Local and log BPS invariants

Transcription

1 Local and log BPS invariants Jinwon Choi and Michel van Garrel joint with S. Katz and N. Takahashi Sookmyung Women s University KIAS August 11, 2016 J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

2 Outline We count curves in X in class β H 2 (X, Z). E K S a smooth elliptic curve. S: del Pezzo surface X = Tot(K S ) local del Pezzo Compare? X = (S, E) log pair Local (BPS) invariant Log (BPS) invariant J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

3 Talk 1 : Local BPS invariants J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

4 Physicists interpretation ( 98) M p V β q pt Physicists claim M is the space of pairs (C, L). C : curve in class β, L : line bundle of degree p a (β). V β is the space of curves in class β. p, q are forgetting maps. H : a cohomology theory on M. Lefschetz actions relative to p and q define an sl 2 sl 2 -action on H. To count genus g curve on X, we count cohomology of genus g Jacobian in H get a number n g β the BPS invariant. J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

5 It is not clear at all! Example. S = P 2, β = 3H. M p P 9 q pt M is space of (C, p), C a cubic curve and p C. P 9 is the space of cubic curves. M is isomorphic to a P 8 -bundle over P 2. M P(K), rank(k) = 9 H (M) = C[H, ξ]/(h 3, ξ 9 3ξ 8 H + 9ξ 7 H 2 ). H = c 1 (O P 2(1)), ξ = c 1 (O P(K) (1)). But, we cannot find the expected sl 2 sl 2 -representation structure on H (M). J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

6 What is M? In physics, n g β is the count of D-branes. In mathematics, D-branes are sheaves! M β = {F : F stable sheaf on X with [F ] = β, χ(f ) = 1}. F is stable w.r.t. L(= K S ) if 1 F is pure (no 0-dim subsheaves) 2 For a proper G F, χ(g) r(g) < χ(f ) r(f ). where r(f ) = L [F ] is the linear coefficient of the Hilbert polynomial. On a smooth curve (having class β), such sheaves are degree g line bundles, where g is the genus of the curve. M β is equipped with a symmetric obstruction theory. Roughly, BPS invariants n g β count the genus g Jacobian in M β. J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

7 Mathematicians rigorous answer ( 12) Kiem-Li s beautiful masterpiece arxiv: M β p V β q pt M β is a critical virtual manifold (locally looks like a critical locus of a hol. fcn.). There is a global perverse shaef P on M. Locally, P is a perverse sheaf of vanishing cycle. χ(h (M, P)) recovers the virtual invariant. t ti dim H i (M, P) is the refined DT invariant. H (M, P) has relative Hard Lefschetz. Hard Lefschetz on p (q) defines the left (right) sl 2 -action. Therefore, H (M, P) is sl 2 sl 2 -representation as we wanted. Cor. n 0 β = DT (M β) := deg[m β ] vir. J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

8 sl 2 -representation sl 2 is three ( dimensional ) ( Lie algebra ) ( generated ) by {h =, e =, f = } For each m 0, there is an irreducible representation V m of sl 2 of dimension m + 1. V m := v 0,, v m h(v j ) v j, e(v j ) v j 1, f (v j ) v j+1, V m is denoted by [ m 2 ]. For example, the cohomology ring of the torus T 2 H (T 2, C) = C C 2 C has the Lefschetz decomposition and it is as sl 2 -representation [ 1 2 ] 2[0] J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

9 Gopakumar-Vafa (BPS) invariant Let j L, j R 1 2 Z. [j L, j R ] is an irreducible sl 2 sl 2 -representation. H (M, P) = j L,j R N β j L,j R [j L, j R ] N β j L,j R is the refined BPS index. Consider the decomposition w.r.t. the left action. Tr( 1) F R H (M, P) = j L N β j L,j R ( 1) 2j R (2j R + 1)[j L ] = g n g β ([1 2 ] 2[0]) g. n g β is the Gopakumar-Vafa (BPS) invariant. J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

10 Gromov-Witten invariants M g (X, β)= moduli space of stable maps = {f : C X, f [C] = β, Aut(f ) < } proper Delign-Mumford stack with perfect obstruction theory. ([Li-Tian, Behrend-Fantechi]) virtual cycle [M g (X, β)] vir A vdim (M g (X, β)) vdim = (dim X 3)(1 g) β ω X for a CY 3-fold X, the GW invariant for X is I g β (X ) = deg[m g (X, β)] vir. J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

