general not topological invariants. Among the properties of Hodge numbers are
54:
3016:
5633:
4377:
4334:
are the sum of the Hodge numbers in a given row. A basic application of Hodge theory is then that the odd Betti numbers
3303:{\displaystyle f\,dz_{1}\wedge \cdots \wedge dz_{p}\wedge d{\overline {w_{1}}}\wedge \cdots \wedge d{\overline {w_{q}}}}
66:
5785:
3894:
4097:
1355:
6059:
5993:
5955:
2084:
has a unique harmonic representative. Concretely, the harmonic representative is the unique closed form of minimum
2099:
For example, the Hodge theorem implies that the cohomology groups with real coefficients of a closed manifold are
4396:
3654:
3171:
3079:
There is also a Hodge decomposition in this situation, generalizing the statement above for the de Rham complex.
1773:
3110:
2390:
3129:
5744:
4678:
4585:
4538:
2365:
5466:
5244:
4096:
The Hodge numbers of a smooth complex projective variety (or compact Kähler manifold) can be listed in the
5218:. The difficulty of the Hodge conjecture reflects the lack of understanding of such integrals in general.
4431:
4381:
3121:
2542:
6193:
5759:
5615:
5607:
5588:
5215:
4954:{\displaystyle (H^{2p}(X,\mathbb {Z} )/{\text{torsion}})\cap H^{p,p}(X)\subseteq H^{2p}(X,\mathbb {C} )}
3500:
in cohomology, so the cup product with complex coefficients is compatible with the Hodge decomposition:
652:
5359:
5288:
2387:
as a generalization of the de Rham complex. The Hodge theorem extends to this setting, as follows. Let
5054:
is big. In short, the Hodge conjecture predicts that the possible "shapes" of complex subvarieties of
532:
6122:
4389:
3844:
3331:
3199:, meaning forms that can locally be written as a finite sum of terms, with each term taking the form
2132:
1930:
155:
3969:
1937:
say that the electromagnetic field in a vacuum, i.e. absent any charges, is represented by a 2-form
1287:
404:
of their cohomology classes is zero, and when made explicit, this gave
Lefschetz a new proof of the
4201:
3827:
3748:
2953:
1919:{\displaystyle {\mathcal {H}}_{\Delta }^{k}(M)=\{\alpha \in \Omega ^{k}(M)\mid \Delta \alpha =0\}.}
872:
831:
571:
124:
97:
6174:
5552:
4767:
2103:. (Admittedly, there are other ways to prove this.) Indeed, the operators Δ are elliptic, and the
1156:
5838:
5414:
5203:
3120:, complex projective manifolds are automatically algebraic: they are defined by the vanishing of
2100:
1934:
1814:
This is a second-order linear differential operator, generalizing the
Laplacian for functions on
613:
139:
105:
93:
85:
5510:
5417:
set of components, each of complex dimension 19. The subspace of K3 surfaces with Picard number
5317:
5131:(even integrally, that is, without the need for a positive integral multiple in the statement).
4725:
4632:
3145:
2088:
norm that represents a given cohomology class. The Hodge theorem was proved using the theory of
1320:
inner product is then defined as the integral of the pointwise inner product of a given pair of
159:
101:
146:, and the interaction between differential forms and topology was poorly understood. In 1928,
6152:
6101:
6071:
6030:
6018:
5997:
5963:
5951:
5628:
5207:
5175:
4435:
4068:
3780:
3486:
2089:
2063:
640:
405:
367:
256:{\displaystyle H_{k}(M;\mathbf {R} )\times H_{\text{dR}}^{k}(M;\mathbf {R} )\to \mathbf {R} .}
163:
70:
58:
5098:(the combination of integral cohomology with the Hodge decomposition of complex cohomology).
5024:
3809:
1331:
512:
6142:
6134:
6063:
5939:
5830:
5734:
4582:. Moreover, the resulting class has a special property: its image in the complex cohomology
4447:
3106:
2384:
2120:
2104:
2093:
1677:
1224:
1197:
971:{\displaystyle H^{k}(M,\mathbf {R} )\cong {\frac {\ker d_{k}}{\operatorname {im} d_{k-1}}}.}
579:
371:
43:
6166:
6115:
6085:
6044:
6011:
5977:
3013:
The cohomology of the complex is canonically isomorphic to the space of harmonic sections,
353:{\displaystyle H_{\text{sing}}^{k}(M;\mathbf {R} )\cong H_{\text{dR}}^{k}(M;\mathbf {R} ).}
6162:
6111:
6081:
6040:
6007:
5973:
5754:
5576:
5440:
5211:
4197:
3160:
3088:
2151:
1961:
1950:
839:
621:
567:
267:
109:
89:
47:
5108:
1515:
Naturally the above inner product induces a norm, when that norm is finite on some fixed
3635:{\displaystyle \smile \colon H^{p,q}(X)\times H^{p',q'}(X)\rightarrow H^{p+p',q+q'}(X).}
5985:
5855:
5708:
5688:
5678:
5660:
5640:
5620:
5603:
5583:
of algebraic cycles on a given variety. The Hodge conjecture is about the image of the
5420:
5392:
5224:
5181:
5157:
5137:
5081:
5057:
4813:
4789:
4518:
4495:
4475:
4455:
2643:{\displaystyle 0\to \Gamma (E_{0})\to \Gamma (E_{1})\to \cdots \to \Gamma (E_{N})\to 0}
2447:
2376:
2124:
1146:
591:
147:
116:
5575:
complex projective varieties with a given topological type. The best case is when the
5385:. Their intersection can have rank anywhere between 1 and 20; this rank is called the
6187:
6051:
5739:
5386:
4350:
4090:
3347:
3098:
2112:
1085:
393:
120:
1074:{\displaystyle \Omega ^{k}(M)=\Gamma \left(\bigwedge \nolimits ^{k}T^{*}(M)\right).}
5682:
5410:
4346:
4327:
636:
575:
130:
have given alternative proofs of, or analogous results to, classical Hodge theory.
2080:
is an isomorphism of vector spaces. In other words, each real cohomology class on
5725:, as would arise from a family of varieties which need not be smooth or compact.
2891:{\displaystyle {\mathcal {H}}=\{e\in {\mathcal {E}}^{\bullet }\mid \Delta e=0\}.}
2049:{\displaystyle \varphi :{\mathcal {H}}_{\Delta }^{k}(M)\to H^{k}(M,\mathbf {R} )}
5764:
5613:
A different generalization of Hodge theory to singular varieties is provided by
4513:
4393:
3799:
3497:
633:
401:
31:
3075:, in the sense that each cohomology class has a unique harmonic representative.
6147:
6138:
5584:
5580:
4247:
3355:
2546:
2380:
2235:
143:
6067:
17:
2824:. As in the de Rham case, this yields the vector space of harmonic sections
2092:
partial differential equations, with Hodge's initial arguments completed by
1764:
74:
5798:
3496:
Taking wedge products of these harmonic representatives corresponds to the
1575:{\displaystyle \langle \omega ,\omega \rangle =\|\omega \|^{2}<\infty ,}
2161:. This says that there is a unique decomposition of any differential form
5685:
is a generalization. Roughly speaking, a mixed Hodge module on a variety
5637:
describes how the Hodge structure of a smooth complex projective variety
5199:
142:
was still nascent in the 1920s. It had not yet developed the notion of
5842:
5821:
Lefschetz, Solomon (1927). "Correspondences
Between Algebraic Curves".
3346:
components of a harmonic form are again harmonic. Therefore, for any
5834:
3362:
with complex coefficients as a direct sum of complex vector spaces:
3140:
which has a strong compatibility with the complex structure, making
2165:
on a closed
Riemannian manifold as a sum of three parts in the form
791:
749:
729:
693:
3455:{\displaystyle H^{r}(X,\mathbf {C} )=\bigoplus _{p+q=r}H^{p,q}(X).}
1585:
then the integrand is a real valued, square integrable function on
5767:, a key consequence of Hodge theory for compact Kähler manifolds.
5571:, but for "special" K3 surfaces the intersection can be bigger.)
