222:) is similar to that of algebraic surfaces in characteristic 0, except that there are no Kodaira surfaces or surfaces of type VII, and there are some extra families of Enriques surfaces in characteristic 2, and hyperelliptic surfaces in characteristics 2 and 3, and in Kodaira dimension 1 in characteristics 2 and 3 one also allows quasielliptic fibrations. These extra families can be understood as follows: In characteristic 0 these surfaces are the quotients of surfaces by finite groups, but in finite characteristics it is also possible to take quotients by finite
837:
1174:
634:
36:
5935:. There are a large number of other constructions known. However, there is no known construction that can produce "typical" surfaces of general type for large Chern numbers; in fact it is not even known if there is any reasonable concept of a "typical" surface of general type. There are many other examples that have been found, including most
199:
The
Enriques–Kodaira classification of compact complex surfaces states that every nonsingular minimal compact complex surface is of exactly one of the 10 types listed on this page; in other words, it is one of the rational, ruled (genus > 0), type VII, K3, Enriques, Kodaira, toric, hyperelliptic,
5681:, there is a quasi-projective scheme classifying the surfaces of general type with those Chern numbers. However it is a very difficult problem to describe these schemes explicitly, and there are very few pairs of Chern numbers for which this has been done (except when the scheme is empty!)
970:
832:{\displaystyle {\begin{aligned}\kappa =-\infty &\longleftrightarrow P_{12}=0\\\kappa =0&\longleftrightarrow P_{12}=1\\\kappa =1&\longleftrightarrow P_{12}>1{\text{ and }}K\cdot K=0\\\kappa =2&\longleftrightarrow P_{12}>1{\text{ and }}K\cdot K>0\\\end{aligned}}}
4916:
which are quotients of these by finite groups of orders 2, 3, 4, or 6, and which have non-trivial canonical bundles. The secondary
Kodaira surfaces have the same relation to primary ones that Enriques surfaces have to K3 surfaces, or bielliptic surfaces have to abelian surfaces.
1839:
4217:
192:
3848:
5473:. Conversely, given a genus 1 curve over the function field of a curve, its relative minimal model is an elliptic surface. Kodaira and others have given a fairly complete description of all elliptic surfaces. In particular, Kodaira gave a
454:
there are examples of surfaces that are homeomorphic but have different plurigenera and
Kodaira dimensions. The individual plurigenera are not often used; the most important thing about them is their growth rate, measured by the
4068:
2718:
1169:{\displaystyle {\begin{matrix}&&h^{0,0}&&\\&h^{1,0}&&h^{0,1}&\\h^{2,0}&&h^{1,1}&&h^{0,2}\\&h^{2,1}&&h^{1,2}&\\&&h^{2,2}&&\\\end{matrix}}}
3979:
2549:
183:); it is similar to the characteristic 0 projective case, except that one also gets singular and supersingular Enriques surfaces in characteristic 2, and quasi-hyperelliptic surfaces in characteristics 2 and 3.
4768:. One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic. The algebraic ones are exactly the 2-dimensional
121:. For most of the classes the moduli spaces are well understood, but for the class of surfaces of general type the moduli spaces seem too complicated to describe explicitly, though some components are known.
1599:
1609:
4752:). The moduli space of marked K3 surfaces is connected non-Hausdorff smooth analytic space of dimension 20. The algebraic K3 surfaces form a countable collection of 19-dimensional subvarieties of it.
5920:
432:
3716:
639:
4079:
2423:
2346:
919:
1959:
5854:
2235:
5621:
2130:
450:
showed that for complex manifolds they only depend on the underlying oriented smooth 4-manifold. For non-Kähler surfaces the plurigenera are determined by the fundamental group, but for
5788:
2794:
3657:
318:
5739:
1245:
3711:
1303:
623:
3494:
implies that all minimal class VII surfaces with positive second Betti number are Kato surfaces, which would more or less complete the classification of the type VII surfaces.
552:
1399:
278:
3703:
946:
2031:
1507:
5527:
2994:
2951:
2581:
510:
5469:
such that all but finitely many fibers are smooth irreducible curves of genus 1). The generic fiber in such a fibration is a genus 1 curve over the function field of
1468:
1435:
1340:
481:
2611:
579:
203:
For the 9 classes of surfaces other than general type, there is a fairly complete description of what all the surfaces look like (which for class VII depends on the
2451:
3331:, so the classification of ruled surfaces up to birational equivalence is essentially the same as the classification of curves. A ruled surface not isomorphic to
57:
2965:
blown up twice. So to classify all compact complex surfaces up to birational isomorphism it is (more or less) enough to classify the minimal non-singular ones.
163:) later extended the classification to include non-algebraic compact surfaces. The analogous classification of surfaces in positive characteristic was begun by
4772:. Most of their theory is a special case of the theory of higher-dimensional tori or abelian varieties. Criteria to be a product of two elliptic curves (up to
5506:
1 is an elliptic surface (or a quasielliptic surface in characteristics 2 or 3), but the converse is not true: an elliptic surface can have
Kodaira dimension
6608:
3994:
2815:
There are further invariants of compact complex surfaces that are not used so much in the classification. These include algebraic invariants such as the
5791:
2953:
may be birational to more than one minimal non-singular surface, but it is easy to describe the relation between these minimal surfaces. For example,
207:, still unproved in 2024). For surfaces of general type not much is known about their explicit classification, though many examples have been found.
2622:
3000:> 0 then the map to the Albanese variety has fibers that are projective lines (if the surface is minimal) so the surface is a ruled surface. If
5266:
In characteristic 2 there are some extra families of
Enriques surfaces called singular and supersingular Enriques surfaces; see the article on
5449:
Over fields of characteristics 2 or 3 there are some extra families given by taking quotients by a non-etale group scheme; see the article on
3859:
3319:. They are all algebraic. (The ones of genus 0 are the Hirzebruch surfaces and are rational.) Any ruled surface is birationally equivalent to
6515:
6395:
6047:
6028:
5986:
5284:
Over the complex numbers these are quotients of a product of two elliptic curves by a finite group of automorphisms. The finite group can be
2880:
this point, which means roughly that we replace it by a copy of the projective line. For the purpose of this article, a non-singular surface
4588:. All K3 surfaces are diffeomorphic, and their diffeomorphism class is an important example of a smooth spin simply connected 4-manifold.
2889:
2462:
337:
The most important invariants of a compact complex surfaces used in the classification can be given in terms of the dimensions of various
1519:
There are many invariants that (at least for complex surfaces) can be written as linear combinations of the Hodge numbers, as follows:
6462:
6096:
3491:
204:
89:
1834:{\displaystyle {\begin{cases}b_{0}=b_{4}=1\\b_{1}=b_{3}=h^{1,0}+h^{0,1}=h^{2,1}+h^{1,2}\\b_{2}=h^{2,0}+h^{1,1}+h^{0,2}\end{cases}}}
1531:
2245:
6339:
6301:
6245:
6207:
5557:
is of genus at least 2 always has
Kodaira dimension 1, but the Kodaira dimension can be 1 also for some elliptic surfaces with
2919:. A smooth projective surface has a minimal model in that stronger sense if and only if its Kodaira dimension is nonnegative.)
2873:
Any surface is birational to a non-singular surface, so for most purposes it is enough to classify the non-singular surfaces.
512:(sometimes written −1) if the plurigenera are all 0, and is otherwise the smallest number (0, 1, or 2 for surfaces) such that
5137:= 0 and the canonical line bundle is non-trivial, but has trivial square. Enriques surfaces are all algebraic (and therefore
4908:
These are never algebraic, though they have non-constant meromorphic functions. They are usually divided into two subtypes:
4212:{\displaystyle 8h^{0,1}+2\left(2h^{0,1}-b_{1}\right)+b_{2}={\begin{cases}22&K=0\\10&{\text{otherwise}}\end{cases}}}
3011:
For non-algebraic surfaces
Kodaira found an extra class of surfaces, called type VII, which are still not well understood.
6507:
6428:
5861:
363:
5141:). They are quotients of K3 surfaces by a group of order 2 and their theory is similar to that of algebraic K3 surfaces.
2858:
6598:
2362:
2262:
847:
6603:
1876:
447:
5499:
surfaces, whose fibers may almost all be rational curves with a single node, which are "degenerate elliptic curves".
2896:
has no (−1)-curves (smooth rational curves with self-intersection number −1). (In the more modern terminology of the
6567:. Theorem 4.3 of this article classifies the Hodge numbers of a quasi-hyperelliptic surface in characteristic three.
5931:
The simplest examples are the product of two curves of genus at least 2, and a hypersurface of degree at least 5 in
5799:
2153:
6012:
5956:
5925:
Most pairs of integers satisfying these conditions are the Chern numbers for some complex surface of general type.
5569:
5553:
are elliptic surfaces, and these examples have
Kodaira dimension less than 1. An elliptic surface whose base curve
2831:
2054:
443:
6593:
6560:
1969:
338:
320:, and Igusa showed that even when they are equal they may be greater than the irregularity (the dimension of the
5108:
Take a non-trivial line bundle over an elliptic curve, remove the zero section, then quotient out the fibers by
6124:
5745:
5687:
There are several conditions that the Chern numbers of a minimal complex surface of general type must satisfy:
3664:
3479:
3023:
2739:
71:
6171:(1914), "Sulla classificazione delle superficie algebriche e particolarmente sulle superficie di genere p=1",
6116:
5477:. The theory of elliptic surfaces is analogous to the theory of proper regular models of elliptic curves over
3605:
6580:
is an interactive visualisation of the
Enriques--Kodaira classification, by Pieter Belmans and Johan Commelin
4233:, so three terms on the left are non-negative integers and there are only a few solutions to this equation.
5936:
5658:
These are all algebraic, and in some sense most surfaces are in this class. Gieseker showed that there is a
5653:
5478:
3843:{\displaystyle {\begin{aligned}\chi &=h^{0,0}-h^{0,1}+h^{0,2}\\c_{2}&=2-2b_{1}+b_{2}\end{aligned}}}
283:
5693:
1186:
236:
constructed some surfaces in positive characteristic that are unirational but not rational, derived from
5940:
5534:
5450:
5279:
2897:
1250:
584:
237:
515:
227:
5263:
Marked
Enriques surfaces form a connected 10-dimensional family, which has been described explicitly.
6442:
6133:
5973:, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin,
5659:
3454:
2353:
1859:
1352:
949:
4171:
4022:
2926:
is birational to a minimal non-singular surface, and this minimal non-singular surface is unique if
2631:
1618:
247:
6559:, Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 12 (1979) no. 4, pp. 473-500.
