. A formula for the geometric genus of surface singularities. Tomohiro Okuma

Transcription

1 A formula for the geometric genus of surface singularities Yamagata University Branched Coverings, Degenerations, and Related Topics 2011 TMU, March 7 10, 2011

2 Contents Some known results When is p g topological? 1 Some known results When is p g topological? 2 3

3 Some known results When is p g topological? (X, o) (C n, o): a normal complex surface singularity Σ: the link of (X, o) Σ = X S 2n 1 ε (0 < ε 1) We may assume X is a cone over Σ π : X X: a resolution with exceptional set E := π 1 (o) a) π is proper and X is smooth, b) X \ E X \ {o},

4 Some known results When is p g topological? (X, o) (C n, o): a normal complex surface singularity Σ: the link of (X, o) Σ = X S 2n 1 ε (0 < ε 1) We may assume X is a cone over Σ π : X X: a resolution with exceptional set E := π 1 (o) a) π is proper and X is smooth, b) X \ E X \ {o}, Definition (The geometric genus) p g (X, o) := h 1 ( X, O X ) := dim C H 1 ( X, O X ), independent of the resolution

5 Some known results When is p g topological? (X, o) (C n, o): a normal complex surface singularity Σ: the link of (X, o) Σ = X S 2n 1 ε (0 < ε 1) We may assume X is a cone over Σ π : X X: a resolution with exceptional set E := π 1 (o) a) π is proper and X is smooth, b) X \ E X \ {o}, Definition (The geometric genus) p g (X, o) := h 1 ( X, O X ) := dim C H 1 ( X, O X ), independent of the resolution By duality, H 0 ( X \ E, O X (K X )) p g (X, o) = dim C H 0 ( X, O X (K X ))

6 Some known results When is p g topological? Hypersurfaces (Counting lower monomials ) Suppose X = { f = 0}, f C[x 1, x 2, x 3 ] Z + := N {0} Kimio Watanabe, 1980 f is weighted homogeneous of degree d with weights (q 1, q 2, q 3 ) p g (X, o) = # { (a 1, a 2, a 3 ) (Z + ) 3 3 (a i=1 i + 1)q i d }

7 Some known results When is p g topological? Hypersurfaces (Counting lower monomials ) Suppose X = { f = 0}, f C[x 1, x 2, x 3 ] Z + := N {0} Kimio Watanabe, 1980 f is weighted homogeneous of degree d with weights (q 1, q 2, q 3 ) p g (X, o) = # { (a 1, a 2, a 3 ) (Z + ) 3 3 (a i=1 i + 1)q i d } Merle Teissier, 1980 f is non-degenerate wrt its Newton boundary p g (X, o) = # { a (Z + ) 3 a + (1, 1, 1) Int(Γ + ) }

8 Some known results When is p g topological? Hypersurfaces (Invariants from resolution process) Any double point is defined by a function of type x g(x 2, x 3 ) Horikawa, 1975 f = x 2 + g(x 2, x 3 ) 1 p g (X, o) = 1 n 2 γ mi i=1 i(γ i + 1), where γ i = 1, m i is the 2 multiplicity of curve singularity which is the center of i-th blowing up in the embedded resolution process for {g = 0} C 2

9 Some known results When is p g topological? Hypersurfaces (Invariants from resolution process) Any double point is defined by a function of type x g(x 2, x 3 ) Horikawa, 1975 f = x 2 + g(x 2, x 3 ) 1 p g (X, o) = 1 n 2 γ mi i=1 i(γ i + 1), where γ i = 1, m i is the 2 multiplicity of curve singularity which is the center of i-th blowing up in the embedded resolution process for {g = 0} C 2 Ashikaga (1999) obtained a formula for x n 1 + g(x 2, x 3 ) Tomari (1985) obtained a formula for any hypersurface singularity

10 Some known results When is p g topological? {E u } u V : the irreducible components of E Then H 2 ( X, Z) = u V ZE u Let Z = min {D Z + E u D E u 0, u V }: the Artin cycle Definition For D u V Z + E u p a (D) = 1 h 0 (O D ) + h 1 (O D ) = D (D + 2 K X ) p a (Z) is independent of the choice of resolution

11 Some known results When is p g topological? {E u } u V : the irreducible components of E Then H 2 ( X, Z) = u V ZE u Let Z = min {D Z + E u D E u 0, u V }: the Artin cycle Definition For D u V Z + E u p a (D) = 1 h 0 (O D ) + h 1 (O D ) = D (D + 2 K X ) p a (Z) is independent of the choice of resolution (Artin, 1966) p a (Z) = 0 p g = 0 (Laufer, 1977) Z K X (then p a(z) = 1) p g = 1