1666:{\displaystyle \|\omega (p)\|_{p}:M\to \mathbf {R} \in L^{2}(M).}
3784:
3170:(with complex coefficients) can be written uniquely as a sum of
2549:
sections of these vector bundles, and that the induced sequence
115:
While Hodge theory is intrinsically dependent upon the real and
266:
As originally stated, de Rham's theorem asserts that this is a
6129:, Graduate Texts in Mathematics, vol. 65 (3rd ed.),
5463:. (Thus, for most projective K3 surfaces, the intersection of
594:, William Vallance Douglas Hodge, 17 June 1903 – 7 July 1975,
5198:. In this sense, Hodge theory is related to a basic issue in
4472:
be a smooth complex projective variety. A complex subvariety
2941:{\displaystyle H:{\mathcal {E}}^{\bullet }\to {\mathcal {H}}}
463:
is a non-zero holomorphic form on an algebraic surface, then
3661:
as a complex manifold (not on the choice of Kähler metric):
3044:
2933:
2917:
2856:
2836:
2774:
2757:
2670:
2323:
1991:
1839:
502:{\displaystyle {\sqrt {-1}}\,\omega \wedge {\bar {\omega }}}
451:{\displaystyle {\sqrt {-1}}\,\omega \wedge {\bar {\omega }}}
5154:
describes the integrals of algebraic differential forms on
1753:{\displaystyle \delta :\Omega ^{k+1}(M)\to \Omega ^{k}(M).}
879:
with real coefficients is computed by the de Rham complex:
5862:, Biogr. Mem. Fellows R. Soc., 1976, vol. 22, pp. 169–192.
5860:
William
Vallance Douglas Hodge, 17 June 1903 – 7 July 1975
5202:: there is in general no "formula" for the integral of an
4786:-linear combination of classes of complex subvarieties of
5677:
varies. In geometric terms, this amounts to studying the
6175:
Python code for computing Hodge numbers of hypersurfaces
2531:{\displaystyle L_{i}:\Gamma (E_{i})\to \Gamma (E_{i+1})}
84:
The theory was developed by Hodge in the 1930s to study
5681:
associated to a family of varieties. Saito's theory of
5014:{\displaystyle H^{2p}(X,\mathbb {Z} )/{\text{torsion}}}
4722:
whose image in complex cohomology lies in the subspace
2234:
metric on differential forms, this gives an orthogonal
2208:{\displaystyle \omega =d\alpha +\delta \beta +\gamma ,}
1589:, evaluated at a given point via its point-wise norms,
6094:
3798:
On the other hand, the integral can be written as the
3653:) of the Hodge decomposition can be identified with a
1277:{\displaystyle \bigwedge \nolimits ^{k}(T_{p}^{*}(M))}
1138:{\displaystyle \bigwedge \nolimits ^{k}(T_{p}^{*}(M))}
6056:
Hodge Theory and
Complex Algebraic Geometry (2 vols.)
5711:
5691:
5663:
5643:
5555:
5513:
5469:
5443:
5423:
5395:
5362:
5320:
5291:
5247:
5227:
5184:
5160:
5140:
5111:
5084:
5060:
5027:
4970:
4843:
4816:
4792:
4770:
4728:
4681:
4635:
4629:
lies in the middle piece of the Hodge decomposition,
4588:
4541:
4521:
4498:
4478:
4458:
4399:
3972:
3897:
3847:
3812:
3737:{\displaystyle H^{p,q}(X)\cong H^{q}(X,\Omega ^{p}),}
3670:
3509:
3371:
3208:
3019:
2907:
2833:
2662:
2558:
2467:
2450:, equipped with metrics, on a closed smooth manifold
2393:
2247:
2174:
2066:
1981:
1835:
1776:
1693:
1598:
1528:
1409:
1358:
1334:
1290:
1233:
1200:
1159:
1094:
1003:
888:
663:
535:
515:
469:
418:
279:
179:
5221:
Example: For a smooth complex projective K3 surface
596:
Biographical Memoirs of Fellows of the Royal Society
388:
are holomorphic differentials on an algebraic curve
152:
Sur les nombres de Betti des espaces de groupes clos
104:. Hodge's primary motivation, the study of complex
5074:(as described by cohomology) are determined by the
4764:should have a positive integral multiple that is a
3068:{\displaystyle H(E_{j})\cong {\mathcal {H}}(E_{j})}
2653:is an elliptic complex. Introduce the direct sums:
2364:The Hodge decomposition is a generalization of the
5717:
5697:
5669:
5649:
5563:
5541:
5499:
5455:
5429:
5401:
5377:
5348:
5306:
5277:
5233:
5190:
5166:
5146:
5123:
5090:
5066:
5046:
5013:
4953:
4822:
4798:
4778:
4756:
4714:
4663:
4621:
4574:
4527:
4504:
4484:
4464:
4417:
4034:) means the dimension of the complex vector space
4012:
3958:
3883:
3818:
3736:
3634:
3454:
3302:
3067:
2940:
2890:
2791:
2642:
2530:
2438:
2353:
2207:
2072:
2048:
1918:
1803:
1752:
1665:
1574:
1504:
1392:
1340:
1312:
1276:
1216:
1186:
1137:
1073:
970:
812:
550:
521:
501:
450:
400:has only one complex dimension; consequently, the
352:
255:
5990:The Theory and Applications of Harmonic Integrals
3959:{\displaystyle H^{2n}(X,\mathbb {C} )=H^{n,n}(X)}
3354:, the Hodge theorem gives a decomposition of the
2123:. It follows, for example, that the image of the
2150:is finite (because the group of isometries of a
370:used topological methods to reprove theorems of
166:chains induces, for any compact smooth manifold
1393:{\displaystyle \omega ,\tau \in \Omega ^{k}(M)}
77:operator of the metric. Such forms are called
5962:. Wiley Classics Library. Wiley Interscience.
5784:Chatterji, Srishti; Ojanguren, Manuel (2010),
4434:also give strong restrictions on the possible
96:. It has major applications in two settings:
3083:Hodge theory for complex projective varieties
412:is a non-zero holomorphic differential, then
8:
4102:(shown in the case of complex dimension 2):
2882:
2844:
1975:. As a result, there is a canonical mapping
1910:
1867:
1615:
1599:
1554:
1547:
1541:
1529:
1487:
1456:
1440:
1428:
5105:says that the Hodge conjecture is true for
4535:defines an element of the cohomology group
4418:{\displaystyle \partial {\bar {\partial }}}
6127:Differential Analysis on Complex Manifolds
5705:is a sheaf of mixed Hodge structures over
5591:which are built from the Hodge structure.
4806:. (Such a linear combination is called an
3101:complex projective manifold, meaning that
1804:{\displaystyle \Delta =d\delta +\delta d.}
162:, integration of differential forms along
123:. In arithmetic situations, the tools of
6146:
5710:
5690:
5662:
5642:
5557:
5556:
5554:
5518:
5512:
5490:
5489:
5474:
5468:
5442:
5422:
5394:
5369:
5365:
5364:
5361:
5325:
5319:
5298:
5294:
5293:
5290:
5268:
5267:
5252:
5246:
5226:
5183:
5159:
5139:
5110:
5083:
5059:
5032:
5026:
5006:
5001:
4994:
4993:
4975:
4969:
4964:may be much smaller than the whole group
4944:
4943:
4925:
4897:
4882:
4877:
4870:
4869:
4851:
4842:
4815:
4791:
4772:
4771:
4769:
4733:
4727:
4705:
4704:
4686:
4680:
4640:
4634:
4612:
4611:
4593:
4587:
4565:
4564:
4546:
4540:
4520:
4497:
4477:
4457:
4442:Algebraic cycles and the Hodge conjecture
4404:
4403:
4398:
3977:
3971:
3935:
3921:
3920:
3902:
3896:
3846:
3811:
3722:
3703:
3675:
3669:
3586:
3548:
3520:
3508:
3470:as a complex manifold, whereas the group
3428:
3406:
3391:
3376:
3370:
3289:
3283:
3260:
3254:
3242:
3220:
3212:
3207:
3056:
3043:
3042:
3030:
3018:
2932:
2931:
2922:
2916:
2915:
2906:
2861:
2855:
2854:
2835:
2834:
2832:
2779:
2773:
2772:
2762:
2756:
2755:
2745:
2735:
2708:
2692:
2675:
2669:
2668:
2663:
2661:
2625:
2597:
2575:
2557:
2513:
2491:
2472:
2466:
2439:{\displaystyle E_{0},E_{1},\ldots ,E_{N}}
2430:
2411:
2398:
2392:
2333:
2328:
2322:
2321:
2305:
2280:
2252:
2246:
2173:
2065:
2038:
2023:
2001:
1996:
1990:
1989:
1980:
1880:
1849:
1844:
1838:
1837:
1834:
1775:
1732:
1704:
1692:
1645:
1633:
1618:
1597:
1557:
1527:
1490:
1450:
1408:
1375:
1357:
1333:
1295:
1289:
1256:
1251:
1238:
1232:
1205:
1199:
1169:
1164:
1158:
1117:
1112:
1099:
1093:
1048:
1038:
1008:
1002:
950:
932:
919:
908:
893:
887:
796:
772:
754:
734:
710:
698:
674:
662:
651:. The de Rham complex is the sequence of
537:
536:
534:
514:
488:
487:
480:
470:
468:
437:
436:
429:
419:
417:
339:
324:
319:
304:
289:
284:
278:
245:
234:
219:
214:
199:
184:
178:
3806:and the cohomology class represented by
57:. The key observation is that, given a
5776:
5925:Griffiths & Harris (1994), p. 594.
