Euler CharacterisCs of Crepant ResoluOns of Weierstrass Models

Euler characteriscs of crepant resoluons of Weierstrass models

Jonathan Mboyo Esole Northeastern University Collaborators

• Paolo Aluffi • James Fullwood • Ravi Jagadassan • Patrick Jefferson • Monica Kang • Sabrina Pasterski • Julian Salazar • Shu-Heng Shao • Shing-Tung Yau Topology was first a dream of Leibniz

“I am sll not sasﬁed with algebra, in that it yields neither the shortest proofs nor the most beauful construcons of geometry. Consequently, in view of this, I consider that we need yet another kind of analysis, geometric or linear, which deals directly with posion, as algebra deals with magnitude.” Leibniz’s leer to Huygens, September 8, 1679. Seven bridges of Königsberg

Königsberg is a medieval city on the Pregal river with two big islands on the river and exactly seven bridges connecng the islands and the banks.

Photo credit: Wikipedia. Seven bridges of Königsberg

Ehler introduced the following famous problem to Euler:

Is there a route which crosses every bridges once and only once?

Euler’s negave soluon to the problem gave birth to both graph theory and topology. Photo credit: Wikipedia. Euler formula for regular polyhedra

Euler also gave us the ﬁrst topological invariant, the Euler characterisc.

= E V + F E : = number of edges V := number of verces F := number of faces Euler-Poincaré characterisc

Given a smooth manifold M, its Euler-Poincaré characterisc is a topological invariant given by the alternang sum of Be numbers:

(M):= + + +( 1)n 0 1 2 ··· n

i is the i-th Betti number of M Gauss-Bonnet Theorem

The Gauss-Bonnet Theorem expresses the Euler characterisc Of a manifold by its curvature:

KdM=2⇡(M) Z

It relates a diﬀerenal geometric invariant with a tological one. Gauss-Bonnet-Chern The Euler characterisc is the degree of the total Chern class (in homology):

(Y )= c(Y ). ZY Batyrev’s theorem

Two variees over the complex numbers related by a crepant biraonal have the same Be numbers.

This is parcularly true when the two variees are crepant resoluons of the same underlying singular variety! Applicaons to ellipc fibraons • Ellipc surfaces were studied by Kodaira and Néron who classified their singular fibers.

• Tate’s developed an algorithm to compute the type of singular ﬁbers of an ellipc surface.

• An ellipc ﬁbraon over a smooth base is biraonal to a (possibly singular) Weierstrass model. Applicaons to ellipc ﬁbraons

• Not all singular Weierstrass models have a crepant resoluon.

• When a Weierstrass model has a crepant resoluon, it is not necessary unique.

• Different crepant resoluons of the same Weierstrass model are connected by a finite sequence of flops.

INCIDENCE GEOMETRY IN A WEYL CHAMBER II: SLn 7

6 6, 1 6, 2 6, 3 ? 6, 4 ? 6, 5 { }{}{}{} { } { }

6, 2, 1 6, 3, 1 6, 4, 1 6, 3, 2 6, 3, 2, 1 6, 5, 1 { }{}{ }{ }{ }{ } 6 ESOLE,JACKSON,JAGADEESAN,ANDNOEL¨ 6 ESOLE,JACKSON,JAGADEESAN,ANDNOEL¨

µ2 5, 4, 3 5Examples of ﬂops , 4, 2 5, 3, 2µ2 5, 4, 1 5, 4 4, 3, 2 { }{}{ }{ }{}{} 1 µ1 µ

5, 4, 3, 2, 1 5, 4, 3, 2 5, 4, 3, 1 5, 4, 2, 1 ? 5, 3, 2, 1 ? 4, 3, 2, 1 SU(4) SU(3) { }{ }{ }{ } { } { } O O O O µ1 µ1

2 Figure 6. The 24 chambers of I(sl6,V ). The four non-simplicial 2 chambers of I(sl6,V ) are marked with a star (see Figure 9 on page 9). 2 3 V µ µ2 3 V µ µ

Figure 3. I(sl ,V54321 2)con- Figure 4. I(sl ,V 2)con- 3 2 4 2 sists of two chambers separated Figure 3. I(sistssl3,V of four chambers.)con- The ad- Figure54 4. I(sl4,V )con- V sists of two chambers separated V sists of four chambers. The ad- by a half-line. See [7, Figure5432 1]. jacency542 graph541 of these cham- by a half-line. See [7, FigureV 1]. jacency graph of theseV cham- 543 bers is a linear chain.641 See [7, Figure 2]. bers is a linear chain. See [7, 5431 651 Figure 2].