12 Some known results When is p g topological? {E u } u V : the irreducible components of E Then H 2 ( X, Z) = u V ZE u Let Z = min {D Z + E u D E u 0, u V }: the Artin cycle Definition For D u V Z + E u p a (D) = 1 h 0 (O D ) + h 1 (O D ) = D (D + 2 K X ) p a (Z) is independent of the choice of resolution (Artin, 1966) p a (Z) = 0 p g = 0 (Laufer, 1977) Z K X (then p a(z) = 1) p g = 1 (Némethi, 1999) (X, o) is Gorenstein, p a (Z) = 1, QHS link p g = the length of the elliptic sequence (SS-T Yau introduced the elliptic sequence and gave bounds of these invariants)

13 Some known results When is p g topological? Assume that π : X X is a good resolution, ie, E u are nonsingular, E u E w = 1 or 0 for u w I(E) = (E u E w ): the intersection matrix of E; it is negative-definite Γ: the weighted dual graph of E E u vertex u; E u E w edge connectin u and w; vertex u has weight (E u E u, g(e u )) (if all E u are rational, Γ I(E))

14 Some known results When is p g topological? Assume that π : X X is a good resolution, ie, E u are nonsingular, E u E w = 1 or 0 for u w I(E) = (E u E w ): the intersection matrix of E; it is negative-definite Γ: the weighted dual graph of E E u vertex u; E u E w edge connectin u and w; vertex u has weight (E u E u, g(e u )) (if all E u are rational, Γ I(E)) Ṭhe graph Γ and the link Σ determine each other (Neumann) An invariant of a singularity is topological if it is determined by the resolution graph

15 Some known results When is p g topological? Ịn general, Gorenstein condition and p g are not topological The following graph ( vertex P 1 ) is realized by:

16 Some known results When is p g topological? Ịn general, Gorenstein condition and p g are not topological The following graph ( vertex P 1 ) is realized by: X 1 : x 2 + y 3 + z 13 = 0, p g = 2

17 Some known results When is p g topological? Ịn general, Gorenstein condition and p g are not topological The following graph ( vertex P 1 ) is realized by: X 1 : x 2 + y 3 + z 13 = 0, p g = 2 X 2 (non ( Gorenstein): ) x y z rank y 3w 2 z + w 3 x 2 + 6wy 2w 3 < 2, p g = 1

18 Some known results When is p g topological? Ịn general, Gorenstein condition and p g are not topological The following graph ( vertex P 1 ) is realized by: X 1 : x 2 + y 3 + z 13 = 0, p g = 2 X 2 (non ( Gorenstein): ) x y z rank y 3w 2 z + w 3 x 2 + 6wy 2w 3 < 2, p g = 1 Main question Ẉhen (How) can we compute p g from the graph?

19 Let E u be an arbitrary component of E Let F 1,, F m be the connected components of E E u (X i, x i ): the normal singularity obtained by contracting F i

20 Let E u be an arbitrary component of E Let F 1,, F m be the connected components of E E u (X i, x i ): the normal singularity obtained by contracting F i Notation c( X, u) := p g (X, o) m i=1 p g(x i, x i )

21 Let E u be an arbitrary component of E Let F 1,, F m be the connected components of E E u (X i, x i ): the normal singularity obtained by contracting F i Notation c( X, u) := p g (X, o) m i=1 p g(x i, x i ) Problem Find a class R of surface singularities, which satisfies the following: For any (X, o) in R, there exists a good resolution X X such that c( X, u) can be computed from the graph for any u V, every (X i, x i ) is in R

22 There exisits E u i V QE i satisfying E u E i = δ ui ( i) Let k be a positive integer such that ke u L I n = H 0 (O X ( nke u )) G := n 0 I n/i n+1 H G (t) := n 0(dim G n )t n : the Hilbert series of G

23 There exisits E u i V QE i satisfying E u E i = δ ui ( i) Let k be a positive integer such that ke u L I n = H 0 (O X ( nke u )) G := n 0 I n/i n+1 H G (t) := n 0(dim G n )t n : the Hilbert series of G Definition Let F(t) = i 0 a i t i be a formal power series If ψ(n) := nk 1 i=0 is a polynomial function of n for some k N, then the constant term of ψ(n) is independent of k N This constant is called the periodic constant of F(t) and denote it by pc F a i