4715:{\displaystyle H^{2p}(X,\mathbb {Z} )}
4675:predicts a converse: every element of
4622:{\displaystyle H^{2p}(X,\mathbb {C} )}
4575:{\displaystyle H^{2p}(X,\mathbb {Z} )}
2948:be the orthogonal projection, and let
2157:A variant of the Hodge theorem is the
1684:with respect to these inner products:
5500:{\displaystyle H^{2}(X,\mathbb {Z} )}
5278:{\displaystyle H^{2}(X,\mathbb {Z} )}
4196:For example, every smooth projective
7:
5907:Huybrechts (2005), Corollary 2.6.21.
5898:Huybrechts (2005), Corollary 3.2.12.
5413:of all projective K3 surfaces has a
4430:Hodge theory and extensions such as
119:, it can be applied to questions in
3787:theorem implies that a holomorphic
2732:
2689:
1235:
1096:
1035:
871:). De Rham's theorem says that the
558:must be non-zero. It follows that
509:is positive, so the cup product of
4406:
4400:
3719:
2870:
2698:
2615:
2587:
2565:
2503:
2481:
2372:Hodge theory of elliptic complexes
2329:
2249:
1997:
1898:
1877:
1845:
1777:
1729:
1701:
1566:
1372:
1292:
1026:
1005:
769:
707:
671:
25:
6023:Complex Geometry: An Introduction
5378:{\displaystyle \mathbb {C} ^{20}}
5307:{\displaystyle \mathbb {Z} ^{22}}
5210:of algebraic functions, known as
5134:The Hodge structure of a variety
5960:Principles of Algebraic Geometry
4386:Hodge-Riemann bilinear relations
3966:, we conclude that must lie in
3392:
2039:
1929:The Laplacian appeared first in
1634:
1328:with respect to the volume form
909:
612:The Hodge theory references the
551:{\displaystyle {\bar {\omega }}}
340:
305:
246:
235:
200:
3884:{\displaystyle (p,q)\neq (k,k)}
3485:depends only on the underlying
3136:induces a Riemannian metric on
2810:. Define the elliptic operator
1149:) the inner product induced by
603:Hodge theory for real manifolds
65:, every cohomology class has a
42:, is a method for studying the
5536:
5530:
5494:
5480:
5343:
5337:
5272:
5258:
4998:
4984:
4948:
4934:
4915:
4909:
4887:
4874:
4860:
4844:
4751:
4745:
4709:
4695:
4658:
4652:
4616:
4602:
4569:
4555:
4409:
4013:{\displaystyle H^{n-k,n-k}(X)}
4007:
4001:
3953:
3947:
3925:
3911:
3878:
3866:
3860:
3848:
3767:) is the space of holomorphic
3728:
3709:
3693:
3687:
3626:
3620:
3579:
3576:
3570:
3538:
3532:
3446:
3440:
3396:
3382:
3062:
3049:
3036:
3023:
2928:
2768:
2714:
2701:
2634:
2631:
2618:
2612:
2606:
2603:
2590:
2584:
2581:
2568:
2562:
2525:
2506:
2500:
2497:
2484:
2345:
2339:
2264:
2258:
2115:on the integral cohomology of
2043:
2029:
2016:
2013:
2007:
1892:
1886:
1861:
1855:
1744:
1738:
1725:
1722:
1716:
1657:
1651:
1630:
1611:
1605:
1483:
1477:
1468:
1462:
1425:
1422:
1410:
1387:
1381:
1313:{\displaystyle \Omega ^{k}(M)}
1307:
1301:
1271:
1268:
1262:
1244:
1181:
1175:
1132:
1129:
1123:
1105:
1060:
1054:
1020:
1014:
913:
899:
784:
778:
722:
716:
686:
680:
667:
598:, vol. 22, 1976, pp. 169–192.
542:
493:
442:
344:
330:
309:
295:
242:
239:
225:
204:
190:
88:, and it built on the work of
55:partial differential equations
1:
5889:Wells (2008), Theorem IV.5.2.
5750:Local invariant cycle theorem
4438:of compact Kähler manifolds.
3891:, then we get zero. Because
3657:group, which depends only on
624:. For a non-negative integer
5941:Computing Some Hodge Numbers
5871:Warner (1983), Theorem 6.11.
5787:A glimpse of the de Rham era
5634:variation of Hodge structure
5564:{\displaystyle \mathbb {Z} }
4779:{\displaystyle \mathbb {Z} }
4378:Lefschetz hyperplane theorem
3334:. On a Kähler manifold, the
3295:
3266:
2960:then asserts the following:
1187:{\displaystyle T_{p}^{*}(M)}
396:is necessarily zero because
366:Separately, a 1927 paper of
27:Mathematical manifold theory
5880:Warner (1983), Theorem 6.8.
5021:, even if the Hodge number
4246:For another example, every
1818:. By definition, a form on
986:Choose a Riemannian metric
6210:
6060:Cambridge University Press
5994:Cambridge University Press
5542:{\displaystyle H^{1,1}(X)}
5349:{\displaystyle H^{1,1}(X)}
4757:{\displaystyle H^{p,p}(X)}
4664:{\displaystyle H^{p,p}(X)}
4445:
3130:standard Riemannian metric
3086:
1826:if its Laplacian is zero:
1153:from each cotangent fiber
6139:10.1007/978-0-387-73892-5
3802:of the homology class of
3655:coherent sheaf cohomology
2368:for the de Rham complex.
2111:determines a real-valued
2096:and others in the 1940s.
1352:. Explicitly, given some
982:Operators in Hodge theory
374:. In modern language, if
6068:10.1017/CBO9780511615344
4432:non-abelian Hodge theory
3830:, the homology class of
3111:complex projective space
2454:with a volume form
2073:{\displaystyle \varphi }
1949:on spacetime, viewed as
73:that vanishes under the
67:canonical representative
6092:Warner, Frank (1983) ,
5745:Helmholtz decomposition
5103:Lefschetz (1,1)-theorem
5047:{\displaystyle h^{p,p}}
3819:{\displaystyle \alpha }
3795:is in fact algebraic.)
3151:For a complex manifold
2366:Helmholtz decomposition
1964:Riemannian manifold is
1767:on forms is defined by
1341:{\displaystyle \sigma }
522:{\displaystyle \omega }
5719:
5699:
5671:
5651:
5589:intermediate Jacobians
5565:
5543:
5501:
5457:
5431:
5403:
5379:
5350:
5308:
5279:
5235:
5216:transcendental numbers
5192:
5168:
5148:
5125:
5092:
5068:
5048:
5015:
4955:
4824:
4800:
4780:
4758:
4716:
4665:
4623:
4576:
4529:
4506:
4486:
4466:
4419:
4382:hard Lefschetz theorem
4014:
3960:
3885:
3820:
3738:
3636:
3456:
3304:
3122:homogeneous polynomial
3069:
2942:
2892:
2793:
2644:
2543:differential operators
2532:
2440:
2355:
2209:
2074:
2050:
1920:
1805:
1754:
1667:
1576:
1506:
1394:
1342:
1314:
1278:
1218:
1217:{\displaystyle k^{th}}
1188:
1139:
1075:
972:
814:
653:differential operators
600:
552:
523:
503:
452:
354:
257:
6123:Wells Jr., Raymond O.