11 Multiple cover formula I g β (X ) is not an actual count of curves. Aspinwall-Morrison multiple cover formula Let C X be a rigid smooth rational curve. The contribution of degree k covers of C to Iβ 0 1 (X ) is. k 3 Conjecture (Katz, Hosono-Saito-Takahashi) When genus is zero, I 0 β (X ) = k β n 0 β/k k 3 Proved for local del Pezzo surfaces by [Toda], [Maulik-Nekrasov-Okounkov-Pandharipande] and others. J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

12 GW-BPS correspondence In general, Conjecture (Gopakumar-Vafa) I g β (X )qβ λ 2g 2 = n g 1 β k (2 sin(kλ 2 ))2g 2 q kβ. g,β,k g,β This conjecture is widely open. J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

13 Setup S: a del Pezzo surface. (A surface with ample K S ) S is either P 1 P 1 or S r := blowup of P 2 at r points (0 r 8). β H 2 (S, Z) a curve class. We often consider β as a divisor on S. w = ( K S ) β. p a (β) = 1 2β(β + K) + 1. Fix O S (1) = K S. We consider genus zero invariants. (We will omit the superscript g.) J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

14 Local BPS invariants A stable sheaf on X = Tot(K S ) must be supported on S. M β = Moduli space of stable sheaves F on S with [F ] = β and χ(f ) = 1. (Le Potier) M β is a smooth projective variety of dimension β So, n β = ( 1) β2 +1 χ(m β ) J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

15 Examples For S = P 2 and β = dh. 1 M 1 = {O L } P 2. L : line. n 1 = 3 2 M 2 = {O C } P 5. C : conic. n 2 = 6. 3 M 3 = {F such that 0 O C F O p 0}, C : cubic, p C = the universal cubic curve P 8 -bundle on P 2. n 3 = χ(m 4 ) = 192. ([Sahin, C-Chung, C-Maican]) n 4 = χ(m 5 ) = ([C-Chung, Maican]) n 5 = χ(m 6 ) = ([C-Chung]) n 6 = J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

16 Local BPS numbers for local P 2 d p a (d) n d ( 1) 3d 1 n d /(3d) , , ,454 8, ,228,160 92, ,748,899 1,027, ,012,150 12,000,405 J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

17 Our conjecture Observation : n d for local P 2 is divisible by 3d. 3d = ( K P 2) β. Conjecture For local del Pezzo surface, n β is divisible by w := ( K S ) β. ( 1) w 1 n β /w coincides with the log BPS invariant m β. We proved Conjecture for all β with p a (β) 2. J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

18 Log BPS invariants The log BPS invariant m β roughly count the curves in class β having maximal tangency with a smooth anticanonical curve E at a certain point P. N P β (S, E) is the corresponding GW-type invariants. N P β (S, E) = More details on Talk 2! {k β : P E(β/k)} ( 1) (k 1)w/k k 2 m P β/k. J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

19 Local BPS computation Theorem Let S be a del Pezzo suface and β be a curve class on S. Assume there are at least one irreducible curves in class β. Let h be the number of ( 1)-curves on S not intersecting β. If p a (β) = 0, then n β = ( 1) w 1 w. If p a (β) = 1 and β K S8, then n β = ( 1) w 1 w(χ(s) h). If β = K S8, then n β = ( 1) w 1 w(χ(s 8 ) + 1) = 12. If p a (β) = 2, then n β = ( 1) w 1 w( ( ) χ(s) h 2 + 5). J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

20 Sketch of proof It is enough to consider only very ample β s. Lemma (di Rocco) With a few exceptions, a curve class β is very ample if and only if β l > 0 for all ( 1)-curves l. Lemma Let π : S S be a blowup of a point. Then M π β(s ) M β (S). So, n π β(s ) = n β (S). = By blowing down all ( 1)-curves l with β l = 0, it is enough to consider very ample β s. For very ample β s, we use PT-BPS correspondence (KKV method). J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

21 Pandharipande-Thomas (PT) stable pair invariant Let X be a Calabi-Yau threefold and β H 2 (X ). A PT stable pair (F, s) on X consists of a sheaf F on X with a section s H 0 (F ) such that F is pure of dimension 1 (No zero-dimensional subsheaf). s generates F outside of finitely many points. The moduli space P n (X, β) of stable pairs with [F ] = β, χ(f ) = n can be constructed by GIT. (Le Potier) P n (X, β) admits a symmetric obstruction theory. = Pandharipande-Thomas invariant PT n,β. J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