2862:
3670:
6476:
6432:
6401:
6356:
6318:
6262:
6224:
6157:
5171:
4607:
4299:
Most solutions to these conditions correspond to classes of surfaces, as in the following table:
3034:
2826:
924:
5138:
4585:
4567:
4538:
4458:
4378:
4273:
3460:
1994:
1473:
1346:
451:
17:
6511:
6458:
6391:
6288:
6183:
6168:
6092:
6043:
6024:
5982:
5509:
5503:
3292:
2976:
2933:
2930:
has Kodaira dimension at least 0 or is not algebraic. Algebraic surfaces of Kodaira dimension
2845:
2559:
1850:
492:
485:
456:
132:
128:
1440:
1407:
1312:
466:
6487:
6450:
6383:
6348:
6310:
6278:
6254:
6216:
6202:
6141:
6084:
6016:
6004:
5974:
5550:
5530:
5474:
5462:
5267:
5128:
4920:
Invariants: If the surface is the quotient of a primary Kodaira surface by a group of order
3296:
3019:
3005:
2586:
2140:
1978:
557:
144:
67:
6549:
6525:
6472:
6413:
6368:
6330:
6274:
6236:
6195:
6153:
6070:
6038:
5996:
2436:
6545:
6541:
6521:
6495:
6468:
6409:
6364:
6326:
6270:
6232:
6191:
6149:
6108:
6066:
6054:
6034:
5992:
5546:
5538:
5489:
5465:
is a surface equipped with an elliptic fibration (a surjective holomorphic map to a curve
4903:
4769:
4765:
2916:
2041:
353:
241:
172:
6556:
6299:
Kodaira, Kunihiko (1968a), "On the structure of compact complex analytic surfaces. III",
6077:
Bombieri, E.; Mumford, D. (1977). "Enriques' Classification of Surfaces in Char. P, II".
2853:
and the integral homology and cohomology groups, and invariants of the underlying smooth
6533:
6446:
6137:
5662:
for surfaces of general type; this means that for any fixed values of the Chern numbers
6243:
Kodaira, Kunihiko (1966), "On the structure of compact complex analytic surfaces. II",
5944:
5482:
5113:
4713:
1974:
50:
44:
6283:
191:
6587:
6480:
6375:
6161:
6112:
6078:
6058:
4063:{\displaystyle h^{0,2}={\begin{cases}1&K=0\\0&{\text{otherwise}}\end{cases}}}
3483:
3475:
3288:
3004:= 0 this argument does not work as the Albanese variety is a point, but in this case
2888:
if it cannot be obtained from another non-singular surface by blowing up a point. By
2840:
1181:
233:
164:
111:
4890:
A product of two elliptic curves. The Jacobian of a genus 2 curve. Any quotient of
4761:
3487:
3471:
2817:
2798:
1524:
1309:
ring of the surface, and are invariant under birational transformations except for
957:
321:
223:
118:
6564:
6088:
6337:
Kodaira, Kunihiko (1968b), "On the structure of complex analytic surfaces. IV",
2803:
554:
is bounded. Enriques did not use this definition: instead he used the values of
342:
124:
103:
6405:
5969:
Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004),
6491:
6420:
6387:
5978:
5542:
2877:
2854:
2713:{\displaystyle {\begin{cases}b^{+}+b^{-}=b_{2}\\b^{+}-b^{-}=\tau \end{cases}}}
2583:
are the dimensions of the maximal positive and negative definite subspaces of
2357:
1306:
439:
6020:
6454:
1985:. For complex surfaces (but not always for surfaces of prime characteristic)
6292:
6050:
softcover) – including a more elementary introduction to the classification
4580:
These are the minimal compact complex surfaces of Kodaira dimension 0 with
625:. These determine the Kodaira dimension given the following correspondence:
6011:, London Mathematical Society Student Texts, vol. 34 (2nd ed.),
6437:
5162:
is odd. The fundamental group has order 2. The second cohomology group H(
3974:{\displaystyle 10+12h^{0,2}=8h^{0,1}+2\left(2h^{0,1}-b_{1}\right)+b_{2}}
1305:
The Hodge numbers of a complex surface depend only on the oriented real
6498:; Tjurina, Galina N.; Tjurin, Andrei N. (1967) , "Algebraic surfaces",
6360:
6322:
6266:
6228:
6145:
4773:
3284:
210:
The classification of algebraic surfaces in positive characteristic (
6352:
6314:
6258:
6220:
6205:(1964), "On the structure of compact complex analytic surfaces. I",
2544:{\displaystyle \tau =4\chi -e=\sum \nolimits _{i,j}(-1)^{j}h^{i,j}.}
2433:
of the second cohomology group for complex surfaces is denoted by
3602:
These surfaces are classified by starting with Noether's formula
3095:
The plurigenera are all 0 and the fundamental group is trivial.
6061:(1977), "Enriques' classification of surfaces in char. p. II",
4736:
K3 surface is a K3 surface together with an isomorphism from II
143:) described the classification of complex projective surfaces.
29:
5900:
3029:. These are all algebraic. The minimal rational surfaces are
2249:
of the trivial bundle (usually differs from the Euler number
6378:(1969), "Enriques' classification of surfaces in char p I",
1594:{\displaystyle b_{i}=\dim H^{i}(S),0\leqslant i\leqslant 4.}
302:
6577:
6540:, Lecture Notes in Math., vol. 677, Berlin, New York:
6490:; Averbuh, Boris G.; Vaĭnberg, Ju. R.; Zhizhchenko, A. B.;
6427:, IAS/Park City Math. Ser., vol. 3, Providence, R.I.:
4782:
The plurigenera are all 1. The surface is diffeomorphic to
4205:
4056:
2876:
Given any point on a surface, we can form a new surface by
2706:
1827:
6000:– the standard reference book for compact complex surfaces
5648:
Surfaces of Kodaira dimension 2 (surfaces of general type)
5178:
of dimension 10 and signature −8 and a group of order 2.
2802:, defined as the integrals of various polynomials in the
6117:"Enriques' classification of surfaces in char. p. III."
62:
6534:"On the Enriques classification of algebraic surfaces"
3470:= 0 have been classified by Bogomolov, and are either
2367:
1849: > 0 the Betti numbers are defined using
1342:
which increases by 1 under blowing up a single point.
975:
127:
began the systematic study of algebraic surfaces, and
5864:
5802:
5748:
5696:
5572:
5512:
4461:
over the complex numbers, but need not be algebraic.
4381:
over the complex numbers, but need not be algebraic.
4082:
3997:
3862:
3714:
3673:
3608:
3478:. Examples with positive second Betti number include
2979:
2936:
2742:
2625:
2589:
2562:
2465:
2439:
2365:
2265:
2156:
2057:
1997:
1879:
1612:
1534:
1476:
1443:
1410:
1355:
1315:
1253:
1189:
973:
927:
850:
637:
587:
560:
518:
495:
469:
366:
286:
250:
5915:{\displaystyle c_{1}^{2}+c_{2}\equiv 0{\bmod {1}}2.}
4584:= 0 and trivial canonical line bundle. They are all
442:
invariants, i.e., invariant under blowing up. Using
427:{\displaystyle P_{n}=\dim H^{0}(K^{n}),n\geqslant 1}
6500:
Proceedings of the Steklov Institute of Mathematics
1401:
and there are only three independent Hodge numbers.
200:properly quasi-elliptic, or general type surfaces.
5914:
5848:
5782:
5733:
5615:
5521:
5495:In finite characteristic 2 and 3 one can also get
4776:) were a popular study in the nineteenth century.
4211:
4062:
3973:
3842:
3697:
3651:
2988:
2945:
2788:
2712:
2605:
2575:
2543:
2445:
2418:{\displaystyle {\tfrac {1}{12}}(c_{1}^{2}+c_{2}).}
2417:
2341:{\displaystyle \chi =p_{g}-q+1=h^{0,2}-h^{0,1}+1.}
2340:
2229:
2124:
2025:
1953:
1833:
1593:
1501:
1462:
1429:
1393:
1334:
1297:
1239:
1168:
940:
914:{\displaystyle h^{i,j}=\dim H^{j}(X,\Omega ^{i}),}
913:
831:
617:
573:
546:
504:
475:
426:
312:
272:
6557:"Quasi-elliptic surfaces in characteristic three"
4073:combining this with the previous equation gives:
6425:Complex algebraic geometry (Park City, UT, 1993)
5492:(e.g., the ring of integers of a number field).
1954:{\displaystyle e=b_{0}-b_{1}+b_{2}-b_{3}+b_{4}.}
244:). In positive characteristic Serre showed that
6380:Global Analysis (Papers in Honor of K. Kodaira)
5644:is an elliptic surface of Kodaira dimension 1.
4756:Abelian surfaces and 2-dimensional complex tori
219:
215:
180:
176:
6382:, Tokyo: Univ. Tokyo Press, pp. 325–339,
5849:{\displaystyle 5c_{1}^{2}-c_{2}+36\geqslant 0}
5170:) is isomorphic to the sum of the unique even
3441:The product of any curve of genus > 0 with
2230:{\displaystyle p_{a}=p_{g}-q=h^{0,2}-h^{0,1}.}
131:proved important parts of the classification.
5616:{\displaystyle c_{1}^{2}=0,c_{2}\geqslant 0.}
5475:complete list of the possible singular fibers
8:
2125:{\displaystyle p_{g}=h^{0,2}=h^{2,0}=P_{1}.}
5364:, giving seven families of such surfaces.
5112:acting as multiplication by powers of some
3311:have a smooth morphism to a curve of genus
356:whose sections are the holomorphic 2-forms.
6423:(1997), "Chapters on algebraic surfaces",
3299:. Many of these examples are non-minimal.
345:and the Hodge numbers defined as follows:
6436:
6282:
6065:, Tokyo: Iwanami Shoten, pp. 23–42,
5903:
5899:
5887:
5874:
5869:
5863:
5828:
5815:
5810:
5801:
5783:{\displaystyle c_{1}^{2}\leqslant 3c_{2}}
5774:
5758:
5753:
5747:
5719:
5706:
5701:
5695:
5601:
5582:
5577:
5571:
5511:
5133:These are the complex surfaces such that
4728:, then blowing up the 16 singular points.
4197:
4166:
4157:
4139:
4120:
4090:
4081:
4048:
4017:
4002:
3996:
3965:
3947:
3928:
3898:
3876:
3861:
3830:
3817:
3791:
3771:
3752:
3733:
3715:
3713:
3683:
3678:
3672:
3640:
3635:
3622:
3607:
2978:
2935:
2789:{\displaystyle c_{1}^{2}=K^{2}=12\chi -e}
2765:
2752:
2747:
2741:
2691:
2678:
2664:
2651:
2638:
2626:
2624:
2594:
2588:
2567:
2561:
2526:
2516:
2491:
2464:
2438:
2403:
2390:
2385:
2366:
2364:
2320:
2301:
2276:
2264:
2212:
2193:
2174:
2161:
2155:
2113:
2094:
2075:
2062:
2056:
2008:
1996:
1942:
1929:
1916:
1903:
1890:
1878:
1812:
1793:
1774:
1761:
1741:
1722:
1703:
1684:
1671:
1658:
1638:
1625:
1613:
1611:
1558:
1539:
1533:
1481:
1475:
1448:
1442:
1415:
1409:
1379:
1360:
1354:
1320:
1314:
1277:
1258:
1252:
1213:
1194:
1188:
1148:
1125:
1106:
1085:
1066:
1047:
1026:
1007:
984:
974:
972:
932:
926:
899:
880:
855:
849:
805:
793:
749:
737:
701:
665:
638:
636:
609:
604:
586:
565:
559:
538:
529:
523:
517:
494:
468:
403:
390:
371:
365:
301:
300:
291:
285:
255:
249:
195:Chern numbers of minimal complex surfaces
117:into ten classes, each parametrized by a
90:Learn how and when to remove this message
5119:. This gives a primary Kodaira surface.
