Euler characteris-cs of crepant resolu-ons of Weierstrass models. Jonathan Mboyo Esole Northeastern University

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1 Euler characteris-cs of crepant resolu-ons of Weierstrass models Jonathan Mboyo Esole Northeastern University

2 Paolo Aluffi James Fullwood Ravi Jagadassan Patrick Jefferson Monica Kang Sabrina Pasterski Julian Salazar Shu-Heng Shao Shing-Tung Yau Collaborators

3 Topology was first a dream of Leibniz I am s'll not sa'sfied with algebra, in that it yields neither the shortest proofs nor the most beau'ful construc'ons of geometry. Consequently, in view of this, I consider that we need yet another kind of analysis, geometric or linear, which deals directly with posi'on, as algebra deals with magnitude. Leibniz s leder to Huygens, September 8, 1679.

4 Seven bridges of Königsberg Königsberg is a medieval city on the Pregal river with two big islands on the river and exactly seven bridges connec-ng the islands and the banks. Photo credit: Wikipedia.

5 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 nega-ve solu-on to the problem gave birth to both graph theory and topology. Photo credit: Wikipedia.

6 Euler formula for regular polyhedra Euler also gave us the first topological invariant, the Euler characteris-c. = E V + F E : = number of edges V := number of ver-ces F := number of faces

7 Euler-Poincaré characteris-c Given a smooth manifold M, its Euler-Poincaré characteris-c is a topological invariant given by the alterna-ng sum of BeZ numbers: (M) := ( 1) n n i is the i-th Betti number of M

8 Gauss-Bonnet Theorem The Gauss-Bonnet Theorem expresses the Euler characteris-c Of a manifold by its curvature: Z KdM=2 (M) It relates a differen-al geometric invariant with a tological one.

9 Gauss-Bonnet-Chern The Euler characteris-c is the degree of the total Chern class (in homology): (Y )= Z Y c(y ).

10 Batyrev s theorem Two varie-es over the complex numbers related by a crepant bira-onal have the same BeZ numbers. This is par-cularly true when the two varie-es are crepant resolu-ons of the same underlying singular variety!

11 Applica-ons to ellip-c fibra-ons Ellip-c 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 ellip-c surface. An ellip-c fibra-on over a smooth base is bira-onal to a (possibly singular) Weierstrass model.

12 Applica-ons to ellip-c fibra-ons Not all singular Weierstrass models have a crepant resolu-on. When a Weierstrass model has a crepant resolu-on, it is not necessary unique. Different crepant resolu-ons of the same Weierstrass model are connected by a finite sequence of flops.

13 {6, 2, 1} {6, 3, 1} {6, 4, 1} 6 6 {6, 3, 2} {6, 3, 2, 1} {6, 5, 1} ESOLE, JACKSON, JAGADEESAN, AND NOE L ESOLE, JACKSON, JAGADEESAN, AND NOE L Examples of flops {5, 4, 2} {5, 3, 2}µ2 {5, 4, 1} {5, 4, 3, 2} {5, 4, 3, 1} {5, 4, 2, 1} {5, 4, 3} µ1 µ1 SU(3) {5, 4, 3, 2, 1} µ2 {4, 3, 2} SU(4)?{5, 3, 2, 1}?{4, 3, 2, 1} O O O O {5, 4} µ1 µ1 V2 Figure 6. The 24 chambers of I(sl, V ). The four non-simplicial 6 V2 chambers of I(sl6, V ) are marked with a star (see Figure 9 on page 9). µ32 µ2 µ V V2 Figure 3. I(sl3, V ) configure 4.V2I(sl4, V ) conv four chambers. ) con- The ad3, of sists of two chambers separated Figure 3. I(sl sists separated by a half-line. See [7, Figure 1]. sists of two chambers jacency graph of these cham by a half-line. See [7, Figure 1]. 641 See [7, bers is a linear chain. 543 Figure 2] µ3 V2 Figure 4. I(sl, V ) con4 54 sists of four chambers. The adjacency graph of these chambers is a linear chain. See [7, Figure 2] SU(6) 24 flops V2 5 Figure 5. Adjacency graph of the chambers of I(sl5, V ). See [20, Figure V 2] and [10, Figure 17] for the central hexagon. Each node represents a chamber 2 7. SAdjacency graph 242,chambers of ). The characterized Figure by a subset = {a1 > a2 > > of as }the of {1, 3, 4, 5}. All V thei(sl6, V V2 2 four colored nodes are the non-simplicial chambers. Figure 5. Adjacency graph of the of I(sl5, V ). See [20, Figure chambers are simplicial. The adjacency graph of the chambers of I(sl ) 5, V chambers