22 PT-BPS Correspondence Z PT = n,β PT n,βq n t β. Conjecture(PT-BPS correspondence) : Z PT = ( ( 1 ( q) j+1 t β) jn 0 2g 2 β (1 ( q) g k t β) ( 1) k+g g n β( 2g 2 k ) ), β j=1 g=1 k=0 where n g β is the BPS invariant. Katz-Klemm-Vafa method (cf. [C.-Katz-Klemm]) There is a procedure to compute all n g β (or more generally all refined BPS indices). In particular, nβ 0 = PT 1,β PT 1,β + correction terms, where correction terms are from reducible curves. J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

23 Example For S = P 2 and β = 4H. X = Tot(O P 2( 3)). P 1 (X, 4H) = {(C, Z) deg C = 4, Z C, length Z = 3}. P 1 (X, 4H) is a P 11 -bundle over Hilb 3 (P 2 ). χ(p 1 (X, 4H)) = = 264, PT 1,4H = 264. Similarly, P 1 (X, 4) is a P 13 -bundle over Hilb 1 (P 2 ). χ(p 1 (X, 4H)) = 14 3 = 42, PT 1,4H = 42. The correction term is PT 0,3H PT 1,H = ( 10) 3 = 30. Therefore, n 4H = 264 ( 42) + 30 = 192. J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

24 δ-stable pairs A pair on X is a pair (F, s) of a coherent sheaf F on X and a nonzero section s H 0 (F ). Let δ be a positive rational number, r(f ) be the leading coefficient of χ(f (m)). A pair (F, s) is called δ-(semi)stable if 1 F is pure (no zero dimensional subsheaf). 2 For all proper nonzero subsheaf F of F, we have χ(f ) + ɛ(s, F )δ r(f ) < ( ) χ(f ) + δ, r(f ) where ɛ(s, F ) = 1 if s factors through F and 0 otherwise. M δ n(x, β) : Moduli space of δ-semistable pairs (F, s) on X such that [F ] = β and χ(f ) = n (Le Potier). J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

25 χ(f ) + ɛ(s, F )δ r(f ) Assume β and χ are coprime. For a sufficiently large δ :=, < ( ) χ(f ) + δ, r(f ) M n (X, β) = P n (X, β). For a sufficiently small δ := 0 +, we have the forgetting map M 0+ 1 (X, β) M β. J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

26 PT 1 (X, β) M1 0+ (X, β) M β Then we have χ(m β ) = χ(m1 0+ (X, β)) χ(m0+ 1 (X, β)). Compare with n β = PT 1,β PT 1,β + correction terms, The correction term are from the wall-crossing between M n (X, β) and M 0+ n (X, β) J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

27 Wall-crossing The moduli space remains unchanged except for only finitely many δ. (Such δ is called the walls.) δ is a wall if and only if there exists strictly semistable objects. Let 0 A P B 0 be a destabilizing sequence at a wall. Then P is stable on the one side of the wall, but not stable on the other side, where it is replaced by new objects 0 B P A 0. (Simple wall-crossing) J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

28 BPS numbers for local del Pezzo Lemma 0-very ample=globally generated, 1-very ample = very ample... Let g = p a (β). If β is (g + n 2)-very ample, then P n (X, β) is a P w n -bundle over Hilb g+n 1 (S). When β is very ample and p a (β) 2, the relavant PT spaces are smooth. For p a (β) = 0 or 1, there are no wall-crossing contributions. For non-very ample β with p a (β) = 1, blowing down corresponds to wall-crossing. (cf. compare with log BPS calculation.) For p a (β) = 2, we computed the wall-crossing and calculated n β. J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

29 Remarks If p a (β) 2, the computation is harder. The PT moduli space is not smooth. The PT invariants are not known in general. The wall-crossing is not of simple type. But we may apply Joyce-Song type virtual wall-crossing. In principle, BPS numbers can be obtained by mirror symmetry and B-model. The same calculation applies to the Poincaré polynomials (the refined invariants) of M β. Conj. The Poincaré polynomial of M β is also divisible by (1 + t + + t w 1 ). Q. Is there a corresponding refinement for log BPS invariants? J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

30 BREAK Coming up next: Talk 2 : Log BPS invariants J. Choi (Sookmyung) Local and log BPS invariants Aug / 58

31 Preparation Joint work with J. Choi, S. Katz and N. Takahashi. Setup S = del Pezzo surface. β H 2 (S, Z) curve class. w = β ( K S ) its degree. E = smooth effective anticanonical divisor (an elliptic curve). Goal Enumerative interpretation for the n β (conjectural). Connection to quiver DT invariants. More practice with moduli spaces. M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

