Is there a topological Bogomolov--Miyaoka--Yau inequality?

J\'anos Koll\'ar (Princeton Univ)

REVIEW

Is there a topological Bogomolov--Miyaoka--Yau inequality?

Algebraic Geometry · Submitted 24 Feb 2006 · 78 references

Pith's verdict: A useful conjecture-and-examples note; the Section 5 examples are credible, but the w*=1 density claim needs a cleaner proof.

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CitationJ\'anos Koll\'ar (Princeton Univ). "Is there a topological Bogomolov--Miyaoka--Yau inequality?." arXiv:math/0602562 (2006). https://pith.science/paper/6KZ42BJU

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Editor's note

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This is a conjecture-and-examples note, not a proof paper, and it should be read that way. The new smooth BMY inequality (Conjecture 2) is a clean, well-motivated question that connects the Montgomery-Yang problem and H-cobordisms of Seifert fibered 3-manifolds. The genuinely new content is Section 5: the hypersurface family S(a1,...,a4) with general exponents, and the contracted surfaces S* which are rational, have Q-homology of CP2, and ample canonical class for ai >= 4, with K^2 converging to 1. Those are rare objects and worth having.

The main theorems appear sound. The birationality argument via the torus map of degree D is straightforward; the Milnor-Orlik computation for the homology is explicit; the contraction of the two curves C1, C2 is justified by the MMP. Including Hacking's theorem and Wahl's example is a plus. The self-citations are to prior work that is directly relevant; there is no circularity or data fitting.

Soft spots are real but minor. The density claim in Section 38 that w* = 1 happens in at least 75% of 4-tuples is only sketched. The CRT product gives 60.7%; the parity improvement is compressed and contains a typo: 'wi = n mod 2' should be 'Wi = n mod 2' (wi is not yet defined at that point). With that repair, the 0.75 bound for n = 4 checks out, so this is a fixable presentation issue, not a broken argument. Also, the paper never names an explicit 4-tuple with w* = 1 and min ai >= 4; such tuples exist (e.g., (4,4,4,5)), so this is an omission, not a fatal gap. The n = 4 adjunction details are deferred to Section 43 and not fully written out, which is acceptable for a note but worth asking about.

I do not see a load-bearing flaw. Conjecture 2 is a conjecture, and the paper is honest that it does not prove it. The examples support it but do not come close to settling it. The gap between Conjecture 2 and Montgomery-Yang (allowing 4 nonfree orbits in the (2,3,5,n) cases) is clearly acknowledged.

Who is this for? Algebraic geometers working on quotient singularities and rational homology CP2s, and low-dimensional topologists interested in circle actions and H-cobordisms. The note raises more questions than it answers, but the questions are good ones.

Recommendation: send it to peer review. A referee should ask for a cleaner statement of the density argument and an explicit example or two, but the core construction and conjecture deserve to be in the literature.

Referee report

Significance

If the Section 5 construction is fully correct, it yields a striking family of rational homology CP^2's with quotient singularities and ample canonical class whose K^2 accumulates at 1, giving indirect support to the topological Conjecture 2 and showing that the proposed bound is close to sharp. The paper's strengths are its explicit, checkable algebraic computations: the weight equations (38.1), the birationality via torus maps, the adjunction formulas (42.4), and the Milnor–Orlik computation are all stated with enough precision to be verified directly. The conjectures are clearly separated from results, and the links to Montgomery–Yang and H-cobordism questions are informative. However, the abundance of the key examples rests on the w* = 1 density estimate, which is not fully proved, and the n = 4 ampleness and homology arguments are deferred and compressed.

Major objections (3)

[Theorem 39(5)] Theorem 39(5) states that if min{a_i} ≥ n then K_H is ample and (K_H^{n−2}) → 1 as min{a_i} → ∞. The proof, however, computes (K_H^{n−2}) ∼ w*, and w* need not be 1; for example, when n = 4 and a_1 = a_2 = a_3 = a_4 = a, one computes w* = a − 1, so the limit would be unbounded, not 1. The statement must either include the hypothesis w* = 1 or state the convergence to w*. The convergence claim in Example 27(4) should also be tied explicitly to the gcd condition that forces w* = 1. Without this correction, the strongest claimed consequence of the construction is not proved as stated.
[Section 38 and Example 27] The density assertion that the gcd condition in Example 27 holds 'in at least 75% of all cases' is not proved as written. The parity sentence before (38.4) reads 'then wi ≡ n mod 2', but wi is not defined until (38.4); the intended statement is presumably W_i ≡ n mod 2. Even after that correction, the improvement from 0.607 to 0.75 for even n requires an explicit count of the all-odd case: when n = 4 and all a_i are odd, every W_i and D are even, so w* is even and the p = 2 contribution is not handled by the earlier CRT argument. The paper also gives no explicit 4-tuple with w* = 1 and min a_i ≥ 4. Since this density statement carries the existence of the promised surfaces S* with K^2 → 1, please supply a complete proof of the 0.75 bound and at least one explicit family or table of admissible tuples.
[Section 43 and the proof of Theorem 39] The n = 4 case is a chain of deferrals rather than a complete argument. Theorem 39's proof refers to (43) for the minor modifications when the weighted projective space has one-dimensional singular locus and for the precise identification of H_2(H,Q); in turn, (43) asserts 'If w* = 1 then S is a rational surface with 4 quotient singularities ... with H2(S,Q) ≅ Q^3' and 'Note that if a_1,...,a_4 ≥ 4 then K_{S*} is ample' without derivation. The contraction of C_1 and C_2 is justified by a brief extremal-ray sentence, and no details are given for why the canonical class of the contracted surface is ample or why the second Betti number drops from 3 to 1. These points are load-bearing for Example 27 and should be expanded into complete arguments or replaced by precise references.