4321

5421 432143 65 432 54 43 432 54 431 5321 53 64 SU(6) 431 53 431 63 24 ﬂops

432 62 421 52

532 621 321 51 632 421631 52 521 61 6321 5 321 51 6 521 Figure 5. Adjacency graph of the chambers of I(sl ,V 2). See [20, Figure 5 5 2] and [10, Figure 17] for the central hexagon. Each node represents a chamber 2 Figure 7. Adjacency graph of the 24V chambers of I(sl6,V ). The characterized by a subset S = a1 >a2 > >as of 1, 2, 3, 4, 5 . All the chambers arefour simplicial. colored The nodes adjacency{Figure are the graph 5. non-simplicial···Adjacency of the} chambers{ graph chambers. of I( ofsl the},V chambers2) of I(sl ,V 2). See [20, Figure 5 V5 is realized by explicit resolutions2] of and singularities [10, Figure of a 17]SU(5) for Weierstrassthe central model hexagon. Each node represents a chamber V V in [8]. characterized by a subset S = a1 >a2 > >as of 1, 2, 3, 4, 5 . All the { ··· } { } 2 chambers are simplicial. The adjacency graph of the chambers of I(sl5,V ) is realized by explicit resolutions of singularities of a SU(5) Weierstrass model in [8]. V 8 ESOLE,JACKSON,JAGADEESAN,ANDNOEL¨

765 SU(7) has 58 ﬂops

762 764 763 761 7

753 76

752 754 75 71 652 751 65 654 74 7621 72 651

653 7521 741 721 6541 73

6521

6542 6531 742 731

743

65421 732 6543 6532 643 7421

7432 65431 65321 543 7321 74321

6432 7431 64321 6321 6542 6431

54321 6421

65421 5432 5421 5431 5321

4321

2 Figure 8. Adjacency graph for the 58 chambers of I(sl7,V ). The 18 colored nodes are the non-simplicial chambers. V The Euler characterisc in Physics

• Number of generaons in a Calabi-Yau compacficaon is |χ|/2 • Charge of D3 branes induced by curvature • Anomaly cancellaon condions in D=6 sugra with 8 supercharges • Wien’s index • Morse’s theory • etc It is possible to compute Euler characterisc of crepant resoluons of ellipc fibraons by the method of “pushforward” which allows to express the topological invariants by using only data from the base of the ellipc fibraon.

The method of pushforward is a basic construcon in intersecon theory. The degree is invariant under pushforward c(Y )= f c(Y ) ⇤ ZY ZB Where f is a morphism from Y to B. Theorem. Consider a projecve bundle over B 2 3 ⇡ : X0 = P[OB L ⌦ L ⌦ ] B where is a line bundle over B. L ! Let Q(t) be a formal power series in t with coeﬃcients In the pullback of the Chow ring of B in the projecve bundle:

Q(H) Q(H) Q(0) ⇡ Q(H)= 2 +3 + , ⇤ H2 H2 6L2 H= 2L H= 3L where L = c1(L ) and H = c1(O(1)) is the ﬁrst Chern class of the dual of the tautological line bundle of X0. Euler characterisc of a smooth Weierstrass model.

Theorem (Aluﬃ-ME). The Euler characterisc of a smooth Weierstrass model Y over a base B is

12L (Y )= c(TB) 1+6L ZB

Where L is the ﬁrst Chern class of the fundamental line bundle of the Weierstrass model and c(TB) is the total Chern class of the base B. Euler characterisc of a smooth Weierstrass model. Aer an explicit expansion 12L (Y )= c(TB) 1+6L ZB becomes d i (Y )= 2 ( 6L) cd i(TB) i=1 X where ci(TB)isthei-th Chern class of the tangent bundle of B. 1.2 G-models

In this section, we recall how a Lie group is naturally associated with an elliptic fibration and introduce the notion of a G-model. Our notation for dual graphs and Kodaira fibers is spelled out in §1.1, and Tables 2 and 3. See also Appendix C for the definitions of a fiber type,ageneric fiber, and a geometric generic fiber. Definition 1.2 ( -model). Let be the type of a generic fiber. Let S B be a divisor of a projective variety B. An elliptic fibration ' Y B over B is said to be -model if K K ⊂ 1. The discriminant locus ' contains∶ as an→ irreducible component theK divisor S B.