5760:Hodge-Arakelov theory
5720:
5700:
5672:
5652:
5616:intersection homology
5608:mixed Hodge structure
5566:
5544:
5502:
5458:
5432:
5404:
5380:
5351:
5309:
5280:
5236:
5193:
5169:
5149:
5126:
5093:
5069:
5049:
5016:
4956:
4825:
4801:
4781:
4759:
4717:
4666:
4624:
4577:
4530:
4507:
4487:
4467:
4420:
4015:
3961:
3886:
3821:
3739:
3637:
3457:
3332:holomorphic functions
3317:a C function and the
3305:
3155:and a natural number
3070:
2943:
2893:
2794:
2645:
2533:
2441:
2356:
2210:
2075:
2051:
1921:
1806:
1755:
1668:
1577:
1507:
1395:
1343:
1315:
1279:
1219:
1189:
1140:
1084:The metric yields an
1076:
973:
815:
584:
553:
524:
504:
453:
355:
258:
170:, a bilinear pairing
5709:
5689:
5661:
5641:
5553:
5511:
5467:
5456:{\displaystyle 20-a}
5441:
5421:
5393:
5360:
5318:
5289:
5245:
5225:
5182:
5158:
5138:
5109:
5082:
5058:
5025:
4968:
4841:
4814:
4790:
4768:
4726:
4679:
4633:
4586:
4539:
4519:
4496:
4476:
4456:
4397:
3970:
3895:
3845:
3810:
3747:where Ω denotes the
3668:
3507:
3369:
3206:
3017:
2905:
2831:
2660:
2556:
2465:
2391:
2245:
2172:
2133:general linear group
2064:
1979:
1956:Every harmonic form
1931:mathematical physics
1833:
1774:
1691:
1596:
1526:
1407:
1356:
1332:
1288:
1231:
1198:
1157:
1092:
1001:
886:
661:
533:
513:
467:
416:
408:. Additionally, if
277:
177:
106:projective varieties
98:Riemannian manifolds
5124:{\displaystyle p=1}
4250:has Hodge diamond
3107:complex submanifold
2338:
2159:Hodge decomposition
2006:
1935:Maxwell's equations
1854:
1261:
1174:
1122:
873:singular cohomology
832:exterior derivative
802:
766:
740:
704:
572:Hodge star operator
329:
294:
224:
6148:10338.dmlcz/141778
6019:Huybrechts, Daniel
5952:Griffiths, Phillip
5715:
5695:
5667:
5647:
5600:Mixed Hodge theory
5561:
5539:
5497:
5453:
5427:
5415:countably infinite
5399:
5375:
5346:
5304:
5275:
5231:
5208:definite integrals
5204:algebraic function
5188:
5164:
5144:
5121:
5088:
5064:
5044:
5011:
4951:
4820:
4796:
4776:
4754:
4712:
4661:
4619:
4572:
4525:
4502:
4482:
4462:
4436:fundamental groups
4415:
4207:has Hodge diamond
4010:
3956:
3881:
3816:
3734:
3632:
3452:
3423:
3300:
3065:
2938:
2888:
2806:be the adjoint of
2789:
2787:
2640:
2528:
2436:
2385:elliptic complexes
2351:
2320:
2230:. In terms of the
2205:
2101:finite-dimensional
2070:
2046:
1988:
1916:
1836:
1801:
1750:
1663:
1572:
1502:
1390:
1338:
1310:
1274:
1247:
1214:
1184:
1160:
1145:by extending (see
1135:
1108:
1071:
968:
842:in the sense that
810:
641:differential forms
608:De Rham cohomology
548:
519:
499:
448:
350:
315:
280:
253:
210:
150:published a note,
140:algebraic topology
128:-adic Hodge theory
94:de Rham cohomology
86:algebraic geometry
6158:978-0-387-73891-8
6077:978-0-521-71801-1
6003:978-0-521-35881-1
5823:Ann. of Math. (2)
5793:, working paper,
5718:{\displaystyle X}
5698:{\displaystyle X}
5670:{\displaystyle X}
5650:{\displaystyle X}
5629:Phillip Griffiths
5549:is isomorphic to
5430:{\displaystyle a}
5402:{\displaystyle X}
5356:is isomorphic to
5285:is isomorphic to
5234:{\displaystyle X}
5206:. In particular,
5191:{\displaystyle X}
5167:{\displaystyle X}
5147:{\displaystyle X}
5091:{\displaystyle X}
5067:{\displaystyle X}
5009:
4885:
4823:{\displaystyle X}
4799:{\displaystyle X}
4528:{\displaystyle p}
4505:{\displaystyle X}
4485:{\displaystyle Y}
4465:{\displaystyle X}
4412:
4390:Kähler identities
4324:
4323:
4244:
4243:
4194:
4193:
4069:complex conjugate
3487:topological space
3402:
3298:
3269:
2971:are well-defined.
1933:. In particular,
994:and recall that:
963:
803:
767:
741:
705:
545:
496:
478:
445:
427:
406:Riemann relations
368:Solomon Lefschetz
322:
287:
217:
156:de Rham's theorem
71:differential form
59:Riemannian metric
44:cohomology groups
16:(Redirected from
6201:
6169:
6150:
6118:
6088:
6047:
6014:
5981:
5947:
5946:
5926:
5923:
5917:
5914:
5908:
5905:
5899:
5896:
5890:
5887:
5881:
5878:
5872:
5869:
5863:
5853:
5847:
5846:
5818:
5812:
5811:
5810:
5809:
5803:
5797:, archived from
5792:
5781:
5735:Potential theory
5724:
5722:
5721:
5716:
5704:
5702:
5701:
5696:
5676:
5674:
5673:
5668:
5656:
5654:
5653:
5648:
5570:
5568:
5567:
5562:
5560:
5548:
5546:
5545:
5540:
5529:
5528:
5506:
5504:
5503:
5498:
5493:
5479:
5478:
5462:
5460:
5459:
5454:
5436:
5434:
5433:
5428:
5408:
5406:
5405:
5400:
5384:
5382:
5381:
5376:
5374:
5373:
5368:
5355:
5353:
5352:
5347:
5336:
5335:
5313:
5311:
5310:
5305:
5303:
5302:
5297:
5284:
5282:
5281:
5276:
5271:
5257:
5256:
5240:
5238:
5237:
5232:
5197:
5195:
5194:
5189:
5173:
5171:
5170:
5165:
5153:
5151:
5150:
5145:
5130:
5128:
5127:
5122:
5097:
5095:
5094:
5089:
5073:
5071:
5070:
5065:
5053:
5051:
5050:
5045:
5043:
5042:
5020:
5018:
5017:
5012:
5010:
5007:
5005:
4997:
4983:
4982:
4960:
4958:
4957:
4952:
4947:
4933:
4932:
4908:
4907:
4886:
4883:
4881:
4873:
4859:
4858:
4829:
4827:
4826:
4821:
4805:
4803:
4802:
4797:
4785:
4783:
4782:
4777:
4775:
4763:
4761:
4760:
4755:
4744:
4743:
4721:
4719:
4718:
4713:
4708:
4694:
4693:
4673:Hodge conjecture
4670:
4668:
4667:
4662:
4651:
4650:
4628:
4626:
4625:
4620:
4615:
4601:
4600:
4581:
4579:
4578:
4573:
4568:
4554:
4553:
4534:
4532:
4531:
4526:
4511:
4509:
4508:
4503:
4491:
4489:
4488:
4483:
4471:
4469:
4468:
4463:
4448:Hodge conjecture
4424:
4422:
4421:
4416:
4414:
4413:
4405:
4372:
4362:
4253:
4252:
4210:
4209:
4105:
4104:
4088:
4058:
4019:
4017:
4016:
4011:
4000:
3999:
3965:
3963:
3962:
3957:
3946:
3945:
3924:
3910:
3909:
3890:
3888:
3887:
3882:
3828:Poincaré duality
3825:
3823:
3822:
3817:
3791:-form on all of
3743:
3741:
3740:
3735:
3727:
3726:
3708:
3707:
3686:
3685:
3641:
3639:
3638:
3633:
3619:
3618:
3617:
3600:
3569:
3568:
3567:
3556:
3531:
3530:
3484:
3461:
3459:
3458:
3453:
3439:
3438:
3422:
3395:
3381:
3380:
3350:Kähler manifold
3345:
3309:
3307:
3306:
3301:
3299:
3294:
3293:
3284:
3270:
3265:
3264:
3255:
3247:
3246:
3225:
3224:
3198:
3183:
3074:
3072:
3071:
3066:
3061:
3060:
3048:
3047:
3035:
3034:
2954:Green's operator
2947:
2945:
2944:
2939:
2937:
2936:
2927:
2926:
2921:
2920:
2897:
2895:
2894:
2889:
2866:
2865:
2860:
2859:
2840:
2839:
2823:
2798:
2796:
2795:
2790:
2788:
2784:
2783:
2778:
2777:
2767:
2766:
2761:
2760:
2750:
2749:
2740:
2739:
2713:
2712:
2697:
2696:
2680:
2679:
2674:
2673:
2649:
2647:
2646:
2641:
2630:
2629:
2602:
2601:
2580:
2579:
2537:
2535:
2534:
2529:
2524:
2523:
2496:
2495:
2477:
2476:
2445:
2443:
2442:
2437:
2435:
2434:
2416:
2415:
2403:
2402:
2360:
2358:
2357:
2352:
2337:
2332:
2327:
2326:
2316:
2315:
2291:
2290:
2257:
2256:
2229:
2214:
2212:
2211:
2206:
2149:
2079:
2077:
2076:
2071:
2055:
2053:
2052:
2047:
2042:
2028:
2027:
2005:
2000:
1995:
1994:
1974:
1953:of dimension 4.
