Hilbert Scheme of skew lines on Cubic threefolds and Locus of primitive vanishing cycles
aa r X i v : . [ m a t h . AG ] O c t COMPACTIFICATION OF LOCUS OF PRIMITIVE VANISHING CYCLESON HYPERPLANE SECTIONS OF A GENERAL CUBIC THREEFOLD
YILONG ZHANG
Abstract.
For a general cubic threefold, the local system of integral vanishing cohomology H van over the universal locus U of smooth hyperplane sections determines a covering space T Ñ U . There is a distinguished connected component T , containing a vanishing cycle ofa nodal degeneration. We provide several geometric interpretations of T and show that itadmits natural compactification to certain normal analytic space which is biholomorphic tothe blowup of the theta divisor of the intermediate Jacobian of the cubic threefold at itsisolated singularity. As a corollary, we show that the global monodromy of the covering map T Ñ U recovers the third cohomology of the cubic threefold. Introduction
Let Y be a smooth projective variety of dimension n embedded in some projective space.Let O sm be the parameter space of smooth hyperplane sections, then there is a local system H n ´ van whose stalk at t is isomorphic to the vanishing cohomology H n ´ van p Y t , Z q “ ker p H n ´ p Y, Z q Ñ H n ´ p Y t , Z qq on the hyperplane section Y t “ Y Ş H t .Let T denote the étale space of the local system H n ´ van , then T Ñ O sm is an analyticcovering space, which has possibly infinitely many connected components. Among thesecomponents, there is a distinguished component T containing a vanishing cycle of a nodaldegeneration, which is represented by the fundamental class of a topological sphere specializ-ing to the node. We call a class α t P H n ´ van p Y t , Z q primitive vanishing cycle if it is monodrmoyequivalent to a vanishing cycle, namely, there exists a path l Ď O sm , transporting α t to avanishing cycle near the node. It is well know that the set of all primitive vanishing cyclesgenerate H n ´ van p Y t , Z q and primitive vanishing cycles are conjugate to each other via mon-odromy ([18], Proposition 3.23), so the component T is well-defined, and we call it the locusof primitive vanishing cycles . Note that it depends on the complexity of the monodromy on T Ñ O sm that the set of primitive vanishing cycles maybe a infinite set.Let Y be a smooth cubic threefold, then the vanishing cohomology on its hyperplanesection has type p , q , so the space T coincides with the locus of the Hodge classes definedin [3]. Moreover, we will show a primitive vanishing cycle on a smooth cubic surface Y t corresponds to the cohomology classes r L s ´ r L s , where L and L are some disjoint lines n Y t . Since the monodromy of the lines on cubic surfaces has finite order (monodromygroup is a subgroup of the Weyl group W p E q ), T Ñ O sm is disjoint union of finite coverings.Therefore by a theorem due to Stein (see Lemma 4 below), it extends to disjoint union offinite analytic branched coverings ¯ T Ñ O “ p P q ˚ and ¯ T is normal, which provides a modelto compactify T .Note ¯ T is exactly the compactification space defined by Schnell in Theorem 4.2 from[13]. To compactify T , Schnell extends the total space of Hodge bundle F H van, C to certainanalytic space defined by a D -module and taking closure of the image of each componentsof T . The space ¯ T will be more interesting when T Ñ O sm has infinite monodromy (forexample, when Y is a quartic threefold, where the Hodge bundle is non-trivial), but it lacksan interpretation of the boundary points.In this paper, we focus on Y being a cubic threefold and our goal is to understand thegeometry of the distinguished component ¯ T as compactification of the locus of primitivevanishing cycles T by proving Theorem 1.
Let Y be a general cubic threefold. Then there exists a normal analytic space ¯ T which contains T as an open subspace, together with a finite analytic covering map ¯ T Ñ O .Moreover, ¯ T is biholomorphic to Bl Θ , the blowup of the theta divisor of the intermediateJacobian J Y at its isolated singularity.
Our strategy is to use the Abel-Jacobi map ψ : F ˆ F Ñ J Y studied in [6] and show thatcertain modification of ψ factors through ¯ T , which will be a finite map if every hyperplanesections of Y has only finitely many lines. It will be satisfied if the cubic threefold is smoothand does not contain an Eckardt point.As a corollary, we strengthen the result in [14] by showing that the tube mapping on thecomponent T is surjective in this special case, which was predicted by Herb Clemens thatthe topology of the locus of vanishing cycles T is "complicated enough" to describe themiddle dimensional primitive cohomology of Y . Corollary 1.
Let Y be a general cubic threefold, and α P H van p Y t , Z q a primitive vanishingcycle on a smooth hyperplane section Y t . Then every element of H p Y, Z q can be representedby a "tube" on α , namely the trace of monodromy of the Poincare dual of α via a loop l Ď O sm based at t such that l ˚ α “ α . Acknowledgement : I would like to thank my advisor, Herb Clemens, for introducingme to this topic, for his constant encouragement, and for his patience to answer my lastquestion. Besides, I would like to thank Christian Schnell and Shizhuo Zhang for someuseful conversations. Also, I would like to thank Dennis Tseng for answering my questionand thank Xiaolei Zhao for a useful comment. . Preliminaries
Let Y be a smooth cubic threefold, then its general hyperplane section is a smooth cubicsurface Y t . By the unimodularity of the intersection pairing x¨ , ¨y , the vanishing cohomology H van p Y t , Z q is isomorphic to the subgroup of H p Y t , Z q which is orthogonal to the hyperplaneclass. So the local system H van can be viewed as a local subsystem of H of the second integralcohomology. In this section, we will characterize the locus of primitive vanishing cycles T in different ways and review some basic results on cubic surfaces and cubic threefolds.2.1. Vanishing Cycles and Their Locus.