2. The generic ﬁber over S is( of) type . ⊂

3. Any other ﬁber away from S is irreducible.K

If the dual graph of corresponds to an affine Dynkin diagram of type g˜t, where g is a Lie algebra, then the -model is also called a g-model. K In F-theory,K a Lie group G ' attached to a given elliptic fibration ' Y B depends on the type of generic singular fibers and the Mordell-Weil group MW ' of the elliptic fibration. The Lie Group associated to an ellipc fibraon ( ) ∶ → algebra g associated to the elliptic fibration is then the Langlands dual g i gi of g i gi.If Given an elliptic fibration ' : Y( ) B, we denote by exp g the unique (up to isomorphism) simply connected! compact∨ ∨ group whose Lie we attach the following group = = algebra is g , then the∨ group associated to the elliptic fibration ' Y B is: ∨ ( ) exp g ∶ → G ' U 1 rk MW ' , MWtor∨' ( ) ( ) ( ) ∶= × ( ) where rk MW ' is the rankDefinition. of the Mordell-WeilAn elliptic( group fibration) of'': andY MWB with' is the torsion subgroup ! tor of the Mordell-Weil groupan of associated'. Lie group G = G(')iscalledaG-model. ( ) ( ) Definition 1.3 (G-model). An elliptic fibration ' Y B with an associated Lie group G G ' is called a G-model. ∶ → = ( ) If the reduced discriminant locus has a unique irreducible component S over which the generic fiber is not irreducible, the group G ' is simple. The relevant fiber g˜t can be realized by resolving the singularities of a Weierstrass model derived from Tate’s algorithm. The relation between the ( ) fiber type and the group G ' is not one-to-one. For example, an SU(2)-model can be given by a s ns ns ns divisor S with a fiber of type I2,I2 ,III,IV ,orI3 . For that reason, a given decorated Kodaira ( ) fiber provides a more refined characterization of a G-model. s Example 1.4. For n 4,anSU(n)-model is a In-model with a trivial Mordell-Weil group. For n 0, s aSpin(8+2n)-model is a In -model with a trivial Mordell-Weil group. For n 1,aSpin(7+2n)- ns ≥ ∗ ns ≥ model is an In -model with a trivial Mordell-Weil group. A G2-model is an I0 -model with a trivial ∗ ss ∗ ≥ Mordell-Weil group. A Spin(7)-model is an I0 -model with a trivial Mordell-Weil group. ∗ ns ns s Example 1.5 (See [24]). The SO(3), SO(5), SO(6), and SO(7)-models are respectively I2 ,I4 ,I4, *ss *s and I0 -models with MW=Z 2Z.Forn 0,anSO(8 2n)-model is an In -model with a Mordell- *ns Weil group MW=Z 2Z.Forn 1,anSO(7 2n)-model is an In -model with Mordell-Weil group ≥ + MW=Z 2Z. ≥ + 5 1.1 Conventions

Throughout this paper, we work over the field of complex numbers. A variety is a reduced and irreducible algebraic scheme. We mostly follow the notation and conventions of Fulton [31]. Let V B be a vector bundle over a variety B. We denote the by P V the projective bundle of lines in V . We use Weierstrass models defined with respect to the projective bundle ⇡ X0 → 2 3 ( ) P OB L L B where L is a line bundle of B. We denote the pullback of L with respect ∶ = to ⇡ by ⇡ ⊗L . We⊗ denote by O 1 the canonical line bundle on X0, i.e., the dual of the tautological [ ⊕ ⊕ ] → line bundle∗ of X0 (see [31, Appendix B.5]). The first Chern class of O 1 is denoted H and the first ( ) Chern class of L is denoted L. The Weierstrass model ' Y0 B is defined as the zero-scheme of a ( ) section of O 3 ⇡ L 6. Weierstrass models are studied in more detail in §C.3. The Chow group ∶ → A X of a nonsingular∗ ⊗ variety X is the group of divisors modulo rational equivalence [31, Chap. ( ) ⊗ 1,§1.3]. We use V to refer to the class of a subvariety V in A X . Given a class ↵ A X ,the ∗( ) degree of ↵ is denoted X ↵ (or simply ↵ if X is clear from the∗ context.) Only the zero component∗ [ ] ´ ´ ( ) ∈ ( ) of ↵ is relevant in computing X ↵—see [31, Definition 1.4, p. 13]. We use c X c TX X to refer to the total homological´ Chern class of a nonsingular variety X, and likewise we use ci TX ( ) = ( ) ∩ [ ] to denote the ith Chern class of the tangent bundle TX. Given two varieties X,Y and a proper ( ) morphism f X Y , the proper pushforward associated to f is denoted f .Ifg X Y is a flat 1 morphism, the pullback of g is denoted g and by definition g V g V ∗ , see [31, Chap 1, §1.7]. ∶ → i ∗ n ∗ − ∶ → Given a formal series Q t i 0 Qit , we define t Q t Qn. [ ] = [ ( )] Our conventions for affine∞ Dynkin diagrams are as follows. A projective Dynkin diagram is ( ) = ∑ = [ ] ( ) = denoted Mk where M is A, B, C, D, E, F ,orG,andk is the number of nodes. An affine Dynkin diagram that becomes a projective Dynkin diagram g after removing a node of multiplicity one is denoted g˜. We denote by g˜t the (the possibly twisted) affine Dynkin diagram whose Cartan matrix is the transpose of the Cartan matrix of g˜. The graph of g˜t is obtained by exchanging the directions of all the arrows of g˜. When the extra node is removed, the dual graph of g˜t reduces to the dual graph of the Langlands dual of g. The affine Dynkin diagrams g˜t and g˜ are distinct only when g is t 1 not simply laced (i.e., when g is G2,F4,Bk,andCk). The notation g˜ follows Carter [9, Appendix, p. 540-609] and is equivalent to the notation g˜ used by MacDonald in §5 of [44]. The multiplicities t t ∨ define a zero vector of the extended Cartan matrix. In the notation of Kac, B` (` 3), C` (` 2), t t Twisted affine vs affine Dynkin diagram. 2 2 3 2 G2,andF4 are respectively denoted A2` 1, D` 1, D4 ,andE6 . ̃ ̃ ( ) ( ) ( ) ( ) ≥ ≥ ̃ ̃ ̃ − ̃ + ̃ ̃ 1 1