1948:
1925:
1923:
1922:
1917:
1885:
1884:
1853:
1848:
1843:
1842:
1810:
1808:
1807:
1802:
1759:
1757:
1756:
1751:
1737:
1736:
1715:
1714:
1678:adjoint operator
1672:
1670:
1669:
1664:
1650:
1649:
1637:
1623:
1622:
1581:
1579:
1578:
1573:
1562:
1561:
1511:
1509:
1508:
1503:
1495:
1494:
1455:
1454:
1399:
1397:
1396:
1391:
1380:
1379:
1348:associated with
1347:
1345:
1344:
1339:
1319:
1317:
1316:
1311:
1300:
1299:
1283:
1281:
1280:
1275:
1260:
1255:
1243:
1242:
1225:exterior product
1223:
1221:
1220:
1215:
1213:
1212:
1193:
1191:
1190:
1185:
1173:
1168:
1144:
1142:
1141:
1136:
1121:
1116:
1104:
1103:
1080:
1078:
1077:
1072:
1067:
1063:
1053:
1052:
1043:
1042:
1013:
1012:
977:
975:
974:
969:
964:
962:
961:
960:
938:
937:
936:
920:
912:
898:
897:
870:
863:
819:
817:
816:
811:
801:
800:
787:
777:
776:
765:
764:
745:
739:
738:
725:
715:
714:
703:
702:
689:
679:
678:
580:Kunihiko Kodaira
557:
555:
554:
549:
547:
546:
538:
528:
526:
525:
520:
508:
506:
505:
500:
498:
497:
489:
479:
471:
457:
455:
454:
449:
447:
446:
438:
428:
420:
359:
357:
356:
351:
343:
328:
323:
320:
308:
293:
288:
285:
262:
260:
259:
254:
249:
238:
223:
218:
215:
203:
189:
188:
110:algebraic cycles
102:Kähler manifolds
21:
6209:
6208:
6204:
6203:
6202:
6200:
6199:
6198:
6184:
6183:
6181:
6159:
6121:
6108:
6091:
6078:
6050:
6037:
6017:
6004:
5986:Hodge, W. V. D.
5984:
5970:
5950:
5944:
5938:Arapura, Donu,
5937:
5934:
5929:
5924:
5920:
5915:
5911:
5906:
5902:
5897:
5893:
5888:
5884:
5879:
5875:
5870:
5866:
5854:
5850:
5835:10.2307/1968379
5820:
5819:
5815:
5807:
5805:
5801:
5790:
5783:
5782:
5778:
5774:
5755:Arakelov theory
5731:
5707:
5706:
5687:
5686:
5659:
5658:
5639:
5638:
5631:'s notion of a
5602:, developed by
5597:
5595:Generalizations
5577:Torelli theorem
5551:
5550:
5514:
5509:
5508:
5470:
5465:
5464:
5439:
5438:
5419:
5418:
5391:
5390:
5363:
5358:
5357:
5321:
5316:
5315:
5292:
5287:
5286:
5248:
5243:
5242:
5223:
5222:
5180:
5179:
5156:
5155:
5136:
5135:
5107:
5106:
5080:
5079:
5076:Hodge structure
5056:
5055:
5028:
5023:
5022:
4971:
4966:
4965:
4921:
4893:
4847:
4839:
4838:
4812:
4811:
4808:algebraic cycle
4788:
4787:
4766:
4765:
4729:
4724:
4723:
4682:
4677:
4676:
4636:
4631:
4630:
4589:
4584:
4583:
4542:
4537:
4536:
4517:
4516:
4494:
4493:
4474:
4473:
4454:
4453:
4450:
4444:
4395:
4394:
4370:
4364:
4354:
4344:
4080:
4050:
3973:
3968:
3967:
3931:
3898:
3893:
3892:
3843:
3842:
3808:
3807:
3779:is projective,
3759:. For example,
3751:of holomorphic
3718:
3699:
3671:
3666:
3665:
3610:
3593:
3582:
3560:
3549:
3544:
3516:
3505:
3504:
3471:
3424:
3372:
3367:
3366:
3335:
3330:
3323:
3285:
3256:
3238:
3216:
3204:
3203:
3186:
3173:
3146:Kähler manifold
3091:
3089:Hodge structure
3085:
3052:
3026:
3015:
3014:
2914:
2903:
2902:
2853:
2829:
2828:
2811:
2786:
2785:
2771:
2754:
2741:
2731:
2724:
2718:
2717:
2704:
2688:
2681:
2667:
2658:
2657:
2621:
2593:
2571:
2554:
2553:
2509:
2487:
2468:
2463:
2462:
2458:. Suppose that
2426:
2407:
2394:
2389:
2388:
2374:
2301:
2276:
2248:
2243:
2242:
2238:decomposition:
2223:
2170:
2169:
2135:
2062:
2061:
2019:
1977:
1976:
1969:
1968:, meaning that
1951:Minkowski space
1942:
1876:
1831:
1830:
1772:
1771:
1728:
1700:
1689:
1688:
1641:
1614:
1594:
1593:
1553:
1524:
1523:
1486:
1446:
1405:
1404:
1371:
1354:
1353:
1330:
1329:
1291:
1286:
1285:
1234:
1229:
1228:
1201:
1196:
1195:
1155:
1154:
1095:
1090:
1089:
1044:
1034:
1033:
1029:
1004:
999:
998:
984:
946:
939:
928:
921:
889:
884:
883:
865:
861:
852:
843:
840:cochain complex
828:
792:
768:
750:
730:
706:
694:
670:
659:
658:
622:smooth manifold
614:de Rham complex
610:
605:
568:Riemann surface
531:
530:
511:
510:
465:
464:
414:
413:
387:
380:
275:
274:
268:perfect pairing
180:
175:
174:
160:Stokes' theorem
136:
117:complex numbers
90:Georges de Rham
48:smooth manifold
28:
23:
22:
15:
12:
11:
5:
6207:
6205:
6197:
6196:
6186:
6185:
6179:
6178:
6171:
6170:
6157:
6119:
6106:
6089:
6076:
6052:Voisin, Claire
6048:
6035:
6015:
6002:
5982:
5968:
5956:Harris, Joseph
5948:
5933:
5930:
5928:
5927:
5918:
5909:
5900:
5891:
5882:
5873:
5864:
5856:Michael Atiyah
5848:
5829:(1): 342–354.
