On Hirzebruch invariants of elliptic fibrations
OOn Hirzebruch invariants of elliptic fibrations
James Fullwood ♣ , , Mark van Hoeij ♣ ♣ Mathematics Department, Florida State University, Tallahassee, FL 32306, U.S.A.
Abstract
We compute all Hirzebruch invariants χ q for D , E , E and E elliptic fibrations of everydimension. A single generating series χ ( t, y ) is produced for each family of fibrations such thatthe coefficient of t k y q encodes χ q over a base of dimension k , solely in terms of invariants of thebase of the fibration. ♣ Email: jfullwoo at math.fsu.edu, hoeij at math.fsu.edu a r X i v : . [ m a t h . AG ] A p r Contents
1. Introduction 32. The fibrations under consideration 63. A motivating example 74. The Chern-ext character 95. The proof 105.1. Proof of main result 105.2. Proof of the corollary 116. Discussion 12References 13 Introduction
The prospect of realizing realistic particle physics (such as the Standard Model) ina regime of string theory coined “F-theory” by its originator Cumrun Vafa [1][2] hasprovided a source of attraction for string theorists and their mathematician counterpartsto the study of elliptic fibrations. F-theory was first formulated as a non-perturbativedescription of Type-IIB string theory on a complex n -fold B with an SL ( Z )-invariantcomplex scalar field known to physicists as the axio-dilaton field. Exercising a stringtheorist’s natural penchant for algebro-geometric descriptions of nature, Vafa formulateda geometrization of the SL ( Z )-invariance of the axio-dilaton via a Calabi-Yau ellipticfibration over the type-IIB n -fold B (which describes a theory in 10-2 n real space-timedimensions), interpreting the axio-dilaton as the complex structure modulus of an ellipticcurve. Not only is this formulation of non-perturbative type-IIB string theory aestheticallypleasing from a purely geometric perspective, the physical theory has attractive featuressuch as providing promising avenues for moduli-stabilization and potential realizationof GUT gauge groups which project by definition to the Standard Model gauge groupat lower non-supersymmetric energy levels. To realize the elliptic fibration explicitly,physicists have primarily focused on a Weierstrass fibration, i.e., a hypersurface in a P -bundle over the Type-IIB base B which in its reduced form is defined as the zero-schemeassociated with the locus y z = x + f xz + gz , where f and g are sections of appropriate tensor powers of a line bundle L on B . As inthe theory of curves, every smooth elliptic fibration is birational to a (possibly singular)fibration in Weierstrass form, often referred to in the physics literature as an E ellipticfibration. But crucial to the physical theory associated with an elliptic fibration arethe singular fibers of the fibration, as the singular fibers encode the structure of gaugetheories associated with D-branes wrapping components of the discriminant locus overwhich they appear. And since singular fibers of a fibration are in general not preservedunder a birational transformation, elliptic fibrations not in Weierstrass form enjoy theirown physical relevance.Motivated by tadpole cancellation in F-theory, in [3] Sethi,Vafa and Witten deriveda formula for the Euler characteristic of an elliptically fibered Calabi-Yau fourfold inWeierstrass form solely in terms of the Chern classes of the base of the fibration. Similarformulas for fibrations not in Weierstrass form were derived by Klemm, Lian, Roan andYau in [4]. It was later shown by Aluffi and Esole in [5][6] that these formulas are allnumerical avatars of more general Chern class identities which hold not only without anyCalabi-Yau hypothesis but over a base of arbitrary dimension. In this note we considerfour families of fibrations ϕ : Y → B which are known to the physics community as D , E , E and E elliptic fibrations from a purely mathematical perspective (i.e., with noCalabi-Yau hypothesis or restrictions on the dimension of the base), and pursue similar Technically this is an abuse of language, as it is the whole type-IIB theory which is SL ( Z )-invariant. formulas not for the Chern classes of a given fibration Y but for its Hirzebruch invariants(or arithmetic genera) χ q ( Y ) := (cid:90) ch(Ω qY )td( Y ) , where ch(Ω qY ) denotes the Chern character of the q th exterior power of the cotangentbundle of Y and td( Y ) := td( T Y ) ∩ [ Y ], i.e., the Todd class of the tangent bundle of Y acting on the fundamental class of Y .As integrals are invariant under proper pushforwards of the integrand, we relate χ q ( Y )to invariants