Denote O “ p P q ˚ and O sm Ď O the opensubspace parameterzing the smooth hyperplane sections on Y . Choose a holomorphic disk ∆ Ă O such that ∆ ˚ Ă O sm and ∆ intersects transversely to the boundary divisor at asmooth point.So t Y t u t P ∆ is a one-parameter family of hyperplane sections with Y has a single node and Y t smooth for t ‰ , then take a small neighborhood B p of the node p P Y in the totalspace Ť t P ∆ Y t . When | t | is small enough, the Milnor fiber Y t Ş B p is diffeomorphic to thedisk bundle of the tangent bundle T S of a topological 2-sphere. Moreover, the zero section S specializes to the node p as t moves to . The fundamental class of the zero sectiondefines a cohomology class δ t P H p Y, Z q , which is called a vanishing cycle associated to thedegeneration t Y t u t P ∆ . Definition.
For any t P O sm , the class δ t P H van p Y t , Z q is called a primitive vanishingcycle if it is monodromy equivalent to the vanishing cycle δ t , namely, there is a path l Ď O sm joining t to t which transports δ t to δ t in the local system H van . This notion is well-defined since for different nodal degenerations, local vanishing cyclesare monodromy equivalent to each other ([18], Proposition 3.23).In other words, the local system H van as a topological space is a covering space T Ñ O sm , and it has a distinguished component T containing a local vanishing cycle in a nodaldegeneration.2.2. Nodal Cubic Surface.
Let’s recall the formation of nodal cubic surface. Again weconsider the nodal degeneration t Y t u t P ∆ . Choose a planar representation of Y t as blowupsix points on P , where the six lines are in general position. We denote E i , i “ , ..., theexceptional divisor, F ij , ď i ă j ď , the proper transform of lines joining i -th and j -thpoint and G i the proper transform of conic missing only i -th point. This enumerates all the27 lines on cubic surface Y t . As t goes to , the six blowup points become to lie on a singleconic, E i and G i comes closer and becomes a double line in Y . There are 6 pair of p E i , G i q ,so there are 6 double lines in Y whose common intersection point is the node in Y . Other15 lines specializes to lines in Y disjoint from the node. oreover, a projective model of nodal cubic surface Y is the following, blowup P atthe 6 points lying on a conic Q and denote the new surface by S , then linear system ofanticanonical divisor | K ˚ S | is base point free and defines a morphism φ : S Ñ P and itcontracts the proper transform of Q to the node. S P P φ As S naturally deforms as blowup points move, we can view the family t Y t u t P ∆ as imageof a smooth family t S t u t P ∆ , so the topological cycle defining the same class as the propertransform of Q , namely r H s ´ r E s ´ ¨ ¨ ¨ ´ r E s , which must be the vanishing cycle. Clearly,for each i “ , ..., , r G i s ´ r E i s coincides with such a class. Remark.
By definition, the primitive vanishing cycles have self-intersection -2. Actually,the converse is also true, namely, the primitive vanishing cycles on Y t are the solutions of (1) x α, h y “ , x α, α y “ ´ , for α P H p Y t , Z q and h the hyperplane class. It is known classically that the solution set ofequations p q , together with the "reflections" s α p β q “ β ` x α, β y α, given by Picard-Lefschetz formula, is isomorphic to the root system of the Lie algebra E . Inparticular, T can be viewed as variation of the root system of E . Pair of Disjoint Lines.
Let F denote the Fano variety of lines of the cubic threefold Y . It is a smooth surface of general type [6] naturally embedded in Gr p , q . Let’s considerthe incidence variety Λ “ tp t, L , L q P O ˆ F ˆ F | L , L Ď Y t u i.e., the set of pair of lines contained in the cubic surface Y t “ Y Ş H t arising as hyperplanesections. The general fiber has cardinality ˆ . Proposition 1. Λ contains three irreducible components Λ i , ď i ď , where a generalelement p t, L , L q in(1) Λ is a pair of disjoint lines L Ş L “ H contained in Y t ;(2) Λ is a pair of lines intersect at a single point L Ş L “ t x u contained in Y t ;(3) Λ is a pair of disjoint line L “ L contained in Y t .Proof. First of all, since over the locus O sm where Λ sm { O sm is unbranched, and by Cheng’stheorem [5], monodromy group of lines on smooth cubic surfaces over O sm is the entire p E q , so in particular, the monodromy action is transitive on pair of lines p L , L q hav-ing the same incidence relation, i.e., (1) L and L disjoint, (2) L and L intersect at asingle point, and (3) L “ L . So it follows that the covering space Λ sm Ñ O sm has threecorresponding connected components Λ ˝ i , ď i ď . (cid:3) Denote M “ Λ ˝ parametrizing pair of disjoint lines on the smooth cubic surfaces, then M Ñ O sm is a p ˆ q -sheeted covering map.Fiberwise evaluating the pair p L , L q to the cohomology class r L s´r L s P H p Y t, Z q on thecubic surface Y t . The class lies in the vanishing cohomology H van p Y t , Z q since its intersectionnumber with a hyperplane class is zero. Therefore there is a natural evaluation map(2) e : M Ñ T , p L , L q ÞÑ r L s ´ r L s over O sm . In sum, we have Proposition 2.