5813:
5775:
5773:
5770:
5769:
5768:
5762:
5757:
5752:
5747:
5742:
5737:
5730:
5727:
5714:
5694:
5679:period mapping
5666:
5646:
5621:Morihiko Saito
5604:Pierre Deligne
5596:
5593:
5559:
5538:
5535:
5532:
5527:
5524:
5521:
5517:
5496:
5492:
5488:
5485:
5482:
5477:
5473:
5452:
5449:
5446:
5437:has dimension
5426:
5398:
5372:
5367:
5345:
5342:
5339:
5334:
5331:
5328:
5324:
5301:
5296:
5274:
5270:
5266:
5263:
5260:
5255:
5251:
5230:
5187:
5163:
5143:
5120:
5117:
5114:
5087:
5063:
5041:
5038:
5035:
5031:
5004:
5000:
4996:
4992:
4989:
4986:
4981:
4978:
4974:
4962:
4961:
4950:
4946:
4942:
4939:
4936:
4931:
4928:
4924:
4920:
4917:
4914:
4911:
4906:
4903:
4900:
4896:
4892:
4889:
4880:
4876:
4872:
4868:
4865:
4862:
4857:
4854:
4850:
4846:
4819:
4795:
4774:
4753:
4750:
4747:
4742:
4739:
4736:
4732:
4711:
4707:
4703:
4700:
4697:
4692:
4689:
4685:
4660:
4657:
4654:
4649:
4646:
4643:
4639:
4618:
4614:
4610:
4607:
4604:
4599:
4596:
4592:
4571:
4567:
4563:
4560:
4557:
4552:
4549:
4545:
4524:
4501:
4481:
4461:
4446:Main article:
4443:
4440:
4411:
4408:
4402:
4368:
4363:and hence has
4338:
4322:
4321:
4319:
4317:
4314:
4312:
4309:
4308:
4306:
4303:
4301:
4298:
4295:
4294:
4291:
4289:
4286:
4284:
4280:
4279:
4277:
4274:
4272:
4269:
4266:
4265:
4263:
4261:
4258:
4256:
4242:
4241:
4239:
4236:
4233:
4232:
4227:
4225:
4219:
4218:
4216:
4213:
4192:
4191:
4189:
4187:
4182:
4180:
4177:
4176:
4174:
4169:
4167:
4162:
4159:
4158:
4153:
4151:
4146:
4144:
4138:
4137:
4135:
4130:
4128:
4123:
4120:
4119:
4117:
4115:
4110:
4108:
4048:Hodge symmetry
4009:
4006:
4003:
3998:
3995:
3992:
3989:
3986:
3983:
3980:
3976:
3955:
3952:
3949:
3944:
3941:
3938:
3934:
3930:
3927:
3923:
3919:
3916:
3913:
3908:
3905:
3901:
3880:
3877:
3874:
3871:
3868:
3865:
3862:
3859:
3856:
3853:
3850:
3815:
3745:
3744:
3733:
3730:
3725:
3721:
3717:
3714:
3711:
3706:
3702:
3698:
3695:
3692:
3689:
3684:
3681:
3678:
3674:
3643:
3642:
3631:
3628:
3625:
3622:
3616:
3613:
3609:
3606:
3603:
3599:
3596:
3592:
3589:
3585:
3581:
3578:
3575:
3572:
3566:
3563:
3559:
3555:
3552:
3547:
3543:
3540:
3537:
3534:
3529:
3526:
3523:
3519:
3515:
3512:
3463:
3462:
3451:
3448:
3445:
3442:
3437:
3434:
3431:
3427:
3421:
3418:
3415:
3412:
3409:
3405:
3401:
3398:
3394:
3390:
3387:
3384:
3379:
3375:
3328:
3321:
3311:
3310:
3297:
3292:
3288:
3282:
3279:
3276:
3273:
3268:
3263:
3259:
3253:
3250:
3245:
3241:
3237:
3234:
3231:
3228:
3223:
3219:
3215:
3211:
3118:Chow's theorem
3087:Main article:
3084:
3081:
3077:
3076:
3064:
3059:
3055:
3051:
3046:
3041:
3038:
3033:
3029:
3025:
3022:
3011:
2991:
2972:
2935:
2930:
2925:
2919:
2913:
2910:
2899:
2898:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2864:
2858:
2852:
2849:
2846:
2843:
2838:
2800:
2799:
2782:
2776:
2770:
2765:
2759:
2753:
2748:
2744:
2738:
2734:
2730:
2727:
2725:
2723:
2720:
2719:
2716:
2711:
2707:
2703:
2700:
2695:
2691:
2687:
2684:
2682:
2678:
2672:
2666:
2665:
2651:
2650:
2639:
2636:
2633:
2628:
2624:
2620:
2617:
2614:
2611:
2608:
2605:
2600:
2596:
2592:
2589:
2586:
2583:
2578:
2574:
2570:
2567:
2564:
2561:
2539:
2538:
2527:
2522:
2519:
2516:
2512:
2508:
2505:
2502:
2499:
2494:
2490:
2486:
2483:
2480:
2475:
2471:
2448:vector bundles
2433:
2429:
2425:
2422:
2419:
2414:
2410:
2406:
2401:
2397:
2373:
2370:
2362:
2361:
2350:
2347:
2344:
2341:
2336:
2331:
2325:
2319:
2314:
2311:
2308:
2304:
2300:
2297:
2294:
2289:
2286:
2283:
2279:
2275:
2272:
2269:
2266:
2263:
2260:
2255:
2251:
2216:
2215:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2125:isometry group
2069:
2045:
2041:
2037:
2034:
2031:
2026:
2022:
2018:
2015:
2012:
2009:
2004:
1999:
1993:
1987:
1984:
1927:
1926:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1883:
1879:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1852:
1847:
1841:
1812:
1811:
1800:
1797:
1794:
1791:
1788:
1785:
1782:
1779:
1761:
1760:
1749:
1746:
1743:
1740:
1735:
1731:
1727:
1724:
1721:
1718:
1713:
1710:
1707:
1703:
1699:
1696:
1674:
1673:
1662:
1659:
1656:
1653:
1648:
1644:
1640:
1636:
1632:
1629:
1626:
1621:
1617:
1613:
1610:
1607:
1604:
1601:
1583:
1582:
1571:
1568:
1565:
1560:
1556:
1552:
1549:
1546:
1543:
1540:
1537:
1534:
1531:
1513:
1512:
1501:
1498:
1493:
1489:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1453:
1449:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1389:
1386:
1383:
1378:
1374:
1370:
1367:
1364:
1361:
1337:
1309:
1306:
1303:
1298:
1294:
1273:
1270:
1267:
1264:
1259:
1254:
1250:
1246:
1241:
1237:
1211:
1208:
1204:
1183:
1180:
1177:
1172:
1167:
1163:
1147:Gramian matrix
1134:
1131:
1128:
1125:
1120:
1115:
1111:
1107:
1102:
1098:
1088:on each fiber
1082:
1081:
1070:
1066:
1062:
1059:
1056:
1051:
1047:
1041:
1037:
1032:
1028:
1025:
1022:
1019:
1016:
1011:
1007:
983:
980:
979:
978:
967:
959:
956:
953:
949:
945:
942:
935:
931:
927:
924:
918:
915:
911:
907:
904:
901:
896:
892:
864:(also written
857:
847:
826:
821:
820:
809:
806:
799:
795:
790:
786:
783:
780:
775:
771:
763:
760:
757:
753:
748:
744:
737:
733:
728:
724:
721:
718:
713:
709:
701:
697:
692:
688:
685:
682:
677:
673:
669:
666:
609:
606:
604:
601:
544:
541:
518:
495:
492:
486:
483:
477:
474:
444:
441:
435:
432:
426:
423:
385:
378:
361:
360:
349:
346:
342:
338:
335:
332:
327:
318:
314:
311:
307:
303:
300:
297:
292:
283:
264:
263:
252:
248:
244:
241:
237:
233:
230:
227:
222:
213:
209:
206:
202:
198:
195:
192:
187:
183:
135:
132:
40:W. V. D. Hodge
38:, named after
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6206:
6195:
6192:
6191:
6189:
6182:
6176:
6173:
6172:
6168:
6164:
6160:
6154:
6149:
6144:
6140:
6136:
6132:
6128:
6124:
6120:
6117:
6113:
6109:
6107:0-387-90894-3
6103:
6099:
6095:
6090:
6087:
6083:
6079:
6073:
6069:
6065:
6061:
6057:
6053:
6049:
6046:
6042:
6038:
6036:3-540-21290-6
6032:
6028:
6024:
6020:
6016:
6013:
6009:
6005:
5999:
5995:
5991:
5987:
5983:
5979:
5975:
5971:
5969:0-471-05059-8
5965:
5961:
5957:
5953:
5949:
5943:
5942:
5936:
5935:
5931:
5922:
5919:
5913:
5910:
5904:
5901:
5895:
5892:
5886:
5883:
5877:
5874:
5868:
5865:
5861:
5857:
5852:
5849:
5844:
5840:
5836:
5832:
5828:
5824:
5817:
5814:
5804:on 2023-12-04
5800:
5796:
5789:
5788:
5780:
5777:
5771:
5766:
5763:
5761:
5758:
5756:
5753:
5751:
5748:
5746:
5743:
5741:
5740:Serre duality
5738:
5736:
5733:
5732:
5728:
5726:
5712:
5692:
5684:
5683:Hodge modules
5680:
5664:
5644:
5636:
5635:
5630:
5625:
5622:
5618:
5617:
5611:
5609:
5605:
5601:
5594:
5592:
5590:
5586:
5582:
5578:
5572:
5533:
5525:
5522:
5519:
5515:
5486:
5483:
5475:
5471:
5450:
5447:
5444:
5424:
5416:
5412:
5396:
5388:
5387:Picard number
5370:
5340:
5332:
5329:
5326:
5322:
5299:
5264:
5261:
5253:
5249:
5228:
5219:
5217:
5213:
5209:
5205:
5201:
5185:
5177:
5161:
5141:
5132:
5118:
5115:
5112:
5104:
5099:
5085:
5077:
5061:
5039:
5036:
5033:
5029:
5002:
4990:
4987:
4979:
4976:
4972:
4940:
4937:
4929:
4926:
4922:
4918:
4912:
4904:
4901:
4898:
4894:
4890:
4878:
4866:
4863:
4855:
4852:
4848:
4837:
4836:
4835:
4831:
4817:
4809:
4793:
4748:
4740:
4737:
4734:
4730:
4701:
4698:
4690:
4687:
4683:
4674:
4655:
4647:
4644:
4641:
4637:
4608:
4605:
4597:
4594:
4590:
4561:
4558:
4550:
4547:
4543:
4522:
4515:
4499:
4479:
4459:
4449:
4441:
4439:
4437:
4433:
4428:
4426:
4391:
4387:
4383:
4379:
4374:
4367:
4361:
4357:
4352:
4351:diffeomorphic
4348:
4342:
4337:
4333:
4329:
4328:Betti numbers
4320:
4318:
4315:
4313:
4311:
4310:
4307:
4304:
4302:
4299:
4297:
4296:
4292:
4290:
4287:
4285:
4282:
4281:
4278:
4275:
4273:
4270:
4268:
4267:
4264:
4262:
4259:
4257:
4255:
4254:
4251:
4249:
4240:
4237:
4235:
4234:
4231:
4228:
4226:
4224:
4221:
4220:
4217:
4214:
4212:
4211:
4208:
4206:
4203:
4199:
4190:
4188:
4186:
4183:
4181:
4179:
4178:
4175:
4173:
4170:
4168:
4166:
4163:
4161:
4160:
4157:
4154:
4152:
4150:
4147:
4145:
4143:
4140:
4139:
4136:
4134:
4131:
4129:
4127:
4124:
4122:
4121:
4118:
4116:
4114:
4111:
4109:
4107:
4106:
4103:
4101:
4100:
4099:Hodge diamond
4094:
4092:
4091:Serre duality
4087:
4083:
4078:
4074:
4070:
4066:
4062:
4057:
4053:
4049:
4045:
4041:
4037:
4033:
4029:
4026:
4021:
4004:
3996:
3993:
3990:
3987:
3984:
3981:
3978:
3974:
3950:
3942:
3939:
3936:
3932:
3928:
3917:
3914:
3906:
3903:
3899:
3875:
3872:
3869:
3863:
3857:
3854:
3851:
3839:
3837:
3833:
3829:
3813:
3805:
3801:
3796:
3794:
3790:
3786:
3782:
3778:
3774:
3770:
3766:
3762:
3758:
3754:
3750:
3731:
3723:
3715:
3712:
3704:
3700:
3696:
3690:
3682:
3679:
3676:
3672:
3664:
3663:
3662:
3660:
3656:
3652:
3648:
3629:
3623:
3614:
3611:
3607:
3604:
3601:
3597:
3594:
3590:
3587:
3583:
3573:
3564:
3561:
3557:
3553:
3550:
3545:
3541:
3535:
3527:
3524:
3521:
3517:
3513:
3510:
3503:
3502:
3501:
3499:
3494:
3492:
3488:
3482:
3478:
3474:
3469:
3449:
3443:
3435:
3432:
3429:
3425:
3419:
3416:
3413:
3410:
3407:
3403:
3399:
3388:
3385:
3377:
3373:
3365:
3364:
3363:
3361:
3357:
3353:
3349:
3343:
3339:
3333:
3327:
3320:
3316:
3290:
3286:
3280:
3277:
3274:
3271:
3261:
3257:
3251:
3248:
3243:
3239:
3235:
3232:
3229:
3226:
3221:
3217:
3213:
3209:
3202:
3201:
3200:
3197:
3193:
3189:
3184:
3181:
3177:
3169:
3165:
3162:
3158:
3154:
3149:
3147:
3143:
3139:
3135:
3131:
3127:
3124:equations on
3123:
3119:
3115:
3112:
3108:
3104:
3100:
3096:
3090:
3082:
3080:
3057:
3053:
3039:
3031:
3027:
3020:
3012:
3010:
3006:
3003:
2999:
2995:
2992:
2989:
2985:
2981:
2977:
2973:
2970:
2966:
2963:
2962:
2961:
2959:
2958:Hodge theorem
2955:
2951:
2923:
2911:
2908:
2885:
2879:
2876:
2873:
2867:
2862:
2850:
2847:
2841:
2827:
2826:
2825:
2822:
2819:
2815:
2809:
2805:
2780:
2763:
2751:
2746:
2742:
2736:
2728:
2726:
2721:
2709:
2705:
2693:
2685:
2683:
2676:
2656:
2655:
2654:
2637:
2626:
2622:
2609:
2598:
2594:
2576:
2572:
2559:
2552:
2551:
2550:
2548:
2544:
2520:
2517:
2514:
2510:
2492:
2488:
2478:
2473:
2469:
2461:
2460:
2459:
2457:
2453:
2449:
2431:
2427:
2423:
2420:
2417:
2412:
2408:
2404:
2399:
2395:
2386:
2382:
2378:
2371:
2369:
2367:
2348:
2342:
2334:
2317:
2312:
2309:
2306:
2302:
2298:
2295:
2292:
2287:
2284:
2281:
2277:
2273:
2270:
2267:
2261:
2253:
2241:
2240:
2239:
2237:
2233:
2227:
2222:is harmonic:
2221:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2168:
2167:
2166:
2164:
2160:
2155:
2153:
2147:
2143:
2139:
2134:
2130:
2126:
2122:
2118:
2114:
2113:inner product
2110:
2106:
2102:
2097:
2095:
2091:
2087:
2083:
2067:
2059:
2058:Hodge theorem
2035:
2032:
2024:
2020:
2010:
2002:
1985:
1982:
1972:
1967:
1963:
1959:
1954:
1952:
1946:
1940:
1936:
1932:
1913:
1907:
1904:
1901:
1895:
1889:
1881:
1873:
1870:
1864:
1858:
1850:
1829:
1828:
1827:
1825:
1821:
1817:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1770:
1769:
1768:
1766:
1747:
1741:
1733:
1719:
1711:
1708:
1705:
1697:
1694:
1687:
1686:
1685:
1683:
1679:
1676:Consider the
1660:
1654:
1646:
1642:
1638:
1627:
1624:
1619:
1608:
1602:
1592:
1591:
1590:
1588:
1569:
1563:
1558:
1550:
1544:
1538:
1535:
1532:
1522:
1521:
1520:
1518:
1499:
1496:
1491:
1480:
1474:
1471:
1465:
1459:
1451:
1447:
1443:
1437:
1434:
1431:
1419:
1416:
1413:
1403:
1402:
1401:
1384:
1376:
1368:
1365:
1362:
1359:
1351:
1335:
1327:
1323:
1304:
1296:
1265:
1257:
1252:
1248:
1239:
1226:
1209:
1206:
1202:
1178:
1170:
1165:
1161:
1152:
1148:
1126:
1118:
1113:
1109:
1100:
1087:
1086:inner product
1068:
1064:
1057:
1049:
1045:
1039:
1030:
1023:
1017:
1009:
997:
996:
995:
993:
989:
981:
965:
957:
954:
951:
947:
943:
940:
933:
929:
925:
922:
916:
905:
902:
894:
890:
882:
881:
880:
878:
874:
868:
860:
856:
850:
846:
841:
838:). This is a
837:
833:
829:
807:
804:
797:
793:
788:
781:
773:
761:
758:
755:
751:
746:
742:
735:
731:
726:
719:
711:
699:
695:
690:
683:
675:
664:
657:
656:
655:
654:
650:
646:
642:
638:
635:
631:
627:
623:
619:
615:
607:
602:
599:
597:
593:
588:
583:
581:
577:
573:
569:
563:
561:
539:
516:
490:
484:
481:
475:
472:
462:
439:
433:
430:
424:
421:
411:
407:
403:
399:
395:
394:wedge product
392:, then their
391:
384:
377:
373:
369:
364:
347:
336:
333:
325:
316:
312:
301:
298:
290:
281:
273:
272:
271:
269:
250:
231:
228:
220:
211:
207:
196:
193:
185:
181:
173:
172:
171:
169:
165:
161:
157:
153:
149:
145:
141:
138:The field of
133:
131:
129:
127:
122:
121:number theory
118:
113:
111:
107:
103:
99:
95:
91:
87:
82:
80:
76:
72:
68:
64:
60:
56:
52:
49:
45:
41:
37:
33:
19:
18:Harmonic form
6194:Hodge theory
6180:
6126:
6093:
6055:
6022:
5989:
5959:
5940:
5921:
5912:
5903:
5894:
5885:
5876:
5867:
5859:
5851:
5826:
5822:
5816:
5806:, retrieved
5799:the original
5786:
5779:
5657:varies when
5632:
5626:
5614:
5612:
5599:
5598:
5573:
5411:moduli space
5241:, the group
5220:
5133:
5100:
5075:
4963:
4832:
4807:
4672:
4451:
4429:
4375:
4365:
4359:
4355:
4347:Hopf surface
4340:
4335:
4331:
4325:
4245:
4229:
4222:
4204:
4195:
4184:
4171:
4164:
4155:
4148:
4141:
4132:
4125:
4112:
4098:
4095:
4085:
4081:
4076:
4072:
4064:
4060:
4055:
4051:
4047:
4043:
4039:
4035:
4031:
4027:
4025:Hodge number
4024:
4022:
3840:
3835:
3831:
3803:
3797:
3792:
3788:
3776:
3772:
3768:
3764:
3760:
3756:
3752:
3746:
3658:
3650:
3646:
3644:
3495:
3490:
3480:
3476:
3472:
3467:
3464:
3359:
3351:
3341:
3337:
3325:
3318:
3314:
3312:
3195:
3191:
3187:
3179:
3175:
3167:
3163:
3156:
3152:
3150:
3141:
3137:
3133:
3125:
3113:
3105:is a closed
3102:
3094:
3092:
3078:
3008:
3004:
3001:
2997:
2993:
2987:
2983:
2979:
2975:
2968:
2964:
2957:
2949:
2900:
2820:
2817:
2813:
2807:
2803:
2801:
2652:
2540:
2455:
2451:
2375:
2363:
2231:
2225:
2219:
2217:
2162:
2158:
2156:
2154:is finite).
