563:. 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".
2786:
5563:
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
807:
567:, 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
575:
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
2348:
2648:
1499:
259:, 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:
554:
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
3297:
3629:
447:
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
4948:
1913:
4822:
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
250:
3454:
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
965:
347:
2637:
649:
1068:
2885:
2043:
5612:
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.
1569:
143:, 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
3449:
2233:
2781:{\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}}}
1660:
2935:
496:
445:
1747:
2525:
5008:
2202:
1271:
1132:
3731:
1395:
5595:, 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
3062:
2653:
4365:
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
3953:
3194:
1387:
4334:
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
4412:
1798:
2433:
4709:
4616:
4569:
5494:
5272:
2096:
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
5372:
5301:
545:
4829:
3878:
4007:
1307:
5558:
4773:
1181:
5536:
5343:
4751:
4658:
1821:
2067:
576:
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.
5041:
3813:
1335:
516:
3823:
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
1211:
5616:
A fundamental aspect of complex geometry is that there are continuous families of non-isomorphic complex manifolds (which are all diffeomorphic as real manifolds).
165:
5450:
5118:
874:
265:
5712:
5692:
5664:
5644:
5424:
5396:
5228:
5185:
5161:
5141:
5085:
5061:
4817:
4793:
4522:
4499:
4479:
4459:
3495:
802:{\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,}
2544:
97:, is encompassed by the latter case. Hodge theory has become an important tool in algebraic geometry, particularly through its connection to the study of
989:
2819:
1967:
3106:
4377:. Many of these results follow from fundamental technical tools which may be proven for compact Kähler manifolds using Hodge theory, including the
559:
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
5783:
1514:
3830:
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
2343:{\displaystyle \Omega ^{k}(M)\cong \operatorname {im} d_{k-1}\oplus \operatorname {im} \delta _{k+1}\oplus {\mathcal {H}}_{\Delta }^{k}(M).}
571:
modified Hodge's proof to repair the error. This established Hodge's sought-for isomorphism between harmonic forms and cohomology classes.
5905:
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.
5576:
from Chow groups to ordinary cohomology, but Hodge theory also gives information about the kernel of the cycle map, for example using the
5568:
holds, meaning that the variety is determined up to isomorphism by its Hodge structure. Finally, Hodge theory gives information about the
352:
De Rham's original statement is then a consequence of the fact that over the reals, singular cohomology is the dual of singular homology.
3357:
1584:
2893:
455:
404:
1679:
2453:
1494:{\displaystyle (\omega ,\tau )\mapsto \langle \omega ,\tau \rangle :=\int _{M}\langle \omega (p),\tau (p)\rangle _{p}\sigma .}
6145:
6064:
5990:
4956:
2160:
1219:
1080:
4031:). These are important invariants of a smooth complex projective variety; they do not change when the complex structure of
3656:
551:
itself must represent a non-zero cohomology class, so its periods cannot all be zero. This resolved a question of Severi.
4035:
is varied continuously, and yet they are in general not topological invariants. Among the properties of Hodge numbers are
3005:
1954:
4323:
are the sum of the Hodge numbers in a given row. A basic application of Hodge theory is then that the odd Betti numbers
3292:{\displaystyle f\,dz_{1}\wedge \cdots \wedge dz_{p}\wedge d{\overline {w_{1}}}\wedge \cdots \wedge d{\overline {w_{q}}}}
5774:
5091:
4086:
3883:
6094:
6023:
5956:
1344:
6119:
6086:
6015:
4374:
2073:
has a unique harmonic representative. Concretely, the harmonic representative is the unique closed form of minimum
2088:
For example, the Hodge theorem implies that the cohomology groups with real coefficients of a closed manifold are
4385:
3068:
There is also a Hodge decomposition in this situation, generalizing the statement above for the de Rham complex.
1762:
2379:
1923:
28:
5738:
43:
4667:
4574:
4527:
3486:
5622:
5455:
5233:
4366:
4085:
The Hodge numbers of a smooth complex projective variety (or compact Kähler manifold) can be listed in the
55:
5207:. The difficulty of the Hodge conjecture reflects the lack of understanding of such integrals in general.
4943:{\displaystyle (H^{2p}(X,\mathbb {Z} )/{\text{torsion}})\cap H^{p,p}(X)\subseteq H^{2p}(X,\mathbb {C} )}
3489:
in cohomology, so the cup product with complex coefficients is compatible with the Hodge decomposition:
6048:
5982:
5944:
5348:
5277:
2376:
as a generalization of the de Rham complex. The Hodge theorem extends to this setting, as follows. Let
5043:
is big. In short, the Hodge conjecture predicts that the possible "shapes" of complex subvarieties of
521:
3833:
3643:
3188:, meaning forms that can locally be written as a finite sum of terms, with each term taking the form
3160:
3958:
1926:
say that the electromagnetic field in a vacuum, i.e. absent any charges, is represented by a 2-form
1276:
393:
of their cohomology classes is zero, and when made explicit, this gave
Lefschetz a new proof of the
3099:
1908:{\displaystyle {\mathcal {H}}_{\Delta }^{k}(M)=\{\alpha \in \Omega ^{k}(M)\mid \Delta \alpha =0\}.}
6163:
5541:
4756:
2092:. (Admittedly, there are other ways to prove this.) Indeed, the operators Δ are elliptic, and the
1145:
5733:
3118:
3109:, complex projective manifolds are automatically algebraic: they are defined by the vanishing of
2354:
1803:
This is a second-order linear differential operator, generalizing the
Laplacian for functions on
5499:
5406:
set of components, each of complex dimension 19. The subspace of K3 surfaces with Picard number
5306:
5120:(even integrally, that is, without the need for a positive integral multiple in the statement).
4714:
4621:
2077:
norm that represents a given cohomology class. The Hodge theorem was proved using the theory of
1309:
inner product is then defined as the integral of the pointwise inner product of a given pair of
4420:
4370:
3110:
2531:
135:, and the interaction between differential forms and topology was poorly understood. In 1928,
2052:
245:{\displaystyle H_{k}(M;\mathbf {R} )\times H_{\text{dR}}^{k}(M;\mathbf {R} )\to \mathbf {R} .}
6182:
5748:
5604:
5596:
5577:
5204:
5087:(the combination of integral cohomology with the Hodge decomposition of complex cohomology).
5013:
3798:
1320:
641:
501:
5928:
4571:. Moreover, the resulting class has a special property: its image in the complex cohomology
1186:
960:{\displaystyle H^{k}(M,\mathbf {R} )\cong {\frac {\ker d_{k}}{\operatorname {im} d_{k-1}}}.}
6155:
6111:
6104:
6074:
6033:
6000:
5966:
3320:
3002:
The cohomology of the complex is canonically isomorphic to the space of harmonic sections,
2121:
1919:
342:{\displaystyle H_{\text{sing}}^{k}(M;\mathbf {R} )\cong H_{\text{dR}}^{k}(M;\mathbf {R} ).}
144:
5429:
4378:
8:
5097:
4190:
3737:
2942:
1504:
Naturally the above inner product induces a norm, when that norm is finite on some fixed
861:
820:
560:
113:
86:
3816:
3624:{\displaystyle \smile \colon H^{p,q}(X)\times H^{p',q'}(X)\rightarrow H^{p+p',q+q'}(X).}
5827:
5697:
5677:
5649:
5629:
5572:
of algebraic cycles on a given variety. The Hodge conjecture is about the image of the
5409:
5403:
5381:
5213:
5192:
5170:
5146:
5126:
5070:
5046:
4802:
4778:
4507:
4484:
4464:
4444:
2632:{\displaystyle 0\to \Gamma (E_{0})\to \Gamma (E_{1})\to \cdots \to \Gamma (E_{N})\to 0}
2089:
602:
148:
128:
94:
82:
74:
5564:
complex projective varieties with a given topological type. The best case is when the
5374:. Their intersection can have rank anywhere between 1 and 20; this rank is called the
3134:
90:
6141:
6090:
6060:
6019:
6007:
5986:
5952:
5940:
5617:
5196:
5164:
4424:
4057:
3769:
3475:
2078:
629:
394:
356:
152:
59:
47:
1063:{\displaystyle \Omega ^{k}(M)=\Gamma \left(\bigwedge \nolimits ^{k}T^{*}(M)\right).}
6131:
6123:
6052:
5819:
5723:
4436:
3095:
2373:
2109:
2093:
2082:
1666:
1213:
568:
360:
119:
have given alternative proofs of, or analogous results to, classical Hodge theory.
32:
5573:
2069:
is an isomorphism of vector spaces. In other words, each real cohomology class on
6151:
6100:
6070:
6029:
5996:
5962:
5743:
5714:, as would arise from a family of varieties which need not be smooth or compact.
5565:
5200:
4186:
3149:
3077:
2880:{\displaystyle {\mathcal {H}}=\{e\in {\mathcal {E}}^{\bullet }\mid \Delta e=0\}.}
2140:
2038:{\displaystyle \varphi :{\mathcal {H}}_{\Delta }^{k}(M)\to H^{k}(M,\mathbf {R} )}
1950:
1939:
828:
610:
556:
256:
98:
78:
36:
5602:
A different generalization of Hodge theory to singular varieties is provided by
5974:
5844:
5667:
5609:
5592:
3064:, in the sense that each cohomology class has a unique harmonic representative.