4301:
3652:{\displaystyle 12\chi =c_{2}+c_{1}^{2}.}
3089:blown up at a point so is not minimal.)
2973:Algebraic surfaces of Kodaira dimension
190:
160:
156:
140:
136:
6538:Séminaire Bourbaki, 29e année (1976/77)
6080:Complex Analysis and Algebraic Geometry
6063:Complex analysis and algebraic geometry
4720:an abelian surface by the automorphism
211:
168:
152:
148:
27:Mathematical classification of surfaces
5274:Hyperelliptic (or bielliptic) surfaces
4924:= 1, 2, 3, 4, 6, then the plurigenera
3459:These surfaces are never algebraic or
3008:implies that the surface is rational.
2843:ρ, topological invariants such as the
961:, often arranged in the Hodge diamond:
214:, Mumford & Bombieri
5636:is a curve of genus at least 2, then
2961:blown up at a point is isomorphic to
1853:and need not satisfy these relations.
313:{\displaystyle h^{1}({\mathcal {O}})}
66:. Parenthetical referencing has been
7:
5734:{\displaystyle c_{1}^{2},c_{2}>0}
2892:, this is equivalent to saying that
6609:Mathematical classification systems
4912:with trivial canonical bundle, and
4614:of dimension 22 and signature −16.
4606:) is isomorphic to the unique even
2488:
1973:is defined as the dimension of the
1515:Invariants related to Hodge numbers
1240:{\displaystyle h^{i,j}=h^{2-i,2-j}}
333:Hodge numbers and Kodaira dimension
5516:
4248:is an even integer between 0 and 2
2983:
2940:
1298:{\displaystyle h^{0,0}=h^{2,2}=1.}
929:
896:
651:
618:{\displaystyle K\cdot K=c_{1}^{2}}
499:
264:
25:
3492:global spherical shell conjecture
3065:associated to the sheaf O(0) + O(
2996:can be classified as follows. If
2890:Castelnuovo's contraction theorem
547:{\displaystyle P_{n}/n^{\kappa }}
205:global spherical shell conjecture
175: and David Mumford (
5792:Bogomolov–Miyaoka–Yau inequality
3022:means surface birational to the
2969:Surfaces of Kodaira dimension −∞
2246:holomorphic Euler characteristic
34:
6340:American Journal of Mathematics
6302:American Journal of Mathematics
6246:American Journal of Mathematics
6208:American Journal of Mathematics
5457:Surfaces of Kodaira dimension 1
3665:intersection number with itself
3598:Surfaces of Kodaira dimension 0
1404:If the surface is compact then
1394:{\displaystyle h^{i,j}=h^{j,i}}
341:groups. The basic ones are the
187:Statement of the classification
108:Enriques–Kodaira classification
18:Enriques-Kodaira classification
6190:, Nicola Zanichelli, Bologna,
4258:For compact complex surfaces 2
3303:Ruled surfaces of genus > 0
3051:≥ 2. (The Hirzebruch surface Σ
2900:, a smooth projective surface
2513:
2503:
2409:
2378:
1570:
1564:
905:
886:
786:
730:
694:
658:
409:
396:
307:
297:
273:{\displaystyle h^{0}(\Omega )}
267:
261:
1:
6532:Van de Ven, Antonius (1978),
6508:American Mathematical Society
6429:American Mathematical Society
2908:if its canonical line bundle
2869:Minimal models and blowing up
6089:10.1017/CBO9780511569197.004
4798:so the fundamental group is
4594:The second cohomology group
3698:{\displaystyle c_{1}^{2}=0.}
3507:= 0. All plurigenera are 0.
3353:The plurigenera are all 0.
941:{\displaystyle \Omega ^{i}}
6625:
6013:Cambridge University Press
6009:Complex algebraic surfaces
5957:List of algebraic surfaces
5651:
5277:
5126:
4914:secondary Kodaira surfaces
4901:
4702:Degree 4 hypersurfaces in
4511:Only characteristics 2, 3
3452:
2026:{\displaystyle q=h^{0,1}.}
1502:{\displaystyle h^{0,1}-1.}
6388:10.1515/9781400871230-019
5979:10.1007/978-3-642-57739-0
5632:is an elliptic curve and
3659:For Kodaira dimension 0,
3480:Inoue-Hirzebruch surfaces
2859:Seiberg–Witten invariants
339:coherent sheaf cohomology
6578:le superficie algebriche
6188:Le Superficie Algebriche
6173:Atti. Acc. Lincei V Ser.
6125:Inventiones Mathematicae
6021:10.1017/CBO9780511623936
5971:Compact Complex Surfaces
5937:Hilbert modular surfaces
5856:(the Noether inequality)
5522:{\displaystyle -\infty }
5479:discrete valuation rings
4910:primary Kodaira surfaces
4716:. These are obtained by
4237:For algebraic surfaces 2
3463:. The minimal ones with
3307:Ruled surfaces of genus
3024:complex projective plane
2989:{\displaystyle -\infty }
2946:{\displaystyle -\infty }
2576:{\displaystyle b^{\pm }}
2356:it is also equal to the
505:{\displaystyle -\infty }
5654:Surface of general type
4454:Abelian surfaces, tori
4428:Non-classical Enriques
4407:Any. Always algebraic.
3315:whose fibers are lines
3279:, Hirzebruch surfaces Σ
1463:{\displaystyle h^{0,1}}
1430:{\displaystyle h^{1,0}}
1335:{\displaystyle h^{1,1}}
476:{\displaystyle \kappa }
5941:fake projective planes
5916:
5850:
5784:
5735:
5617:
5535:hyperelliptic surfaces
5523:
5451:hyperelliptic surfaces
4487:Any. Always algebraic
4431:Only characteristic 2
4213:
4064:
3975:
3844:
3699:
3653:
2990:
2947:
2790:
2714:
2607:
2606:{\displaystyle H^{2},}
2577:
2545:
2447:
2419:
2342:
2231:
2126:
2027:
1955:
1835:
1595:
1503:
1464:
1431:
1395:
1336:
1299:
1241:
1170:
942:
915:
833:
619:
575:
574:{\displaystyle P_{12}}
548:
506:
477:
446:, Robert Friedman and
428:
328:Invariants of surfaces
314:
274:
238:inseparable extensions
196:
43:This article includes
5917:
5851:
5785:
5736:
5618:
5524:
5280:hyperelliptic surface
4214:
4065:
3976:
3845:
3700:
3654:
3486:, and more generally
3449:Surfaces of class VII
3339:has a unique ruling (
3226:(Hirzebruch surfaces)
3006:Castelnuovo's theorem
2991:
2948:
2898:minimal model program
2825:) of divisors modulo
2791:
2715:
2608:
2578:
2546:
2448:
2446:{\displaystyle \tau }
2420:
2343:
2232:
2127:
2028:
1956:
1836:
1596:
1504:
1465:
1432:
1396:
1337:
1300:
1242:
1171:
943:
916:
834:
620:
576:
549:
507:
478:
444:Seiberg–Witten theory
429:
354:canonical line bundle
315:
275:
194:
133:Federigo Enriques
6544:, pp. 237–251,
6506:, Providence, R.I.:
6488:Shafarevich, Igor R.
5862:
5800:
5746:
5694:
5660:coarse moduli scheme
5570:
5510:
4760:The two-dimensional
4566:Only complex, never
4537:Only complex, never
4508:Quasi-hyperelliptic
4080:
3995:
3860:
3712:
3671:
3606:
3455:Surface of class VII
2977:
2934:
2863:Donaldson invariants
2740:
2623:
2587:
2560:
2463:
2437:
2363:
2263:
2154:
2055:
1995:
1877:
1860:Euler characteristic
1610:
1532:
1474:
1441:
1408:
1353:
1313:
1251:
1187:
971:
925:
848:
635:
585:
558:
516:
493:
467:
364:
284:
248:
145:Kunihiko Kodaira
6599:Birational geometry
6565:10.24033/asens.1373
6496:Moishezon, Boris G.
6455:10.1090/pcms/003/02
6447:1996alg.geom..2006R
6138:1976InMat..35..197B
5879:
5820:
5763:
5711:
5587:
5481:(e.g., the ring of
4404:Classical Enriques
3688:
3645:
3327:for a unique curve
3035:Hirzebruch surfaces
2829:, its quotient the
2757:
2395:
614:
173:Enrico Bombieri
171:) and completed by
72:shortened footnotes
6604:Algebraic surfaces
6431:, pp. 3–159,
6184:Enriques, Federigo
6169:Enriques, Federigo
6146:10.1007/BF01390138
6083:. pp. 23–42.
5912:
5865:
5846:
5806:
5780:
5749:
5731:
5697:
5613:
5573:
5519:
5172:unimodular lattice
4608:unimodular lattice
4563:Secondary Kodaira
4209:
4204:
4060:
4055:
3971:
3840:
3838:
3695:
3674:
3649:
3631:
3293:del Pezzo surfaces
3148:(Projective plane)
2986:
2943:
2832:Néron–Severi group
2827:linear equivalence
2806:over the manifold.
2786:
2743:
2710:
2705:
2603:
2573:
2541:
2443:
2415:
2381:
2376:
2338:
2227:
2122:
2023:
1951:
1845:In characteristic
1831:
1826:
1591:
1499:
1460:
1427:
1391:
1345:If the surface is
1332:
1295:
1237:
1166:
1164:
938:
911:
829:
827:
615:
600:
571:
544:
502:
473:
424:
310:
270:
197:
51:properly formatted
6555:Lang, William E.
6517:978-0-8218-1875-6
6397:978-1-4008-7123-0
6203:Kodaira, Kunihiko
6048:978-0-521-49842-5
6030:978-0-521-49510-3
6005:Beauville, Arnaud
5988:978-3-540-00832-3
5561:of genus 0 or 1.
5551:rational surfaces
5531:Enriques surfaces
5504:Kodaira dimension
5502:Every surface of
5445:
5444:
5268:Enriques surfaces
5259:
5258:
5158:is even and 0 if
5123:Enriques surfaces
5101:
5100:
5023:
5022:
4939:and 0 otherwise.