2145:
2141:
2137:
2128:
2116:
2108:
2098:
2085:
2081:
2060:states that
2057:
1970:
1957:
1955:
1944:
1938:
1928:
1823:
1819:
1815:
1813:
1762:
1681:
1675:
1586:
1584:
1516:
1514:
1349:
1325:
1324:-forms over
1321:
1150:
1083:
991:
987:
985:
876:
866:
858:
854:
848:
844:
835:
830:denotes the
824:
822:
648:
644:
637:vector space
629:
625:
617:
611:
595:
592:M. F. Atiyah
589:
585:
576:Hermann Weyl
564:
559:
460:
409:
397:
389:
382:
375:
365:
362:
265:
167:
151:
137:
125:
114:
83:
78:
62:
50:
36:Hodge theory
35:
29:
5765:ddbar lemma
5178:classes in
4514:codimension
4349:, which is
3800:cap product
3498:cup product
2956:for Δ. The
2541:are linear
402:cup product
148:Élie Cartan
32:mathematics
5932:References
5808:2018-10-15
5619:. Namely,
5581:Chow group
4384:, and the
4248:K3 surface
3771:-forms on
3755:-forms on
3645:The piece
3356:cohomology
2545:acting on
2236:direct sum
1941:such that
643:of degree
639:of smooth
144:cohomology
6177:on GitHub
6125:(2008) ,
6054:(2007) ,
5958:(1994) .
5585:cycle map
5448:−
5214:, can be
4919:⊆
4891:∩
4410:¯
4407:∂
4401:∂
4067:) is the
4059:(because
3994:−
3982:−
3864:≠
3814:α
3720:Ω
3697:≅
3580:→
3542:×
3514::
3511:⌣
3404:⨁
3296:¯
3278:∧
3275:⋯
3272:∧
3267:¯
3249:∧
3233:∧
3230:⋯
3227:∧
3172:forms of
3166:-form on
3040:≅
2929:→
2924:∙
2871:Δ
2868:∣
2863:∙
2851:∈
2781:∙
2769:→
2764:∙
2733:⨁
2699:Γ
2690:⨁
2677:∙
2635:→
2616:Γ
2613:→
2610:⋯
2607:→
2588:Γ
2585:→
2566:Γ
2563:→
2504:Γ
2501:→
2482:Γ
2421:…
2330:Δ
2318:⊕
2303:δ
2299:
2293:⊕
2285:−
2274:
2268:≅
2250:Ω
2218:in which
2200:γ
2194:β
2191:δ
2185:α
2176:ω
2068:φ
2017:→
1998:Δ
1983:φ
1902:α
1899:Δ
1896:∣
1878:Ω
1874:∈
1871:α
1846:Δ
1793:δ
1787:δ
1778:Δ
1765:Laplacian
1763:Then the
1730:Ω
1726:→
1702:Ω
1695:δ
1639:∈
1631:→
1616:‖
1603:ω
1600:‖
1567:∞
1555:‖
1551:ω
1548:‖
1542:⟩
1539:ω
1533:ω
1530:⟨
1497:σ
1488:⟩
1475:τ
1460:ω
1457:⟨
1448:∫
1441:⟩
1438:τ
1432:ω
1429:⟨
1426:↦
1420:τ
1414:ω
1373:Ω
1369:∈
1366:τ
1360:ω
1336:σ
1293:Ω
1258:∗
1236:⋀
1171:∗
1119:∗
1097:⋀
1050:∗
1036:⋀
1027:Γ
1006:Ω
955:−
944:
926:
917:≅
770:Ω
759:−
743:⋯
708:Ω
672:Ω
668:→
632:) be the
543:¯
540:ω
517:ω
494:¯
491:ω
485:∧
482:ω
473:−
443:¯
440:ω
434:∧
431:ω
422:−
313:≅
243:→
208:×
75:Laplacian
6188:Category
6131:Springer
6098:Springer
6027:Springer
6021:(2005),
5988:(1941),
5729:See also
5200:calculus
5176:homology
4392:and the
3615:′
3598:′
3565:′
3554:′
3159:, every
3109:of some
2802:and let
2383:defined
2090:elliptic
1824:harmonic
1400:we have
789:→
747:→
727:→
691:→
628:, let Ω(
164:singular
79:harmonic
6167:2359489
6116:0722297
6086:1967689
6045:2093043
6012:0003947
5978:0507725
5843:1968379
5212:periods
5008:torsion
4884:torsion
4079:)) and
3348:compact
2952:be the
2152:lattice
2131:in the
2121:torsion
2119:modulo
2094:Kodaira
1519:-form:
1194:to its
372:Riemann
134:History
6165:
6155:
6114:
6104:
6084:
6074:
6043:
6033:
6010:
6000:
5976:
5966:
5841:
5409:. The
5314:, and
4671:. The
4425:-lemma
4380:, the
3826:. By
3775:. (If
3174:type (
3128:. The
3099:smooth
2377:Atiyah
2105:kernel
2056:. The
1966:closed
1962:closed
1284:. The
823:where
616:. Let
158:. By
53:using
5945:(PDF)
5839:JSTOR
5802:(PDF)
5791:(PDF)
5772:Notes
5507:with
5174:over
4202:genus
4198:curve
3781:Serre
3749:sheaf
3313:with
3185:with
3116:. By
3097:be a
2974:Id =
1960:on a
834:on Ω(
620:be a
46:of a
6153:ISBN
6102:ISBN
6072:ISBN
6031:ISBN
5998:ISBN
5964:ISBN
5795:EPFL
5101:The
4452:Let
4326:The
4089:(by
4023:The
3785:GAGA
3324:and
3093:Let
2967:and
2901:Let
2812:Δ =
2381:Bott
2379:and
1564:<
634:real
529:and
381:and
286:sing
100:and
69:, a
6143:hdl
6135:doi
6064:doi
5831:doi
5389:of
5078:of
4830:.)
4810:on
4512:of
4492:in
4371:= 1
4353:to
4330:of
4200:of
4093:).
4071:of
3783:'s
3489:of
3358:of
3132:on
2978:+ Δ
2446:be
2228:= 0
2136:GL(
2127:of
1973:= 0
1947:= 0
1822:is
1680:of
990:on
923:ker
875:of
869:= 0
862:= 0
647:on
92:on
61:on
30:In
6190::
6163:MR
6161:,
6151:,
6141:,
6133:,
6112:MR
6110:,
6100:,
6096:,
6082:MR
6080:,
6070:,
6062:,
6058:,
6041:MR
6039:,
6029:,
6025:,
6008:MR
6006:,
5996:,
5992:,
5974:MR
5972:.
5954:;
5858:,
5837:.
5827:28
5825:.
5610:.
5445:20
5371:20
5300:22
4427:.
4373:.
4358:×
4343:+1
4288:20
4084:=
4054:=
4020:.
3838:.
3493:.
3479:,
3340:,
3194:=
3190:+
3178:,
3148:.
3144:a
3134:CP
3126:CP
3114:CP
3009:GL
3007:=
3000:,
2998:GL
2996:=
2994:LG
2986:+
2982:=
2816:+
2814:LL
2456:dV
2296:im
2271:im
2148:))
2144:,
1971:dα
1444::=
1227::
941:im
853:∘
851:+1
321:dR
216:dR
112:.
81:.
34:,
6145::
6137::
6066::
5980:.
5845:.
5833::
5713:X
5693:X
5665:X
5645:X
5558:Z
5537:)
5534:X
5531:(
5526:1
5523:,
5520:1
5516:H
5495:)
5491:Z
5487:,
5484:X
5481:(
5476:2
5472:H
5451:a
5425:a
5397:X
5366:C
5344:)
5341:X
5338:(
5333:1
5330:,
5327:1
5323:H
5295:Z
5273:)
5269:Z
5265:,
5262:X
5259:(
5254:2
5250:H
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