Minor objections (4)

[Theorem 39(1)] The statement of Theorem 39(1) writes P(w_0,...,w_n); the coordinates are indexed 1 through n, so this should be P(w_1,...,w_n).
[Proof of Theorem 39(5)] The sentence 'In fact it is easy to see that d − ∑ w_i > 0 if min{a_i} ≥ n' is an unproved inequality used for ampleness; a one-line verification would make the proof self-contained.
[Example 21] There is a typo in the phrase 'boun day' near the beginning of Example 21; it should read 'boundary'.
[General presentation] The opening title contains irregular spacing ('Bogomolov–Miy Aoka–Y au'); this is a cosmetic issue but should be cleaned up in the final version.

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The paper

The paper claims

This paper proposes a topological form of the Bogomolov–Miyaoka–Yau inequality for smooth four-manifolds with spherical boundary, and builds algebraic surfaces that come close to the limit.

The reading

The paper asks whether the Bogomolov–Miyaoka–Yau inequality, which for normal projective surfaces with quotient singularities reads $\sum_{s\in\operatorname{Sing} S}(1-1/|\pi_1(L_s)|)\le 3$, is actually a topological statement about smooth compact 4-manifolds whose boundary components are spherical. The proposed Conjecture 2 asserts that for any such $M$ with $H_1(M,\mathbb Z)=0$ and $H_2(M,\mathbb Z)\cong\mathbb Z$, the same sum over the boundary components is $<3$. The paper shows that a proof of this inequality would nearly settle the Montgomery–Yang problem on circle actions on the 5-sphere and would constrain which Seifert-fibered 3-manifolds bound a 4-dimensional homology ball. On the algebraic side it constructs, inside a family of weighted projective hypersurfaces $x_1^{a_1}x_2+\cdots+x_4^{a_4}x_1=0$, rational surfaces with quotient singularities that have the rational homology of $\mathbb{CP}^2$, ample canonical class, and $(K_{S^*})^2$ approaching 1 as the exponents grow.

What carries the argument

The machinery has four pieces. First, the orbifold Euler characteristic $e_{\mathrm{orb}}(S)=e(S)-\sum_s(1-1/|\pi_1(L_s)|)$ turns the algebraic Bogomolov–Miyaoka–Yau inequality into the boundary sum (1.3), and the topological conjecture is the assertion that this sum is a smooth invariant bounded by 3. Second, the correspondence of Theorem 8 between pseudo-free circle actions on rational homology 5-spheres and 4-manifolds with lens-space boundary converts that bound into a statement about nonfree orbit stabilizers. Third, the family $H(a_1,\ldots,a_n)\subset\mathbb P(w_1,\ldots,w_n)$ defined by $\sum x_i^{a_i}x_{i+1}=0$ has weights determined by the linear system $a_iw_i+w_{i+1}=d$, and the ratio condition $w^*=\gcd(W_1,\ldots,W_n)=1$ makes the linear system $|H|$ a birational map to $\mathbb{CP}^{n-2}$ via the torus homomorphism $\eta^*y_i=x_i^{a_i}x_{i+1}$. Fourth, the Milnor–Orlik formula computes the rational homology of these hypersurfaces, and the adjunction formula for log divisors allows the contraction of the two disjoint rational curves in the $n=4$ case.

Extended reading notes

Core claim

The central claim is Conjecture 2: no smooth compact 4-manifold with $H_1(M,\mathbb Z)=0$ and $H_2(M,\mathbb Z)\cong\mathbb Z$ can have spherical boundary components $L_i$ with $\sum_i(1-1/|\pi_1(L_i)|)\ge 3$; in particular such an $M$ has at most five boundary components. The author does not prove this conjecture, but shows it would imply the Montgomery–Yang bound that pseudo-free circle actions on rational homology 5-spheres have at most four nonfree orbits, and he gives a short list of possible exceptional stabilizer patterns. The proved contribution is a family of examples: for exponent tuples with the numerical condition $w^*=1$, the hypersurface $S(a_1,a_2,a_3,a_4)$ is rational, has four quotient singularities, and after contracting two disjoint rational curves becomes $S^*(a_1,\ldots,a_4)$, a rational surface with quotient singularities, rational homology of $\mathbb{CP}^2$, ample canonical class when all $a_i\ge 4$, and $K_{S^*}^2\to 1$. These are, in the paper's view, illustrations that the topological inequality sits at the edge of what algebraic surfaces can do.

Load-bearing premise

The abundance of examples rests on the numerical gcd condition $w^*=1$, which the paper cannot characterize; the supporting claim that this condition, and the similar gcd in Example 27, holds for at least 75 percent of exponent tuples is asserted without proof.

Editorial extensions

If this is right

A proof of Conjecture 2 would nearly settle the Montgomery–Yang problem, reducing it to checking four explicit stabilizer patterns: $k\le 3$ nonfree orbits, $(2,3,5,n)$ with $(n,30)=1$, $(2,3,7,n)$ for $n$ in a short list, or $(2,3,11,13)$.
For Seifert fibered rational homology spheres that are rationally H-cobordant to zero, the same bound would force $\sum(1-1/d_i)<3$; for integral homology spheres $\Sigma(d_1,\ldots,d_k)$ H-cobordant to zero, it would force $k\le 3$.
The constructed surfaces $S^*(a_1,\ldots,a_4)$ would be rational homology $\mathbb{CP}^2$ surfaces with ample canonical class and $K^2$ arbitrarily close to 1, showing that the algebraic inequality (1.3) can be nearly saturated with four quotient singularities.
The bound would be genuinely smooth-topological: with topological homology cells, one can build homology $\mathbb{CP}^2$"s with arbitrarily many singular points, so the smooth hypothesis is essential.
If the $w^*=1$ condition holds, the example family supplies infinitely many rational surfaces that are birational to $\mathbb{CP}^2$ yet have ample canonical class, separating birational type from Kodaira dimension in the singular setting.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The $w^*=1$ condition is a concrete number-theoretic obstruction, so a finite computer search over exponent tuples could determine its actual density and test the paper's asymptotic 0.75 heuristic.
If Conjecture 2 is true, the boundary defect $\sum(1-1/|\pi_1(L_i)|)$ would function as a new smooth invariant of 4-manifolds with spherical boundary, and one could ask how it behaves under surgery and connected sums.
The paper's suggested generalization (2.2) to arbitrary $H_2(M,\mathbb Q)$ could first be tested on algebraic surfaces with larger Picard number, such as the Hacking examples, before being taken as a topological principle.
Applying the same $w^*$ machinery to the $n=3$ case may yield new Seifert fibered rational homology spheres that are rationally H-cobordant to zero, extending the link between algebraic hypersurfaces and the H-cobordism problem.