t 2 2 2 1 2 2 2 2 B3 ` B3 ` 1 1 ̃ + ̃ +

t 1 1 1 1 1 1 1 2 2 2 2 1 C2 ` C2 ` ̃ + ̃ + t 1 2 3 2 1 1 2 3 4 2 F4 F4 ̃ ̃ t 1 2 1 1 2 3 G2 G2 ̃ ̃ 1 There is a typo on page 570 of [9]intheﬁrstDynkindiagramofB` on the top of the page, where the arrow is in the wrong direction but correctly oriented in the rest of the page. ̃

4 New Pushforward Theorem

Theorem (ME, Kang, Jeﬀerson).

Let E be the class be the exceponal divisor of the blowup of the f : X X ! complete intersecon of d variees Zi Then we have the following pushforward: e d Zm f Q(E)= Q(Z`)M`, where M` = . ⇤ Zm Z` `=1 m=` X Y6

Examples

2L +3LS S2 6 c(B) SU(2) (1 + S)(1 + 6L 2S) L +2SL S2 12 c(B) SU(3), G2, USp(4) (1 + S)(1 + 6L 3S) L +3SL 2S2 12 c(B) Spin(8), F4 model (1 + S)(1 + 6L 4S) L +6LS 5S2 E8 12 c(B) (1 + S)(1 + 6L 5S) Models χ Y3 ,Eulercharacteristic ( ) Smooth Weierstrass 12L c1 − 6L ( 2 ) 2 SU 2 6 2c1L − 12L + 5LS − S ( ) ( 2 2 ) SU 3 or USp 4 or G2 12 c1L − 6L + 4LS − S ( ) ( ) ( 2 )2 SU 4 or Spin 7 4 3c1L − 18L + 16LS − 5S ( ) ( ) ( 2 2 ) Spin 8 or F4 12 c1L − 6L + 6LS − 2S ( ) ( 2 )2 SU 5 2 6c1L − 36L + 40LS − 15S ( ) ( 2 2 ) Spin 10 4 3c1L − 18L + 21LS − 8S ( ) ( 2 2) E6 6 2c1L − 12L + 15LS − 6S ( 2 2) E7 2 6c1L − 36L + 49LS − 21S ( 2 2 ) E8 12 c1L − 6L + 10LS − 5S ( 2 ) SO 3 12L c2 − 4c1L + 16L ( ) ( 2 ) SO 5 12L 20L − 8c1L + 3c2 ( ) ( 2 ) SO 6 12 4L − 2Lc1 + c2 L ( ) ( ) Table 8: Euler characteristic for elliptic threefolds