of the base by pushing forward ch(Ω qY )td( Y ) via ϕ ∗ (the pushforward mapassociated with ϕ : Y → B ). By the celebrated Hirzebruch-Riemann-Roch theorem (latergeneralized by Grothendieck), χ q ( Y ) = (cid:88) i ( − i dim H i ( Y, Ω qY ) = h q, − h q, + · · · + ( − q h q,q , thus Hirzebruch invariants yield linear relations among the Hodge numbers of Y . For ageneral smooth complex projective variety X of fixed dimension, the standard approachto computing Hirzebruch invariants of X is to encode them in a generating series( † ) χ ( y ) = (cid:88) q χ q y q = (cid:90) X dim( X ) (cid:89) i =1 (1 + ye − λ i ) λ i − e − λ i , where the λ i s are the Chern roots of the tangent bundle of X . Given an elliptic fibration ϕ : Y → B of type D , E , E or E over a base B of arbitrary dimension , what we achievein this note is a single generating series χ ( t, y ) for each family where the coefficient of t k y q encodes χ q for the given family of elliptic fibrations over a base of dimension k , solelyin terms of Chern classes of the base and the first Chern class of a line bundle L → B (i.e., one can see the Hirzebruch invariants of the fibration as functions of invariants ofthe base).Let B be a smooth compact complex projective variety of arbitrary dimension endowedwith a line bundle L → B . The elliptic fibrations we consider will all be subvarietiesof an ambient projective bundle P ( E ) → B (each fibration will be precisely defined in § E is a vector bundle over B that is constructed by taking direct sums oftensor powers of L . Before stating the main result of this note, let us make the followingdefinitions:Let X be a smooth variety. We define the Hirzebruch series of X to be H y ( X ) := H ( X ) + H ( X ) y + H ( X ) y + · · · , where H q ( X ) := ch(Ω qX )td( X ) is the q th Hirzebruch characteristic class of X . Thengiven a proper morphism ϕ : X → B we define ϕ ∗ ( H y ( X )) in the obvious way: ϕ ∗ ( H y ( X )) := ϕ ∗ ( H ( X )) + ϕ ∗ ( H ( X )) y + ϕ ∗ ( H ( X )) y + · · · , where ϕ ∗ is the proper pushforward associated with the morphism ϕ . Our main result isthe following Theorem 1.1.
Let ϕ : Y → B be an elliptic fibration of type D , E , E or E and let U = e − c ( L ) . Then ϕ ∗ ( H y ( Y )) = Q · H y ( B ) , where Q = − y + ( y +1)( yU − yU +1) − U ( y +1) ( yU +1) for Y a D fibration − y + ( y +1)( yU − U − yU +1) for Y an E fibration − y + ( y +1)( yU − U − yU +1) for Y an E fibration − y + ( y +1)( yU − U − yU +1) for Y an E fibration The proof of this result is considerably streamlined via the use of
Chern-ext characters ,which we first introduce and define in §
4. We note that the numbers 4, 3, 2 and 1 in theexpressions for Q coincide with the number of sections of the given fibration. Reading offthe coefficient of y q in Q · H y ( B ) immediately yields χ q ( Y ) = (cid:90) B ( P H q ( B ) + P H q − ( B ) + · · · + P q td( B )) , where the P i s are polynomials in U = e − c ( L ) (we list all P i s in § χ ( t, y ) for each family of fibrations, where thecoefficient of t k y q encodes χ q for the given family of fibrations over a base of dimension k . Before unveiling the χ ( t, y ) we need the following definitions:Let R be a commutative ring with unity. For two series f ( t ) = a + a t + a t + · · · and g ( t ) = b + b t + b t + · · · in R [[ t ]], we recall the Hadamard product of f and g isdefined to be f (cid:12) g := a b + a b t + a b t + · · · . Furthermore, let [ t d ] : R [[ t ]] → R denote the map given by [ t d ][ f ] := a d . Now let X be asmooth projective variety of dimension d , let g = (1 + ye − t ) t − e − t and let f = ln( g ). As aconsequence of Lemma 5.1 we show H y ( X ) = (1 + y ) d · [ t d ][exp ( f (cid:12) ( − tC (cid:48) /C ))] , where C = 1 − c t + c t − c t + · · · ∈ R [[ t ]] with R = Z [ c , c , . . . , ] . As ϕ ∗ ( H y ( Y )) = As f (cid:12) ( − tC (cid:48) /C ) is independent of d , the c i s appearing in the definition of C are countably manyformal variables which acquire their familiar meaning as Chern classes of the tangent bundle of X inconcrete examples. Q · H y ( B ) by Theorem 1.1, the generating series χ ( t, y ) is constructed by replacing U = e − c ( L ) by U t = e − c ( L ) t in Q and then constructing a series in R [[ t ]] with R = Z [ c , c , . . . ]such that the coefficient of t d is precisely (1 + y ) d · [ t d ]exp ( f (cid:12) ( − tC (cid:48) /C )), which weinterpret as the Hirzebruch series for a base of dimension d . All Hirzebruch invariants χ q for D , E , E and E fibrations of all dimensions are then contained in the following Corollary 1.2.