4883:
4882:
4770:abelian varieties
4693:
4692:
4573:
4572:
4200:
4051:
3593:
3592:
3434:
3433:
3257:
3256:
3179:
3178:
3085:is isomorphic to
3073:is isomorphic to
3015:Rational surfaces
2846:fundamental group
2375:
2354:Noether's formula
1851:l-adic cohomology
808:
752:
486:Kodaira dimension
457:Kodaira dimension
165:David Mumford
129:Guido Castelnuovo
100:
99:
92:
16:(Redirected from
6616:
6594:Complex surfaces
6552:
6528:
6483:
6440:
6438:alg-geom/9602006
6416:
6371:
6347:(4): 1048–1066,
6333:
6295:
6286:
6239:
6198:
6179:
6164:
6121:
6109:Bombieri, Enrico
6102:
6073:
6055:Bombieri, Enrico
6041:
5999:
5921:
5919:
5918:
5913:
5908:
5907:
5892:
5891:
5878:
5873:
5855:
5853:
5852:
5847:
5833:
5832:
5819:
5814:
5789:
5787:
5786:
5781:
5779:
5778:
5762:
5757:
5740:
5738:
5737:
5732:
5724:
5723:
5710:
5705:
5673:
5672:
5622:
5620:
5619:
5614:
5606:
5605:
5586:
5581:
5547:abelian surfaces
5539:Kodaira surfaces
5528:
5526:
5525:
5520:
5490:Dedekind domains
5463:elliptic surface
5374:
5373:
5188:
5187:
5147:The plurigenera
5129:Enriques surface
5027:
5026:
4949:
4948:
4935:is divisible by
4898:Kodaira surfaces
4812:
4811:
4766:abelian surfaces
4622:
4621:
4586:Kähler manifolds
4534:Primary Kodaira
4302:
4218:
4216:
4215:
4210:
4208:
4207:
4201:
4198:
4162:
4161:
4149:
4145:
4144:
4143:
4131:
4130:
4101:
4100:
4069:
4067:
4066:
4061:
4059:
4058:
4052:
4049:
4013:
4012:
3980:
3978:
3977:
3972:
3970:
3969:
3957:
3953:
3952:
3951:
3939:
3938:
3909:
3908:
3887:
3886:
3849:
3847:
3846:
3841:
3839:
3835:
3834:
3822:
3821:
3796:
3795:
3782:
3781:
3763:
3762:
3744:
3743:
3704:
3702:
3701:
3696:
3687:
3682:
3658:
3656:
3655:
3650:
3644:
3639:
3627:
3626:
3517:
3516:
3363:
3362:
3297:Veronese surface
3183:
3182:
3105:
3104:
3069:). The surface Σ
3020:Rational surface
2995:
2993:
2992:
2987:
2952:
2950:
2949:
2944:
2904:would be called
2839:) with rank the
2811:Other invariants
2795:
2793:
2792:
2787:
2770:
2769:
2756:
2751:
2719:
2717:
2716:
2711:
2709:
2708:
2696:
2695:
2683:
2682:
2669:
2668:
2656:
2655:
2643:
2642:
2612:
2610:
2609:
2604:
2599:
2598:
2582:
2580:
2579:
2574:
2572:
2571:
2550:
2548:
2547:
2542:
2537:
2536:
2521:
2520:
2502:
2501:
2452:
2450:
2449:
2444:
2424:
2422:
2421:
2416:
2408:
2407:
2394:
2389:
2377:
2368:
2347:
2345:
2344:
2339:
2331:
2330:
2312:
2311:
2281:
2280:
2236:
2234:
2233:
2228:
2223:
2222:
2204:
2203:
2179:
2178:
2166:
2165:
2141:arithmetic genus
2131:
2129:
2128:
2123:
2118:
2117:
2105:
2104:
2086:
2085:
2067:
2066:
2032:
2030:
2029:
2024:
2019:
2018:
1979:Albanese variety
1960:
1958:
1957:
1952:
1947:
1946:
1934:
1933:
1921:
1920:
1908:
1907:
1895:
1894:
1840:
1838:
1837:
1832:
1830:
1829:
1823:
1822:
1804:
1803:
1785:
1784:
1766:
1765:
1752:
1751:
1733:
1732:
1714:
1713:
1695:
1694:
1676:
1675:
1663:
1662:
1643:
1642:
1630:
1629:
1600:
1598:
1597:
1592:
1563:
1562:
1544:
1543:
1508:
1506:
1505:
1500:
1492:
1491:
1469:
1467:
1466:
1461:
1459:
1458:
1436:
1434:
1433:
1428:
1426:
1425:
1400:
1398:
1397:
1392:
1390:
1389:
1371:
1370:
1341:
1339:
1338:
1333:
1331:
1330:
1304:
1302:
1301:
1296:
1288:
1287:
1269:
1268:
1246:
1244:
1243:
1238:
1236:
1235:
1205:
1204:
1175:
1173:
1172:
1167:
1165:
1162:
1161:
1159:
1158:
1142:
1141:
1138:
1136:
1135:
1119:
1117:
1116:
1100:
1096:
1095:
1079:
1077:
1076:
1060:
1058:
1057:
1039:
1037:
1036:
1020:
1018:
1017:
1001:
998:
997:
995:
994:
978:
977:
955:-forms, are the
948:is the sheaf of
947:
945:
944:
939:
937:
936:
920:
918:
917:
912:
904:
903:
885:
884:
866:
865:
838:
836:
835:
830:
828:
809:
806:
798:
797:
753:
750:
742:
741:
706:
705:
670:
669:
624:
622:
621:
616:
613:
608:
580:
578:
577:
572:
570:
569:
553:
551:
550:
545:
543:
542:
533:
528:
527:
511:
509:
508:
503:
482:
480:
479:
474:
433:
431:
430:
425:
408:
407:
395:
394:
376:
375:
319:
317:
316:
311:
306:
305:
296:
295:
280:may differ from
279:
277:
276:
271:
260:
259:
242:Zariski surfaces
115:complex surfaces
95:
88:
84:
81:
75:
65:
60:this article by
45:inline citations
38:
37:
30:
21:
6624:
6623:
6619:
6618:
6617:
6615:
6614:
6613:
6584:
6583:
6574:
6542:Springer-Verlag
6531:
6518:
6486:
6465:
6419:
6406:j.ctt13x10qw.21
6398:
6374:
6353:10.2307/2373289
6336:
6315:10.2307/2373426
6298:
6259:10.2307/2373150
6242:
6221:10.2307/2373157
6201:
6182:
6167:
6119:
6107:
6099:
6076:
6053:
6031:
6003:
5989:
5968:
5965:
5953:
5945:Barlow surfaces
5883:
5860:
5859:
5824:
5798:
5797:
5770:
5744:
5743:
5715:
5692:
5691:
5680:
5671:
5668:
5667:
5666:
5656:
5650:
5597:
5568:
5567:
5529:, 0, or 1. All
5508:
5507:
5459:
5282:
5276:
5177:
5152:
5131:
5125:
4929:
4906:
4904:Kodaira surface
4900:
4758:
4739:
4718:quotienting out
4714:Kummer surfaces
4613:
4578:
4330:
4318:
4310:
4286:
4274:Kähler surfaces
4268:
4253:
4247:
4232:
4203:
4202:
4195:
4189:
4188:
4177:
4167:
4153:
4135:
4116:
4112:
4108:
4086:
4078:
4077:
4054:
4053:
4046:
4040:
4039:
4028:
4018:
3998:
3993:
3992:
3984:Moreover since
3961:
3943:
3924:
3920:
3916:
3894:
3872:
3858:
3857:
3837:
3836:
3826:
3813:
3797:
3787:
3784:
3783:
3767:
3748:
3729:
3722:
3710:
3709:
3669:
3668:
3618:
3604:
3603:
3600:
3557:
3469:
3457:
3451:
3305:
3282:
3278:
3084:
3072:
3056:
3042:
3033:itself and the
3017:
2975:
2974:
2971:
2932:
2931:
2913:
2871:
2852:
2813:
2761:
2738:
2737:
2731:
2704:
2703:
2687:
2674:
2671:
2670:
2660:
2647:
2634:
2627:
2621:
2620:
2590:
2585:
2584:
2563:
2558:
2557:
2522:
2512:
2487:
2461:
2460:
2435:
2434:
2399:
2361:
2360:
2316:
2297:
2272:
2261:
2260:
2253:defined above):
2208:
2189:
2170:
2157:
2152:
2151:
2109:
2090:
2071:
2058:
2053:
2052:
2042:geometric genus
2004:
1993:
1992:
1981:and denoted by
1938:
1925:
1912:
1899:
1886:
1875:
1874:
1825:
1824:
1808:
1789:
1770:
1757:
1754:
1753:
1737:
1718:
1699:
1680:
1667:
1654:
1651:
1650:
1634:
1621:
1614:
1608:
1607:
1554:
1535:
1530:
1529:
1517:
1477:
1472:
1471:
1444:
1439:
1438:
1411:
1406:
1405:
1375:
1356:
1351:
1350:
1316:
1311:
1310:
1273:
1254:
1249:
1248:
1209:
1190:
1185:
1184:
1163:
1160:
1144:
1139:
1137:
1121:
1118:
1102:
1098:
1097:
1081:
1078:
1062:
1059:
1043:
1040:
1038:
1022:
1019:
1003:
999:
996:
980:
969:
968:
928:
923:
922:
895:
876:
851:
846:
845:
826:
825:
807: and
789:
782:
770:
769:
751: and
733:
726:
714:
713:
697:
690:
678:
677:
661:
654:
633:
632:
583:
582:
561:
556:
555:
534:
519:
514:
513:
491:
490:
465:
464:
452:Kähler surfaces
434:are called the
399:
386:
367:
362:
361:
335:
330:
287:
282:
281:
251:
246:
245:
189:
96:
85:
79:
76:
63:correcting them
61:
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
6622:
6620:
6612:
6611:
6606:
6601:
6596:
6586:
6585:
6582:
6581:
6573:
6572:External links
6570:
6569:
6568:
6553:
6529:
6516:
6492:Manin, Yuri I.
6484:
6463:
6417:
6396:
6376:Mumford, David
6372:
6334:
6296:
6253:(3): 682–721,
6240:
6215:(4): 751–798,
6199:
6180:
6165:
6113:Mumford, David
6105:
6104:
6103:
6097:
6059:Mumford, David
6051:
6029:
6001:
5987:
5964:
5961:
5960:
5959:
5952:
5949:
5923:
5922:
5911:
5906:
5902:
5898:
5895:
5890:
5886:
5882:
5877:
5872:
5868:
5857:
5845:
5842:
5839:
5836:
5831:
5827:
5823:
5818:
5813:
5809:
5805:
5795:
5777:
5773:
5769:
5766:
5761:
5756:
5752:
5741:
5730:
5727:
5722:
5718:
5714:
5709:
5704:
5700:
5678:
5669:
5652:Main article:
5649:
5646:
5612:
5609:
5604:
5600:
5596:
5593:
5590:
5585:
5580:
5576:
5518:
5515:
5497:quasi-elliptic
5486:-adic integers
5458:
5455:
5447:
5446:
5443:
5442:
5440:
5438:
5435:
5433:
5430:
5429:
5427:
5424:
5422:
5419:
5416:
5415:
5412:
5410:
5407:
5405:
5401:
5400:
5398:
5395:
5393:
5390:
5387:
5386:
5384:
5382:
5379:
5377:
5368:Hodge diamond:
5278:Main article:
5275:
5272:
5261:
5260:
5257:
5256:
5254:
5252:
5249:
5247:
5244:
5243:
5241:
5238:
5236:
5233:
5230:
5229:
5226:
5224:
5221:
5219:
5215:
5214:
5212:
5209:
5207:
5204:
5201:
5200:
5198:
5196:
5193:
5191:
5182:Hodge diamond:
5175:
5150:
5127:Main article:
5124:
5121:
5114:complex number
5103:
5102:
5099:
5098:
5096:
5094:
5091:
5089:
5086:
5085:
5083:
5080:
5078:
5075:
5072:
5071:
5068:
5065:
5063:
5060:
5058:
5054:
5053:
5051:
5048:
5046:
5043:
5040:
5039:
5037:
5035:
5032:
5030:
5024:
5021:
5020:
5018:
5016:
5013:
5011:
5008:
5007:
5005:
5002:
5000:
4997:
4994:
4993:
4990:
4987:
4985:
4982:
4980:
4976:
4975:
4973:
4970:
4968:
4965:
4962:
4961:
4959:
4957:
4954:
4952:
4943:Hodge diamond:
4927:
4902:Main article:
4899:
4896:
4894:by a lattice.