2436:
2365:
2113:
1135:
580:
105:
6136:
6127:
136:
6176:
6056:
6040:
5728:
5375:
4339:
4079:
3336:
3087:
2101:
1074:
382:
109:
2813:. As in the de Rham case, this yields the vector space of harmonic sections
2081:
partial differential equations, with Hodge's initial arguments completed by
5787:
5671:
5399:
4335:
4316:
3485:
Taking wedge products of these harmonic representatives corresponds to the
1564:{\displaystyle \langle \omega ,\omega \rangle =\|\omega \|^{2}<\infty ,}
625:
564:
2150:. This says that there is a unique decomposition of any differential form
5753:
5674:
is a generalization. Roughly speaking, a mixed Hodge module on a variety
5626:
describes how the Hodge structure of a smooth complex projective variety
4502:
4382:
3788:
622:
390:
20:
131:
was still nascent in the 1920s. It had not yet developed the notion of
5831:
5810:
Lefschetz, Solomon (1927). "Correspondences
Between Algebraic Curves".
5569:
4236:
3344:
2535:
2369:
2224:
132:
3335:
components of a harmonic form are again harmonic. Therefore, for any
1753:
63:
5823:
3351:
with complex coefficients as a direct sum of complex vector spaces:
3129:
which has a strong compatibility with the complex structure, making
2154:
on a closed
Riemannian manifold as a sum of three parts in the form
780:
738:
718:
682:
5188:
3444:{\displaystyle H^{r}(X,\mathbf {C} )=\bigoplus _{p+q=r}H^{p,q}(X).}
1574:
then the integrand is a real valued, square integrable function on
5756:, a key consequence of Hodge theory for compact Kähler manifolds.
5560:, but for "special" K3 surfaces the intersection can be bigger.)
1655:{\displaystyle \|\omega (p)\|_{p}:M\to \mathbf {R} \in L^{2}(M).}
3159:(with complex coefficients) can be written uniquely as a sum of
3071:
2538:
sections of these vector bundles, and that the induced sequence
104:
While Hodge theory is intrinsically dependent upon the real and
255:
As originally stated, de Rham's theorem asserts that this is a
6118:, Graduate Texts in Mathematics, vol. 65 (3rd ed.),
5452:. (Thus, for most projective K3 surfaces, the intersection of
583:, William Vallance Douglas Hodge, 17 June 1903 – 7 July 1975,
5187:. In this sense, Hodge theory is related to a basic issue in
4461:
be a smooth complex projective variety. A complex subvariety
2930:{\displaystyle H:{\mathcal {E}}^{\bullet }\to {\mathcal {H}}}
452:
is a non-zero holomorphic form on an algebraic surface, then
4430:
3650:
as a complex manifold (not on the choice of Kähler metric):
491:{\displaystyle {\sqrt {-1}}\,\omega \wedge {\bar {\omega }}}
440:{\displaystyle {\sqrt {-1}}\,\omega \wedge {\bar {\omega }}}
5143:
describes the integrals of algebraic differential forms on
3773:
1742:{\displaystyle \delta :\Omega ^{k+1}(M)\to \Omega ^{k}(M).}
868:
with real coefficients is computed by the de Rham complex:
5851:, Biogr. Mem. Fellows R. Soc., 1976, vol. 22, pp. 169–192.
5849:
William
Vallance Douglas Hodge, 17 June 1903 – 7 July 1975
5191:: there is in general no "formula" for the integral of an
4775:-linear combination of classes of complex subvarieties of
5666:
varies. In geometric terms, this amounts to studying the
6164:
Python code for computing Hodge numbers of hypersurfaces
2520:{\displaystyle L_{i}:\Gamma (E_{i})\to \Gamma (E_{i+1})}
73:
The theory was developed by Hodge in the 1930s to study
5670:
associated to a family of varieties. Saito's theory of
5003:{\displaystyle H^{2p}(X,\mathbb {Z} )/{\text{torsion}}}
4711:
whose image in complex cohomology lies in the subspace
2223:
metric on differential forms, this gives an orthogonal
2197:{\displaystyle \omega =d\alpha +\delta \beta +\gamma ,}
1578:, evaluated at a given point via its point-wise norms,
6083:
3787:
On the other hand, the integral can be written as the
3642:) of the Hodge decomposition can be identified with a
1266:{\displaystyle \bigwedge \nolimits ^{k}(T_{p}^{*}(M))}
1127:{\displaystyle \bigwedge \nolimits ^{k}(T_{p}^{*}(M))}
6045:
Hodge Theory and
Complex Algebraic Geometry (2 vols.)
5700:
5680:
5652:
5632:
5544:
5502:
5458:
5432:
5412:
5384:
5351:
5309:
5280:
5236:
5216:
5173:
5149:
5129:
5100:
5073:
5049:
5016:
4959:
4832:
4805:
4781:
4759:
4717:
4670:
4624:
4618:
lies in the middle piece of the Hodge decomposition,
4577:
4530:
4510:
4487:
4467:
4447:
4388:
3961:
3886:
3836:
3801:
3726:{\displaystyle H^{p,q}(X)\cong H^{q}(X,\Omega ^{p}),}
3659:
3498:
3360:
3197:
3008:
2896:
2822:
2651:
2547:
2456:
2439:, equipped with metrics, on a closed smooth manifold
2382:
2360:
2236:
2163:
2055:
1970:
1824:
1765:
1682:
1587:
1517:
1398:
1347:
1323:
1279:
1222:
1189:
1148:
1083:
992:
877:
652:
524:
504:
458:
407:
268:
168:
5210:
Example: For a smooth complex projective K3 surface
585:
Biographical Memoirs of Fellows of the Royal Society
377:
are holomorphic differentials on an algebraic curve
141:
Sur les nombres de Betti des espaces de groupes clos
93:. Hodge's primary motivation, the study of complex
5063:(as described by cohomology) are determined by the
4753:should have a positive integral multiple that is a
3057:{\displaystyle H(E_{j})\cong {\mathcal {H}}(E_{j})}
2642:is an elliptic complex. Introduce the direct sums:
2353:The Hodge decomposition is a generalization of the
5772:
5706:
5686:
5658:
5638:
5552:
5530:
5488:
5444:
5418:
5390:
5366:
5337:
5295:
5266:
5222:
5179:
5155:
5135:
5112:
5079:
5055:
5035:
5002:
4942:
4811:
4787:
4767:
4745:
4703:
4652:
4610:
4563:
4516:
4493:
4473:
4453:
4406:
4023:) means the dimension of the complex vector space
4001:
3947:
3872:
3807:
3725:
3623:
3443:
3291:
3056:
2929:
2879:
2780:
2631:
2519:
2427:
2342:
2196:
2061:
2037:
1907:
1792:
1741:
1654:
1563:
1493:
1381:
1329:
1301:
1265:
1205:
1175:
1126:
1062:
959:
801:
591:
539:
510:
490:
439:
389:has only one complex dimension; consequently, the
341:
244:
5979:The Theory and Applications of Harmonic Integrals
3948:{\displaystyle H^{2n}(X,\mathbb {C} )=H^{n,n}(X)}
3343:, the Hodge theorem gives a decomposition of the
2112:. It follows, for example, that the image of the
6174:
2139:is finite (because the group of isometries of a
359:used topological methods to reprove theorems of
155:chains induces, for any compact smooth manifold
5939:
1382:{\displaystyle \omega ,\tau \in \Omega ^{k}(M)}
66:operator of the metric. Such forms are called
5951:. Wiley Classics Library. Wiley Interscience.
5773:Chatterji, Srishti; Ojanguren, Manuel (2010),
4423:also give strong restrictions on the possible
85:. It has major applications in two settings:
3072:Hodge theory for complex projective varieties
401:is a non-zero holomorphic differential, then
4091:(shown in the case of complex dimension 2):
2871:
2833:
1964:. As a result, there is a canonical mapping
1899:
1856:
1604:
1588:
1543:
1536:
1530:
1518:
1476:
1445:
1429:
1417:
970:
5094:says that the Hodge conjecture is true for
4524:defines an element of the cohomology group
4407:{\displaystyle \partial {\bar {\partial }}}
6116:Differential Analysis on Complex Manifolds
6006:
5694:is a sheaf of mixed Hodge structures over
5580:which are built from the Hodge structure.
4795:. (Such a linear combination is called an
3090:complex projective manifold, meaning that
1793:{\displaystyle \Delta =d\delta +\delta d.}
151:, integration of differential forms along
112:. In arithmetic situations, the tools of
6135:
6110:
5809:
5546:
5479:
5354:
5283:
5257:
4983:
4953:may be much smaller than the whole group
4933:
4859:
4761:
4694:
4601:
4554:
4431:Algebraic cycles and the Hodge conjecture
3910:
3459:as a complex manifold, whereas the group
3201:
2428:{\displaystyle E_{0},E_{1},\ldots ,E_{N}}
640:. The de Rham complex is the sequence of
469:
418:
3795:and the cohomology class represented by
46:. The key observation is that, given a
6175:
6080:
6039:
5914:Griffiths & Harris (1994), p. 594.