Let Q t = Q ( e − c ( L ) t ) , where Q is defined as in Theorem 1.1 and let ∼ χ ( t, y ) = Q t · exp (cid:18) ln (cid:18) (1 + ye − t ) t − e − t (cid:19) (cid:12) − tC (cid:48) C (cid:19) , where C = 1 − c t + c t − · · · ∈ R [[ t ]] with R = Z [ c , c , . . . ] and C (cid:48) = ddt C . Then χ ( t, y ) := ∼ χ ( t (1 + y ) , y ) is a generating series for Hirzebruch invariants of D , E , E and E fibrations as thedefinition of Q varies according to Theorem 1.1, i.e., the integral of the coefficient of t k y q over a base of dimension k is precisely χ q for the given family of fibrations. As an illustration, the coefficient of t y in χ ( t, y ) in the E case is − L (1729 L − c L + ( − c + 193 c ) L + 5 c c − c ), thus χ of an E fibration over a base B ofdimension four is (for a computer implementation see [11]) (cid:90) B − L
12 (1729 L − c ( B ) L + ( − c ( B ) + 193 c ( B )) L + 5 c ( B ) c ( B ) − c ( B )) . The fibrations under consideration
We now formally define the objects under consideration, namely D , E , E and E elliptic fibrations (the names and definitions we use are all lifted from the physics literature[4]). We work over C though everything we say is equally valid over an algebraically closedfield of characteristic zero. All fibrations are constructed by taking equations of classicalelliptic curves and promoting their coefficients from scalars (or sections of line bundles overa point) to sections of line bundles over some smooth positive dimensional base variety B .As such, let B be some smooth compact complex projective variety of arbitrary dimensionendowed with (a suitably ample) line bundle L → B . This will be the base assumptionin each of the D , E , E and E cases.Now let E = O ⊕ L ⊕ L ⊕ L . A D elliptic fibration Y D is defined to be a smooth com-plete intersection in P ( E ) (here we take the projective bundle of lines in P ( E )) associatedwith the locus Y D : (cid:40) x − y − z ( az + cw ) = 0 w − x − z ( dz + ex + f y ) = 0 , where z is a section of O (1) (the dual of the tautological line bundle on P ( E )), and x , y and w are sections of O (1) ⊗ π ∗ L , where π is the projection π : P ( E ) → B . Then we take a , c , d , e and f to be sections of (minimal) appropriate tensor powers of π ∗ L such thateach of the defining equations for Y D is a well defined section of a line bundle on P ( E ).Taking a and d to be sections of π ∗ L and c , e and f to be sections of π ∗ L then defines Y D as a variety of class (2 H + 2 L ) ∈ A ∗ P ( E ), where H := c ( O (1)) and L := c ( π ∗ L ).Such a locus naturally determines an elliptic fibration ϕ : Y D → B , with generic fiber anelliptic curve in P . Such fibrations contain fibers not on the list of Kodaira and werestudied extensively in [7] from both a mathematical and physical perspective. We notethat as our results are topological in nature, they depend only on the class [ Y ] ∈ A ∗ P ( E )of the given fibration, i.e., the explicit equations which define the fibration are essentiallyirrelevant. The equations are included for concreteness and to not completely sever our-selves from the physical theories with which they are associated. The definitions of E , E and E fibrations are summarized in the following table:equation ambient projective bundle class in A ∗ P ( E ) E x + y = dxyz + exz + f yz + gz P ( O ⊕ π ∗ L ⊕ π ∗ L ) 3 H + 3 LE y = x + ex z + f xz + gz P , , ( O ⊕ π ∗ L ⊕ π ∗ L ) 4 H + 4 LE y z = x + f xz + gz P ( O ⊕ π ∗ L ⊕ π ∗ L ) 3 H + 6 L The coefficients of each fibration are chosen to be suitably generic sections of tensorpowers of π ∗ L such that the fibration is a smooth divisor of the indicated class in A ∗ P ( E ).We point out that the total space of an E fibration is defined as a hypersurface ina weighted projective bundle, which is isomorphic to a bundle of quadric cones. Toavoid dealing with any singularities of the ambient projective bundle while performingintersection theoretic computations, we embed P , , ( E ) in a P -bundle and then realizethe total space of the E fibration as a complete intersection of the image of P , , ( E ) viaits embedding in the P -bundle with another hypersurface.3. A motivating example