Original abstract

The aim of this paper is to consider a possible extension of the Bogomolov--Miyaoka--Yau inequality to differentiable orbifolds. The conjectured extension is related to the Montgomery--Yang problem about circle actions on the 5--sphere and also to the H--cobordism of Seifert fibered 3--manifolds. Related conjectures on algebraic surfaces with quotient singularities which have the same rational homology as the projective plane are also considered. Finally we give such examples which are birational to the projective plane yet have ample canonical class.

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What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.17 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.18 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The aim of this paper is to consider a possible extension of the Bogomolov--Miyaoka--Yau inequality to differentiable orbifolds. The conjectured extension is related to the Montgomery--Yang problem about circle actions on the 5--sphere and also to the H--cobordism of Seifert fibered 3--manifolds. Related conjectures on algebraic surfaces with quotient singularities which have the same rational homology as the projective plane are also considered. Finally we give such examples which are birational to the projective plane yet have ample canonical class.

background contribution method result interpretation main claim

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s0	The aim of this paper is to consider a possible extension of the Bogomolov--Miyaoka--Yau inequality to differentiable orbifolds.	contribution 0.76	0.39	0.05	hedged possibility 0.63
s1	The conjectured extension is related to the Montgomery--Yang problem about circle actions on the 5--sphere and also to the H--cobordism of Seifert fibered 3--manifolds.	background 0.76	0.14	0.04	asserted 0.71
s2	Related conjectures on algebraic surfaces with quotient singularities which have the same rational homology as the projective plane are also considered.	contribution 0.75	0.51	0.04	asserted 0.80
s3	Finally we give such examples which are birational to the projective plane yet have ample canonical class.	result 0.65	0.91	0.35	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.51 · evidence from proof or derivation 0.55.

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References

Works cited (78)

[1]

Selman Akbulut and Robion Kirby, Mazur manifolds, Michigan Math. J. 26 (1979), no. 3, 259--284. MR544597 (80h:57004)

View on Pith 1979
[2]

M. Artin, Wildly ramified Z/2 actions in dimension two , Proc. Amer. Math. Soc. 52 (1975), 60--64. MR0374136 (51 \#10336)

View on Pith 1975
[3]

D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365--385. MR0184241 (32 \#1714)

View on Pith 1965
[4]

Gottfried Barthel and Alexandru Dimca, On complex projective hypersurfaces which are homology- P n 's , Singularities (Lille, 1991), London Math. Soc. Lecture Note Ser., vol. 201, Cambridge Univ. Press, Cambridge, 1994, pp. 1--27. MR1295070 (95j:14060)

View on Pith 1991
[5]

Charles P. Boyer, Krzysztof Galicki, and J \'a nos Koll \'a r, E instein metrics on spheres , Annals of Mathematics. to appear 162 (2005), no. 1, 1--24

View on Pith 2005
[6]

F. A. Bogomolov, Holomorphic tensors and vector bundles on projective manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 6, 1227--1287, 1439. MR522939 (80j:14014)

View on Pith 1978
[7]

Egbert Brieskorn, Beispiele zur D ifferentialtopologie von S ingularit\"aten , Invent. Math. 2 (1966), 1--14. 34 \#6788

View on Pith 1966
[8]

, Rationale S ingularit\"aten komplexer F l\"achen , Invent. Math. 4 (1967/1968), 336--358. 36 \#5136

View on Pith 1967
[9]

Andrew J. Casson and John L. Harer, Some homology lens spaces which bound rational homology balls, Pacific J. Math. 96 (1981), no. 1, 23--36. MR634760 (83h:57013)

View on Pith 1981
[10]

Alessio Corti, Adjunction of log divisors, Flips and abundance for algebraic threefolds (J \'a nos Koll \'a r, ed.), Soci\'et\'e Math\'ematique de France, 1992, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Ast\'erisque No. 211 (1992), pp. 171--182. 94f:14013

View on Pith 1992
[11]

Michel Demazure, Anneaux gradu\'es normaux, Introduction \`a la th\'eorie des singularit\'es, II, Travaux en Cours, vol. 37, Hermann, Paris, 1988, pp. 35--68. 91k:14004

View on Pith 1988
[12]

I. V. Dolgachev, Automorphic forms, and quasihomogeneous singularities, Funkcional. Anal. i Prilo zen. 9 (1975), no. 2, 67--68. 58 \#27958

View on Pith 1975
[13]

Daniel Daigle and Peter Russell, Affine rulings of normal rational surfaces, Osaka J. Math. 38 (2001), no. 1, 37--100. MR1824902 (2002g:14096)

View on Pith 2001
[14]

, On weighted projective planes and their affine rulings, Osaka J. Math. 38 (2001), no. 1, 101--150. MR1824903 (2002h:14062)

View on Pith 2001
[15]

, On log Q -homology planes and weighted projective planes , Canad. J. Math. 56 (2004), no. 6, 1145--1189. MR2102629 (2005i:14077)

View on Pith 2004
[16]

Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no. 3, 357--453. MR679066 (84b:57006)

View on Pith 1982
[17]

Ronald Fintushel and Ronald J. Stern, Pseudofree orbifolds, Ann. of Math. (2) 122 (1985), no. 2, 335--364. MR808222 (87a:57027)

View on Pith 1985
[18]