Models χ Y4 ,Eulercharacteristic ( ) 2 Smooth Weierstrass 12L −6c1L + c2 + 36L 2 (2 3 ) 2 2 3 SU 2 6 −12c1L + 5c1LS − c1S + 2c2L + 72L − 54L S + 15LS − S ( ) ( 2 2 3 2 2 3) SU 3 or USp 4 or G2 12 −6c1L + 4c1LS − c1S + c2L + 36L − 42L S + 17LS − 2S ( ) ( ) ( 2 2 3 2 2 ) 3 SU 4 or Spin 7 4 −18c1L + 16c1LS − 5c1S + 3c2L + 108L − 166L S + 89LS − 15S ( ) ( ) ( 2 2 3 2 2 )3 SU 5 −72c1L + 80c1LS − 30c1S + 12c2L + 432L − 830L S + 555LS − 120S ( ) 2 2 3 2 2 3 Spin 10 4 −18c1L + 21c1LS − 8c1S + 3c2L + 108L − 210L S + 140LS − 30S ( ) ( 2 3 2 2 2 3 ) Spin 8 or F4 12 −6c1L + c2L + 36L + 6c1LS − 2c1S − 60L S + 34LS − 6S ( ) ( 2 2 3 2 2 ) 3 E6 3 −24c1L + 30c1LS − 12c1S + 4c2L + 144L − 288L S + 195LS − 42S ( 2 2 3 2 2 3) E7 2 −36c1L + 49c1LS − 21c1S + 6c2L + 216L − 454L S + 321LS − 72S ( 2 2 3 2 2 3 ) E8 12 −6c1L + 10c1LS − 5c1S + c2L + 36L − 90L S + 75LS − 20S ( 2 3 ) SO 3 12L c3 − 4c2L + 16c1L − 64L ( ) ( 3 2 ) SO 5 4L −48L + 20L c1 − 8Lc2 + 3c3 ( ) ( 3 2 ) SO 6 12L −8L + 4L c1 − 2Lc2 + c3 ( ) ( ) Table 9: Euler characteristic for elliptic fourfolds

Models χ Y4 ,Eulercharacteristic ( ) 3 Smooth Weierstrass 12c1c2 + 360c1 3 2 2 3 SU 2 6 2c1c2 + 60c1 − 49c1S + 14c1S − S ( ) ( 3 2 2 3) SU 3 or USp 4 or G2 12 c1c2 + 30c1 − 38c1S + 16c1S − 2S ( ) ( ) ( 3 2 2 3) SU 4 or Spin 7 12 3c1c2 + 30c1 − 50c1S + 28c1S − 5S ( ) ( ) ( 3 2 2 3)) Spin 8 or F4 12 c1c2 + 30c1 − 54c1S + 32c1S − 6S ( ) ( 3 2 2 ) 3 SU 5 3 4c1c2 + 120c1 − 250c1S + 175c1S − 40S ( ) ( 3 2 2 3 ) Spin 10 12 c1c2 + 30c1 − 63c1S + 44c1S − 10S ( ) ( 3 2 2 )3 E6 3 4c1c2 + 120c1 − 258c1S + 183c1S − 42S ( 3 2 2 3 ) E7 6 2c1c2 + 60c1 − 135c1S + 100c1S − 24S ( 3 2 2 3 ) E8 12 c1c2 + 30c1 − 80c1S + 70c1S − 20S ( 3 ) SO 3 12c1 c3 − 48c1 + −4c1c2 ( ) ( 3 ) SO 5 4c1 3c3 − 28c1 − 8c1c2 ( ) ( 3 ) SO 6 12c1 −4c1 − 2c1c2 + c3 ( ) ( ) Table 10: Euler characteristic for Calabi-Yau elliptic fourfolds where c1 L. = 28 Models χ Y3 ,Eulercharacteristic ( ) Smooth Weierstrass 12L c1 − 6L ( 2 ) 2 SU 2 6 2c1L − 12L + 5LS − S ( ) ( 2 2 ) SU 3 or USp 4 or G2 12 c1L − 6L + 4LS − S ( ) ( ) ( 2 )2 SU 4 or Spin 7 4 3c1L − 18L + 16LS − 5S ( ) ( ) ( 2 2 ) Spin 8 or F4 12 c1L − 6L + 6LS − 2S ( ) ( 2 )2 SU 5 2 6c1L − 36L + 40LS − 15S ( ) ( 2 2 ) Spin 10 4 3c1L − 18L + 21LS − 8S ( ) ( 2 2) E6 6 2c1L − 12L + 15LS − 6S ( 2 2) E7 2 6c1L − 36L + 49LS − 21S ( 2 2 ) E8 12 c1L − 6L + 10LS − 5S ( 2 ) SO 3 12L c2 − 4c1L + 16L ( ) ( 2 ) SO 5 12L 20L − 8c1L + 3c2 ( ) ( 2 ) SO 6 12 4L − 2Lc1 + c2 L ( ) ( ) Table 8: Euler characteristic for elliptic threefolds

Thank you!

Euler Characteris*Cs of Crepant Resolu*Ons of Weierstrass Models

Euler CharacterisCs of Crepant ResoluOns of Weierstrass Models