Let B be a non-singular compact complex algebraic variety of arbitrary dimensionendowed with a (suitably ample) line bundle L . We recall the definition of an E ellipticfibration, i.e., a surjective proper morphism ϕ : Y → B , whose total space Y is realizedas a hypersurface of class 3 H + 6 L in the the Chow ring of a projective bundle P ( E ) π → B ,where E = O ⊕ L ⊕ L , H := c ( O (1)) and L we non-reluctantly use to denote both c ( L ) and π ∗ c ( L ). As one can show that ϕ ∗ c ( Y ) = L L · c ( B ) [5], we exploit thefact that (cid:82) Y c ( Y ) = (cid:82) B ϕ ∗ c ( Y ) and compute the topological Euler characteristic χ ( Y ) byintegrating the coefficient of t dim ( B ) in the formal series χ dim ( B ) ( t ), where we define χ N ( t )for general N ∈ N to be More precisely, over the locus a = c = d = e = f = 0 in the base the fibers consist of four P s meetingat a point. Here and throughout, terms not expanded in a series such as Lt Lt in the series above are a shorthandfor their associated series expansions about t . χ N ( t ) := 12 Lt Lt · (1 + c t + c t + · · · + c N t N ) . As the series χ n ( t ) and χ n +1 ( t ) are identical up to order n (for any n ), we notice that theformal series χ E ( t ) = 12 Lt Lt · (1 + c t + c t + · · · + c m t m + · · · )serves as a generating series for the topological Euler characteristic for Y of all possibledimensions (i.e., the coefficient of t k encodes the Euler characteristic of Y over a base ofdimension k , solely in terms of L and Chern classes of B ). The c i s are then temporarilyformal objects (as their subscripts tend towards infinity), acquiring their familiar meaningwhence integrated upon. For example, over a base of dimension 3 the coefficient of t in χ E ( t ) is 12 L ( c − Lc + 36 L ), so χ ( Y ) over a base B of dimension 3 is (cid:90) B L ( c ( B ) − Lc ( B ) + 36 L ) . Though admittingly these observations are all rather trivial, what is striking is that thepushforward ϕ ∗ c ( Y ) is manifestly independent of the base of the fibration, i.e., over a baseof dimension k the actual class in A ∗ B associated with ϕ ∗ c ( Y ) is obtained by truncatinga formal powers eries at order k . As such, we deem this a “motivating example” as itexhibits general features which can be abstracted to cases of other invariants of ellipticfibrations of the form (cid:82) α . In particular, the first step at arriving at such base independentexpressions for ϕ ∗ c ( Y ) is deriving a factorization of c ( Y ) (which plays the role of α inour more general considerations) as g ( L, H ) · π ∗ c ( B ), where g is a rational expressiondepending only on L and H (as defined above). Once we have such a factorization, weget that ϕ ∗ c ( Y ) = π ∗ ( g ( L, H )) c ( B ) = 12 L L c ( B ) , an expression which depends in no way on the dimension of B . Essential to the baseindependence of the formula above is a pushforward formula which computes π ∗ ( g ( L, H ))in terms of a rational expression in L whose associated series is truncated at the dimensionof the base to obtain the given class. Such a pushforward formula was recently obtained in[10]. More generally, for a given invariant of the form I Y = (cid:82) α for some subvariety Y of aprojective bundle, we seek an analogous factorization α = g ( L, H ) π ∗ ( I B ), where the classassociated with g is obtained by truncating a formal series in L and H at the dimension of Y . Then by applying the pushforward formula of [10] to g ( L, H ) we immediately arrive ata base independent expression for the pushforward of α , which lends itself naturally to agenerating series which encodes the invariant I Y for Y over bases of arbitrary dimension.In what follows we successfully carry out this program for Hirzebruch invariants of D , E , E and E elliptic fibrations. The Chern-ext character
We now define a series which plays a key role in our analysis:
Definition 4.1.