4885:
4884:
4881:
4880:
4878:
4876:
4873:
4871:
4868:
4867:
4865:
4862:
4860:
4857:
4854:
4853:
4850:
4848:
4845:
4843:
4839:
4838:
4836:
4833:
4831:
4828:
4825:
4824:
4822:
4820:
4817:
4815:
4806:Hodge diamond:
4757:
4754:
4737:
4730:
4729:
4711:
4691:
4690:
4688:
4686:
4683:
4681:
4678:
4677:
4675:
4672:
4670:
4667:
4664:
4663:
4660:
4658:
4655:
4653:
4649:
4648:
4646:
4643:
4641:
4638:
4635:
4634:
4632:
4630:
4627:
4625:
4618:Hodge diamond:
4611:
4577:
4574:
4571:
4570:
4564:
4561:
4558:
4555:
4552:
4549:
4546:
4542:
4541:
4535:
4532:
4529:
4526:
4523:
4520:
4517:
4513:
4512:
4509:
4506:
4504:
4502:
4499:
4496:
4493:
4489:
4488:
4485:
4484:Hyperelliptic
4482:
4479:
4476:
4473:
4470:
4467:
4463:
4462:
4455:
4452:
4449:
4446:
4443:
4440:
4437:
4433:
4432:
4429:
4426:
4424:
4422:
4419:
4416:
4413:
4409:
4408:
4405:
4402:
4399:
4396:
4393:
4390:
4387:
4383:
4382:
4375:
4372:
4369:
4366:
4363:
4360:
4357:
4353:
4352:
4349:
4346:
4341:
4336:
4328:
4324:
4319:
4316:
4311:
4308:
4297:
4296:
4284:
4270:
4266:
4256:
4251:
4245:
4230:
4220:
4219:
4206:
4196:
4194:
4191:
4190:
4187:
4184:
4181:
4178:
4176:
4173:
4172:
4170:
4165:
4160:
4156:
4152:
4148:
4142:
4138:
4134:
4129:
4126:
4123:
4119:
4115:
4111:
4107:
4104:
4099:
4096:
4093:
4089:
4085:
4071:
4070:
4057:
4047:
4045:
4042:
4041:
4038:
4035:
4032:
4029:
4027:
4024:
4023:
4021:
4016:
4011:
4008:
4005:
4001:
3982:
3981:
3968:
3964:
3960:
3956:
3950:
3946:
3942:
3937:
3934:
3931:
3927:
3923:
3919:
3915:
3912:
3907:
3904:
3901:
3897:
3893:
3890:
3885:
3882:
3879:
3875:
3871:
3868:
3865:
3853:we arrive at:
3851:
3850:
3833:
3829:
3825:
3820:
3816:
3812:
3809:
3806:
3803:
3800:
3798:
3794:
3790:
3786:
3785:
3780:
3777:
3774:
3770:
3766:
3761:
3758:
3755:
3751:
3747:
3742:
3739:
3736:
3732:
3728:
3725:
3723:
3721:
3718:
3717:
3694:
3691:
3686:
3681:
3677:
3648:
3643:
3638:
3634:
3630:
3625:
3621:
3617:
3614:
3611:
3599:
3596:
3595:
3594:
3591:
3590:
3588:
3586:
3583:
3581:
3578:
3577:
3575:
3572:
3570:
3567:
3564:
3563:
3560:
3558:
3555:
3550:
3548:
3544:
3543:
3541:
3538:
3536:
3533:
3530:
3529:
3527:
3525:
3522:
3520:
3511:Hodge diamond:
3484:Enoki surfaces
3476:Inoue surfaces
3467:
3453:Main article:
3450:
3447:
3436:
3435:
3432:
3431:
3429:
3427:
3424:
3422:
3419:
3418:
3416:
3413:
3411:
3408:
3405:
3404:
3401:
3399:
3396:
3394:
3390:
3389:
3387:
3384:
3382:
3379:
3376:
3375:
3373:
3371:
3368:
3366:
3357:Hodge diamond:
3304:
3301:
3289:cubic surfaces
3280:
3276:
3259:
3258:
3255:
3254:
3252:
3250:
3247:
3245:
3242:
3241:
3239:
3236:
3234:
3231:
3228:
3227:
3224:
3221:
3219:
3216:
3214:
3210:
3209:
3207:
3204:
3202:
3199:
3196:
3195:
3193:
3191:
3188:
3186:
3180:
3177:
3176:
3174:
3172:
3169:
3167:
3164:
3163:
3161:
3158:
3156:
3153:
3150:
3149:
3146:
3143:
3141:
3138:
3136:
3132:
3131:
3129:
3126:
3124:
3121:
3118:
3117:
3115:
3113:
3110:
3108:
3099:Hodge diamond:
3082:
3070:
3052:
3038:
3016:
3013:
2985:
2982:
2970:
2967:
2942:
2939:
2922:Every surface
2911:
2870:
2867:
2850:
2812:
2809:
2808:
2807:
2785:
2782:
2779:
2776:
2773:
2768:
2764:
2760:
2755:
2750:
2746:
2729:
2723:
2722:
2721:
2720:
2707:
2702:
2699:
2694:
2690:
2686:
2681:
2677:
2673:
2672:
2667:
2663:
2659:
2654:
2650:
2646:
2641:
2637:
2633:
2632:
2630:
2615:
2614:
2602:
2597:
2593:
2570:
2566:
2554:
2553:
2552:
2551:
2540:
2535:
2532:
2529:
2525:
2519:
2515:
2511:
2508:
2505:
2500:
2497:
2494:
2490:
2486:
2483:
2480:
2477:
2474:
2471:
2468:
2455:
2454:
2442:
2426:
2425:
2414:
2411:
2406:
2402:
2398:
2393:
2388:
2384:
2380:
2374:
2371:
2350:
2349:
2348:
2337:
2334:
2329:
2326:
2323:
2319:
2315:
2310:
2307:
2304:
2300:
2296:
2293:
2290:
2287:
2284:
2279:
2275:
2271:
2268:
2255:
2254:
2240:
2239:
2238:
2237:
2226:
2221:
2218:
2215:
2211:
2207:
2202:
2199:
2196:
2192:
2188:
2185:
2182:
2177:
2173:
2169:
2164:
2160:
2146:
2145:
2135:
2134:
2133:
2132:
2121:
2116:
2112:
2108:
2103:
2100:
2097:
2093:
2089:
2084:
2081:
2078:
2074:
2070:
2065:
2061:
2047:
2046:
2036:
2035:
2034:
2033:
2022:
2017:
2014:
2011:
2007:
2003:
2000:
1987:
1986:
1975:Picard variety
1964:
1963:
1962:
1961:
1950:
1945:
1941:
1937:
1932:
1928:
1924:
1919:
1915:
1911:
1906:
1902:
1898:
1893:
1889:
1885:
1882:
1869:
1868:
1855:
1854:
1843:
1842:
1841:
1828:
1821:
1818:
1815:
1811:
1807:
1802:
1799:
1796:
1792:
1788:
1783:
1780:
1777:
1773:
1769:
1764:
1760:
1756:
1755:
1750:
1747:
1744:
1740:
1736:
1731:
1728:
1725:
1721:
1717:
1712:
1709:
1706:
1702:
1698:
1693:
1690:
1687:
1683:
1679:
1674:
1670:
1666:
1661:
1657:
1653:
1652:
1649:
1646:
1641:
1637:
1633:
1628:
1624:
1620:
1619:
1617:
1602:
1601:
1590:
1587:
1584:
1581:
1578:
1575:
1572:
1569:
1566:
1561:
1557:
1553:
1550:
1547:
1542:
1538:
1516:
1513:
1512:
1511:
1510:
1509:
1498:
1495:
1490:
1487:
1484:
1480:
1457:
1454:
1451:
1447:
1424:
1421:
1418:
1414:
1402:
1388:
1385:
1382:
1378:
1374:
1369:
1366:
1363:
1359:
1329:
1326:
1323:
1319:
1294:
1291:
1286:
1283:
1280:
1276:
1272:
1267:
1264:
1261:
1257:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1212:
1208:
1203:
1200:
1197:
1193:
1178:
1177:
1176:
1157:
1154:
1151:
1147:
1143:
1140:
1134:
1131:
1128:
1124:
1120:
1115:
1112:
1109:
1105:
1101:
1099:
1094:
1091:
1088:
1084:
1080:
1075:
1072:
1069:
1065:
1061:
1056:
1053:
1050:
1046:
1042:
1041:
1035:
1032:
1029:
1025:
1021:
1016:
1013:
1010:
1006:
1002:
1000:
993:
990:
987:
983:
979:
976:
963:
962:
935:
931:
910:
907:
902:
898:
894:
891:
888:
883:
879:
875:
872:
869:
864:
861:
858:
854:
842:
841:
840:
839:
824:
821:
818:
815:
812:
804:
801:
796:
792:
788:
785:
783:
781:
778:
775:
772:
771:
768:
765:
762:
759:
756:
748:
745:
740:
736:
732:
729:
727:
725:
722:
719:
716:
715:
712:
709:
704:
700:
696:
693:
691:
689:
686:
683:
680:
679:
676:
673:
668:
664:
660:
657:
655:
653:
650:
647:
644:
641:
640:
627:
626:
612:
607:
603:
599:
596:
593:
590:
568:
564:
541:
537:
532:
526:
522:
501:
498:
472:
461:
460:
423:
420:
417:
414:
411:
406:
402:
398:
393:
389:
385:
382:
379:
374:
370:
358:
357:
334:
331:
329:
326:
322:Picard variety
309:
304:
299:
294:
290:
269:
266:
263:
258:
254:
188:
185:
98:
97:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
6621:
6610:
6607:
6605:
6602:
6600:
6597:
6595:
6592:
6591:
6589:
6579:
6576:
6575:
6571:
6566:
6562:
6558:
6554:
6551:
6547:
6543:
6539:
6535:
6530:
6527:
6523:
6519:
6513:
6509:
6505:
6501:
6497:
6493:
6489:
6485:
6482:
6478:
6474:
6470:
6466:
6464:9780821811450
6460:
6456:
6452:
6448:
6444:
6439:
6434:
6430:
6426:
6422:
6418:
6415:
6411:
6407:
6403:
6399:
6393:
6389:
6385:
6381:
6377:
6373:
6370:
6366:
6362:
6358:
6354:
6350:
6346:
6342:
6341:
6335:
6332:
6328:
6324:
6320:
6316:
6312:
6308:
6304:
6303:
6297:
6294:
6290:
6285:
6280:
6276:
6272:
6268:
6264:
6260:
6256:
6252:
6248:
6247:
6241:
6238:
6234:
6230:
6226:
6222:
6218:
6214:
6210:
6209:
6204:
6200:
6197:
6193:
6189:
6185:
6181:
6178:
6174:
6170:
6166:
6163:
6159:
6155:
6151:
6147:
6143:
6139:
6135:
6131:
6127:
6126:
6118:
6114:
6110:
6106:
6100:
6098:9780521217774
6094:
6090:
6086:
6082:
6081:
6075:
6074:
6072:
6068:
6064:
6060:
6056:
6052:
6049:
6045:
6040:
6036:
6032:
6026:
6022:
6018:
6014:
6010:
6006:
6002:
5998:
5994:
5990:
5984:
5980:
5976:
5972:
5967:
5966:
5962:
5958:
5955:
5954:
5950:
5948:
5947:, and so on.