32 Local Gromov-Witten invariants of S Recall that for the local CY 3-fold X = Tot(K S ), the genus 0 degree β GW invariant is I β (X ) = deg[m 0 (X, β)] vir. They are virtually counting genus 0 stable maps f : C Tot(K S ) of degree f [C] = β. M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

33 Relative g=0 GW invariants of (S, E) of maximal tangency N β (S, E) := # vir of g=0 relative stable maps C S meeting E in exactly one point of maximal tangency. Generic image curve: The theory (moduli space and obstruction theory) was developed by J. Li. For a generalization, see Gross-Siebert. M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

34 Think of E = boundary. We counts maps A 1 S E extending to P 1 S, meeting E in one point. Virtual dimension = 0 by example The space of conics in P 2 has dimension 5. A generic conic meets E in 6 points: M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

35 Each time 2 intersection points are moved together, intersection multiplicity at that point +1 and vdim 1. M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

36 M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

37 In the end, after 5 identifications, vdim = 0. M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

38 A local-relative correspondence theorem Theorem (Gathmann, Graber-Hassett, van Garrel-Graber-Ruddat) ( 1) w 1 w I β (K S ) = N β (S, E). K S local CY S (S, E) log CY I β (K S ) N β (S, E) M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

39 A local-relative correspondence theorem Theorem (Gathmann, Graber-Hassett, van Garrel-Graber-Ruddat) ( 1) w 1 w I β (K S ) = N β (S, E). K S local CY S (S, E) log CY I β (K S ) N β (S, E) n β M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

40 A local-relative correspondence theorem Theorem (Gathmann, Graber-Hassett, van Garrel-Graber-Ruddat) ( 1) w 1 w I β (K S ) = N β (S, E). K S local CY S (S, E) log CY I β (K S ) n β N β (S, E)? m β M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

41 Why relative GW? 2 Major Advantages Finite number of possible image curves in each degree. Interplay with the arithmetic of E. E(β) := {P E : C of degree β s.t. C E = wp}, finite set. The relevant moduli space M β (S, E) decomposes as M β (S, E) = M P β (S, E), and hence N β (S, E) = P E(β) Nβ P P E(β) (S, E). M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

42 Degree 3 over primitive P in P 2 M P 3H(P 2, E) = 3 isolated points. N P 3H (P2, E) = 3. M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

43 Degree 3 over a flex P - generic E M P 3H(P 2, E) = [{C1}] [{C2}] R, where R = {3 : 1 relative covers C L} is 2-dimensional. N P 3H (P2, E) = (Gross-Pandharipande-Siebert). M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

44 Degree 3 over a flex P - special E M P 3H(P 2, E) = [{cusp} 2 ] R. N P 3H (P2, E) = = M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

45 Recall: n β/k k β k 3 Identities, after a suggestion of P. Bousseau ( 1) w 1 w n β = k β = I β (X ) and ( 1) w 1 w I β (K S ) = N β (S, E). = k β = P E(β) Definition, after P. Bousseau m P β := {k β : P E(β/k)} ( 1) (k 1)w/k k 2 µ(k) N β/k (S, E) }{{} ( 1) (k 1)w/k k 2 m P β, where: µ(k) ( 1) (k 1)w/k k 2 P E(β/k) Nβ/k P (S, E) µ(k) Nβ/k P (S, E). M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

46 Definition m P β := {k β : P E(β/k)} ( 1) (k 1)w/k k 2 µ(k) Nβ/k P (S, E). Equivalently, N P β (S, E) = {k β : P E(β/k)} ( 1) (k 1)w/k k 2 m P β/k. M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

47 Degree 3 over a flex P - generic case, revisited. For Q non-flex, N Q 3H (P2, E) = 3 = m Q 3H. For P a flex, N P 3H (P2, E) = m P 3H = 1 NP 3H (P2, E) 1 9 NP H (P2, E) = = 3. It follows from [N. Takahashi, unpublished] that the m P β agree for P2 in degree 4, leading to: M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

48 Conjecture A (very hard) P, P E(β), m P β = mp β. Since E(β) = w 2, Conj. A = ( 1) w 1 w n β = mβ P = w 2 m Q β P E(β) for any Q E(β). Conjecture B (weaker) Choose P E(β) to be β-primitive (definition later). Then ( 1) w 1 n β w = mp β. N. Takahashi 96 implies that Conj. B holds for P 2 and degree 8. M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