Let E be a vector bundle, then we define the Chern-ext character of E to be ch ext ( E ) := 1 + ch( E ) y + ch(Λ E ) y + · · · . We note that for any commutative ring R with unity, any element of the form f ( y ) =1 + a y + a y + · · · is a unit in R [[ y ]], thus f ( y ) is well defined. If E is of rank r withChern roots ( λ , λ , · · · , λ r ), then ch ext ( E ) is a polynomial in y of degree r which factorsas ch ext ( E ) = r (cid:89) i =1 (1 + y · exp( λ i )) . However, due to the dimension-independent nature of our results we prefer to think of ch ext as a series, which also makes more evident the invertability of the Chern-ext character.Furthermore, given a smooth projective variety X we note that the Hirzebruch series of X is simply H y ( X ) = ch ext (Ω X )td( X ) . Armed with such a series, we now prove the following
Lemma 4.2.
Let → A → B → C → be an exact sequence of vector bundles. Then ch ext ( B ) = ch ext ( A ) · ch ext ( C ) Proof. ¿From the λ -ring identity ([8], pg.2) λ p ( x + y ) = p (cid:88) i =0 λ i ( x ) λ p − i ( y ) , along with nice properties of the Chern character ([9], example 3.2.3), we getch(Λ p B ) = p (cid:88) i =0 ch(Λ i A ) · ch(Λ p − i C ) . The lemma immediately follows. (cid:3) The proof
Proof of main result.
Let ϕ : Y → B be a D , E , E or E elliptic fibration asdefined in § N denote both the normal bundle of Y in P ( E ) and the bundle on P ( E ) which restricts to it. Using the exact sequences (we use a superscript “ ∨ ” to denoteduals) 0 → N ∨ → i ∗ Ω P ( E ) → Ω Y → → π ∗ Ω B → Ω P ( E ) → Ω ( E ) /B → → Ω P ( E ) /B → ( π ∗ E ⊗ O (1)) ∨ → O P ( E ) → , along with Lemma 4.2 we get i ∗ (ch ext (Ω Y )td( Y )) = (cid:18) ch ext ( F ∨ )ch ext ( N ∨ )(1 + y ) · α (cid:19) π ∗ (ch ext (Ω B )td( B )) , where F we use to denote π ∗ E ⊗ O (1), α = F td( N ) ∩ [ Y ], i : Y (cid:44) → P ( E ) is the inclusionand we use the fact that ch ext ( O P ( E ) ) = 1 + y . Furthermore, π ∗ (and π ∗ ) act on Chern-extcharacters in the obvious manner. Now we apply π ∗ to the equation above yielding ϕ ∗ (ch ext (Ω Y )td( Y )) = π ∗ (cid:18) ch ext ( F ∨ )ch ext ( N ∨ )(1 + y ) · α (cid:19) ch ext (Ω B )td( B )by the projection formula. Thus computing ϕ ∗ (ch ext (Ω Y )td( Y )) amounts to computing π ∗ (cid:18) ch ext ( F ∨ )ch ext ( N ∨ )(1 + y ) · α (cid:19) . We spell out the details of this computation in the case of D only, as the other casesdiffer inasmuch as the Chern roots of F and N vary from case to case. For D the Chernroots of F and N are ( H, H + L, H + L, H + L ) and (2 H + 2 L, H + 2 L ) respectively,where H := c ( O (1)) and L we will use to denote both c ( L ) and π ∗ c ( L ). Putting thisall together we getch ext ( F ∨ )ch ext ( N ∨ )(1 + y ) · α = (1 + y · e − H )(1 + y · e − H − L ) (1 + y · e − H − L ) (1 + y ) · H ( H + L ) (1 − e − H − L ) (1 − e − H )(1 − e − H − L ) , where we have cancelled a factor of (2 H + 2 L ) from the numerator and denominator of α . Now let D = (1+ y · e − H )(1+ y · e − H − L ) (1+ y · e − H − L ) (1+ y ) · H ( H + L ) (1 − e − H − L ) (1 − e − H )(1 − e − H − L ) . By the pushforward formula of[10] we get π ∗ ( D ) = 12 d dH (cid:18) D − ( a + a H + a H ) H (cid:19) | H = − L = 4 − y + ( y + 1)( yU − yU + 1) − U ( y + 1) ( yU + 1) , where U = e − L and the a i s are expressions in L and y obtained by expanding D asa series in H . Identifying ch ext (Ω Y )td( Y ) and ch ext (Ω B )td( B ) with