5766:
4704:{\displaystyle H^{2p}(X,\mathbb {Z} )}
4664:predicts a converse: every element of
4611:{\displaystyle H^{2p}(X,\mathbb {C} )}
4564:{\displaystyle H^{2p}(X,\mathbb {Z} )}
2937:be the orthogonal projection, and let
2146:A variant of the Hodge theorem is the
1673:with respect to these inner products:
5973:
5489:{\displaystyle H^{2}(X,\mathbb {Z} )}
5267:{\displaystyle H^{2}(X,\mathbb {Z} )}
4185:For example, every smooth projective
596:
5896:Huybrechts (2005), Corollary 2.6.21.
5887:Huybrechts (2005), Corollary 3.2.12.
5402:of all projective K3 surfaces has a
4419:Hodge theory and extensions such as
108:, it can be applied to questions in
5926:
3776:theorem implies that a holomorphic
2721:
2678:
1224:
1085:
1024:
860:). De Rham's theorem says that the
547:must be non-zero. It follows that
498:is positive, so the cup product of
13:
5583:
4395:
4389:
3708:
3033:
2922:
2906:
2859:
2845:
2825:
2763:
2746:
2687:
2659:
2604:
2576:
2554:
2492:
2470:
2361:Hodge theory of elliptic complexes
2318:
2312:
2238:
1986:
1980:
1887:
1866:
1834:
1828:
1766:
1718:
1690:
1555:
1361:
1281:
1015:
994:
758:
696:
660:
14:
6194:
6012:Complex Geometry: An Introduction
5367:{\displaystyle \mathbb {C} ^{20}}
5296:{\displaystyle \mathbb {Z} ^{22}}
5199:of algebraic functions, known as
5123:The Hodge structure of a variety
5949:Principles of Algebraic Geometry
4375:Hodge-Riemann bilinear relations
3955:, we conclude that must lie in
3381:
2028:
1918:The Laplacian appeared first in
1623:
1317:with respect to the volume form
898:
601:The Hodge theory references the
540:{\displaystyle {\bar {\omega }}}
329:
294:
235:
224:
189:
5908:
3873:{\displaystyle (p,q)\neq (k,k)}
3474:depends only on the underlying
3125:induces a Riemannian metric on
2799:. Define the elliptic operator
1138:) the inner product induced by
592:Hodge theory for real manifolds
54:, every cohomology class has a
31:, is a method for studying the
5899:
5890:
5881:
5872:
5863:
5854:
5838:
5803:
5525:
5519:
5483:
5469:
5332:
5326:
5261:
5247:
4987:
4973:
4937:
4923:
4904:
4898:
4876:
4863:
4849:
4833:
4740:
4734:
4698:
4684:
4647:
4641:
4605:
4591:
4558:
4544:
4398:
4002:{\displaystyle H^{n-k,n-k}(X)}
3996:
3990:
3942:
3936:
3914:
3900:
3867:
3855:
3849:
3837:
3756:) is the space of holomorphic
3717:
3698:
3682:
3676:
3615:
3609:
3568:
3565:
3559:
3527:
3521:
3435:
3429:
3385:
3371:
3051:
3038:
3025:
3012:
2917:
2757:
2703:
2690:
2623:
2620:
2607:
2601:
2595:
2592:
2579:
2573:
2570:
2557:
2551:
2514:
2495:
2489:
2486:
2473:
2334:
2328:
2253:
2247:
2104:on the integral cohomology of
2032:
2018:
2005:
2002:
1996:
1881:
1875:
1850:
1844:
1733:
1727:
1714:
1711:
1705:
1646:
1640:
1619:
1600:
1594:
1472:
1466:
1457:
1451:
1414:
1411:
1399:
1376:
1370:
1302:{\displaystyle \Omega ^{k}(M)}
1296:
1290:
1260:
1257:
1251:
1233:
1170:
1164:
1121:
1118:
1112:
1094:
1049:
1043:
1009:
1003:
902:
888:
773:
767:
711:
705:
675:
669:
656:
587:, vol. 22, 1976, pp. 169–192.
531:
482:
431:
333:
319:
298:
284:
231:
228:
214:
193:
179:
77:, and it built on the work of
44:partial differential equations
1:
5920:
5878:Wells (2008), Theorem IV.5.2.
5739:Local invariant cycle theorem
4427:of compact Kähler manifolds.
3880:, then we get zero. Because
3646:group, which depends only on
613:. For a non-negative integer
5930:Computing Some Hodge Numbers
5860:Warner (1983), Theorem 6.11.
5776:A glimpse of the de Rham era
5623:variation of Hodge structure
5553:{\displaystyle \mathbb {Z} }
4768:{\displaystyle \mathbb {Z} }
4367:Lefschetz hyperplane theorem
3323:. On a Kähler manifold, the
3284:
3255:
2949:then asserts the following:
1176:{\displaystyle T_{p}^{*}(M)}
385:is necessarily zero because
355:Separately, a 1927 paper of
16:Mathematical manifold theory
7:
5869:Warner (1983), Theorem 6.8.
5717:
5010:, even if the Hodge number
4235:For another example, every
1807:. By definition, a form on
975:Choose a Riemannian metric
10:
6201:
6049:Cambridge University Press
5983:Cambridge University Press
5531:{\displaystyle H^{1,1}(X)}
5338:{\displaystyle H^{1,1}(X)}
4746:{\displaystyle H^{p,p}(X)}
4653:{\displaystyle H^{p,p}(X)}
4434:
3119:standard Riemannian metric
3075:
1815:if its Laplacian is zero:
1142:from each cotangent fiber
122:
6128:10.1007/978-0-387-73892-5
3791:of the homology class of
3644:coherent sheaf cohomology
2357:for the de Rham complex.
2100:determines a real-valued
2085:and others in the 1940s.
1341:. Explicitly, given some
971:Operators in Hodge theory
363:. In modern language, if
6057:10.1017/CBO9780511615344
5760:
4421:non-abelian Hodge theory
3819:, the homology class of
3100:complex projective space
2443:with a volume form
2062:{\displaystyle \varphi }
1938:on spacetime, viewed as
62:that vanishes under the
56:canonical representative
6081:Warner, Frank (1983) ,
5734:Helmholtz decomposition
5092:Lefschetz (1,1)-theorem
5036:{\displaystyle h^{p,p}}
3808:{\displaystyle \alpha }
3784:is in fact algebraic.)
3140:For a complex manifold
2355:Helmholtz decomposition
1953:Riemannian manifold is
1756:on forms is defined by
1330:{\displaystyle \sigma }
511:{\displaystyle \omega }
5708:
5688:
5660:
5640:
5578:intermediate Jacobians
5554:
5532:
5490:
5446:
5420:
5392:
5368:
5339:
5297:
5268:
5224:
5205:transcendental numbers
5181:
5157:
5137:
5114:
5081:
5057:
5037:
5004:
4944:
4813:
4789:
4769:
4747:
4705:
4654:
4612:
4565:
4518:
4495:
4475:
4455:
4408:
4371:hard Lefschetz theorem
4003:
3949:
3874:
3809:
3727:
3625:
3445:
3293:
3111:homogeneous polynomial
3058:
2931:
2881:
2782:
2633:
2532:differential operators
2521:
2429:
2344:
2198:
2063:
2039:
1909:
1794:
1743:
1656:
1565:
1495:
1383:
1331:
1303:
1267:
1207:
1206:{\displaystyle k^{th}}
1177:
1128:
1064:
961:
803:
642:differential operators
589:
541:
512:
492:
441:
343:
246:
6112:Wells Jr., Raymond O.