24 There exisits E u i V QE i satisfying E u E i = δ ui ( i) Let k be a positive integer such that ke u L I n = H 0 (O X ( nke u )) G := n 0 I n/i n+1 H G (t) := n 0(dim G n )t n : the Hilbert series of G Definition Let F(t) = i 0 a i t i be a formal power series If ψ(n) := nk 1 i=0 is a polynomial function of n for some k N, then the constant term of ψ(n) is independent of k N This constant is called the periodic constant of F(t) and denote it by pc F Lemma Assume F(t) = p(t) + r(t)/q(t), where p, q, and r are polynomials with deg r < deg q Then pc F = p(1) a i

25 Ụnder a certain condition, c( X, u) = pc H G

26 Ụnder a certain condition, c( X, u) = pc H G Let E V be the set of ends Definition Let i E We say that X satisfies the weak end curve condition (weak ECC) at E i if there exist n N and f H 0 (O X ) such that H = div( f) ne has no component of E i

27 Ụnder a certain condition, c( X, u) = pc H G Let E V be the set of ends Definition Let i E We say that X satisfies the weak end curve condition (weak ECC) at E i if there exist n N and f H 0 (O X ) such that H = div( f) ne has no component of E i We say that X satisfies the ECC at E i if it satisfies the weak ECC at E i and the divisor H can be chosen as a divisor with irreducible support We say simply that X satisfies the (weak) ECC if it satisfies the (weak) ECC at E i for every i E

28 Proposition Assume that X satisfies the weak ECC Then 1 p g (X, o) = pc H G + m i=1 g(x i, x i ) (ie, c( X, u) = pc H G ) 2 X i satisfies the weak ECC

29 Proposition Assume that X satisfies the weak ECC Then 1 p g (X, o) = pc H G + m i=1 g(x i, x i ) (ie, c( X, u) = pc H G ) 2 X i satisfies the weak ECC The point of (1) is that L n := O X ( nke u ) (n 0) has no base points in E If π : X X is the contraction of E E u, then 0 = H 1 (π L n) H 1 (L n ) H 0 (R 1 π L n ) 0 From the exact sequence we have 0 L n O X O nke u 0 p g (X, o) = dim I 0 /I n χ(o nke u ) + h 1 (L n )

30 Corollary (Tomari Watanabe, 1989) Suppose X is a hypersurface with multiplicity d Then p g (X, o) = 1 d(d 1)(d 2) if the blowing up of X at o has only 6 rational singularities (Eg superisolated singularities)

31 Corollary (Tomari Watanabe, 1989) Suppose X is a hypersurface with multiplicity d Then p g (X, o) = 1 d(d 1)(d 2) if the blowing up of X at o has only 6 rational singularities (Eg superisolated singularities) The Hilbert series of the tangent cone is F(t) = 1 td (1 t) ( ) 3 d pc F = = 1 d(d 1)(d 2) 3 6

32 Ạssume that H 1 (Σ, Q) = 0 ( E u P 1 and Γ is a tree)

33 Ạssume that H 1 (Σ, Q) = 0 ( E u P 1 and Γ is a tree) Theorem (End Curve Theorem (Neumann-Wahl) ) (X, o) is a splice quotient The minimal good resolution of (X, o) satisfies the ECC Definition Let i E X satisfies the weak ECC at E i if there exist n N and f H 0 (O X ) such that H = div( f) ne has no component i of E X satisfies the ECC at E i if it satisfies the weak end curve condition at E i and the divisor H can be chosen as a divisor with irreducible support X satisfies the ECC if it satisfies the ECC at E i for every i E

34 Splice quotients includes weighted homogeneous singularities with QHS links Rational singularities are splice quotients Minimally elliptic singularities with QHS links

35 Definition L := u V ZE u L := u V ZE u (E w E u = δ wu) H := L /L Then H H 1 (Σ, Z), H = det I(E)

36 Definition L := u V ZE u L := u V ZE u (E w E u = δ wu) H := L /L Then H H 1 (Σ, Z), H = det I(E) For h H, θ(h, i) := exp(2π 1 h E i ) where is (L /L) L Q/Z (m i j ) = H ( I(E)) 1 Z SW (t) = 1 (1 θ(h, i) t m wi ) δi 2 Γ,w H h H i V δ i = (E E i ) E i

37 Assume (X, o) is splice quotient Theorem (O, 2008) 1 p g (X, o) = pc Z SW + m p Γ,w i=1 g(x i, x i ) 2 Each (X i, x i ) is a splice quotient Corollary If Γ is a star-shaped graph with center u, then p g (X, o) = pc Z SW Γ,w

38 Example The following graph is realized by a Gorenstein splice quotient 4 2 u p a (Z) = 6

39 Example The following graph is realized by a Gorenstein splice quotient 4 2 u p a (Z) = 6 The polynomial part of Z SW (t) is Γ,w 6 + t 5 + t 10