, O (2) actions on the 5-sphere , Invent. Math. 87 (1987), no. 3, 457--476. MR874031 (88e:57032)

View on Pith 1987
[19]

Ronald Fintushel and Ronald J. Stern, Double node neighborhoods and families of simply connected 4-manifolds with b^+=1 , arXiv:math.GT/0412126, 2004

View on Pith arXiv 2004
[20]

Takao Fujita, On the topology of noncomplete algebraic surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 3, 503--566. MR687591 (84g:14035)

View on Pith 1982
[21]

M. Furuta, Monopole equation and the 11 8 -conjecture , Math. Res. Lett. 8 (2001), no. 3, 279--291. MR1839478 (2003e:57042)

View on Pith 2001
[22]

Hubert Flenner and Mikhail Zaidenberg, Q -acyclic surfaces and their deformations , Classification of algebraic varieties (L'Aquila, 1992), Contemp. Math., vol. 162, Amer. Math. Soc., Providence, RI, 1994, pp. 143--208. MR1272698 (95b:14024)

View on Pith 1992
[23]

, Normal affine surfaces with C -actions , Osaka J. Math. 40 (2003), no. 4, 981--1009. 2 020 670

View on Pith 2003
[24]

R. V. Gurjar and C. R. Pradeep, Q -homology planes are rational. III , Osaka J. Math. 36 (1999), no. 2, 259--335. MR1736480 (2000m:14070)

View on Pith 1999
[25]

R. V. Gurjar and A. R. Shastri, On the rationality of complex homology 2 -cells. I , J. Math. Soc. Japan 41 (1989), no. 1, 37--56. MR972163 (89k:14035)

View on Pith 1989
[26]

, On the rationality of complex homology 2 -cells. II , J. Math. Soc. Japan 41 (1989), no. 2, 175--212. MR984747 (90b:14045)

View on Pith 1989
[27]

Paul Hacking and Yuri Prokhorov, Degenerations of del P ezzo surfaces I , arXiv:math.AG/0509529, 2005

View on Pith arXiv 2005
[28]

JongHae Keum, A rationality criterion for projective surfaces---partial solution to K oll\'ar's conjecture , Algebraic geometry, Contemp. Math., vol. 422, Amer. Math. Soc., Providence, RI, 2007, pp. 75--87. MR2296433

View on Pith 2007
[29]

J \'a nos Koll \'a r and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. MR1658959 (2000b:14018)

View on Pith 1998
[30]

Se \'a n Keel and James McKernan, Rational curves on quasi-projective surfaces, Mem. Amer. Math. Soc. 140 (1999), no. 669, viii+153. MR1610249 (99m:14068)

View on Pith 1999
[31]

Ryoichi Kobayashi, Shu Nakamura, and Fumio Sakai, A numerical characterization of ball quotients for normal surfaces with branch loci, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 7, 238--241. MR1030189 (90k:32034)

View on Pith 1989
[32]

J \'a nos Koll \'a r, E instein metrics on 5--dimensional S eifert bundles , Jour. Geom. Anal. 15 (2005), no. 3, 463--495

View on Pith 2005
[33]

, Circle actions on simply connected 5-manifolds, Topology 45 (2006), no. 3, 643--671. MR2218760 (2006m:57044)

View on Pith 2006
[34]

A. G. Kouchnirenko, Poly\`edres de N ewton et nombres de M ilnor , Invent. Math. 32 (1976), no. 1, 1--31. MR0419433 (54 \#7454)

View on Pith 1976
[35]

J. Koll \'a r and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), no. 2, 299--338. MR922803 (88m:14022)

View on Pith 1988
[36]

A. S. Libgober, A geometrical procedure for killing the middle dimensional homology groups of algebraic hypersurfaces, Proc. Amer. Math. Soc. 63 (1977), no. 2, 198--202. MR0440576 (55 \#13450)

View on Pith 1977
[37]

Paolo Lisca, Lens spaces, rational balls and the ribbon conjecture, Geom. Topol. 11 (2007), 429--472. MR2302495

View on Pith 2007
[38]

Marco Manetti, Normal degenerations of the complex projective plane, J. Reine Angew. Math. 419 (1991), 89--118. MR1116920 (92f:14028)

View on Pith 1991
[39]

Yukio Matsumoto, On the bounding genus of homology 3 -spheres , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 2, 287--318. MR672065 (84g:57010)

View on Pith 1982
[40]

G \'a bor Megyesi, C hern classes of Q -sheaves , Flips and abundance for algebraic threefolds (J \'a nos Koll \'a r, ed.), Soci\'et\'e Math\'ematique de France, 1992, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Ast\'erisque No. 211 (1992), pp. 115--126. 94f:14013

View on Pith 1992
[41]

Yoichi Miyaoka, On the C hern numbers of surfaces of general type , Invent. Math. 42 (1977), 225--237. MR0460343 (57 \#337)

View on Pith 1977
[42]

, The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268 (1984), no. 2, 159--171. MR744605 (85j:14060)

View on Pith 1984
[43]

Masayoshi Miyanishi, Open algebraic surfaces, CRM Monograph Series, vol. 12, American Mathematical Society, Providence, RI, 2001. 2002e:14101

View on Pith 2001
[44]

John Milnor and Peter Orlik, Isolated singularities defined by weighted homogeneous polynomials, Topology 9 (1970), 385--393. MR0293680 (45 \#2757)

View on Pith 1970
[45]

M. Miyanishi and S. Tsunoda, Open algebraic surfaces with K odaira dimension - , Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 435--450. MR927966 (89d:14051)

View on Pith 1985
[46]

Deane Montgomery and C. T. Yang, Differentiable pseudo-free circle actions on homotopy seven spheres, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971), Part I (Berlin), Springer, 1972, pp. 41--101. Lecture Notes in Math., Vol. 298. MR0362383 (50 \#14825)

View on Pith 1971
[47]

Walter D. Neumann, An invariant of plumbed homology spheres, Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), Lecture Notes in Math., vol. 788, Springer, Berlin, 1980, pp. 125--144. MR585657 (82j:57033)