5749:Hodge-Arakelov theory
5709:
5689:
5661:
5641:
5605:intersection homology
5597:mixed Hodge structure
5555:
5533:
5491:
5447:
5421:
5393:
5369:
5340:
5298:
5269:
5225:
5182:
5158:
5138:
5115:
5082:
5058:
5038:
5005:
4945:
4814:
4790:
4770:
4748:
4706:
4655:
4613:
4566:
4519:
4496:
4476:
4456:
4409:
4004:
3950:
3875:
3810:
3728:
3626:
3446:
3321:holomorphic functions
3306:a C function and the
3294:
3144:and a natural number
3059:
2932:
2882:
2783:
2634:
2522:
2430:
2345:
2199:
2064:
2040:
1910:
1795:
1744:
1657:
1566:
1496:
1384:
1332:
1304:
1268:
1208:
1178:
1129:
1073:The metric yields an
1065:
962:
804:
573:
542:
513:
493:
442:
344:
247:
159:, a bilinear pairing
5698:
5678:
5650:
5630:
5542:
5500:
5456:
5445:{\displaystyle 20-a}
5430:
5410:
5382:
5349:
5307:
5278:
5234:
5214:
5171:
5147:
5127:
5098:
5071:
5047:
5014:
4957:
4830:
4803:
4779:
4757:
4715:
4668:
4622:
4575:
4528:
4508:
4485:
4465:
4445:
4386:
3959:
3884:
3834:
3799:
3736:where Ω denotes the
3657:
3496:
3358:
3195:
3006:
2894:
2820:
2649:
2545:
2454:
2380:
2234:
2161:
2122:general linear group
2053:
1968:
1945:Every harmonic form
1920:mathematical physics
1822:
1763:
1680:
1585:
1515:
1396:
1345:
1321:
1277:
1220:
1187:
1146:
1081:
990:
875:
650:
522:
502:
456:
405:
397:. Additionally, if
266:
166:
95:projective varieties
87:Riemannian manifolds
5113:{\displaystyle p=1}
4239:has Hodge diamond
3096:complex submanifold
2327:
2148:Hodge decomposition
1995:
1924:Maxwell's equations
1843:
1250:
1163:
1111:
862:singular cohomology
821:exterior derivative
791:
755:
729:
693:
561:Hodge star operator
318:
283:
213:
6137:10338.dmlcz/141778
6008:Huybrechts, Daniel
5941:Griffiths, Phillip
5704:
5684:
5656:
5636:
5589:Mixed Hodge theory
5550:
5528:
5486:
5442:
5416:
5404:countably infinite
5388:
5364:
5335:
5293:
5264:
5220:
5197:definite integrals
5193:algebraic function
5177:
5153:
5133:
5110:
5077:
5053:
5033:
5000:
4940:
4809:
4785:
4765:
4743:
4701:
4650:
4608:
4561:
4514:
4491:
4471:
4451:
4425:fundamental groups
4404:
4196:has Hodge diamond
3999:
3945:
3870:
3805:
3723:
3621:
3441:
3412:
3289:
3054:
2927:
2877:
2795:be the adjoint of
2778:
2776:
2629:
2517:
2425:
2374:elliptic complexes
2340:
2309:
2219:. In terms of the
2194:
2090:finite-dimensional
2059:
2035:
1977:
1905:
1825:
1790:
1739:
1652:
1561:
1491:
1379:
1327:
1299:
1263:
1236:
1203:
1173:
1149:
1134:by extending (see
1124:
1097:
1060:
957:
831:in the sense that
799:
630:differential forms
597:De Rham cohomology
537:
508:
488:
437:
339:
304:
269:
242:
199:
139:published a note,
129:algebraic topology
117:-adic Hodge theory
83:de Rham cohomology
75:algebraic geometry
6147:978-0-387-73891-8
6066:978-0-521-71801-1
5992:978-0-521-35881-1
5812:Ann. of Math. (2)
5782:, working paper,
5707:{\displaystyle X}
5687:{\displaystyle X}
5659:{\displaystyle X}
5639:{\displaystyle X}
5618:Phillip Griffiths
5538:is isomorphic to
5419:{\displaystyle a}
5391:{\displaystyle X}
5345:is isomorphic to
5274:is isomorphic to
5223:{\displaystyle X}
5195:. In particular,
5180:{\displaystyle X}
5156:{\displaystyle X}
5136:{\displaystyle X}
5080:{\displaystyle X}
5056:{\displaystyle X}
4998:
4874:
4812:{\displaystyle X}
4788:{\displaystyle X}
4517:{\displaystyle p}
4494:{\displaystyle X}
4474:{\displaystyle Y}
4454:{\displaystyle X}
4401:
4379:Kähler identities
4313:
4312:
4233:
4232:
4183:
4182:
4058:complex conjugate
3476:topological space
3391:
3287:
3258:
2960:are well-defined.
1922:. In particular,
983:and recall that:
952:
792:
756:
730:
694:
534:
485:
467:
434:
416:
395:Riemann relations
357:Solomon Lefschetz
311:
276:
206:
145:de Rham's theorem
60:differential form
48:Riemannian metric
33:cohomology groups
6190:
6158:
6139:
6107:
6077:
6036:
6003:
5970:
5936:
5935:
5915:
5912:
5906:
5903:
5897:
5894:
5888:
5885:
5879:
5876:
5870:
5867:
5861:
5858:
5852:
5842:
5836:
5835:
5807:
5801:
5800:
5799:
5798:
5792:
5786:, archived from
5781:
5770:
5724:Potential theory
5713:
5711:
5710:
5705:
5693:
5691:
5690:
5685:
5665:
5663:
5662:
5657:
5645:
5643:
5642:
5637:
5559:
5557:
5556:
5551:
5549:
5537:
5535:
5534:
5529:
5518:
5517:
5495:
5493:
5492:
5487:
5482:
5468:
5467:
5451:
5449:
5448:
5443:
5425:
5423:
5422:
5417:
5397:
5395:
5394:
5389:
5373:
5371:
5370:
5365:
5363:
5362:
5357:
5344:
5342:
5341:
5336:
5325:
5324:
5302:
5300:
5299:
5294:
5292:
5291:
5286:
5273:
5271:
5270:
5265:
5260:
5246:
5245:
5229:
5227:
5226:
5221:
5186:
5184:
5183:
5178:
5162:
5160:
5159:
5154:
5142:
5140:
5139:
5134:
5119:
5117:
5116:
5111:
5086:
5084:
5083:
5078:
5062:
5060:
5059:
5054:
5042:
5040:
5039:
5034:
5032:
5031:
5009:
5007:
5006:
5001:
4999:
4996:
4994:
4986:
4972:
4971:
4949:
4947:
4946:
4941:
4936:
4922:
4921:
4897:
4896:
4875:
4872:
4870:
4862:
4848:
4847:
4818:
4816:
4815:
4810:
4794:
4792:
4791:
4786:
4774:
4772:
4771:
4766:
4764:
4752:
4750:
4749:
4744:
4733:
4732:
4710:
4708:
4707:
4702:
4697:
4683:
4682:
4662:Hodge conjecture
4659:
4657:
4656:
4651:
4640:
4639:
4617:
4615:
4614:
4609:
4604:
4590:
4589:
4570:
4568:
4567:
4562:
4557:
4543:
4542:
4523:
4521:
4520:
4515:
4500:
4498:
4497:
4492:
4480:
4478:
4477:
4472:
4460:
4458:
4457:
4452:
4437:Hodge conjecture
4413:
4411:
4410:
4405:
4403:
4402:
4394:
4361:
4351:
4242:
4241:
4199:
4198:
4094:
4093:
4077:
4047:
4008:
4006:
4005:
4000:
3989:
3988:
3954:
3952:
3951:
3946:
3935:
3934:
3913:
3899:
3898:
3879:
3877:
3876:
3871:
3817:Poincaré duality
3814:
3812:
3811:
3806:
3780:-form on all of
3732:
3730:
3729:
3724:
3716:
3715:
3697:
3696:
3675:
3674:
3630:
3628:
3627:
3622:
3608:
3607:
3606:
3589:
3558:
3557:
3556:
3545:
3520:
3519:
3473:
3450:
3448:
3447:
3442:
3428:
3427:
3411:
3384:
3370:
3369:
3339:Kähler manifold
3334:
3298:
3296:
3295:
3290:
3288:
3283:
3282:
3273:
3259:
3254:
3253:
3244:
3236:
3235:
3214:
3213:
3187:
3172:
3063:
3061:
3060:
3055:
3050:
3049:
3037:
3036:
3024:
3023:
2943:Green's operator
2936:
2934:
2933:
2928:
2926:
2925:
2916:
2915:
2910:
2909:
2886:
2884:
2883:
2878:
2855:
2854:
2849:
2848:
2829:
2828:
2812:
2787:
2785:
2784:
2779:
2777:
2773:
2772:
2767:
2766:
2756:
2755:
2750:
2749:
2739:
2738:
2729:
2728:
2702:
2701:
2686:
2685:
2669:
2668:
2663:
2662:
2638:
2636:
2635:
2630:
2619:
2618:
2591:
2590:
2569:
2568:
2526:
2524:
2523:
2518:
2513:
2512:
2485:
2484:
2466:
2465:
2434:
2432:
2431:
2426:
2424:
2423:
2405:
2404:
2392:
2391:
2349:
2347:
2346:
2341:
2326:
2321:
2316:
2315:
2305:
2304:
2280:
2279:
2246:
2245:
2218:
2203:
2201:
2200:
2195:
2138:
2068:
2066:
2065:
2060:
2044:
2042:
2041:
2036:
2031:
2017:
2016:
1994:
1989:
1984:
1983:
1963:
1942:of dimension 4.