There is a commutative diagram (3) M T O smeπ π consisting of of covering maps. Moreover, the degrees of π , π and e are , and ,respectively.Proof. First π and π are covering maps by construction. Next, the evaluation map e isa covering map because the deformation of lines corresponds to the deformation of thiscohomology class. It suffices to show that deg p e q “ . This dues to our analysis of rep-resentatives of a local vanishing cycle in a nodal degeneration in the the previous section,where we found r G i s ´ r E i s “ r H s ´ r E s ´ ¨ ¨ ¨ ´ r E s , i “ , ..., are the same class, and byknowledge of the H p Y t , Z q generated by cohomology classes of lines, together with incidencerelation between lines, we conclude that these 6 classes are the only classes coincide with r H s ´ r E s ´ ¨ ¨ ¨ ´ r E s . So deg p e q “ , and it follows that deg p π q “ . (cid:3) Proposition 3.
Let ∆ Ď O be a disk parameterizing a nodal degeneration. The restrictionof Λ over ∆ Λ | ∆ Ñ ∆ is a 27-to-1 branched covering map branched over simply and there are exactly 6 ramifica-tion points over . .4. Abel-Jacobi map.
Let
J Y “ F H p Y, C q ˚ { H p Y, Z q be the intermediate Jacobian ofcubic threefold Y . There is an Abel-Jacobi map(4) M Ñ J Y, p L , L , t q ÞÑ ż L L , by integrating along a -chain Γ with B Γ “ L ´ L . Lemma 1.
The map p q factors through T . In other words, there is a commutative diagram (5) T M J Y φeψ
Proof.
This is purely a topological fact. If r L s´r L s and r L s´r L s are the same cohomologyclass in Y t , so they differ by a 3 chain Γ t supported on Y t . Now let ω be a -form on Y representing a class in F H p X, C q , since H p X q is primitive, the restriction ω t of ω to Y isexact, moreover by B ¯ B -lemma, there is a p , q -form σ on Y t such that dσ “ ω t , so by Stokestheorem ż Γ t ω “ ż B Γ t σ “ ż p L ´ L q´p L ´ L q σ “ , since L i and L i are algebraic cycles, so an integral of a p , q form against them vanishes. (cid:3) We call φ the topological Abel-Jacobi map , which coincides with and is defined in a muchgeneral setting in [19]. Remark.
Actually these cycles are all rationally equivalent, so cohomologous to each otherand their Abel-Jacobi images are the same. To see it, if L and L are disjoint on a smoothcubic surface Y t , there are exactly 5 lines J , ..., J incident to both of the lines, in particular,for i “ , ..., , L and J i form a plane, which intersects Y t along a residual line K i , andsimilarly we get K i which is disjoint from K i . So L ` J i ` K i is rational equivalent to L ` J i ` K i , so r L s ´ r L s “ r K i s ´ r K i s as homology class. General cubic threefolds
Our goal is to prove the following argument:
Proposition 4.
For Y being a general cubic threefold, every hyperplane section of Y containsat most 27 lines. In this case, Λ Ñ O is finite. or smooth cubic surfaces, there are exactly 27 lines. For cubic surfaces with "mild"singularities, the number of lines is less than 27 and such number depends on the type ofsingularities as well as how lines passing through these singularities. However, for cubicsurfaces is "too singular", they contain infinitely many lines. The following result will makethese more precise: Lemma 2.
Let S be a cubic surface, then there are three possibilities:(i) S contains at worse rational double point singularities (RDPs);(ii) S is a cone over a plane cubic curve;(iii) S is projective equivalent to S : t t ` t t “ , or S : t t t ` t t ` t “ .In case (i), the cubic surface S contains at most 27 lines. In case (ii) and (iii), S containsinfinitely many lines.Proof. First assume S is normal, so it has only isolated singularities. By the classificationtheorem [2], S is either in case (i) (actually with at worst E ) or in case (ii). For non-normalcubic surfaces, they must be projective equivalent to S or S , due to Theorem 9.2.1 byDolgachev [8].For the second half of the statement about number of lines, by [8], section 9.2.2, all cubicsurfaces with at worst RDPs have at most 27 lines. Cone over cubic curves contains a one-parameter family of lines. Finally, S and S are both singular along the line L “ t t “ t “ u . So any plane in P containing L intersects S i along a residual line, for i “ , , so theycontains at least a P -family of lines. (cid:3) Based on the lemma above, in order to prove Proposition 4, we need to prove that ageneral cubic threefold does not have hyperplane section of type (ii) and (iii).If a cubic threefold Y contains S or S as a hyperplane section, then Y has definingequation F i p t , ..., t q ` t Q p t , ..., t q “ . with F i p t , ..., t q the defining equation of S i and Q p t , ..., t q a homogeneous quadric. Thenby taking the partial derivatives and restrict to the line, one finds that Y is singular at theintersection between the line L and the quadric surface Q p t , ..., t , q “ . In particular, asmooth cubic threefold does not contain S or S as a hyperplane section. Proof of Proposition 4.