5946:
5942:
5938:
5934:
5930:
5926:
5909:
5904:
5896:
5893:
5888:
5884:
5880:
5875:
5870:
5866:
5858:
5843:
5840:
5837:
5834:
5829:
5825:
5821:
5816:
5811:
5807:
5803:
5796:
5793:
5775:
5771:
5767:
5764:
5759:
5754:
5750:
5742:
5728:
5725:
5720:
5716:
5712:
5707:
5702:
5698:
5690:
5689:
5688:
5686:
5682:
5677:
5665:
5661:
5655:
5647:
5645:
5643:
5639:
5635:
5631:
5627:
5623:
5610:
5607:
5602:
5598:
5594:
5591:
5588:
5583:
5578:
5574:
5566:
5562:
5560:
5556:
5552:
5548:
5544:
5540:
5536:
5532:
5513:
5505:
5500:
5498:
5493:
5491:
5487:
5485:
5480:
5476:
5472:
5468:
5464:
5456:
5454:
5453:for details.
5452:
5441:
5439:
5436:
5434:
5432:
5431:
5428:
5425:
5423:
5420:
5418:
5417:
5413:
5411:
5408:
5406:
5403:
5402:
5399:
5396:
5394:
5391:
5389:
5388:
5385:
5383:
5380:
5378:
5376:
5375:
5372:
5371:
5370:
5369:
5365:
5363:
5359:
5355:
5351:
5348: +
5347:
5343:
5339:
5335:
5331:
5327:
5324: +
5323:
5319:
5315:
5311:
5307:
5303:
5300: +
5299:
5295:
5291:
5287:
5281:
5273:
5271:
5270:for details.
5269:
5264:
5255:
5253:
5250:
5248:
5246:
5245:
5242:
5239:
5237:
5234:
5232:
5231:
5227:
5225:
5222:
5220:
5217:
5216:
5213:
5210:
5208:
5205:
5203:
5202:
5199:
5197:
5194:
5192:
5190:
5189:
5186:
5185:
5184:
5183:
5179:
5173:
5169:
5165:
5161:
5157:
5153:
5146:
5142:
5140:
5136:
5130:
5122:
5120:
5118:
5115:
5111:
5107:
5097:
5095:
5092:
5090:
5088:
5087:
5084:
5081:
5079:
5076:
5074:
5073:
5069:
5066:
5064:
5061:
5059:
5056:
5055:
5052:
5049:
5047:
5044:
5042:
5041:
5038:
5036:
5033:
5031:
5029:
5028:
5025:
5019:
5017:
5014:
5012:
5010:
5009:
5006:
5003:
5001:
4998:
4996:
4995:
4991:
4988:
4986:
4983:
4981:
4978:
4977:
4974:
4971:
4969:
4966:
4964:
4963:
4960:
4958:
4955:
4953:
4951:
4950:
4947:
4946:
4945:
4944:
4940:
4938:
4934:
4930:
4923:
4918:
4915:
4911:
4905:
4897:
4895:
4893:
4889:
4879:
4877:
4874:
4872:
4870:
4869:
4866:
4863:
4861:
4858:
4856:
4855:
4851:
4849:
4846:
4844:
4841:
4840:
4837:
4834:
4832:
4829:
4827:
4826:
4823:
4821:
4818:
4816:
4814:
4813:
4810:
4809:
4808:
4807:
4803:
4801:
4797:
4793:
4789:
4785:
4781:
4777:
4775:
4771:
4767:
4763:
4755:
4753:
4751:
4747:
4743:
4735:
4727:
4723:
4719:
4715:
4712:
4709:
4705:
4701:
4700:
4699:
4697:
4689:
4687:
4684:
4682:
4680:
4679:
4676:
4673:
4671:
4668:
4666:
4665:
4661:
4659:
4656:
4654:
4651:
4650:
4647:
4644:
4642:
4639:
4637:
4636:
4633:
4631:
4628:
4626:
4624:
4623:
4620:
4619:
4615:
4609:
4605:
4601:
4597:
4593:
4589:
4587:
4583:
4575:
4569:
4565:
4562:
4559:
4556:
4553:
4550:
4547:
4544:
4543:
4540:
4536:
4533:
4530:
4527:
4524:
4521:
4518:
4515:
4514:
4510:
4507:
4505:
4503:
4500:
4497:
4494:
4491:
4490:
4486:
4483:
4480:
4477:
4474:
4471:
4468:
4465:
4464:
4460:
4456:
4453:
4450:
4447:
4444:
4441:
4438:
4435:
4434:
4430:
4427:
4425:
4423:
4420:
4417:
4414:
4411:
4410:
4406:
4403:
4400:
4397:
4394:
4391:
4388:
4385:
4384:
4380:
4376:
4373:
4370:
4367:
4364:
4361:
4358:
4355:
4354:
4350:
4347:
4345:
4342:
4340:
4337:
4335:
4331:
4325:
4323:
4320:
4315:
4312:
4307:
4304:
4303:
4300:
4294:
4290:
4283:
4279:
4275:
4271:
4265:
4261:
4257:
4254:
4244:
4240:
4236:
4235:
4234:
4229:
4225:
4192:
4185:
4182:
4179:
4174:
4168:
4163:
4158:
4154:
4150:
4146:
4140:
4136:
4132:
4127:
4124:
4121:
4117:
4113:
4109:
4105:
4102:
4097:
4094:
4091:
4087:
4083:
4076:
4075:
4074:
4043:
4036:
4033:
4030:
4025:
4019:
4014:
4009:
4006:
4003:
3999:
3991:
3990:
3989:
3988:= 0 we have:
3987:
3966:
3962:
3958:
3954:
3948:
3944:
3940:
3935:
3932:
3929:
3925:
3921:
3917:
3913:
3910:
3905:
3902:
3899:
3895:
3891:
3888:
3883:
3880:
3877:
3873:
3869:
3866:
3863:
3856:
3855:
3854:
3831:
3827:
3823:
3818:
3814:
3810:
3807:
3804:
3801:
3799:
3792:
3788:
3778:
3775:
3772:
3768:
3764:
3759:
3756:
3753:
3749:
3745:
3740:
3737:
3734:
3730:
3726:
3724:
3719:
3708:
3707:
3706:
3692:
3689:
3684:
3679:
3675:
3666:
3662:
3646:
3641:
3636:
3632:
3628:
3623:
3619:
3615:
3612:
3609:
3597:
3589:
3587:
3584:
3582:
3580:
3579:
3576:
3573:
3571:
3568:
3566:
3565:
3561:
3559:
3554:
3551:
3549:
3546:
3545:
3542:
3539:
3537:
3534:
3532:
3531:
3528:
3526:
3523:
3521:
3519:
3518:
3515:
3514:
3513:
3512:
3508:
3506:
3502:
3499:
3495:
3493:
3489:
3488:Kato surfaces
3485:
3481:
3477:
3473:
3472:Hopf surfaces
3466:
3462:
3456:
3448:
3446:
3444:
3440:
3430:
3428:
3425:
3423:
3421:
3420:
3417:
3414:
3412:
3409:
3407:
3406:
3402:
3400:
3397:
3395:
3392:
3391:
3388:
3385:
3383:
3380:
3378:
3377:
3374:
3372:
3369:
3367:
3365:
3364:
3361:
3360:
3359:
3358:
3354:
3352:
3348:
3346:
3342:
3338:
3334:
3330:
3326:
3322:
3318:
3314:
3310:
3302:
3300:
3298:
3294:
3290:
3286:
3274:
3270:
3266:
3263:
3253:
3251:
3248:
3246:
3244:
3243:
3240:
3237:
3235:
3232:
3230:
3229:
3225:
3222:
3220:
3217:
3215:
3212:
3211:
3208:
3205:
3203:
3200:
3198:
3197:
3194:
3192:
3189:
3187:
3185:
3184:
3181:
3175:
3173:
3170:
3168:
3166:
3165:
3162:
3159:
3157:
3154:
3152:
3151:
3147:
3144:
3142:
3139:
3137:
3134:
3133:
3130:
3127:
3125:
3122:
3120:
3119:
3116:
3114:
3111:
3109:
3107:
3106:
3103:
3102:
3101:
3100:
3096:
3094:
3090:
3088:
3080:
3076:
3068:
3064:
3060:
3055:
3050:
3046:
3041:
3036:
3032:
3028:
3025:
3021:
3014:
3012:
3009:
3007:
3003:
2999:
2980:
2968:
2966:
2964:
2960:
2956:
2937:
2929:
2925:
2920:
2918:
2914:
2907:
2903:
2899:
2895:
2891:
2887:
2883:
2879:
2874:
2868:
2866:
2864:
2860:
2856:
2848:
2847:
2842:
2841:Picard number
2838:
2834:
2833:
2828:
2824:
2820:
2819:
2810:
2805:
2804:Chern classes
2801:
2800:
2799:Chern numbers
2783:
2780:
2777:
2774:
2771:
2766:
2762:
2758:
2753:
2748:
2744:
2735:
2728:
2725:
2724:
2700:
2697:
2692:
2688:
2684:
2679:
2675:
2665:
2661:
2657:
2652:
2648:
2644:
2639:
2635:
2628:
2619:
2618:
2617:
2616:
2600:
2595:
2591:
2568:
2564:
2556:
2555:
2538:
2533:
2530:
2527:
2523:
2517:
2509:
2506:
2498:
2495:
2492:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2459:
2458:
2457:
2456:
2440:
2432:
2428:
2427:
2412:
2404:
2400:
2396:
2391:
2386:
2382:
2372:
2369:
2359:
2355:
2351:
2335:
2332:
2327:
2324:
2321:
2317:
2313:
2308:
2305:
2302:
2298:
2294:
2291:
2288:
2285:
2282:
2277:
2273:
2269:
2266:
2259:
2258:
2257:
2256:
2252:
2248:
2247:
2242:
2241:
2224:
2219:
2216:
2213:
2209:
2205:
2200:
2197:
2194:
2190:
2186:
2183:
2180:
2175:
2171:
2167:
2162:
2158:
2150:
2149:
2148:
2147:
2143:
2142:
2137:
2136:
2119:
2114:
2110:
2106:
2101:
2098:
2095:
2091:
2087:
2082:
2079:
2076:
2072:
2068:
2063:
2059:
2051:
2050:
2049:
2048:
2044:
2043:
2038:
2037:
2020:
2015:
2012:
2009:
2005:
2001:
1998:
1991:
1990:
1989:
1988:
1984:
1980:
1976:
1972:
1971:
1966:
1965:
1948:
1943:
1939:
1935:
1930:
1926:
1922:
1917:
1913:
1909:
1904:
1900:
1896:
1891:
1887:
1883:
1880:
1873:
1872:
1871:
1870:
1866:
1862:
1861:
1857:
1856:
1852:
1848:
1844:
1819:
1816:
1813:
1809:
1805:
1800:
1797:
1794:
1790:
1786:
1781:
1778:
1775:
1771:
1767:
1762:
1758:
1748:
1745:
1742:
1738:
1734:
1729:
1726:
1723:
1719:
1715:
1710:
1707:
1704:
1700:
1696:
1691:
1688:
1685:
1681:
1677:
1672:
1668:
1664:
1659:
1655:
1647:
1644:
1639:
1635:
1631:
1626:
1622:
1615:
1606:
1605:
1604:
1603:
1588:
1585:
1582:
1579:
1576:
1573:
1567:
1559:
1555:
1551:
1548:
1545:
1540:
1536:
1528:: defined by
1527:
1526:
1525:Betti numbers
1522:
1521:
1520:
1514:
1496:
1493:
1488:
1485:
1482:
1478:
1455:
1452:
1449:
1445:
1422:
1419:
1416:
1412:
1403:
1386:
1383:
1380:
1376:
1372:
1367:
1364:
1361:
1357:
1348:
1344:
1343:
1327:
1324:
1321:
1317:
1308:
1292:
1289:
1284:
1281:
1278:
1274:
1270:
1265:
1262:
1259:
1255:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1210:
1206:
1201:
1198:
1195:
1191:
1183:
1182:Serre duality
1179:
1155:
1152:
1149:
1145:
1132:
1129:
1126:
1122:
1113:
1110:
1107:
1103:
1092:
1089:
1086:
1082:
1073:
1070:
1067:
1063:
1054:
1051:
1048:
1044:
1033:
1030:
1027:
1023:
1014:
1011:
1008:
1004:
991:
988:
985:
981:
967:
966:
965:
964:
960:
959:
958:Hodge numbers
954:
951:
933:
908:
900:
892:
889:
881:
877:
873:
870:
867:
862:
859:
856:
852:
844:
843:
822:
819:
816:
813:
810:
802:
799:
794:
790:
784:
779:
776:
773:
766:
763:
760:
757:
754:
746:
743:
738:
734:
728:
723:
720:
717:
710:
707:
702:
698:
692:
687:
684:
681:
674:
671:
666:
662:
656:
648:
645:
642:
631:
630:
629:
628:
610:
605:
601:
597:
594:
591:
588:
566:
562:
539:
535:
530:
524:
520:
496:
488:
487:
470:
463:
462:
458:
453:
449:
445:
441:
437:
421:
418:
415:
412:
404:
400:
391:
387:
383:
380:
377:
372:
368:
360:
359:
355:
351:
348:
347:
346:
344:
340:
332:
327:
325:
323:
292:
288:
256:
252:
243:
239:
235:
234:Oscar Zariski
231:
229:
226:that are not
225:
224:group schemes
221:
217:
213:
208:
206:
201:
193:
186:
184:
182:
178:
174:
170:
166:
162:
158:
154:
150:
146:
142:
138:
134:
130:
126:
122:
120:
116:
113:
109:
105:
94:
91:
83:
73:
70:; convert to
69:
64:
59:
54:
52:
49:they are not
46:
41:
32:
31:
19:
6537:
6503:
6499:
6424:
6379:
6344:
6338:
6309:(1): 55–83,
6306:
6300:
6250:
6244:
6212:
6206:
6187:
6176:
6172:
6129:
6123:
6079:
6062:
6008:
5970:
5932:
5928:
5927:
5924:
5684:
5683:
5675:
5663:
5657:
5641:
5637:
5633:
5629:
5625:
5624:
5564:
5563:
5558:
5554:
5501:
5496:
5494:
5483:
5470:
5466:
5460:
5448:
5367:
5366:
5361:
5357:
5353:
5349:
5345:
5341:
5337:
5333:
5329:
5325:
5321:
5317:
5313:
5309:
5305:
5301:
5297:
5293:
5289:
5285:
5283:
5265:
5262:
5181:
5180:
5167:
5163:
5159:
5155:
5148:
5144:
5143:
5134:
5132:
5116:
5109:
5105:
5104:
4942:
4941:
4936:
4932:
4925:
4921:
4919:
4913:
4909:
4907:
4891:
4887:
4886:
4805:
4804:
4799:
4795:
4791:
4787:
4783:
4779:
4778:
4764:include the
4762:complex tori
4759:
4749:
4745:
4741:
4733:
4731:
4725:
4721:
4717:
4707:
4703:
4695:
4694:
4617:
4616:
4603:
4599:
4595:
4591:
4590:
4581:
4579:
4457:Any. Always
4377:Any. Always
4343:
4338:
4333:
4326:
4321:
4313:
4305:
4298:
4292:
4288:
4281:
4277:
4263:
4259:
4249:
4242:
4238:
4227:
4223:
4222:In general 2
4221:
4072:
3985:
3983:
3852:
3660:
3601:
3552:
3510:
3509:
3504:
3500:
3497:
3496:
3464:
3458:
3442:
3438:
3437:
3356:
3355:
3350:
3349:
3344:
3340:
3336:
3332:
3328:
3324:
3320:
3316:
3312:
3308:
3306:
3272:
3268:
3264:
3261:
3260:
3098:
3097:
3092:
3091:
3086:
3078:
3074:
3066:
3062:
3061:bundle over
3058:
3053:
3048:
3044:
3039:
3030:
3026:
3018:
3010:
3001:
2997:
2972:
2962:
2958:
2954:
2927:
2923:
2921:
2909:
2905:
2901:
2893:
2885:
2881:
2875:
2872:
2857:such as the
2844:
2836:
2830:
2822:
2818:Picard group
2816:
2814:
2797:
2733:
2726:
2430:
2250:
2244:
2139:
2040:
1982:
1970:irregularity
1968:
1865:Euler number
1864:
1858:
1846:
1523:
1518:
956:
952:
484:
435:
349:
336:
232:
212:Mumford 1969
209:
202:
198:
123:
119:moduli space
114:
107:
101:
86:
77:
48:
6421:Reid, Miles
6132:: 197–232,
5685:Invariants:
5565:Invariants:
5549:, and some
5543:K3 surfaces
5145:Invariants:
5070:(Secondary)
4780:Invariants:
4592:Invariants:
4576:K3 surfaces
3498:Invariants:
3351:Invariants:
3093:Invariants:
950:holomorphic
448:John Morgan
438:. They are
436:plurigenera
343:plurigenera
125:Max Noether
104:mathematics
6588:Categories
5963:References
3347:has two).
2884:is called
2878:blowing up
2855:4-manifold
2358:Todd genus
1307:cohomology
440:birational
80:March 2022
68:deprecated
6510:: 1–215,
6481:116933286
6162:122816845
5929:Examples:
5894:≡
5841:⩾
5822:−
5765:⩽
5608:⩾
5517:∞
5514:−
5154:are 1 if
5106:Examples:
4992:(Primary)
4931:are 1 if
4888:Examples:
4348:Surfaces
4269:= 0 or 1.
4199:otherwise
4133:−
4050:otherwise
3941:−
3808:−
3746:−
3720:χ
3663:has zero
3613:χ
3439:Examples:
3262:Examples:
2984:∞
2981:−
2941:∞
2938:−
2781:−
2778:χ
2701:τ
2693:−
2685:−
2653:−
2569:±
2507:−
2489:∑
2479:−
2476:χ
2467:τ
2441:τ
2431:signature
2314:−
2283:−
2267:χ
2206:−
2181:−
1923:−
1897:−
1586:⩽
1580:⩽
1552:
1494:−
1230:−
1218:−
930:Ω
897:Ω
874:
814:⋅
787:⟷
774:κ
758:⋅
731:⟷
718:κ
695:⟷
682:κ
659:⟷
652:∞
649:−
643:κ
592:⋅
540:κ
500:∞
497:−
471:κ
419:⩾
384:
265:Ω
6563: :
6293:16578569
6186:(1949),
6115:(1976),
6007:(1996),
5951:See also
5626:Example:
5340:,
5332:,
5316:,
5292:,
4696:Examples
4287:= 0 and
3285:quadrics
2796:are the
1977:and the
489:: it is
6550:0521772
6526:0190143
6473:1442522
6443:Bibcode
6414:0254053
6369:0239114
6361:2373289
6331:0228019
6323:2373426
6275:0205280
6267:2373150
6237:0187255
6229:2373157
6196:0031770
6154:0491720
6134:Bibcode
6071:0491719
6039:1406314
5997:2030225
5545:, some
5541:, some
4774:isogeny
4501:0 or 1
4498:1 or 2
4351:Fields
3081:, and Σ
3057:is the
3047:= 0 or
2906:minimal
2886:minimal
1437:equals
483:is the
352:is the
167: (
147: (
135: (
112:compact
110:groups
58:improve
56:Please
6548:
6524:
6514:
6479:
6471:
6461:
6412:
6404:
6394:
6367:
6359:
6329:
6321:
6291:
6284:300219
6281:
6273:
6265:
6235:
6227:
6194:
6160:
6152:
6095:
6069:
6046:
6037:
6027:
5995:
5985:
5537:, all
5533:, all
5488:) and
5139:Kähler
4734:marked
4568:Kähler
4539:Kähler
4459:Kähler
4379:Kähler
3705:Using
3490:. The
3461:Kähler
1347:Kähler
921:where
106:, the
47:, but
6477:S2CID
6433:arXiv
6402:JSTOR
6357:JSTOR
6319:JSTOR
6263:JSTOR
6225:JSTOR
6158:S2CID
6120:(PDF)
5790:(the
5356:, or
3667:, so
3503:= 1,
1349:then
228:étale
161:1968b
157:1968a
6512:ISBN
6459:ISBN
6392:ISBN
6289:PMID
6093:ISBN
6044:ISBN
6025:ISBN
5983:ISBN
5726:>
5674:and
4738:3,19
4612:3,19
4272:For
3043:for
2861:and
2821:Pic(
2736:and
2429:The
2243:The
2138:The
2039:The
1967:The
1247:and
820:>
800:>
744:>
581:and
220:1977
216:1976
181:1977
177:1976
169:1969
153:1966
149:1964
141:1949
137:1914
6561:doi
6451:doi
6384:doi
6349:doi
6311:doi
6279:PMC
6255:doi
6217:doi
6142:doi
6085:doi
6042:; (
6017:doi
5975:doi
5901:mod
5628:If
5461:An
5176:1,9
4740:to
4724:→ −
4412:10
4401:10
4386:10
4374:K3
4371:20
4356:22
3474:or
3275:= Σ
2917:nef
2915:is
2835:NS(
2613:so:
2352:By
1863:or
1549:dim
1470:or
1180:By
871:dim
381:dim
324:).