1937:
1914:
1912:
1911:
1906:
1874:
1873:
1842:
1837:
1832:
1831:
1799:
1797:
1796:
1791:
1748:
1746:
1745:
1740:
1726:
1725:
1704:
1703:
1667:adjoint operator
1661:
1659:
1658:
1653:
1639:
1638:
1626:
1612:
1611:
1570:
1568:
1567:
1562:
1551:
1550:
1500:
1498:
1497:
1492:
1484:
1483:
1444:
1443:
1388:
1386:
1385:
1380:
1369:
1368:
1337:associated with
1336:
1334:
1333:
1328:
1308:
1306:
1305:
1300:
1289:
1288:
1272:
1270:
1269:
1264:
1249:
1244:
1232:
1231:
1214:exterior product
1212:
1210:
1209:
1204:
1202:
1201:
1182:
1180:
1179:
1174:
1162:
1157:
1133:
1131:
1130:
1125:
1110:
1105:
1093:
1092:
1069:
1067:
1066:
1061:
1056:
1052:
1042:
1041:
1032:
1031:
1002:
1001:
966:
964:
963:
958:
953:
951:
950:
949:
927:
926:
925:
909:
901:
887:
886:
859:
852:
808:
806:
805:
800:
790:
789:
776:
766:
765:
754:
753:
734:
728:
727:
714:
704:
703:
692:
691:
678:
668:
667:
569:Kunihiko Kodaira
546:
544:
543:
538:
536:
535:
527:
517:
515:
514:
509:
497:
495:
494:
489:
487:
486:
478:
468:
460:
446:
444:
443:
438:
436:
435:
427:
417:
409:
348:
346:
345:
340:
332:
317:
312:
309:
297:
282:
277:
274:
251:
249:
248:
243:
238:
227:
212:
207:
204:
192:
178:
177:
99:algebraic cycles
91:Kähler manifolds
6200:
6199:
6193:
6192:
6191:
6189:
6188:
6187:
6173:
6172:
6170:
6148:
6097:
6067:
6026:
5993:
5975:Hodge, W. V. D.
5959:
5933:
5927:Arapura, Donu,
5923:
5918:
5913:
5909:
5904:
5900:
5895:
5891:
5886:
5882:
5877:
5873:
5868:
5864:
5859:
5855:
5843:
5839:
5824:10.2307/1968379
5808:
5804:
5796:
5794:
5790:
5779:
5771:
5767:
5763:
5744:Arakelov theory
5720:
5699:
5696:
5695:
5679:
5676:
5675:
5651:
5648:
5647:
5631:
5628:
5627:
5620:'s notion of a
5591:, developed by
5586:
5584:Generalizations
5566:Torelli theorem
5545:
5543:
5540:
5539:
5507:
5503:
5501:
5498:
5497:
5478:
5463:
5459:
5457:
5454:
5453:
5431:
5428:
5427:
5411:
5408:
5407:
5383:
5380:
5379:
5358:
5353:
5352:
5350:
5347:
5346:
5314:
5310:
5308:
5305:
5304:
5287:
5282:
5281:
5279:
5276:
5275:
5256:
5241:
5237:
5235:
5232:
5231:
5215:
5212:
5211:
5172:
5169:
5168:
5148:
5145:
5144:
5128:
5125:
5124:
5099:
5096:
5095:
5072:
5069:
5068:
5065:Hodge structure
5048:
5045:
5044:
5021:
5017:
5015:
5012:
5011:
4995:
4990:
4982:
4964:
4960:
4958:
4955:
4954:
4932:
4914:
4910:
4886:
4882:
4871:
4866:
4858:
4840:
4836:
4831:
4828:
4827:
4804:
4801:
4800:
4797:algebraic cycle
4780:
4777:
4776:
4760:
4758:
4755:
4754:
4722:
4718:
4716:
4713:
4712:
4693:
4675:
4671:
4669:
4666:
4665:
4629:
4625:
4623:
4620:
4619:
4600:
4582:
4578:
4576:
4573:
4572:
4553:
4535:
4531:
4529:
4526:
4525:
4509:
4506:
4505:
4486:
4483:
4482:
4466:
4463:
4462:
4446:
4443:
4442:
4439:
4433:
4393:
4392:
4387:
4384:
4383:
4359:
4353:
4343:
4333:
4069:
4039:
3966:
3962:
3960:
3957:
3956:
3924:
3920:
3909:
3891:
3887:
3885:
3882:
3881:
3835:
3832:
3831:
3800:
3797:
3796:
3768:is projective,
3748:. For example,
3740:of holomorphic
3711:
3707:
3692:
3688:
3664:
3660:
3658:
3655:
3654:
3599:
3582:
3575:
3571:
3549:
3538:
3537:
3533:
3509:
3505:
3497:
3494:
3493:
3460:
3417:
3413:
3395:
3380:
3365:
3361:
3359:
3356:
3355:
3324:
3319:
3312:
3278:
3274:
3272:
3249:
3245:
3243:
3231:
3227:
3209:
3205:
3196:
3193:
3192:
3175:
3162:
3135:Kähler manifold
3080:
3078:Hodge structure
3074:
3045:
3041:
3032:
3031:
3019:
3015:
3007:
3004:
3003:
2921:
2920:
2911:
2905:
2904:
2903:
2895:
2892:
2891:
2850:
2844:
2843:
2842:
2824:
2823:
2821:
2818:
2817:
2800:
2775:
2774:
2768:
2762:
2761:
2760:
2751:
2745:
2744:
2743:
2734:
2730:
2724:
2720:
2713:
2707:
2706:
2697:
2693:
2681:
2677:
2670:
2664:
2658:
2657:
2656:
2652:
2650:
2647:
2646:
2614:
2610:
2586:
2582:
2564:
2560:
2546:
2543:
2542:
2502:
2498:
2480:
2476:
2461:
2457:
2455:
2452:
2451:
2447:. Suppose that
2419:
2415:
2400:
2396:
2387:
2383:
2381:
2378:
2377:
2363:
2322:
2317:
2311:
2310:
2294:
2290:
2269:
2265:
2241:
2237:
2235:
2232:
2231:
2227:decomposition:
2212:
2162:
2159:
2158:
2124:
2054:
2051:
2050:
2027:
2012:
2008:
1990:
1985:
1979:
1978:
1969:
1966:
1965:
1958:
1957:, meaning that
1940:Minkowski space
1931:
1869:
1865:
1838:
1833:
1827:
1826:
1823:
1820:
1819:
1764:
1761:
1760:
1721:
1717:
1693:
1689:
1681:
1678:
1677:
1634:
1630:
1622:
1607:
1603:
1586:
1583:
1582:
1546:
1542:
1516:
1513:
1512:
1479:
1475:
1439:
1435:
1397:
1394:
1393:
1364:
1360:
1346:
1343:
1342:
1322:
1319:
1318:
1284:
1280:
1278:
1275:
1274:
1245:
1240:
1227:
1223:
1221:
1218:
1217:
1194:
1190:
1188:
1185:
1184:
1158:
1153:
1147:
1144:
1143:
1106:
1101:
1088:
1084:
1082:
1079:
1078:
1037:
1033:
1027:
1023:
1022:
1018:
997:
993:
991:
988:
987:
973:
939:
935:
928:
921:
917:
910:
908:
897:
882:
878:
876:
873:
872:
854:
850:
841:
832:
829:cochain complex
817:
785:
781:
761:
757:
743:
739:
723:
719:
699:
695:
687:
683:
663:
659:
651:
648:
647:
611:smooth manifold
603:de Rham complex
599:
594:
557:Riemann surface
526:
525:
523:
520:
519:
503:
500:
499:
477:
476:
459:
457:
454:
453:
426:
425:
408:
406:
403:
402:
376:
369:
328:
313:
308:
293:
278:
273:
267:
264:
263:
257:perfect pairing
234:
223:
208:
203:
188:
173:
169:
167:
164:
163:
149:Stokes' theorem
125:
106:complex numbers
79:Georges de Rham
37:smooth manifold
17:
12:
11:
5:
6198:
6197:
6186:
6185:
6168:
6167:
6160:
6159:
6146:
6108:
6095:
6078:
6065:
6041:Voisin, Claire
6037:
6024:
6004:
5991:
5971:
5957:
5945:Harris, Joseph
5937:
5922:
5919:
5917:
5916:
5907:
5898:
5889:
5880:
5871:
5862:
5853:
5845:Michael Atiyah
5837:
5818:(1): 342–354.