Based on the discussion above, it suffices to show a general cubicthreefold does not contain cone over a plane cubic curve as a hyperplane section.Denote by C the subvariety that parameterizes cone over cubic curves in the universalfamily P of cubic surfaces. Choose a plane in P , there is ` ` ˘ ´ “ dimensionalcubics in it and there are dimensional planes in P . So C is a P -bundle over P , so it hasdimension . onsider the incidence variety I “ tp S, H q P P ˆ p P q ˚ | S is a cubic surface contained in H u which is a projective bundle over p P q ˚ and C Ď I be the locus of cone over cubic curves,namely as the preimage of C under the second projection. Set P to be the space of allcubic threefolds. Then there is a map f : P ˆ p P q ˚ Ñ I p Y, H q ÞÑ p Y č H, H q by sending a cubic threefold to a hyperplane section. This map is surjective and has constantfiber dimension 15. (Say the coordinate on P is x , ..., x , and H is given by x “ , then theset of all cubic threefolds containing a fixed cubic surface F p x , ..., x q “ in the hyperplane x “ is F p x , ..., x q ` x Q p x , ..., x q =0, where Q p x , ..., x q is a quadric in variables, itfollows that there are ` ` ´ ˘ “ dimensional such quadrics.) Since C has codimension 7in P , the locus of C and therefore its preimage f ´ p C q has codimension as well. It followsthat its image in P under the projection to the first coordinate has codimension at least 3,which completes the proof. (cid:3) Remark.
According to [6] , Lemma 8.1, a smooth cubic threefold Y contains a cone overplane cubic curve as hyperplane section if and only if Y has an Eckardt point. Therefore tosay Y is "general" in Proposition 4 is the same as saying Y is smooth and has no Eckardtpoint. Remark.
Based on our assumption that the cubic threefold Y is general, its relative Hilbertscheme of lines F is flat over O , and has length on each fiber (also see the introductionof [16] and Example 1.1 (b) of [17] ). Classically, the lines for cubic surfaces with RDPs werestudied by Cayley [4] , and the number "27" is interpreted as the number of lines counted withmultiplicities, and the multiplicity of a line depends on type of singularities it passes through.In this way, Λ can be regarded as the reduced scheme associated to F ˆ O F , and Λ is thereduced scheme associated to the Hilbert scheme whose general element parameterizes pair ofdisjoint lines. Λ -component Recall that Λ is a proper space with general element a triple p L , L , t q where L and L are disjoint lines on Y t . For Y a cubic threefold that is general, the projection to the lastcoordinate Λ Ñ O is finite.In this section, we study the geometry of Λ by projecting it to the first and secondcoordinates: : Λ Ñ F ˆ F, p t, L , L q ÞÑ p L , L q . It is surjective and the fiber is the set of t where the hyperplane H t Ď P containing L and L . So when the two lines are disjoint, the fiber is a point; when the two lines intersectat a single point, the fiber is a P ; and when two lines overlap, the fiber is a P .Let ∆ Ď F ˆ F be the diagonal, and let E Ď F ˆ F be the incidence divisor consisting ofthe pair of lines p L , L q such that L Ş L ‰ H . We argue that Proposition 5.
Restricting f to each irreducible components endows fibration structure onthree components:(1) Λ is isomorphic to Bl ∆ p F ˆ F q ;(2) Λ is generically a P -bundle over E ;(3) Λ is generically a P -bundle over ∆ .Proof. By specialization principle, the configuration of two lines can become more specialover the boundary point O z O sm , i.e., the pair of incident lines can become overlap in thelimit and the pair of disjoint lines can become incident, or even overlap in the limit, but notthe other way around.Clearly, f p Λ q “ ∆ , and (3) will follows from the fact that, for a general line L P F ,the general element in P – f | ´ p L, L q determines a smooth hyperplane section. Similarly, f p Λ q “ E , and (2) follows from that, for a general p L , L q P E , the general element of thefiber P – f | ´ p L , L q determines a smooth hyperlane section. This proves (2) and (3).Now, let’s prove (1). First of all, denote f : “ f | Λ , so f : Λ Ñ F ˆ F a is birationalmorphism which is an isomorphsim over the open subset U Ď F ˆ F , where L and L aredisjoint, and Span p L , L q Ş Y is smooth. Second, according to Lemma 12.16 of [6], theinverse map Φ : F ˆ F Λ is regular on the complement of the diagonal, defined by(6) Φ : F ˆ F z ∆ Q p L , L q ÞÑ $&% Span p L , L q , if L Ş L “ H ; T y Y, if L Ş L “ t y u . In other words, let V be the Hilbert scheme with general element parameters pair ofdisjoint lines in Y . It is an irreducible component of Hilb t ` p Y q , the Hilbert scheme ofsubschemes of Y with Hilbert polynomial t ` . Then the argument says Φ factors throughan Zariski open dense subset V ˝ of V . V ˝ has a divisor parametrizing pair of (distinct)incident lines with a nilpotent of length one at the intersection. The pair of incident lines,together with the normal direction determined by the nilpotent, spans a hyperplane in P ,which is exactly the tangent space of Y at the incident point. o the birational center of f is supported in the diagonal. So f is blowup of an idealsheaf supported on the diagonal.On the other hand, by definition, Λ is the closure of the graph of the rational map p q .However, according to [6], the rational map Φ factors through Abel-Jacobi maps:(7) F ˆ F Θ p P q ˚ , ψ G where Θ is the theta divisor of the intermediate Jacobian J p Y q , with a single singularity at , ψ is the Abel-Jacobi map by sending p p, q q to the linear functional ş L q L p and G : Θ p P q ˚ is the Gauss map, which associate each smooth point of Θ to its tangent hyperplane in thedual projective tangent bundle P p T ˚ J q – J ˆ p P q ˚ followed by the projection to the secondcoordinate.Note that the fundamental locus of the rational map Φ “ G ˝ ψ is the diagonal ∆ . Byfactorization p q , the image of the diagonal ∆ under the Abel-Jacobi map ψ is the singularpoint of P Θ , so by [10], II 7.15, ψ induces morphism ˜ ψ : Bl ∆ p F ˆ F q Ñ Bl Θ .Finally, denote K the exceptional divisor in Bl Θ , then by a Torelli theorem in [1], isthe only singularity of Θ and K is isomorphic to the cubic threefold Y . Therefore, the Gaussmap extends to a regular map on the blowup, ˜ G : Bl Θ Ñ p P q ˚ by sending y P Y – K tothe tangent hyperplane T y Y P p P q ˚ .It follows that the rational map Φ as composite p q extends to a composite of regular map ˜Φ : Bl ∆ p F ˆ F q ˜ ψ ÝÑ Bl Θ ˜ G ÝÑ p P q ˚ , by blowing up the diagonal. Now the argument follows from the following lemma. (cid:3) Lemma 3.
