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

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Euler Characteris*Cs of Crepant Resolu*Ons of Weierstrass Models Euler characteris-cs of crepant resolu-ons 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 s'll not sa'sfied with algebra, in that it yields neither the shortest proofs nor the most beau'ful 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 posi'on, as algebra deals with magnitude.” Leibniz’s leDer 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 connec-ng 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 solu-on 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 characteris-c. χ = E V + F − E : = number of edges V := number of ver-ces F := number of faces Euler-Poincaré characteris-c Given a smooth manifold M, its Euler-Poincaré characteris-c is a topological invariant given by the alternang sum of Be numbers: χ(M):=β β + β + +( 1)nβ 0 − 1 2 ··· − n βi is the i-th Betti number of M Gauss-Bonnet Theorem The Gauss-Bonnet Theorem expresses the Euler characteris-c Of a manifold by its curvature: KdM=2⇡(M) Z It relates a differen-al geometric invariant with a tological one. Gauss-Bonnet-Chern The Euler characteris-c is the degree of the total Chern class (in homology): χ(Y )= c(Y ). ZY Batyrev’s theorem Two varie-es over the complex numbers related by a crepant biraonal have the same Be numbers. This is par-cularly true when the two varie-es are crepant resolu-ons of the same underlying singular variety! Applicaons to ellip-c fibraons • 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 fibraon over a smooth base is biraonal to a (possibly singular) Weierstrass model. Applicaons to ellip-c fibraons • 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. 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 characteris-c in Physics • Number of generaons in a Calabi-Yau compac-ficaon is |χ|/2 • Charge of D3 branes induced by curvature • Anomaly cancellaon condi-ons in D=6 sugra with 8 supercharges • Wimen’s index • Morse’s theory • etc It is possible to compute Euler characteris-c of crepant resolu-ons of ellip-c fibraons by the method of “pushforward” which allows to express the topological invariants by using only data from the base of the ellip-c fibraon. The method of pushforward is a basic construc-on in intersec-on 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 projec-ve 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 projec-ve 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 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 12L χ(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 characteris-c of a smooth Weierstrass model. Arer an explicit expansion 12L χ(Y )= c(TB) 1+6L ZB becomes d i χ(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 ellip-c 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.
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