40 Example The following graph is realized by a Gorenstein splice quotient 4 2 u p a (Z) = 6 The polynomial part of Z SW (t) is Γ,w 6 + t 5 + t 10 p g = = 8

41 Casson Invariant Conjecture Assume (X, o) is complete intersection F: the Milnor fiber of (X, o) (Then Σ F) σ(f): the signature λ(σ): the Casson invariant Casson Invariant Conjecture (Neumann-Wahl) Ạssume that H 1 (Σ, Z) = 0 Then λ(σ) = σ(f)/8

42 Casson Invariant Conjecture Assume (X, o) is complete intersection F: the Milnor fiber of (X, o) (Then Σ F) σ(f): the signature λ(σ): the Casson invariant Casson Invariant Conjecture (Neumann-Wahl) Ạssume that H 1 (Σ, Z) = 0 Then λ(σ) = σ(f)/8 By Laufer Durfee, CIC p g (X, o) + λ(σ) + K2 + s 8 K = K X, s = #V = 0

43 A generalization of CIC In the following, we do not assume that H = {0}, the existence of smoothing Seiberg Witten Invariant Conjecture (Némethi Nicolaescu) p g (X, o) + sw(σ) + K2 + s 0 8 If (X, o) is Q-Gorenstein, then = holds Ḥ 1 (Σ, Z) = 0 sw(σ) = λ(σ) (Casson invariant)

44 A generalization of CIC In the following, we do not assume that H = {0}, the existence of smoothing Seiberg Witten Invariant Conjecture (Némethi Nicolaescu) p g (X, o) + sw(σ) + K2 + s 0 8 If (X, o) is Q-Gorenstein, then = holds Ḥ 1 (Σ, Z) = 0 sw(σ) = λ(σ) (Casson invariant) Theorem (Némethi Nicolaescu) SWIC is true for quotient singularities, weighted homogeneous singularities, and z n = g(x, y)

45 For (X i, x i ), define Σ i, s i, K 2 as Σ, s, K 2 i Theorem (Braun Némethi, 2010) sw(σ) + K2 + s m = pc Z SW 8 + Γ,w sw(σ i) + i=1 K 2 i + s i 8

46 For (X i, x i ), define Σ i, s i, K 2 as Σ, s, K 2 i Theorem (Braun Némethi, 2010) sw(σ) + K2 + s m = pc Z SW 8 + Γ,w sw(σ i) + i=1 p g (X, o) = pc Z SW Γ,w + m i=1 p g(x i, x i ) Corollary ṢWIC and CIC are true for splice quotients K 2 i + s i 8

47 For (X i, x i ), define Σ i, s i, K 2 as Σ, s, K 2 i Theorem (Braun Némethi, 2010) sw(σ) + K2 + s m = pc Z SW 8 + Γ,w sw(σ i) + i=1 p g (X, o) = pc Z SW Γ,w + m i=1 p g(x i, x i ) Corollary ṢWIC and CIC are true for splice quotients K 2 i + s i 8 Ṭhere are counterexamples for SWIC; no for CIC

48 A counterexample for SWIC (Luengo-Velasco, Melle-Hernández, and Némethi) The following graph is realized by a superisolated singulaity 4 2 u Then Z 2 = 5, 5(5 1)(5 2) p g = = 10 6 (p g = 8 if it is splice quotient)

49 A counterexample for SWIC (Luengo-Velasco, Melle-Hernández, and Némethi) The following graph is realized by a superisolated singulaity 4 2 u Then Z 2 = 5, 5(5 1)(5 2) p g = = 10 6 (p g = 8 if it is splice quotient) Problem Characterize singularities satisfying SWIC (or pc Z SW = pc H G ) CIC is true?

50 Non splice quotient The following graph is not realized by a splice quotient, but realized by a hypersurface singularity with p a (Z) = 1, its minimal good resolution satisfies the weak ECC E E 3 1 E 1 6 E 4 4 E 6 E 5 3

51 Non splice quotient The following graph is not realized by a splice quotient, but realized by a hypersurface singularity with p a (Z) = 1, its minimal good resolution satisfies the weak ECC E E 3 1 E 1 6 E 4 4 E 6 E 5 3 p g = pc Z SW Γ,5 In particular, SWIC holds = pc ZSW Γ,6 = 2

52 Non splice quotient The following graph is not realized by a splice quotient, but realized by a hypersurface singularity with p a (Z) = 1, its minimal good resolution satisfies the weak ECC E E 3 1 E 1 6 E 4 4 E 6 E 5 3 p g = pc Z SW Γ,5 In particular, SWIC holds = pc ZSW Γ,6 = 2 The class of singularities satisfying SWIC is bigger than the class of splice quotients