49 Important Remarks Nβ P(S, E) is further decomposed and mp β further refined according to the disjoint components of M P β (S, E). Adjusted multiple cover contributions of components of M P β (S, E) yield quiver DT invariant contributions to m P β. In the previous example, m3h P = = 3, N = DT 3 2 is the 2-loop quiver DT invariant of degree 3: M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

50 More generally Define: C = rational curve in S, fully tangent to E at P. Contr d (C) := contribution of d : 1 relative covers C C to Nd[C] P (S, E). By GPS, Contr d (C) = 1 ( ) d([c] E 1) 1 d 2. d 1 Recall: m P β = {k β : P E(β/k)} ( 1) (k 1)w/k k 2 µ(k) Nβ/k P (S, E). Contr(C, m P d[c] ) := k d ( 1) (k 1)d[C] E/k k 2 µ(k) Contr d/k (C), the contribution of covers of C to m P d[c]. M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

51 Multiple cover contributions are DT invariants Theorem (relying on Reineke) Contr(C, md[c] P ([C] E 1) ) = DT d. Here, DT (m) n is the degree n DT invariant of the m-loop quiver: Question What about relative covers over reducible curves? M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

52 β-primitivity Definition P E is said to be β-primitive if β E wp, but there is no decomposition into pseudo-effective curve classes β = β + β, with β, β 0 and β E w P. M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

53 Log BPS state counts For a β-primitive P, set M P rel (S, β) := {S C rational : [C] = β, C E = wp in one branch}. Theorem (Enumerative interpretation) explicit m : Mrel P (S, β) N such that mβ P = m(c). C Mrel P (β) Moreover, m(nodal) = 1 and m(cusp) = 2. Definition For P a β-primitive point, the log BPS numbers of degree β are m β := m P β. M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

54 Conjecture B, restated ( 1) w 1 n β w = m β. Theorem Conjecture B holds for all S and for classes β of arithmetic genus p a (β) := 1 2 β(β + K S) Recall from the previous talk: Theorem h = # of ( 1)-curves on S not intersecting β. If p a (β) = 0, then n β = ( 1) w 1 w. If p a (β) = 1 and β K S8, then n β = ( 1) w 1 w(χ(s) h). If β = K S8, then n β = ( 1) w 1 w(χ(s 8 ) + 1) = 12. If p a (β) = 2, then n β = ( 1) w 1 w( ( ) χ(s) h 2 + 5). M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

55 Proposition Assume irreducible reduced curve C of class β such that C E wp for some P E. Then there is a S.E.S. 0 H 0 (O S (C E)) H 0 (O S (C)) res H 0 (O E (wp)) 0. Moreover, dim ( H 0 (O S (C E)) ) = p a (β). Arithmetic genus 0 Assume that p a (β) = 0. By the Proposition, H 0 (O S (C)) res = H 0 (O E (wp)). Up to scalar multiple, there is exactly one section s H 0 (O E (wp)) vanishing at P with multiplicity w. Hence m β = 1 as predicted. M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

56 Arithmetic genus 1 Assume that p a (β) = 1 and that β K S8. By the Proposition, H 0 (O S (C)) res H 0 (O E (wp)), with 1-dimensional kernel. Let s H 0 (O E (wp)) be the only, up to C, section vanishing at P with multiplicity w. Consider the linear system res 1 (C s). Step 1: Blow down all the ( 1)-curves on S that do not intersect β. Step 2: Adding in E, we obtain a pencil Λ. There is one curve D in the pencil that is nodal at P. Step 3: Blow up the strict transform of P a number w of times. M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

57 Keeping track of the blowups Exercise We get the universal family U P 1 of the pencil and the proper transform of D is a cycle of w P 1 s. For a smooth curve C of the pencil, χ(c) = 0. If C is nodal, χ(c) = 1. If C is cuspidal, χ(c) = 2. There are no worse singularities. M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

58 By the cut and paste properties of χ( ), χ(u) = 0 χ(p 1 ) + # { nodal fibers } + 2 # { cuspidal fibers } + w. On the other hand, χ(u) = χ(s) + # { blow ups } # { blow downs}. Hence, m β = # { nodal fibers } + 2 # { cuspidal fibers } = χ(s) h, as predicted. Similarly for β = K S8. M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

59 Arithmetic genus 2 Unfortunately waaaaaaay too technical for the time remaining. Hence... Have a nice lunch and... We hope you enjoyed our talks and... M. van Garrel (KIAS) Local and log BPS invariants Aug / 58

60 Thank you! M. van Garrel (KIAS) Local and log BPS invariants Aug / 58