Let f : M N be a rational map between smooth projective varieties withfundamental locus B Ď M smooth and has codimension at least two. Assume that theinduced rational map on the blowup ˜ f : Bl B M N is regular, then Γ p f q – Bl B M .Proof. Γ p f q is the closure of the graph of f over M z B inside M ˆ N , then the first projection π : Γ p f q Ñ M is projective and birational, so according to II. 7.17 for [10], it is the blowupalong an ideal sheaf I supported on B . So by the universal property of blowup (II. 7.14 from[10]), π factors though Bl B M :(8) Γ p f q Bl B MM φπ σ On the other hand, the assumption implies ˜ f : Bl B M Ñ N is a morphism, then togetherwith the blowup map σ : Bl B M Ñ M , it gives a map g : Bl B M Ñ M ˆ N . Its image is closed nd irreducible and contains the graph of f over M z B as a Zariski open dense subset, sothe image g p Bl B M q coincides with the graph closure Γ p f q . Moreover, there is factorization σ “ π ˝ g , together with p q , we have σ ˝ φ ˝ g “ σ , which implies φ ˝ g is an isomorphismby the universal property, similarly g ˝ φ is an isomorphism, so Γ p f q – Bl B M . (cid:3) compactification of T’ In this section, we construct the compactification space ¯ T of T using a classical theorembased on work by Dethloff, Grauert, Remmert and Stein on extension analytic branchedcovering maps.Recall that a finite surjective map f : M Ñ N between reduced complex space is calledan analytic branched covering of N if there exists a nowhere dense analytic subset Z of N with the following properties:(1) The set f ´ p Z q is a nowhere dense analytic subset of M .(2) The induced map f : M z f ´ p Z q Ñ N z Z is locally biholomorphic. Moreover Z is calledthe critical locus of f .The following result is due to Stein [15], can be found in [7], section 3. Lemma 4.
Let N Ď P n be a normal projective variety, and let B be a nowhere dense complexanalytic subset. Let h : X Ñ p N z B q be an analytically branched covering with critical locus A . Assume further that A Ť B is analytic. Then there exists a normal variety ¯ X and acompact analytic cover ¯ h : ¯ X Ñ N extending h : X Ñ p N z B q . Extension across the nodes.
Let O dp be the locus where the hyperplane section has asingle ordinary double point, which also coincides with the smooth points of the dual varietyof Y . Denote ˆ O “ O sm Ť O dp . Then ˆ O is open whose complement is closed of codimensiontwo. We look for extension of the diagram p q to a diagram over ˆ O . Proposition 6.