View on Pith 1979
[48]

Andr \'a s N \'e methi and Liviu I. Nicolaescu, Seiberg- W itten invariants and surface singularities , Geom. Topol. 6 (2002), 269--328 (electronic). MR1914570 (2003i:14048)

View on Pith 2002
[49]

, Seiberg- W itten invariants and surface singularities. II . S ingularities with good C * -action , J. London Math. Soc. (2) 69 (2004), no. 3, 593--607. MR2050035 (2005g:14070)

View on Pith 2004
[50]

, Seiberg- W itten invariants and surface singularities: splicings and cyclic covers , Selecta Math. (N.S.) 11 (2005), no. 3-4, 399--451. MR2215260

View on Pith 2005
[51]

Walter Neumann and Jonathan Wahl, Casson invariant of links of singularities, Comment. Math. Helv. 65 (1990), no. 1, 58--78. MR1036128 (91c:57022)

View on Pith 1990
[52]

P. Orlik and R. Randell, The structure of weighted homogeneous polynomials, Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975), Amer. Math. Soc., Providence, R. I., 1977, pp. 57--64. MR0447229 (56 \#5544)

View on Pith 1975
[53]

S. Yu. Orevkov, Acyclic algebraic surfaces bounded by S eifert spheres , Osaka J. Math. 34 (1997), no. 2, 457--480. MR1483860 (99b:14033)

View on Pith 1997
[54]

Peter Orlik and Philip Wagreich, Seifert n -manifolds , Invent. Math. 28 (1975), 137--159. MR0361150 (50 \#13596)

View on Pith 1975
[55]

Keiji Oguiso and De-Qi Zhang, On the complete classification of extremal log E nriques surfaces , Math. Z. 231 (1999), no. 1, 23--50. MR1696755 (2000e:14059)

View on Pith 1999
[56]

Ted Petrie, Equivariant quasi-equivalence, transversality and normal cobordism, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, Canad. Math. Congress, Montreal, Que., 1975, pp. 537--541. MR0431234 (55 \#4235)

View on Pith 1974
[57]

H. Pinkham, Normal surface singularities with C * action , Math. Ann. 227 (1977), no. 2, 183--193. 55 \#5623

View on Pith 1977
[58]

Jongil Park, Andr \'a s I. Stipsicz, and Zolt \'a n Szab \'o , Exotic smooth structures on CP 2\#5 CP 2 , Math. Res. Lett. 12 (2005), no. 5-6, 701--712. MR2189231 (2006i:57059)

View on Pith 2005
[59]

Gopal Prasad and Sai-Kee Yeung, Fake projective planes, Invent. Math. 168 (2007), no. 2, 321--370. MR2289867

View on Pith 2007
[60]

Miles Reid, Bogomolov's theorem c 1 2 4c 2 , Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) (Tokyo), Kinokuniya Book Store, 1978, pp. 623--642. MR578877 (82b:14014)

View on Pith 1977
[61]

Fumio Sakai, Semistable curves on algebraic surfaces and logarithmic pluricanonical maps, Math. Ann. 254 (1980), no. 2, 89--120. MR597076 (82f:14031)

View on Pith 1980
[62]

Nikolai Saveliev, Invariants for homology 3 -spheres , Encyclopaedia of Mathematical Sciences, vol. 140, Springer-Verlag, Berlin, 2002. MR1941324 (2004c:57026)

View on Pith 2002
[63]

Peter Scott, The geometries of 3 -manifolds , Bull. London Math. Soc. 15 (1983), no. 5, 401--487. MR705527 (84m:57009)

View on Pith 1983
[64]

Herbert Seifert, Topologie dreidimensionaler gefaserte R \"a ume , Acta Math. 60 (1932), 148--238

View on Pith 1932
[65]

V. V. Shokurov, Complements on surfaces, J. Math. Sci. (New York) 102 (2000), no. 2, 3876--3932. 2002c:14030

View on Pith 2000
[66]

Stephen Smale, On the structure of 5 -manifolds , Ann. of Math. (2) 75 (1962), 38--46. 25 \#4544

View on Pith 1962
[67]

Andr \'a s I. Stipsicz and Zolt \'a n Szab \'o , An exotic smooth structure on C P 2\#6 C P 2 , Geom. Topol. 9 (2005), 813--832 (electronic). MR2140993

View on Pith 2005
[68]

Andr \'a s I Stipsicz, Zolt \'a n Szab \'o , and Jonathan Wahl, Rational blow-downs and smoothings of surface singularities, arXiv:math.GT/0611157, 2006

View on Pith arXiv 2006
[69]

Herbert Seifert and William Threlfall, Seifert and T hrelfall: a textbook of topology , Pure and Applied Mathematics, vol. 89, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980. 82b:55001

View on Pith 1980
[70]

Ronald J. Stern, Some more B rieskorn spheres which bound contractible manifolds , Notices Amer. Math. Soc. 25 (1978), A448

View on Pith 1978
[71]

Tammo tom Dieck and Ted Petrie, Homology planes: an announcement and survey, Topological methods in algebraic transformation groups (New Brunswick, NJ, 1988), Progr. Math., vol. 80, Birkh\"auser Boston, Boston, MA, 1989, pp. 27--48. MR1040856 (91d:14016)

View on Pith 1988
[72]

Jonathan Wahl, Elliptic deformations of minimally elliptic singularities, Math. Ann. 253 (1980), no. 3, 241--262. MR597833 (82d:14004)

View on Pith 1980
[73]

, Smoothings of normal surface singularities, Topology 20 (1981), no. 3, 219--246. MR608599 (83h:14029)

View on Pith 1981
[74]

, letter to W . N eumann , unpublished, 1983

View on Pith 1983
[75]

, A characteristic number for links of surface singularities, J. Amer. Math. Soc. 3 (1990), no. 3, 625--637. MR1044058 (91c:14043)

View on Pith 1990
[76]

, Miyaoka- Y au inequality for normal surfaces and local analogues , Classification of algebraic varieties (L'Aquila, 1992), Contemp. Math., vol. 162, Amer. Math. Soc., Providence, RI, 1994, pp. 381--402. MR1272710 (95b:14025)

View on Pith 1992
[77]

Joseph A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York, 1967. MR0217740 (36 \#829)

View on Pith 1967
[78]

Shing Tung Yau, Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 5, 1798--1799. MR0451180 (56 \#9467)

View on Pith 1977

How the literature uses it

1 arXiv paper cite this in 1 sentence

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it

Builds on it 1 paper

This solves [ref] in the case that the canonical class is negative.