5802:
5764:
5762:
5759:
5758:
5757:
5751:
5746:
5741:
5736:
5731:
5726:
5719:
5716:
5703:
5683:
5668:period mapping
5655:
5635:
5610:Morihiko Saito
5593:Pierre Deligne
5585:
5582:
5548:
5527:
5524:
5521:
5516:
5513:
5510:
5506:
5485:
5481:
5477:
5474:
5471:
5466:
5462:
5441:
5438:
5435:
5426:has dimension
5415:
5387:
5361:
5356:
5334:
5331:
5328:
5323:
5320:
5317:
5313:
5290:
5285:
5263:
5259:
5255:
5252:
5249:
5244:
5240:
5219:
5176:
5152:
5132:
5109:
5106:
5103:
5076:
5052:
5030:
5027:
5024:
5020:
4993:
4989:
4985:
4981:
4978:
4975:
4970:
4967:
4963:
4951:
4950:
4939:
4935:
4931:
4928:
4925:
4920:
4917:
4913:
4909:
4906:
4903:
4900:
4895:
4892:
4889:
4885:
4881:
4878:
4869:
4865:
4861:
4857:
4854:
4851:
4846:
4843:
4839:
4835:
4808:
4784:
4763:
4742:
4739:
4736:
4731:
4728:
4725:
4721:
4700:
4696:
4692:
4689:
4686:
4681:
4678:
4674:
4649:
4646:
4643:
4638:
4635:
4632:
4628:
4607:
4603:
4599:
4596:
4593:
4588:
4585:
4581:
4560:
4556:
4552:
4549:
4546:
4541:
4538:
4534:
4513:
4490:
4470:
4450:
4435:Main article:
4432:
4429:
4400:
4397:
4391:
4357:
4352:and hence has
4327:
4311:
4310:
4308:
4306:
4303:
4301:
4298:
4297:
4295:
4292:
4290:
4287:
4284:
4283:
4280:
4278:
4275:
4273:
4269:
4268:
4266:
4263:
4261:
4258:
4255:
4254:
4252:
4250:
4247:
4245:
4231:
4230:
4228:
4225:
4222:
4221:
4216:
4214:
4208:
4207:
4205:
4202:
4181:
4180:
4178:
4176:
4171:
4169:
4166:
4165:
4163:
4158:
4156:
4151:
4148:
4147:
4142:
4140:
4135:
4133:
4127:
4126:
4124:
4119:
4117:
4112:
4109:
4108:
4106:
4104:
4099:
4097:
4037:Hodge symmetry
3998:
3995:
3992:
3987:
3984:
3981:
3978:
3975:
3972:
3969:
3965:
3944:
3941:
3938:
3933:
3930:
3927:
3923:
3919:
3916:
3912:
3908:
3905:
3902:
3897:
3894:
3890:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3804:
3734:
3733:
3722:
3719:
3714:
3710:
3706:
3703:
3700:
3695:
3691:
3687:
3684:
3681:
3678:
3673:
3670:
3667:
3663:
3632:
3631:
3620:
3617:
3614:
3611:
3605:
3602:
3598:
3595:
3592:
3588:
3585:
3581:
3578:
3574:
3570:
3567:
3564:
3561:
3555:
3552:
3548:
3544:
3541:
3536:
3532:
3529:
3526:
3523:
3518:
3515:
3512:
3508:
3504:
3501:
3452:
3451:
3440:
3437:
3434:
3431:
3426:
3423:
3420:
3416:
3410:
3407:
3404:
3401:
3398:
3394:
3390:
3387:
3383:
3379:
3376:
3373:
3368:
3364:
3317:
3310:
3300:
3299:
3286:
3281:
3277:
3271:
3268:
3265:
3262:
3257:
3252:
3248:
3242:
3239:
3234:
3230:
3226:
3223:
3220:
3217:
3212:
3208:
3204:
3200:
3107:Chow's theorem
3076:Main article:
3073:
3070:
3066:
3065:
3053:
3048:
3044:
3040:
3035:
3030:
3027:
3022:
3018:
3014:
3011:
3000:
2980:
2961:
2924:
2919:
2914:
2908:
2902:
2899:
2888:
2887:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2853:
2847:
2841:
2838:
2835:
2832:
2827:
2789:
2788:
2771:
2765:
2759:
2754:
2748:
2742:
2737:
2733:
2727:
2723:
2719:
2716:
2714:
2712:
2709:
2708:
2705:
2700:
2696:
2692:
2689:
2684:
2680:
2676:
2673:
2671:
2667:
2661:
2655:
2654:
2640:
2639:
2628:
2625:
2622:
2617:
2613:
2609:
2606:
2603:
2600:
2597:
2594:
2589:
2585:
2581:
2578:
2575:
2572:
2567:
2563:
2559:
2556:
2553:
2550:
2528:
2527:
2516:
2511:
2508:
2505:
2501:
2497:
2494:
2491:
2488:
2483:
2479:
2475:
2472:
2469:
2464:
2460:
2437:vector bundles
2422:
2418:
2414:
2411:
2408:
2403:
2399:
2395:
2390:
2386:
2362:
2359:
2351:
2350:
2339:
2336:
2333:
2330:
2325:
2320:
2314:
2308:
2303:
2300:
2297:
2293:
2289:
2286:
2283:
2278:
2275:
2272:
2268:
2264:
2261:
2258:
2255:
2252:
2249:
2244:
2240:
2205:
2204:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2114:isometry group
2058:
2034:
2030:
2026:
2023:
2020:
2015:
2011:
2007:
2004:
2001:
1998:
1993:
1988:
1982:
1976:
1973:
1916:
1915:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1872:
1868:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1841:
1836:
1830:
1801:
1800:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1750:
1749:
1738:
1735:
1732:
1729:
1724:
1720:
1716:
1713:
1710:
1707:
1702:
1699:
1696:
1692:
1688:
1685:
1663:
1662:
1651:
1648:
1645:
1642:
1637:
1633:
1629:
1625:
1621:
1618:
1615:
1610:
1606:
1602:
1599:
1596:
1593:
1590:
1572:
1571:
1560:
1557:
1554:
1549:
1545:
1541:
1538:
1535:
1532:
1529:
1526:
1523:
1520:
1502:
1501:
1490:
1487:
1482:
1478:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1442:
1438:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1378:
1375:
1372:
1367:
1363:
1359:
1356:
1353:
1350:
1326:
1298:
1295:
1292:
1287:
1283:
1262:
1259:
1256:
1253:
1248:
1243:
1239:
1235:
1230:
1226:
1200:
1197:
1193:
1172:
1169:
1166:
1161:
1156:
1152:
1136:Gramian matrix
1123:
1120:
1117:
1114:
1109:
1104:
1100:
1096:
1091:
1087:
1077:on each fiber
1071:
1070:
1059:
1055:
1051:
1048:
1045:
1040:
1036:
1030:
1026:
1021:
1017:
1014:
1011:
1008:
1005:
1000:
996:
972:
969:
968:
967:
956:
948:
945:
942:
938:
934:
931:
924:
920:
916:
913:
907:
904:
900:
896:
893:
890:
885:
881:
853:(also written
846:
836:
815:
810:
809:
798:
795:
788:
784:
779:
775:
772:
769:
764:
760:
752:
749:
746:
742:
737:
733:
726:
722:
717:
713:
710:
707:
702:
698:
690:
686:
681:
677:
674:
671:
666:
662:
658:
655:
598:
595:
593:
590:
533:
530:
507:
484:
481:
475:
472:
466:
463:
433:
430:
424:
421:
415:
412:
374:
367:
350:
349:
338:
335:
331:
327:
324:
321:
316:
307:
303:
300:
296:
292:
289:
286:
281:
272:
253:
252:
241:
237:
233:
230:
226:
222:
219:
216:
211:
202:
198:
195:
191:
187:
184:
181:
176:
172:
124:
121:
29:W. V. D. Hodge
27:, named after
15:
9:
6:
4:
3:
2:
6196:
6195:
6184:
6181:
6180:
6178:
6171:
6165:
6162:
6161:
6157:
6153:
6149:
6143:
6138:
6133:
6129:
6125:
6121:
6117:
6113:
6109:
6106:
6102:
6098:
6096:0-387-90894-3
6092:
6088:
6084:
6079:
6076:
6072:
6068:
6062:
6058:
6054:
6050:
6046:
6042:
6038:
6035:
6031:
6027:
6025:3-540-21290-6
6021:
6017:
6013:
6009:
6005:
6002:
5998:
5994:
5988:
5984:
5980:
5976:
5972:
5968:
5964:
5960:
5958:0-471-05059-8
5954:
5950:
5946:
5942:
5938:
5932:
5931:
5925:
5924:
5911:
5902:
5893:
5884:
5875:
5866:
5857:
5850:
5846:
5841:
5833:
5829:
5825:
5821:
5817:
5813:
5806:
5793:on 2023-12-04
5789:
5785:
5778:
5777:
5769:
5765:
5755:
5752:
5750:
5747:
5745:
5742:
5740:
5737:
5735:
5732:
5730:
5729:Serre duality
5727:
5725:
5722:
5721:
5715:
5701:
5681:
5673:
5672:Hodge modules
5669:
5653:
5633:
5625:
5624:
5619:
5614:
5611:
5607:
5606:
5600:
5598:
5594:
5590:
5581:
5579:
5575:
5571:
5567:
5561:
5522:
5514:
5511:
5508:
5504:
5475:
5472:
5464:
5460:
5439:
5436:
5433:
5413:
5405:
5401:
5385:
5377:
5376:Picard number
5359:
5329:
5321:
5318:
5315:
5311:
5288:
5253:
5250:
5242:
5238:
5217:
5208:
5206:
5202:
5198:
5194:
5190:
5174:
5166:
5150:
5130:
5121:
5107:
5104:
5101:
5093:
5088:
5074:
5066:
5050:
5028:
5025:
5022:
5018:
4991:
4979:
4976:
4968:
4965:
4961:
4929:
4926:
4918:
4915:
4911:
4907:
4901:
4893:
4890:
4887:
4883:
4879:
4867:
4855:
4852:
4844:
4841:
4837:
4826:
4825:
4824:
4820:
4806:
4798:
4782:
4737:
4729:
4726:
4723:
4719:
4690:
4687:
4679:
4676:
4672:
4663:
4644:
4636:
4633:
4630:
4626:
4597:
4594:
4586:
4583:
4579:
4550:
4547:
4539:
4536:
4532:
4511:
4504:
4488:
4468:
4448:
4438:
4428:
4426:
4422:
4417:
4415:
4380:
4376:
4372:
4368:
4363:
4356:
4350:
4346:
4341:
4340:diffeomorphic
4337:
4331:
4326:
4322:
4318:
4317:Betti numbers
4309:
4307:
4304:
4302:
4300:
4299:
4296:
4293:
4291:
4288:
4286:
4285:
4281:
4279:
4276:
4274:
4271:
4270:
4267:
4264:
4262:
4259:
4257:
4256:
4253:
4251:
4248:
4246:
4244:
4243:
4240:
4238:
4229:
4226:
4224:
4223:
4220:
4217:
4215:
4213:
4210:
4209:
4206:
4203:
4201:
4200:
4197:
4195:
4192:
4188:
4179:
4177:
4175:
4172:
4170:
4168:
4167:
4164:
4162:
4159:
4157:
4155:
4152:
4150:
4149:
4146:
4143:
4141:
4139:
4136:
4134:
4132:
4129:
4128:
4125:
4123:
4120:
4118:
4116:
4113:
4111:
4110:
4107:
4105:
4103:
4100:
4098:
4096:
4095:
4092:
4090:
4089:
4088:Hodge diamond
4083:
4081:
4080:Serre duality
4076:
4072:
4067:
4063:
4059:
4055:
4051:
4046:
4042:
4038:
4034:
4030:
4026:
4022:
4018:
4015:
4010:
3993:
3985:
3982:
3979:
3976:
3973:
3970:
3967:
3963:
3939:
3931:
3928:
3925:
3921:
3917:
3906:
3903:
3895:
3892:
3888:
3864:
3861:
3858:
3852:
3846:
3843:
3840:
3828:
3826:
3822:
3818:
3802:
3794:
3790:
3785:
3783:
3779:
3775:
3771:
3767:
3763:
3759:
3755:
3751:
3747:
3743:
3739:
3720:
3712:
3704:
3701:
3693:
3689:
3685:
3679:
3671:
3668:
3665:
3661:
3653:
3652:
3651:
3649:
3645:
3641:
3637:
3618:
3612:
3603:
3600:
3596:
3593:
3590:
3586:
3583:
3579:
3576:
3572:
3562:
3553:
3550:
3546:
3542:
3539:
3534:
3530:
3524:
3516:
3513:
3510:
3506:
3502:
3499:
3492:
3491:
3490:
3488:
3483:
3481:
3477:
3471:
3467:
3463:
3458:
3438:
3432:
3424:
3421:
3418:
3414:
3408:
3405:
3402:
3399:
3396:
3392:
3388:
3377:
3374:
3366:
3362:
3354:
3353:
3352:
3350:
3346:
3342:
3338:
3332:
3328:
3322:
3316:
3309:
3305:
3279:
3275:
3269:
3266:
3263:
3260:
3250:
3246:
3240:
3237:
3232:
3228:
3224:
3221:
3218:
3215:
3210:
3206:
3202:
3198:
3191:
3190:
3189:
3186:
3182:
3178:
3173:
3170:
3166:
3158:
3154:
3151:
3147:
3143:
3138:
3136:
3132:
3128:
3124:
3120:
3116:
3113:equations on
3112:
3108:
3104:
3101:
3097:
3093:
3089:
3085:
3079:
3069:
3046:
3042:
3028:
3020:
3016:
3009:
3001:
2999:
2995:
2992:
2988:
2984:
2981:
2978:
2974:
2970:
2966:
2962:
2959:
2955:
2952:
2951:
2950:
2948:
2947:Hodge theorem
2944:
2940:
2912:
2900:
2897:
2874:
2868:
2865:
2862:
2856:
2851:
2839:
2836:
2830:
2816:
2815:
2814:
2811:
2808:
2804:
2798:
2794:
2769:
2752:
2740:
2735:
2731:
2725:
2717:
2715:
2710:
2698:
2694:
2682:
2674:
2672:
2665:
2645:
2644:
2643:
2626:
2615:
2611:
2598:
2587:
2583:
2565:
2561:
2548:
2541:
2540:
2539:
2537:
2533:
2509:
2506:
2503:
2499:
2481:
2477:
2467:
2462:
2458:
2450:
2449:
2448:
2446:
2442:
2438:
2420:
2416:
2412:
2409:
2406:
2401:
2397:
2393:
2388:
2384:
2375:
2371:
2367:
2358:
2356:
2337:
2331:
2323:
2306:
2301:
2298:
2295:
2291:
2287:
2284:
2281:
2276:
2273:
2270:
2266:
2262:
2259:
2256:
2250:
2242:
2230:
2229:
2228:
2226:
2222:
2216:
2211:is harmonic:
2210:
2191:
2188:
2185:
2182:
2179:
2176:
2173:
2170:
2167:
2164:
2157:
2156:
2155:
2153:
2149:
2144:
2142:
2136:
2132:
2128:
2123:
2119:
2115:
2111:
2107:
2103:
2102:inner product
2099:
2095:
2091:
2086:
2084:
2080:
2076:
2072:
2056:
2048:
2047:Hodge theorem
2024:
2021:
2013:
2009:
1999:
1991:
1974:
1971:
1961:
1956:
1952:
1948:
1943:
1941:
1935:
1929:
1925:
1921:
1902:
1896:
1893:
1890:
1884:
1878:
1870:
1862:
1859:
1853:
1847:
1839:
1818:
1817:
1816:
1814:
1810:
1806:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1759:
1758:
1757:
1755:
1736:
1730:
1722:
1708:
1700:
1697:
1694:
1686:
1683:
1676:
1675:
1674:
1672:
1668:
1665:Consider the
1649:
1643:
1635:
1631:
1627:
1616:
1613:
1608:
1597:
1591:
1581:
1580:
1579:
1577:
1558:
1552:
1547:
1539:
1533:
1527:
1524:
1521:
1511:
1510:
1509:
1507:
1488:
1485:
1480:
1469:
1463:
1460:
1454:
1448:
1440:
1436:
1432:
1426:
1423:
1420:
1408:
1405:
1402:
1392:
1391:
1390:
1373:
1365:
1357:
1354:
1351:
1348:
1340:
1324:
1316:
1312:
1293:
1285:
1254:
1246:
1241:
1237:
1228:
1215:
1198:
1195:
1191:
1167:
1159:
1154:
1150:
1141:
1137:
1115:
1107:
1102:
1098:
1089:
1076:
1075:inner product
1057:
1053:
1046:
1038:
1034:
1028:
1019:
1012:
1006:
998:
986:
985:
984:
982:
978:
954:
946:
943:
940:
936:
932:
929:
922:
918:
914:
911:
905:
894:
891:
883:
879:
871:
870:
869:
867:
863:
857:
849:
845:
839:
835:
830:
827:). This is a
826:
822:
818:
796:
793:
786:
782:
777:
770:
762:
750:
747:
744:
740:
735:
731:
724:
720:
715:
708:
700:
688:
684:
679:
672:
664:
653:
646:
645:
644:
643:
639:
635:
631:
627:
624:
620:
616:
612:
608:
604:
588:
586:
582:
577:
572:
570:
566:
562:
558:
552:
550:
528:
505:
479:
473:
470:
464:
461:
451:
428:
422:
419:
413:
410:
400:
396:
392:
388:
384:
383:wedge product
381:, then their
380:
373:
366:
362:
358:
353:
336:
325:
322:
314:
305:
301:
290:
287:
279:
270:
262:
261:
260:
258:
239:
220:
217:
209:
200:
196:
185:
182:
174:
170:
162:
161:
160:
158:
154:
150:
146:
142:
138:
134:
130:
127:The field of
120:
118:
116:
111:
110:number theory
107:
102:
100:
96:
92:
88:
84:
80:
76:
71:
69:
65:
61:
57:
53:
49:
45:
41:
38:
34:
30:
26:
22:
6183:Hodge theory
6169:
6115:
6082:
6044:
6011:
5978:
5948:
5929:
5910:
5901:
5892:
5883:
5874:
5865:
5856:
5848:
5840:
5815:
5811:
5805:
5795:, retrieved
5788:the original
5775:
5768:
5646:varies when
5621:
5615:
5603:
5601:
5588:
5587:
5562:
5400:moduli space
5230:, the group
5209:
5122:
5089:
5064:
4952:
4821:
4796:
4661:
4440:
4418:
4364:
4354:
4348:
4344:
4336:Hopf surface
4329:
4324:
4320:
4314:
4234:
4218:
4211:
4193:
4184:
4173:
4160:
4153:
4144:
4137:
4130:
4121:
4114:
4101:
4087:
4084:
4074:
4070:
4065:
4061:
4053:
4049:
4044:
4040:
4036:
4032:
4028:
4024:
4020:
4016:
4014:Hodge number
4013:
4011:
3829:
3824:
3820:
3792:
3786:
3781:
3777:
3765:
3761:
3757:
3753:
3749:
3745:
3741:
3735:
3647:
3639:
3635:
3633:
3484:
3479:
3469:
3465:
3461:
3456:
3453:
3348:
3340:
3330:
3326:
3314:
3307:
3303:
3301:
3184:
3180:
3176:
3168:
3164:
3156:
3152:
3145:
3141:
3139:
3130:
3126:
3122:
3114:
3102:
3094:is a closed
3091:
3083:
3081:
3067:
2997:
2993:
2990:
2986:
2982:
2976:
2972:
2968:
2964:
2957:
2953:
2946:
2938:
2889:
2809:
2806:
2802:
2796:
2792:
2790:
2641:
2529:
2444:
2440:
2364:
2352:
2220:
2214:
2208:
2206:
2151:
2147:
2145:
2143:is finite).