There are a smooth analytic space ˆ M (resp. ˆ T ) extending M (resp. T )together with commutative diagram ˆ M ˆ T ˆ O ˆ e ˆ π ˆ π which extends the diagram p q . Moreover, the horizontal map is étale, and both ˆ π and ˆ π aresimple branched covering maps. There are ramification points over each branching pointof ˆ π .Proof. First we have a natural candidate for ˆ M , namely the preimage of ˆ O of the projection Λ Ñ O . So the resulting morphism ˆ π : ˆ M Ñ ˆ O is a ramified covering, which can be escribed more geometrically as follows: In a normal direction of O dp , a smooth cubic surface Y t degenerate into Y as a cubic surface with 1 node. Recall that as we have mentioned,there are 12 lines appearing in pairs specialize to passing the node in the limit. We definethe involution σ on the set of 27 lines by sending a line to its pair if it specializes to passthe node, and identity otherwise. For those pair of disjoint lines p L , L q on Y t , there are 4cases:(1). Both L and L are disjoint from node in the specialization (e.g., L “ F , L “ F ),so its limit at t “ is still unramified over O dp . There are ˆ such pairs.(2). Both L and L pass through the node in the limit, and r L s´r L s does not representsthe vanishing cycle (e.g., L “ E , L “ E ). In the limit, the p L , L q is identified with p σ p L q , σ p L qq , unramified over O dp . There are ˆ such pairs.(3). Both L and L pass through the node in the limit, and r L s ´ r L s represents thevanishing cycle (e.g., L “ E , L “ G ). In the limit, the p L , L q is identified with p L , L q ,simply ramified over O dp . There are ˆ such pairs.(4). One of L and L passes through the node in Y (e.g., L “ E , L “ F ), so inthe limit, the p L , L q is identified with p σ p L q , σ p L qq , simply ramified over O dp . There are ˆ ˆ such pairs.So only case (3) and (4) contribute ramification and ˆ π : ˆ M Ñ ˆ O is simply ramified along O dp with ramification points.Now, we describe the extension ˆ T of T . As there is no natural ambient space for T to takeclosure in as M does, we need to consider the following: As the dual variety Y _ “ O z O sm haseven degree (degree is 108), the line bundle O p Y _ q admits a square root and there is a analyticspace W and a double cover λ : W Ñ O branched along Y _ . Denote W sm “ λ ´ p O sm q , and ˆ W “ λ ´ p ˆ O q , then ˆ W Ñ ˆ O is branched along p Y _ q sm and ˆ W is smooth, while W sm Ñ O sm is unbranched double cover. Now pullback the local system H van to W sm , then it has triviallocal monodromy along ˆ W z W sm – p Y _ q sm . Then we can naturally extend the local systemto ˆ W and there is still a distinguished component T W containing a vanishing cycle. Its openlocus over W sm admits Z -étale cover, and the Z -action extends to the entire T W . Now wedefine ˆ T to be the analytic space T W { Z .The morphism ˆ e is given by application of Lemma 4 together with the commutativity ofthe diagram (3).Finally, we will show ˆ e is a unbranched, by restricting to a one-parameter family over ∆ . Let p t, α t q P T with t ‰ , then the specialization of α t to t “ is ramified if theintersection number x α t , δ t y ‰ , and the specialization stays simple if x α t , δ t y “ , where δ t is the vanishing cycle of the nodal degeneration, which is defined up to a sign. This featureis exactly the same by looking at the local monodromy of the 6 preimages e ´ pp t, α t qq , sincethe intersection number only depends on the cohomology class. Therefore p t, α t q is ramified stays simple, resp.) if and only if the 6 preimages are ramified (stays simple, resp.) at t “ .Therefore ¯ e is a unbranched. (cid:3) Compactification.
The following is our main theorem in this section:
Theorem 2. (i) There is a normal analytic space ¯ T together with an analytic branchedcovering map ¯ π : ¯ T Ñ O . Moreover, the restriction p ¯ π q p ˆ O q Ñ ˆ O is analytic equivalentto the branched covering ˆ T Ñ ˆ O that we defined in Proposition 6.(ii)Assume the cubic threefold Y is general, then e extends to an analytic branched coveringmap ¯ e : Λ { O sm Ñ ¯ T { O .In other words, there is a commutative diagram Λ ¯ T Λ ˝ T ˆ T OO sm ˆ O ¯ e ¯ π ¯ π eπ π ˆ π extending the diagram p q .Proof. The first statement dues to application to π of Lemma 4. The restriction p ¯ π q p ˆ O q Ñ ˆ O is also an analytic branched covering, so by uniqueness, the extension factors through ˆ T { ˆ O .For the second statement, apply Lemma 4 again to the covering maps π and e , and use thecommutativity π ˝ e “ π , there is an analytic space Λ ˝ , together with morphisms ¯ e : Λ ˝ Ñ ¯ T and ¯ π : Λ ˝ Ñ O extending e and π such that ¯ π ˝ ¯ e “ ¯ π . Finally, use the fact that Λ ˝ isopen dense in Λ and ¯ π : Λ Ñ O is finite under the assumption that the cubic threefold Y being general. So by uniqueness of the extension, there is an analytic equivalent Λ ˝ – Λ preserving the projections to O . So the diagram commutes. (cid:3) Lifting of Abel-Jacobi map and Proof of Theorem 1
According to Theorem 2 and Proposition 5 (1), there is a finite analytic branched covering Bl ∆ p F ˆ F q – Λ e ÝÑ ¯ T . On the other hand, there is the Abel-Jacobi map ψ : F ˆ F Ñ J, p L , L q ÞÑ ż L L , hose image is the theta divisor [6] with the diagonal sending to as an isolated triple pointsingularity of Θ . Moreover, we there is the induced map on blowup ˜ ψ : Bl ∆ p F ˆ F q Ñ Bl Θ . On the other hand, as we have seen, T admits a holomorphic map to J Y descendedfrom the Abel-Jacobi map ψ : Λ ˝ Ñ J Y . Since ¯ T is normal, by the Riemann extensiontheorem (by locally choose a lifting in C and by the Abel-Jacobi map is bounded), there isan extension (also see [9], Example 3.)(9) ¯ φ : ¯ T Ñ J Y together with a commutative diagram(10) Λ ¯ T Bl ∆ p F ˆ F q Bl Θ J YF ˆ F Θ ¯ e – f φ ¯ φ ˜ ψσ ψ The diagram commutes due to uniqueness of Riemann extension.
Proposition 7.