Smoothable del Pezzo surfaces with quotient singularities · Introduction · 0.92

Discussion

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[1] [1]

Selman Akbulut and Robion Kirby, Mazur manifolds, Michigan Math. J. 26 (1979), no. 3, 259--284. MR544597 (80h:57004)

View on Pith 1979

[2] [2]

M. Artin, Wildly ramified Z/2 actions in dimension two , Proc. Amer. Math. Soc. 52 (1975), 60--64. MR0374136 (51 \#10336)

View on Pith 1975

[3] [3]

D. Barden, Simply connected five-manifolds, Ann. of Math. (2) 82 (1965), 365--385. MR0184241 (32 \#1714)

View on Pith 1965

[4] [4]

Gottfried Barthel and Alexandru Dimca, On complex projective hypersurfaces which are homology- P n 's , Singularities (Lille, 1991), London Math. Soc. Lecture Note Ser., vol. 201, Cambridge Univ. Press, Cambridge, 1994, pp. 1--27. MR1295070 (95j:14060)

View on Pith 1991

[5] [5]

Charles P. Boyer, Krzysztof Galicki, and J \'a nos Koll \'a r, E instein metrics on spheres , Annals of Mathematics. to appear 162 (2005), no. 1, 1--24

View on Pith 2005

[6] [6]

F. A. Bogomolov, Holomorphic tensors and vector bundles on projective manifolds, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 6, 1227--1287, 1439. MR522939 (80j:14014)

View on Pith 1978

[7] [7]

Egbert Brieskorn, Beispiele zur D ifferentialtopologie von S ingularit\"aten , Invent. Math. 2 (1966), 1--14. 34 \#6788

View on Pith 1966

[8] [8]

, Rationale S ingularit\"aten komplexer F l\"achen , Invent. Math. 4 (1967/1968), 336--358. 36 \#5136

View on Pith 1967

[9] [9]

Andrew J. Casson and John L. Harer, Some homology lens spaces which bound rational homology balls, Pacific J. Math. 96 (1981), no. 1, 23--36. MR634760 (83h:57013)

View on Pith 1981

[10] [10]

Alessio Corti, Adjunction of log divisors, Flips and abundance for algebraic threefolds (J \'a nos Koll \'a r, ed.), Soci\'et\'e Math\'ematique de France, 1992, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Ast\'erisque No. 211 (1992), pp. 171--182. 94f:14013

View on Pith 1992

[11] [11]

Michel Demazure, Anneaux gradu\'es normaux, Introduction \`a la th\'eorie des singularit\'es, II, Travaux en Cours, vol. 37, Hermann, Paris, 1988, pp. 35--68. 91k:14004

View on Pith 1988

[12] [12]

I. V. Dolgachev, Automorphic forms, and quasihomogeneous singularities, Funkcional. Anal. i Prilo zen. 9 (1975), no. 2, 67--68. 58 \#27958

View on Pith 1975

[13] [13]

Daniel Daigle and Peter Russell, Affine rulings of normal rational surfaces, Osaka J. Math. 38 (2001), no. 1, 37--100. MR1824902 (2002g:14096)

View on Pith 2001

[14] [14]

, On weighted projective planes and their affine rulings, Osaka J. Math. 38 (2001), no. 1, 101--150. MR1824903 (2002h:14062)

View on Pith 2001

[15] [15]

, On log Q -homology planes and weighted projective planes , Canad. J. Math. 56 (2004), no. 6, 1145--1189. MR2102629 (2005i:14077)

View on Pith 2004

[16] [16]

Michael Hartley Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no. 3, 357--453. MR679066 (84b:57006)

View on Pith 1982

[17] [17]

Ronald Fintushel and Ronald J. Stern, Pseudofree orbifolds, Ann. of Math. (2) 122 (1985), no. 2, 335--364. MR808222 (87a:57027)

View on Pith 1985

[18] [18]

, O (2) actions on the 5-sphere , Invent. Math. 87 (1987), no. 3, 457--476. MR874031 (88e:57032)

View on Pith 1987

[19] [19]

Ronald Fintushel and Ronald J. Stern, Double node neighborhoods and families of simply connected 4-manifolds with b^+=1 , arXiv:math.GT/0412126, 2004

View on Pith arXiv 2004

[20] [20]

Takao Fujita, On the topology of noncomplete algebraic surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 3, 503--566. MR687591 (84g:14035)

View on Pith 1982

[21] [21]

M. Furuta, Monopole equation and the 11 8 -conjecture , Math. Res. Lett. 8 (2001), no. 3, 279--291. MR1839478 (2003e:57042)

View on Pith 2001

[22] [22]

Hubert Flenner and Mikhail Zaidenberg, Q -acyclic surfaces and their deformations , Classification of algebraic varieties (L'Aquila, 1992), Contemp. Math., vol. 162, Amer. Math. Soc., Providence, RI, 1994, pp. 143--208. MR1272698 (95b:14024)

View on Pith 1992

[23] [23]

, Normal affine surfaces with C -actions , Osaka J. Math. 40 (2003), no. 4, 981--1009. 2 020 670

View on Pith 2003

[24] [24]

R. V. Gurjar and C. R. Pradeep, Q -homology planes are rational. III , Osaka J. Math. 36 (1999), no. 2, 259--335. MR1736480 (2000m:14070)