H y ( Y ) and H y ( B )respectively yields the conclusion of Theorem 1.1. We note that reading off the coefficientof y q in the series ϕ ∗ ( H y ( Y )) gives us ϕ ∗ ( H q ( Y )) = q (cid:88) i =0 P q − i H i ( B ) , where we recall that H q ( X ) := ch(Ω qX )td( X ) denotes the q th Hirzebruch characteristicclass of a smooth variety X and the P i s are polynomials in U = e − L . We list the explicitform of the P i s for each case below: P P P n for n > D − U U + 3 U − U − − U (( n + 1) U − n + 2)( U − U + 1) ( − U ) n − E − U U + 2 U + U − U − − U ( U − U + 1) ( − U ) n − E − U U + U + U − U − − U ( U − U + U + 1)( − U ) n − E − U U + U − U − − U ( U − U + 1)( − U ) n − The fact that P is the same in all cases is a consequence of the fact that K Y = ϕ ∗ ( c ( L ) − c ( B )) for all the fibrations considered (see the appendix of [7]). With theexception of the D case, all roots of P n for n > S ∪ ⊂ C (for the D casean “anomolous” root of n − n +1 appears). We note that the length of our original proof wassubstantially greater, as we computed each χ q individually and then proved a recursiverelation between them. As it turned out, it was much easier to compute all of the χ q sat once, which we were easily able to do once armed with the Chern-ext character and acomputer implementation of π ∗ [11].5.2. Proof of the corollary.
We first need some definitions. Let R be a commutativering with 1, let f = a + a t + a t + · · · ∈ R [[ t ]] and let [ t d ] : R [[ t ]] → R be defined as in §
1. If λ , . . . , λ d ∈ R , we use the notation p i := λ i + · · · λ id and we let C := d (cid:89) i =1 (1 − λ i t ) = 1 − c t + c t − c t + · · · Then c i is the i th symmetric polynomial , and p i is the i th power polynomial of λ , . . . , λ d . Lemma 5.1.
Let f ( t ) = a + a t + · · · ∈ R [[ t ]] . Then d (cid:88) i =1 f ( λ i t ) = f (cid:12) ( d + p t + p t + · · · ) = da + f (cid:12) ( − tC (cid:48) /C ) , where (cid:12) denotes the Hadamard product as defined in § The first equality follows from the definition of the p i . For the second, note that − tC (cid:48) /C is well defined because the polynomial C has a constant term of 1. The equation − tC (cid:48) /C = p t + p t + · · · is obvious for d = 1 (geometric series). For d >
1, recall thatlogarithmic derivatives turn products into sums: ( CD ) (cid:48) / ( CD ) = C (cid:48) /C + D (cid:48) /D . (cid:3) Now let Y be a D , E , E or E elliptic fibration over a base B of some fixed dimension d . Then by Theorem 1.1 and equation ( † ) in § ϕ ∗ H y ( Y ) = Q · H y ( B ) = Q · d (cid:89) i =1 g ( λ i ) , where the λ i s are the Chern roots of the tangent bundle of B and g = (1 + ye − t ) t − e − t .Now let f = ln( g ) = a + a t + · · · (note that a = ln(1 + y )) . Lemma 5.1 then yields H y ( B ) = d (cid:89) i =1 g ( λ i ) = [ t d ][ d (cid:89) i =1 g ( λ i t )]= [ t d ][exp( d (cid:88) i =1 f ( λ i t ))]= [ t d ][exp( da + f (cid:12) ( − tC (cid:48) /C ))]= (1 + y ) d · [ t d ][exp ( f (cid:12) ( − tC (cid:48) /C ))] . The conclusion of Theorem 1.1 then states that over a base B of dimension d we have( †† ) ϕ ∗ H y ( Y ) = Q · (1 + y ) d · [ t d ][exp ( f (cid:12) ( − tC (cid:48) /C ))] . But the right hand side of ( †† ) is just [ t d ] ∼ χ ( t, y ) | t = t (1+ y ) , with ∼ χ ( t, y ) as defined in Corol-lary 1.2. The corollary then follows. 6. Discussion