Denote φ the rational map as the composite of ¯ φ and Θ Bl Θ . Then φ is a morphism.Proof. Let’s study the monodromy of p T q dp “ ˆ T z T over the locus O dp “ p Y _ q sm where thecubic surfaces have a single ordinary node, as we have described in Proposition 6, p T q dp { O dp is a covering space with 4 connected components with degree:(1) degree 20,(2) degree 10,(3) degree 1,(4) degree 20.In particular, we have a section O dp Ñ T dp which sends to the degree 1 component, soby denoting S the image of the section, it is the locus where a primitive vanishing cyclespecializes to a node. It is the locus where a primitive vanishing cycle specializes to a node. S is also dominated by the locus of double lines p L, L, t q P Λ as specialization of a pair ofdisjoint lines p L , L , t q where the difference r L s ´ r L s is a local vanishing cycle. Therefore,by the commutativity of the diagram (10), the image of S , therefore its closure ¯ S in ¯ T , iszero under the extended topological Abel-Jacobi map ¯ φ . ince ¯ S is a divisor, by the universal property of the blowup ([11], Prop. 2.2), thereis a unique morphism ¯ T Ñ Bl Θ making the diagram commute, which extends φ to amorphism. (cid:3) Finally, to prove our main theorem, the Theorem 1, it suffices to show the following result:
Theorem 3.
Assume the cubic threefold Y is general, then φ : ¯ T Ñ Bl Θ is a biholomor-phism.Proof. First, by p q , ¯ e is finite of degree 6. If we can show that ˜ ψ is also finite of degree 6,then ¯ φ has to be finite of degree one, so according to an analytic version of Zariski’s maintheorem ([12], Prop. 14.7), φ must be a biholomorphism.Nest, let’s show ˜ ψ is finite. Since ˜ ψ is proper, it suffices to show it is quasi-finite, namelythe cardinality of each fiber is finite. According to [6], the composite map F ˆ F z ∆ Ñ Θ zt u G ÝÑ p P q ˚ coincides with the map defined in p q , where G is Gauss map coincides with Φ defined in p q .So in particular, if p L , L q R ∆ and p L , L q R ∆ have the same Abel-Jacobi image, thenthey have to lie in the same hyperplane section of Y . However, by our assumption on Y ,every hyperplane section has only finitely many lines (a cubic surface with only RDPs has atmost 27 lines), it follows that there are at most ` ˘ such preimages, so F ˆ F z ∆ Ñ Θ zt u is quasi-finite.Next, denote N the exceptional divisor on Bl ∆ p F ˆ F q and K – Y the exceptional divisorover P Θ . Let’s show that N Ñ K is quasi-finite. Under identification N – P p T ∆ q – P p T F q , according to Proposition 12.31 in [6], the map T F Ñ K Ď P is identified with sending each the projective tangent space P p T l F q to the projective line l Ď P , under certain natural identification of C and therefore their projectivizations. Onthe other hand, we know the fiber at y P K – Y of this map is the set of lines in Y passing y ,which is finite unless there is an Eckardt point on Y , so by choosing a general cubic threefoldthis would not occur.Finally, by an argument computing the coefficients of cycle class (page 348 of [6]), one canshow that ψ ˚ p F ˆ F q “ as divisor (alternative approaches were given in [1]), so ˜ ψ has tobe of degree 6. (cid:3) Relation to Schnell’s results
Completion of vanishing cycles component.
In [13], for a polarized variation ofHodge structure p H , Q q of even weight over a quasi-projective variety B , as a Zariski open ubset of a smooth projective variety B , Schnell constructed a completion of T Z , the étalespace of the local system H Z .More explicitly, assume that H has weight n . The data p H , Q q consists of a Z -local systemover B , a flat connection ∇ on H C “ H ˆ Z O B , Hodge bundles F p H C and a nondegeneratepairing Q : H Q ˆ H Q Ñ Q satisfying Hodge-Riemann conditions.Consider F n H the associated Hodge bundle, i.e., the subbundle whose fiber at p P B is F n H p , the n -th Hodge filtration of the complex vector space H p . Then it is shown in Lemma3.1 from [13] that for each connected component T λ { B of T Z { B , the natural mapping T λ Ñ T p F n H q α ÞÑ Q p α, ¨q is finite, where T p F n H q is the underlying analytic space of the Hodge bundle.Moreover, according to Saito’s Mixed Hodge Modules theory, there is a Hodge module M underlying a filtered D B -module p M , F ‚ M q supported on B , as the minimal extension of p H , ∇ q .Schnell considered the space T p F n ´ M q as analytic spectrum of the n ´ -the filtration of M and showed that the analytic closure of the image of the composite of T λ Ñ T p F n H q Ñ T p F n ´ M q is still analytic, therefore it extends to a finite analytic covering by Grauert’s theorem andso there is a normal analytic space ¯ T λ extending T λ . Schnell defines ¯ T Z as the union Ť λ ¯ T λ In the case when the variation of Hodge structure comes from vanishing cohomologyof hyperplane sections on cubic threefold. All H on the general hyperplane section isconcentrated in p , q part, so F H is trivial and so is M , therefore the analytic spectrum isnothing but the base space O . So the completion of vanishing cycle component T is exactlythe ¯ T space we are discussing. In short, we have Corollary 2.