View on Pith 1999

[25] [25]

R. V. Gurjar and A. R. Shastri, On the rationality of complex homology 2 -cells. I , J. Math. Soc. Japan 41 (1989), no. 1, 37--56. MR972163 (89k:14035)

View on Pith 1989

[26] [26]

, On the rationality of complex homology 2 -cells. II , J. Math. Soc. Japan 41 (1989), no. 2, 175--212. MR984747 (90b:14045)

View on Pith 1989

[27] [27]

Paul Hacking and Yuri Prokhorov, Degenerations of del P ezzo surfaces I , arXiv:math.AG/0509529, 2005

View on Pith arXiv 2005

[28] [28]

JongHae Keum, A rationality criterion for projective surfaces---partial solution to K oll\'ar's conjecture , Algebraic geometry, Contemp. Math., vol. 422, Amer. Math. Soc., Providence, RI, 2007, pp. 75--87. MR2296433

View on Pith 2007

[29] [29]

J \'a nos Koll \'a r and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. MR1658959 (2000b:14018)

View on Pith 1998

[30] [30]

Se \'a n Keel and James McKernan, Rational curves on quasi-projective surfaces, Mem. Amer. Math. Soc. 140 (1999), no. 669, viii+153. MR1610249 (99m:14068)

View on Pith 1999

[31] [31]

Ryoichi Kobayashi, Shu Nakamura, and Fumio Sakai, A numerical characterization of ball quotients for normal surfaces with branch loci, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 7, 238--241. MR1030189 (90k:32034)

View on Pith 1989

[32] [32]

J \'a nos Koll \'a r, E instein metrics on 5--dimensional S eifert bundles , Jour. Geom. Anal. 15 (2005), no. 3, 463--495

View on Pith 2005

[33] [33]

, Circle actions on simply connected 5-manifolds, Topology 45 (2006), no. 3, 643--671. MR2218760 (2006m:57044)

View on Pith 2006

[34] [34]

A. G. Kouchnirenko, Poly\`edres de N ewton et nombres de M ilnor , Invent. Math. 32 (1976), no. 1, 1--31. MR0419433 (54 \#7454)

View on Pith 1976

[35] [35]

J. Koll \'a r and N. I. Shepherd-Barron, Threefolds and deformations of surface singularities, Invent. Math. 91 (1988), no. 2, 299--338. MR922803 (88m:14022)

View on Pith 1988

[36] [36]

A. S. Libgober, A geometrical procedure for killing the middle dimensional homology groups of algebraic hypersurfaces, Proc. Amer. Math. Soc. 63 (1977), no. 2, 198--202. MR0440576 (55 \#13450)

View on Pith 1977

[37] [37]

Paolo Lisca, Lens spaces, rational balls and the ribbon conjecture, Geom. Topol. 11 (2007), 429--472. MR2302495

View on Pith 2007

[38] [38]

Marco Manetti, Normal degenerations of the complex projective plane, J. Reine Angew. Math. 419 (1991), 89--118. MR1116920 (92f:14028)

View on Pith 1991

[39] [39]

Yukio Matsumoto, On the bounding genus of homology 3 -spheres , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982), no. 2, 287--318. MR672065 (84g:57010)

View on Pith 1982

[40] [40]

G \'a bor Megyesi, C hern classes of Q -sheaves , Flips and abundance for algebraic threefolds (J \'a nos Koll \'a r, ed.), Soci\'et\'e Math\'ematique de France, 1992, Papers from the Second Summer Seminar on Algebraic Geometry held at the University of Utah, Salt Lake City, Utah, August 1991, Ast\'erisque No. 211 (1992), pp. 115--126. 94f:14013

View on Pith 1992

[41] [41]

Yoichi Miyaoka, On the C hern numbers of surfaces of general type , Invent. Math. 42 (1977), 225--237. MR0460343 (57 \#337)

View on Pith 1977

[42] [42]

, The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268 (1984), no. 2, 159--171. MR744605 (85j:14060)

View on Pith 1984

[43] [43]

Masayoshi Miyanishi, Open algebraic surfaces, CRM Monograph Series, vol. 12, American Mathematical Society, Providence, RI, 2001. 2002e:14101

View on Pith 2001

[44] [44]

John Milnor and Peter Orlik, Isolated singularities defined by weighted homogeneous polynomials, Topology 9 (1970), 385--393. MR0293680 (45 \#2757)

View on Pith 1970

[45] [45]

M. Miyanishi and S. Tsunoda, Open algebraic surfaces with K odaira dimension - , Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 435--450. MR927966 (89d:14051)

View on Pith 1985

[46] [46]

Deane Montgomery and C. T. Yang, Differentiable pseudo-free circle actions on homotopy seven spheres, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971), Part I (Berlin), Springer, 1972, pp. 41--101. Lecture Notes in Math., Vol. 298. MR0362383 (50 \#14825)

View on Pith 1971

[47] [47]

Walter D. Neumann, An invariant of plumbed homology spheres, Topology Symposium, Siegen 1979 (Proc. Sympos., Univ. Siegen, Siegen, 1979), Lecture Notes in Math., vol. 788, Springer, Berlin, 1980, pp. 125--144. MR585657 (82j:57033)

View on Pith 1979

[48] [48]

Andr \'a s N \'e methi and Liviu I. Nicolaescu, Seiberg- W itten invariants and surface singularities , Geom. Topol. 6 (2002), 269--328 (electronic). MR1914570 (2003i:14048)

View on Pith 2002

[49] [49]

, Seiberg- W itten invariants and surface singularities. II . S ingularities with good C * -action , J. London Math. Soc. (2) 69 (2004), no. 3, 593--607. MR2050035 (2005g:14070)

View on Pith 2004

[50] [50]

, Seiberg- W itten invariants and surface singularities: splicings and cyclic covers , Selecta Math. (N.S.) 11 (2005), no. 3-4, 399--451. MR2215260