After going through the proof of the main result, one immediately notices that the onlydata needed from the elliptic fibrations under consideration were the Chern roots of itsnormal bundle in P ( E ) along with the Chern roots of the relative tangent bundle T P ( E ) /B of the ambient projective bundle P ( E ). As such, our program can be carried out verbatimfor any smooth subvariety of P ( E ), long as E is a direct summand of tensor powers of afixed line bundle on the base (this assumption is needed to apply the pushforward formulafrom [10]). More precisely, take any smooth complete intersection X in some projectivespace P n given by equations X : ( F = F = · · · = F m = 0), promote the coefficients ofthe F i to appropriate sections of tensor powers of a fixed line bundle on some smooth basevariety B and we will have then constructed a fibration ϕ : Y → B such that the genericfiber is a complete intersection which is rationally equivalent to X . Then substituting the n + 1 Chern roots of the relative tangent bundle of the ambient P n -bundle where Y residesalong with the m Chern roots of the normal bundle to Y into our calculations above willyield analogous results for the “ X fibration” ϕ : Y → B . Our results are thus genuinelymore general than the title of this note suggests (see [11] for more details).We conclude by noting that a true culmination of these results will not be achievedwithout a Lefschetz hyperplane type theorem for varieties in projective bundles, as hy-persurfaces in projective bundles are almost never ample divisors (which is the key as-sumption of the Lefschetz hyperplane theorem). Once such a theorem is unveiled, onlythe middle cohomology will be unique to a hypersurface in a projective bundle thus ren-dering only (cid:100) d (cid:101) of its Hodge numbers as non-trivial (where d is the dimension of thehypersurface). The Hirzebruch invariants could then be used for the determination of thenon-trivial Hodge numbers. As the cohomology of a projective bundle can be related toits base via the projective bundle theorem, a Lefschetz hyperplane type theorem alongwith the results in this paper would then relate all Hodge numbers of a hypersurface (andso complete intersections) in a projective bundle to invariants of the base. References [1] C. Vafa, “Evidence for F theory,” Nucl. Phys.
B469 , 403-418 (1996). [hep-th/9602022].[2] F. Denef. Les Houches Lectures on Constructing String Vacua arXiv:0803.1194.[3] S. Sethi, C. Vafa, and E. Witten. Constraints on low-dimensional string compactifications.
NuclearPhys. B , 480(1-2):213–224, 1996.[4] A. Klemm, B. Lian, S. S. Roan, S. -T. Yau, “Calabi-Yau fourfolds for M theory and F theorycompactifications,” Nucl. Phys.
B518 , 515-574 (1998). [hep-th/9701023].[5] P. Aluffi, M. Esole, “Chern class identities from tadpole matching in type IIB and F-theory,”JHEP , 032 (2009). [arXiv:0710.2544 [hep-th]].[6] P. Aluffi, M. Esole, “New Orientifold Weak Coupling Limits in F-theory,” JHEP , 020 (2010).[arXiv:0908.1572 [hep-th]].[7] M. Esole, J. Fullwood, S.T. Yau. “ D elliptic fibrations: Non-Kodaira fibers and new orientifoldlimits of F-theory” [arXiv:1110.6177 [hep-th]][8] W. Fulton, S. Lang. Riemann-Roch algebra , volume 277 of
Grundlehren der Mathematischen Wis-senschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, New York, 1985.[9] W. Fulton.
Intersection theory , volume 2 of
Ergebnisse der Mathematik und ihrer Grenzgebiete (3)[Results in Mathematics and Related Areas (3)] ∼∼