The completion space ¯ T Z defined in [13] contains an irreducible componentwhich is biholomorphic to Bl Θ . Tube Mapping.
In [14], Schnell studied the relationship between the primitive homol-ogy H n p Y, Z q prim of a smooth projective variety Y Ď P N of dimension n and the vanishinghomology H n ´ p S, Z q van of a smooth hyperplane section S “ Y Ş H . Let U Ď p P N q ˚ be theopen set of smooth hyperplanes, and l Ď U be a loop based at t , and α P H n ´ p S, Z q van , if l ˚ α “ α , then the trace of α is a topological n -chain on Y with boundary α ´ l ˚ α “ so isa n -cycle which is well-defined in the primitive homology. Since the n -cycle is a "tube" on over the loop l , such map is called tube mapping . Schnell proved that if H n ´ van p S, Z q ‰ ,then the tube map tpr l s , α q P π p U, t q ˆ H n ´ p S, Z q van | l ˚ α “ α u Ñ H n p Y, Z q prim has cofinite image. Equivalently, the set of tubes on vanishing cohomology classes generatethe middle dimensional primitive cohomology on Y over Q .Herb Clemens conjectured that the theorem is still true by restricting the tube map totubes on a single vanishing cycle α (a cycle deformable to a class representing a topologicalsphere when the hyperplane section is close to acquire a node), namely, Clemens conjecturedthat Conjecture 1. under the same hypothesis, the image of (11) tpr l s , α q P π p U, t q| l ˚ α “ α u Ñ H n p Y, Z q prim is cofinite. We will prove this conjecture when Y is a general cubic 3-fold. Theorem 4.
Let Y be a general cubic threefold, then the conjecture is true.Proof. First note that H p Y, Z q “ H p Y, Z q prim due to H p Y q “ . Second, a vanishing cycle α P H p Y t , Z q is represented by difference of two lines, so p t, α q is a point on T . Also, recallthat T is a finite-sheet covering space of U “ O sm . A loop l Ď U such that l ˚ α “ α basedat t corresponds to a loop ˜ l Ď T based at p t, α q , so by abusing the notation ˚ as the basepoint, the map p q is the same as(12) π p T , ˚q Ñ H p Y, Z q . The following result is proved in [19], p.26 in a more general setting. For reader’s conve-nience, we provide a self-contained proof here.
Proposition 8.
The map p q is induced on fundamental group by the topological Abel-Jacobimap φ : T Ñ J Y , followed by the isomorphism π p J Y, ˚q – H p Y, Z q .Proof. Let ˜ l Ď T be a loop based at p t, α q , then its Abel-Jacobi image is determined by afamily of 3-chains Γ t indexed by t P r , s modulo 3-cycles on Y , so we can choose Γ t to bethe union Γ Ť Γ t where Γ t “ Ť s Pr ,t s α s as trace of vanishing cycles along the path r , s . Itfollows that Γ is a 3-chain such that B Γ “ B Γ “ α , so the induced map on π sends ˜ l tothe image of the 3-cycle Γ ´ Γ “ Ť t Pr , s α t in H p Y, Z q . (cid:3) Finally, the theorem follows from the following argument, which is equivalent to Corollary1 that we stated in the Introduction. (cid:3) roposition 9. The map (12) induced by φ is surjective.Proof. First of all, φ : T Ñ J Y factors through the inclusion T Ď ¯ T . Moreover, T Ď ¯ T is a complement of a divisor in a smooth complex manifold, as a smooth loop based can bedeformed to be disjoint from a real codimension-two set, there is a surjection π p T , ˚q ։ π p ¯ T , ˚q . Therefore, it suffices to show that π p ¯ T , ˚q Ñ π p J Y q induced by ¯ φ in p q issurjective.Next, choose p P F such that its corresponding line L p is of second type on Y and let D p be the divisor of lines that are incident to L p . So by Lemma 10.7 of [6], p P D p , andit follows that t p u ˆ F z D p is disjoint from the diagonal and in particular, by denoting σ : Bl ∆ p F ˆ F q Ñ F ˆ F be the blowup map, the restriction of σ ´ to the domain of ψ p isan isomorphism. We define the restricted Abel-Jacobi map(13) ψ p : t p u ˆ F z D p Ñ J Y.
It follows from the commutativity of the diagram p q that the restricted Abel-Jacobi map p q coincides with the composite ¯ φ ˝ ¯ e ˝ f ´ ˝ σ ´ of morphisms, so ¯ φ ˚ p π p ¯ T , ˚qq contains p ψ p q ˚ p π pt p u ˆ F z D p , ˚qq as a subgroup. So it suffices to show that ψ p induces surjectivityon fundamental groups.To show this, note that ψ p factors through the inclusion t p u ˆ F z D p Ď t p u ˆ F whichinduces surjective map on fundamental group for the same reason as in the first paragraphin the proof. Moreover the map t p u ˆ F – F Ñ J Y factors through the Albanese map(14)
F J YAlb p F q ψalb – together with the isomorphism Alb p F q – ÝÑ J Y [6]. It follows that ψ induces an isomorphismbetween fundamental groups, therefore so does ψ p . (cid:3) References [1] Beauville, Arnaud
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Thesis (Ph.D.)-The Universityof Michigan. 2015.
Department of Mathematics, Ohio State University, Columbus, Ohio 43210
Email address , Y. Zhang: [email protected]@osu.edu