View on Pith 2005

[51] [51]

Walter Neumann and Jonathan Wahl, Casson invariant of links of singularities, Comment. Math. Helv. 65 (1990), no. 1, 58--78. MR1036128 (91c:57022)

View on Pith 1990

[52] [52]

P. Orlik and R. Randell, The structure of weighted homogeneous polynomials, Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975), Amer. Math. Soc., Providence, R. I., 1977, pp. 57--64. MR0447229 (56 \#5544)

View on Pith 1975

[53] [53]

S. Yu. Orevkov, Acyclic algebraic surfaces bounded by S eifert spheres , Osaka J. Math. 34 (1997), no. 2, 457--480. MR1483860 (99b:14033)

View on Pith 1997

[54] [54]

Peter Orlik and Philip Wagreich, Seifert n -manifolds , Invent. Math. 28 (1975), 137--159. MR0361150 (50 \#13596)

View on Pith 1975

[55] [55]

Keiji Oguiso and De-Qi Zhang, On the complete classification of extremal log E nriques surfaces , Math. Z. 231 (1999), no. 1, 23--50. MR1696755 (2000e:14059)

View on Pith 1999

[56] [56]

Ted Petrie, Equivariant quasi-equivalence, transversality and normal cobordism, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1, Canad. Math. Congress, Montreal, Que., 1975, pp. 537--541. MR0431234 (55 \#4235)

View on Pith 1974

[57] [57]

H. Pinkham, Normal surface singularities with C * action , Math. Ann. 227 (1977), no. 2, 183--193. 55 \#5623

View on Pith 1977

[58] [58]

Jongil Park, Andr \'a s I. Stipsicz, and Zolt \'a n Szab \'o , Exotic smooth structures on CP 2\#5 CP 2 , Math. Res. Lett. 12 (2005), no. 5-6, 701--712. MR2189231 (2006i:57059)

View on Pith 2005

[59] [59]

Gopal Prasad and Sai-Kee Yeung, Fake projective planes, Invent. Math. 168 (2007), no. 2, 321--370. MR2289867

View on Pith 2007

[60] [60]

Miles Reid, Bogomolov's theorem c 1 2 4c 2 , Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) (Tokyo), Kinokuniya Book Store, 1978, pp. 623--642. MR578877 (82b:14014)

View on Pith 1977

[61] [61]

Fumio Sakai, Semistable curves on algebraic surfaces and logarithmic pluricanonical maps, Math. Ann. 254 (1980), no. 2, 89--120. MR597076 (82f:14031)

View on Pith 1980

[62] [62]

Nikolai Saveliev, Invariants for homology 3 -spheres , Encyclopaedia of Mathematical Sciences, vol. 140, Springer-Verlag, Berlin, 2002. MR1941324 (2004c:57026)

View on Pith 2002

[63] [63]

Peter Scott, The geometries of 3 -manifolds , Bull. London Math. Soc. 15 (1983), no. 5, 401--487. MR705527 (84m:57009)

View on Pith 1983

[64] [64]

Herbert Seifert, Topologie dreidimensionaler gefaserte R \"a ume , Acta Math. 60 (1932), 148--238

View on Pith 1932

[65] [65]

V. V. Shokurov, Complements on surfaces, J. Math. Sci. (New York) 102 (2000), no. 2, 3876--3932. 2002c:14030

View on Pith 2000

[66] [66]

Stephen Smale, On the structure of 5 -manifolds , Ann. of Math. (2) 75 (1962), 38--46. 25 \#4544

View on Pith 1962

[67] [67]

Andr \'a s I. Stipsicz and Zolt \'a n Szab \'o , An exotic smooth structure on C P 2\#6 C P 2 , Geom. Topol. 9 (2005), 813--832 (electronic). MR2140993

View on Pith 2005

[68] [68]

Andr \'a s I Stipsicz, Zolt \'a n Szab \'o , and Jonathan Wahl, Rational blow-downs and smoothings of surface singularities, arXiv:math.GT/0611157, 2006

View on Pith arXiv 2006

[69] [69]

Herbert Seifert and William Threlfall, Seifert and T hrelfall: a textbook of topology , Pure and Applied Mathematics, vol. 89, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1980. 82b:55001

View on Pith 1980

[70] [70]

Ronald J. Stern, Some more B rieskorn spheres which bound contractible manifolds , Notices Amer. Math. Soc. 25 (1978), A448

View on Pith 1978

[71] [71]

Tammo tom Dieck and Ted Petrie, Homology planes: an announcement and survey, Topological methods in algebraic transformation groups (New Brunswick, NJ, 1988), Progr. Math., vol. 80, Birkh\"auser Boston, Boston, MA, 1989, pp. 27--48. MR1040856 (91d:14016)

View on Pith 1988

[72] [72]

Jonathan Wahl, Elliptic deformations of minimally elliptic singularities, Math. Ann. 253 (1980), no. 3, 241--262. MR597833 (82d:14004)

View on Pith 1980

[73] [73]

, Smoothings of normal surface singularities, Topology 20 (1981), no. 3, 219--246. MR608599 (83h:14029)

View on Pith 1981

[74] [74]

, letter to W . N eumann , unpublished, 1983

View on Pith 1983

[75] [75]

, A characteristic number for links of surface singularities, J. Amer. Math. Soc. 3 (1990), no. 3, 625--637. MR1044058 (91c:14043)

View on Pith 1990

[76] [76]

, Miyaoka- Y au inequality for normal surfaces and local analogues , Classification of algebraic varieties (L'Aquila, 1992), Contemp. Math., vol. 162, Amer. Math. Soc., Providence, RI, 1994, pp. 381--402. MR1272710 (95b:14025)

View on Pith 1992

[77] [77]

Joseph A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York, 1967. MR0217740 (36 \#829)

View on Pith 1967

[78] [78]

Shing Tung Yau, Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 5, 1798--1799. MR0451180 (56 \#9467)

View on Pith 1977