102:In
6590::
6546:MR
6536:,
6522:MR
6520:,
6504:75
6502:,
6494:;
6475:,
6469:MR
6467:,
6457:,
6449:,
6441:,
6410:MR
6408:,
6400:,
6390:,
6365:MR
6363:,
6355:,
6345:90
6343:,
6327:MR
6325:,
6317:,
6307:90
6305:,
6287:,
6277:,
6271:MR
6269:,
6261:,
6251:88
6249:,
6233:MR
6231:,
6223:,
6213:86
6211:,
6192:MR
6177:23
6175:,
6156:,
6150:MR
6148:,
6140:,
6130:35
6128:,
6122:,
6111:;
6091:.
6067:MR
6057:;
6035:MR
6033:,
6023:,
6015:,
5993:MR
5991:,
5981:,
5943:,
5939:,
5910:2.
5838:36
5611:0.
5360:/6
5352:/2
5344:/4
5336:/4
5328:/3
5320:/3
5312:/3
5308:,
5304:/2
5296:/2
5288:/2
5223:10
5174:II
5166:,
4802:.
4794:×
4790:×
4786:×
4748:,
4732:A
4698::
4657:20
4610:II
4602:,
4560:0
4557:0
4554:0
4551:1
4548:1
4545:0
4531:2
4528:1
4525:1
4522:2
4519:3
4516:4
4495:2
4492:2
4481:2
4478:1
4475:0
4472:1
4469:2
4466:2
4451:4
4448:2
4445:1
4442:2
4439:4
4436:6
4421:1
4418:1
4415:0
4398:0
4395:0
4392:0
4389:0
4368:0
4365:1
4362:0
4359:0
4332:=
4291:=
4280:−
4262:−
4241:−
4226:≥
4193:10
4175:22
3870:12
3864:10
3693:0.
3610:12
3482:,
3445:.
3343:×
3335:×
3323:×
3295:,
3291:,
3287:,
3283:,
3271:×
3267:,
3077:×
2957:×
2865:.
2775:12
2732:=
2373:12
2336:1.
1589:4.
1497:1.
1293:1.
795:12
739:12
703:12
667:12
567:12
230:.
218:,
179:,
159:,
155:,
151:,
139:,
6453::
6445::
6435::
6386::
6351::
6313::
6257::
6219::
6144::
6136::
6101:.
6087::
6019::
5977::
5933:P
5905:1
5897:0
5889:2
5885:c
5881:+
5876:2
5871:1
5867:c
5844:0
5835:+
5830:2
5826:c
5817:2
5812:1
5808:c
5804:5
5794:)
5776:2
5772:c
5768:3
5760:2
5755:1
5751:c
5729:0
5721:2
5717:c
5713:,
5708:2
5703:1
5699:c
5679:2
5676:c
5670:1
5664:c
5642:B
5640:×
5638:E
5634:B
5630:E
5603:2
5599:c
5595:,
5592:0
5589:=
5584:2
5579:1
5575:c
5559:B
5555:B
5484:p
5471:B
5467:B
5437:1
5426:1
5421:1
5414:0
5409:2
5404:0
5397:1
5392:1
5381:1
5362:Z
5358:Z
5354:Z
5350:Z
5346:Z
5342:Z
5338:Z
5334:Z
5330:Z
5326:Z
5322:Z
5318:Z
5314:Z
5310:Z
5306:Z
5302:Z
5298:Z
5294:Z
5290:Z
5286:Z
5251:1
5240:0
5235:0
5228:0
5218:0
5211:0
5206:0
5195:1
5168:Z
5164:X
5160:n
5156:n
5151:n
5149:P
5135:q
5117:z
5110:Z
5093:1
5082:0
5077:1
5067:0
5062:0
5057:0
5050:1
5045:0
5034:1
5015:1
5004:1
4999:2
4989:1
4984:2
4979:1
4972:2
4967:1
4956:1
4937:k
4933:n
4928:n
4926:P
4922:k
4892:C
4875:1
4864:2
4859:2
4852:1
4847:4
4842:1
4835:2
4830:2
4819:1
4800:Z
4796:S
4792:S
4788:S
4784:S
4750:Z
4746:X
4744:(
4742:H
4726:a
4722:a
4710:)
4708:C
4706:(
4704:P
4685:1
4674:0
4669:0
4662:1
4652:1
4645:0
4640:0
4629:1
4604:Z
4600:X
4598:(
4596:H
4582:q
4344:h
4339:h
4334:h
4329:g
4327:p
4322:h
4317:1
4314:b
4309:2
4306:b
4295:.
4293:h
4289:h
4285:1
4282:b
4278:h
4276:2
4267:1
4264:b
4260:h
4255:.
4252:g
4250:p
4246:1
4243:b
4239:h
4231:1
4228:b
4224:h
4186:0
4183:=
4180:K
4169:{
4164:=
4159:2
4155:b
4151:+
4147:)
4141:1
4137:b
4128:1
4125:,
4122:0
4118:h
4114:2
4110:(
4106:2
4103:+
4098:1
4095:,
4092:0
4088:h
4084:8
4044:0
4037:0
4034:=
4031:K
4026:1
4020:{
4015:=
4010:2
4007:,
4004:0
4000:h
3986:κ
3967:2
3963:b
3959:+
3955:)
3949:1
3945:b
3936:1
3933:,
3930:0
3926:h
3922:2
3918:(
3914:2
3911:+
3906:1
3903:,
3900:0
3896:h
3892:8
3889:=
3884:2
3881:,
3878:0
3874:h
3867:+
3832:2
3828:b
3824:+
3819:1
3815:b
3811:2
3805:2
3802:=
3793:2
3789:c
3779:2
3776:,
3773:0
3769:h
3765:+
3760:1
3757:,
3754:0
3750:h
3741:0
3738:,
3735:0
3731:h
3727:=
3690:=
3685:2
3680:1
3676:c
3661:K
3647:.
3642:2
3637:1
3633:c
3629:+
3624:2
3620:c
3616:=
3585:1
3574:0
3569:1
3562:0
3556:2
3553:b
3547:0
3540:1
3535:0
3524:1
3505:h
3501:q
3468:2
3465:b
3443:P
3426:1
3415:g
3410:g
3403:0
3398:2
3393:0
3386:g
3381:g
3370:1
3345:P
3341:P
3337:P
3333:P
3329:C
3325:C
3321:P
3317:P
3313:g
3309:g
3281:n
3277:0
3273:P
3269:P
3265:P
3249:1
3238:0
3233:0
3223:0
3218:2
3213:0
3206:0
3201:0
3190:1
3171:1
3160:0
3155:0
3145:0
3140:1
3135:0
3128:0
3123:0
3112:1
3087:P
3083:1
3079:P
3075:P
3071:0
3067:n
3063:P
3059:P
3054:n
3049:n
3045:n
3040:n
3037:Σ
3031:P
3027:P
3002:q
2998:q
2963:P
2959:P
2955:P
2928:X
2924:X
2912:X
2910:K
2902:X
2894:X
2882:X
2851:1
2849:π
2837:X
2823:X
2784:e
2772:=
2767:2
2763:K
2759:=
2754:2
2749:1
2745:c
2734:e
2730:2
2727:c
2698:=
2689:b
2680:+
2676:b
2666:2
2662:b
2658:=
2649:b
2645:+
2640:+
2636:b
2629:{
2601:,
2596:2
2592:H
2565:b
2539:.
2534:j
2531:,
2528:i
2524:h
2518:j
2514:)
2510:1
2504:(
2499:j
2496:,
2493:i
2485:=
2482:e
2473:4
2470:=
2453::
2413:.
2410:)
2405:2
2401:c
2397:+
2392:2
2387:1
2383:c
2379:(
2370:1
2333:+
2328:1
2325:,
2322:0
2318:h
2309:2
2306:,
2303:0
2299:h
2295:=
2292:1
2289:+
2286:q
2278:g
2274:p
2270:=
2251:e
2225:.
2220:1
2217:,
2214:0
2210:h
2201:2
2198:,
2195:0
2191:h
2187:=
2184:q
2176:g
2172:p
2168:=
2163:a
2159:p
2144::
2120:.
2115:1
2111:P
2107:=
2102:0
2099:,
2096:2
2092:h
2088:=
2083:2
2080:,
2077:0
2073:h
2069:=
2064:g
2060:p
2045::
2021:.
2016:1
2013:,
2010:0
2006:h
2002:=
1999:q
1983:q
1949:.
1944:4
1940:b
1936:+
1931:3
1927:b
1918:2
1914:b
1910:+
1905:1
1901:b
1892:0
1888:b
1884:=
1881:e
1867::
1847:p
1820:2
1817:,
1814:0
1810:h
1806:+
1801:1
1798:,
1795:1
1791:h
1787:+
1782:0
1779:,
1776:2
1772:h
1768:=
1763:2
1759:b
1749:2
1746:,
1743:1
1739:h
1735:+
1730:1
1727:,
1724:2
1720:h
1716:=
1711:1
1708:,
1705:0
1701:h
1697:+
1692:0
1689:,
1686:1
1682:h
1678:=
1673:3
1669:b
1665:=
1660:1
1656:b
1648:1
1645:=
1640:4
1636:b
1632:=
1627:0
1623:b
1616:{
1583:i
1577:0
1574:,
1571:)
1568:S
1565:(
1560:i
1556:H
1546:=
1541:i
1537:b
1489:1
1486:,
1483:0
1479:h
1456:1
1453:,
1450:0
1446:h
1423:0
1420:,
1417:1
1413:h
1387:i
1384:,
1381:j
1377:h
1373:=
1368:j
1365:,
1362:i
1358:h
1328:1
1325:,
1322:1
1318:h
1290:=
1285:2
1282:,
1279:2
1275:h
1271:=
1266:0
1263:,
1260:0
1256:h
1233:j
1227:2
1224:,
1221:i
1215:2
1211:h
1207:=
1202:j
1199:,
1196:i
1192:h
1156:2
1153:,
1150:2
1146:h
1133:2
1130:,
1127:1
1123:h
1114:1
1111:,
1108:2
1104:h
1093:2
1090:,
1087:0
1083:h
1074:1
1071:,
1068:1
1064:h
1055:0
1052:,
1049:2
1045:h
1034:1
1031:,
1028:0
1024:h
1015:0
1012:,
1009:1
1005:h
992:0
989:,
986:0
982:h
953:i
934:i
909:,
906:)
901:i
893:,
890:X
887:(
882:j
878:H
868:=
863:j
860:,
857:i
853:h
823:0
817:K
811:K
803:1
791:P
780:2
777:=
767:0
764:=
761:K
755:K
747:1
735:P
724:1
721:=
711:1
708:=
699:P
688:0
685:=
675:0
672:=
663:P
646:=
611:2
606:1
602:c
598:=
595:K
589:K
563:P
536:n
531:/
525:n
521:P
459:.
422:1
416:n
413:,
410:)
405:n
401:K
397:(
392:0
388:H
378:=
373:n
369:P
350:K
308:)
303:O
298:(
293:1
289:h
268:)
262:(
257:0
253:h
240:(
93:)
87:(
82:)
78:(
74:.
53:.
20:)
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