2134:
2130:
2126:
2117:
2105:
2097:
2087:
2074:
2070:
2049:states that
2046:
1959:
1946:
1944:
1933:
1927:
1917:
1812:
1808:
1804:
1802:
1751:
1670:
1664:
1575:
1573:
1505:
1503:
1338:
1314:
1313:-forms over
1310:
1139:
1072:
980:
976:
974:
865:
855:
847:
843:
837:
833:
824:
819:denotes the
813:
811:
637:
633:
626:vector space
618:
614:
606:
600:
584:
581:M. F. Atiyah
578:
574:
565:Hermann Weyl
553:
548:
449:
398:
386:
378:
371:
364:
354:
351:
254:
156:
140:
126:
114:
103:
72:
67:
51:
39:
25:Hodge theory
24:
18:
5754:ddbar lemma
5167:classes in
4503:codimension
4338:, which is
3789:cap product
3487:cup product
2945:for Δ. The
2530:are linear
391:cup product
137:Élie Cartan
21:mathematics
5921:References
5797:2018-10-15
5608:. Namely,
5570:Chow group
4373:, and the
4237:K3 surface
3760:-forms on
3744:-forms on
3634:The piece
3345:cohomology
2534:acting on
2225:direct sum
1930:such that
632:of degree
628:of smooth
133:cohomology
6166:on GitHub
6114:(2008) ,
6043:(2007) ,
5947:(1994) .
5574:cycle map
5437:−
5203:, can be
4908:⊆
4880:∩
4399:¯
4396:∂
4390:∂
4056:) is the
4048:(because
3983:−
3971:−
3853:≠
3803:α
3709:Ω
3686:≅
3569:→
3531:×
3503::
3500:⌣
3393:⨁
3285:¯
3267:∧
3264:⋯
3261:∧
3256:¯
3238:∧
3222:∧
3219:⋯
3216:∧
3161:forms of
3155:-form on
3029:≅
2918:→
2913:∙
2860:Δ
2857:∣
2852:∙
2840:∈
2770:∙
2758:→
2753:∙
2722:⨁
2688:Γ
2679:⨁
2666:∙
2624:→
2605:Γ
2602:→
2599:⋯
2596:→
2577:Γ
2574:→
2555:Γ
2552:→
2493:Γ
2490:→
2471:Γ
2410:…
2319:Δ
2307:⊕
2292:δ
2288:
2282:⊕
2274:−
2263:
2257:≅
2239:Ω
2207:in which
2189:γ
2183:β
2180:δ
2174:α
2165:ω
2057:φ
2006:→
1987:Δ
1972:φ
1891:α
1888:Δ
1885:∣
1867:Ω
1863:∈
1860:α
1835:Δ
1782:δ
1776:δ
1767:Δ
1754:Laplacian
1752:Then the
1719:Ω
1715:→
1691:Ω
1684:δ
1628:∈
1620:→
1605:‖
1592:ω
1589:‖
1556:∞
1544:‖
1540:ω
1537:‖
1531:⟩
1528:ω
1522:ω
1519:⟨
1486:σ
1477:⟩
1464:τ
1449:ω
1446:⟨
1437:∫
1430:⟩
1427:τ
1421:ω
1418:⟨
1415:↦
1409:τ
1403:ω
1362:Ω
1358:∈
1355:τ
1349:ω
1325:σ
1282:Ω
1247:∗
1225:⋀
1160:∗
1108:∗
1086:⋀
1039:∗
1025:⋀
1016:Γ
995:Ω
944:−
933:
915:
906:≅
759:Ω
748:−
732:⋯
697:Ω
661:Ω
657:→
621:) be the
532:¯
529:ω
506:ω
483:¯
480:ω
474:∧
471:ω
462:−
432:¯
429:ω
423:∧
420:ω
411:−
302:≅
232:→
197:×
64:Laplacian
6177:Category
6120:Springer
6087:Springer
6016:Springer
6010:(2005),
5977:(1941),
5718:See also
5189:calculus
5165:homology
4381:and the
3604:′
3587:′
3554:′
3543:′
3148:, every
3098:of some
2791:and let
2372:defined
2079:elliptic
1813:harmonic
1389:we have
778:→
736:→
716:→
680:→
617:, let Ω(
153:singular
68:harmonic
6156:2359489
6105:0722297
6075:1967689
6034:2093043
6001:0003947
5967:0507725
5832:1968379
5201:periods
4997:torsion
4873:torsion
4068:)) and
3337:compact
2941:be the
2141:lattice
2120:in the
2110:torsion
2108:modulo
2083:Kodaira
1508:-form:
1183:to its
361:Riemann
123:History
6154:
6144:
6103:
6093:
6073:
6063:
6032:
6022:
5999:
5989:
5965:
5955:
5830:
5398:. The
5303:, and
4660:. The
4414:-lemma
4369:, the
3815:. By
3764:. (If
3163:type (
3117:. The
3088:smooth
2366:Atiyah
2094:kernel
2045:. The
1955:closed
1951:closed
1273:. The
812:where
605:. Let
147:. By
42:using
5934:(PDF)
5828:JSTOR
5791:(PDF)
5780:(PDF)
5761:Notes
5496:with
5163:over
4191:genus
4187:curve
3770:Serre
3738:sheaf
3302:with
3174:with
3105:. By
3086:be a
2963:Id =
1949:on a
823:on Ω(
609:be a
35:of a
6142:ISBN
6091:ISBN
6061:ISBN
6020:ISBN
5987:ISBN
5953:ISBN
5784:EPFL
5090:The
4441:Let
4315:The
4078:(by
4012:The
3774:GAGA
3313:and
3082:Let
2956:and
2890:Let
2801:Δ =
2370:Bott
2368:and
1553:<
623:real
518:and
370:and
275:sing
89:and
58:, a
6132:hdl
6124:doi
6053:doi
5820:doi
5378:of
5067:of
4819:.)
4799:on
4501:of
4481:in
4360:= 1
4342:to
4319:of
4189:of
4082:).
4060:of
3772:'s
3478:of
3347:of
3121:on
2967:+ Δ
2435:be
2217:= 0
2125:GL(
2116:of
1962:= 0
1936:= 0
1811:is
1669:of
979:on
912:ker
864:of
858:= 0
851:= 0
636:on
81:on
50:on
19:In
6179::
6152:MR
6150:,
6140:,
6130:,
6122:,
6101:MR
6099:,
6089:,
6085:,
6071:MR
6069:,
6059:,
6051:,
6047:,
6030:MR
6028:,
6018:,
6014:,
5997:MR
5995:,
5985:,
5981:,
5963:MR
5961:.
5943:;
5847:,
5826:.
5816:28
5814:.
5599:.
5434:20
5360:20
5289:22
4416:.
4362:.
4347:×
4332:+1
4277:20
4073:=
4043:=
4009:.
3827:.
3482:.
3468:,
3329:,
3183:=
3179:+
3167:,
3137:.
3133:a
3123:CP
3115:CP
3103:CP
2998:GL
2996:=
2989:,
2987:GL
2985:=
2983:LG
2975:+
2971:=
2805:+
2803:LL
2445:dV
2285:im
2260:im
2137:))
2133:,
1960:dα
1433::=
1216::
930:im
842:∘
840:+1
310:dR
205:dR
101:.
70:.
23:,
6134::
6126::
6055::
5969:.
5834:.
5822::
5702:X
5682:X
5654:X
5634:X
5547:Z
5526:)
5523:X
5520:(
5515:1
5512:,
5509:1
5505:H
5484:)
5480:Z
5476:,
5473:X
5470:(
5465:2
5461:H
5440:a
5414:a
5386:X
5355:C
5333:)
5330:X
5327:(
5322:1
5319:,
5316:1
5312:H
5284:Z
5262:)
5258:Z
5254:,
5251:X
5248:(
5243:2
5239:H
5218:X
5175:X
5151:X
5131:X
5108:1
5105:=
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