14 V SU(7) has 58 flops

15 The Euler characteris-c in Physics Number of genera-ons in a Calabi-Yau compac-fica-on is χ /2 Charge of D3 branes induced by curvature Anomaly cancella-on condi-ons in D=6 sugra with 8 supercharges Wimen s index Morse s theory etc

16 It is possible to compute Euler characteris-c of crepant resolu-ons of ellip-c fibra-ons by the method of pushforward which allows to express the topological invariants by using only data from the base of the ellip-c fibra-on. The method of pushforward is a basic construc-on in intersec-on theory. The degree is invariant under pushforward Z Z Y c(y )= B f c(y ) Where f is a morphism from Y to B.

17 Theorem. Consider a projec-ve bundle over B : X 0 = P[O B L 2 L 3 ]! B L where is a line bundle over B. Let Q(t) be a formal power series in t with coefficients In the pullback of the Chow ring of B in the projec-ve bundle: Q(H) = 2 Q(H) H 2 H= 2L +3 Q(H) H 2 H= 3L + Q(0) 6L 2, where L = c 1 (L ) and H = c 1 (O(1)) is the first Chern class of the dual of the tautological line bundle of X 0.

18 Euler characteris-c of a smooth Weierstrass model. Theorem (Aluffi-ME). The Euler characteris-c of a smooth Weierstrass model Y over a base B is (Y )= Z B 12L 1+6L c(tb) 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.

19 Euler characteris-c of a smooth Weierstrass model. Arer an explicit expansion (Y )= Z B 12L 1+6L c(tb) becomes (Y )= 2 dx ( 6L) i c d i (TB) i=1 where c i (TB)isthei-th Chern class of the tangent bundle of B.

20 ( ) Group associated to an ellip-c fibra-on ( ) Given an elliptic fibration ' : Y! B, we attach the following group G(') = exp(g ) MW tor (') U(1)rk MW('), Definition. An elliptic fibration ' : Y! B with an associated Lie group G = G(') iscalledag-model.

21 Twisted affine ( ) vs ( affine ) ( Dynkin ) diagram. ( ) B t 3+` B 3+` C t 2+` C 2+` F t F 4 G t G 2

22 New Pushforward Theorem Theorem (ME, Kang, Jefferson). Let E be the class be the excep-onal divisor of the blowup f : X e! X of the complete intersec-on of d varie-es Z i Then we have the following pushforward: f Q(E) = dx `=1 Q(Z`)M`, where M` = Y m6=` Z m Z m Z`.

23 Examples SU(2) SU(3), G2, USp(4) Spin(8), F4 model L +3LS S 2 (1 + S)(1 + 6L 2S) c(b) L +2SL S 2 (1 + S)(1 + 6L 3S) c(b) L +3SL 2S 2 (1 + S)(1 + 6L 4S) c(b) E8 12 L +6LS 5S 2 (1 + S)(1 + 6L 5S) c(b)

24 Models χ(y 3 ),Eulercharacteristic Smooth Weierstrass 12L(c 1 6L) SU(2) 6(2c 1 L 12L 2 + 5LS S 2 ) SU(3) or USp(4) or G 2 12(c 1 L 6L 2 + 4LS S 2 ) SU(4) or Spin(7) 4(3c 1 L 18L LS 5S 2 ) Spin(8) or F 4 12(c 1 L 6L 2 + 6LS 2S 2 ) SU(5) 2(6c 1 L 36L LS 15S 2 ) Spin(10) 4(3c 1 L 18L LS 8S 2 ) E 6 6(2c 1 L 12L LS 6S 2 ) E 7 2(6c 1 L 36L LS 21S 2 ) E 8 12(c 1 L 6L LS 5S 2 ) SO(3) 12L(c 2 4c 1 L + 16L 2 ) SO(5) 12L(20L 2 8c 1 L + 3c 2 ) SO(6) 12(4L 2 2Lc 1 + c 2 )L

25 Models χ(y 4 ),Eulercharacteristic Smooth Weierstrass 12L( 6c 1 L + c L 2 ) SU(2) 6( 12c 1 L 2 + 5c 1 LS c 1 S 2 + 2c 2 L + 72L 3 54L 2 S + 15LS 2 S 3 ) SU(3) or USp(4) or G 2 12( 6c 1 L 2 + 4c 1 LS c 1 S 2 + c 2 L + 36L 3 42L 2 S + 17LS 2 2S 3 ) SU(4) or Spin(7) 4( 18c 1 L c 1 LS 5c 1 S 2 + 3c 2 L + 108L 3 166L 2 S + 89LS 2 15S 3 ) SU(5) 72c 1 L c 1 LS 30c 1 S c 2 L + 432L 3 830L 2 S + 555LS 2 120S 3 Spin(10) 4( 18c 1 L c 1 LS 8c 1 S 2 + 3c 2 L + 108L 3 210L 2 S + 140LS 2 30S 3 ) Spin(8) or F 4 12( 6c 1 L 2 + c 2 L + 36L 3 + 6c 1 LS 2c 1 S 2 60L 2 S + 34LS 2 6S 3 ) E 6 3( 24c 1 L c 1 LS 12c 1 S 2 + 4c 2 L + 144L 3 288L 2 S + 195LS 2 42S 3 ) E 7 2( 36c 1 L c 1 LS 21c 1 S 2 + 6c 2 L + 216L 3 454L 2 S + 321LS 2 72S 3 ) E 8 12( 6c 1 L c 1 LS 5c 1 S 2 + c 2 L + 36L 3 90L 2 S + 75LS 2 20S 3 ) SO(3) 12L(c 3 4c 2 L + 16c 1 L 2 64L 3 ) SO(5) 4L( 48L L 2 c 1 8Lc 2 + 3c 3 ) SO(6) 12L( 8L 3 + 4L 2 c 1 2Lc 2 + c 3 ) ( )

26 Models χ(y 4 ),Eulercharacteristic Smooth Weierstrass 12c 1 c c 3 1 SU(2) 6(2c 1 c c c2 1 S + 14c 1S 2 S 3 ) SU(3) or USp(4) or G 2 12(c 1 c c c2 1 S + 16c 1S 2 2S 3 ) SU(4) or Spin(7) 12(3c 1 c c c2 1 S + 28c 1S 2 5S 3 )) Spin(8) or F 4 12(c 1 c c c2 1 S + 32c 1S 2 6S 3 ) SU(5) 3(4c 1 c c c2 1 S + 175c 1S 2 40S 3 ) Spin(10) 12(c 1 c c c2 1 S + 44c 1S 2 10S 3 ) E 6 3(4c 1 c c c2 1 S + 183c 1S 2 42S 3 ) E 7 6(2c 1 c c c2 1 S + 100c 1S 2 24S 3 ) E 8 12(c 1 c c c2 1 S + 70c 1S 2 20S 3 ) SO(3) 12c 1 (c 3 48c c 1c 2 ) SO(5) 4c 1 (3c 3 28c 3 1 8c 1c 2 ) SO(6) 12c 1 ( 4c 3 1 2c 1c 2 + c 3 ) = Prove a conjecture of Blumenhagen-Grimm-Jurke-Weigand.

27 Thank you!