An Almost Complex Castelnuovo De Franchis Theorem

arXiv:1502.06205v2 [math.CV] 13 Apr 2016 tflosta hr r ope audsot functions smooth valued complex are there that follows it wherever 2.1. eu tleast at genus ω hnteeeit mohams ooopi map holomorphic almost smooth a exists there Then soitst niorpcsbpc of subspace isotropic an to associates orsodnebtensubspaces between correspondence X ento 1.1. Definition ea lotcmlxsrcueon structure complex almost an be usaeof subspace position hoe 1.2. Theorem structu complex almost an only extract have We we when further. content through go topological to this theorem emphasize to is note Let group. i h oooia aueo hstermwsbogtotb Catan by out brought was theorem this of nature topological The . ie mohcmlxpoetv variety projective complex smooth a Given 2000 eaenwi oiint tt h antermo hsnote. this of theorem main the state to position a in now are We follows, what In e od n phrases. and words Key = 1 h zero-sets the (1) 2 hs submanifolds these (2) ω efspace. Leaf 1 • • f , ahmtc ujc Classification. Subject Mathematics ∗ · · · ω r mohebde umnfls and submanifolds, embedded smooth are each NAMS OPE ATLUV EFACI THEOREM FRANCHIS DE CASTELNUOVO COMPLEX ALMOST AN e o it. for rem Abstract. η if i i ω r ngnrlpsto with position general in are j o all for ω , H .Hnetecollection the Hence 0. 6= ω g 1 i g ( ecoe complex closed be X, so type of is n hr r ieryidpnetholomorphic independent linearly are there and , Let esalsyta olcino lsdcmlx1–forms complex closed of collection a that say shall We i suethat Assume ie opc lotcmlxmnfl,w rv Castelnu a prove we manifold, complex almost compact a Given . C M M fdimension of ) ilb opc mohmnfl fdmnin2 dimension of manifold smooth compact a be will eacmatsmooth compact a be lotcmlxsrcue atluv–eFaci theore Franchis Castelnuovo–de structure, complex Almost (1 2. Z , ( 0) efsaeadams ope blow-up complex almost and space Leaf ω meaning , i ω nesc transversally. intersect ) Z i 1 ( NRNLBSA N AA MJ MAHAN AND BISWAS INDRANIL frson –forms r om si hoe ..Since 1.2. Theorem in as forms are ω U e M g i := ) ≥ meaning , ⊂ H 26,14F45. 32Q60, ω 0 i n rainlfirtoson fibrations irrational and 1 H ( ∧ J ω X, W { 1. 1 M i ω x ( ( 2 X, J j Ω k M Introduction = ∈ ω : M mnfl qipdwt nams ope structure complex almost an with equipped –manifold 0 = X 1 X i C M T ( J M { fdmningetrta n nirtoa eclon pencil irrational an one than greater dimension of ) ieryidpnetover independent linearly v = fteform the of ) h lsia atluv-eFaci hoe 2 4] [2, theorem Franchis Castelnuovo-de classical the , M ω )= )) 1 1 o all for f | , ◦ −→ i,j · · · ω J f i M ω ( √ ω , : j x M T − 0 = ) = f M j , i g i,j 1 } − · . uhthat such −→ ω Id U ⊂ } i ( M T ⊕ v oooia yohssta lo the allow that hypotheses topological ) 1 C U . –forms o all for M re. s 3,woetbihdabijective a established who [3], ese X oacmatReansraeof surface Riemann compact a to with ω i fgenus of v–eFaci yetheo- type Franchis ovo–de k ω ∧ Let . 1 C ,Reansrae fundamental surface, Riemann m, ω v U , η with , · · · j 1 ∈ en aia isotropic maximal a being o l 1 all for 0 = , · · · ω , M T g n h ups fthis of purpose The . g η , and , on ≥ g on M 2 uhthat such , r in are C ≤ uhthat such , j , i general ≤ J (1) M g . , 2 INDRANIL BISWAS AND MAHAN MJ determines a complex line subbundle of the complexified cotangent bundle (T ∗M) C over the open subset ⊗ g V := M Z(ω ) M. \ i ⊂ i\=1

Lemma 2.1. Let := v T V ωi(v)=0 1 i g T V be the distribution on V deﬁned by . Then is integrableF { and∈ deﬁnes| a foliation∀ of real≤ codimension≤ } ⊂ two on V . W F Proof. For any x M and 1 i g, consider the R–linear homomorphism ∈ ≤ ≤ ωx : T M C , v ω (x)(v) . i x −→ 7−→ i x x x Since ωi is of type (1 , 0), if ωi(x) = 0, then ωi is surjective. Also, ωj is a scalar multiple of ωi because ω ω = 0. Therefore, we conclude6 that the distribution on V is of real codimension two. i ∧ j F Since dw = 0 for all i, it follows that the distribution is integrable. i F Remark 2.2. Since Z(ωi) are smooth embedded submanifolds by hypothesis, and ωi are of type (1 , 0), it follows that Z(ωi) are almost complex submanifolds of M, meaning JM preserves the tangent subbundle TZ(ωi) (TM) Z(ωi). Consequently, all the intersections of the Z(ωi)’s are also almost complex submanifolds⊂ and are| therefore even dimensional.

Clearly, each Z(ωi) has complex codimension at least one (real codimension at least two) as an almost complex submanifold. Therefore, the complement M V has complex codimension at least two. \ 2.2. Almost complex blow-up. We shall need an appropriate notion of blow-up in our context. Sup- pose K M is a smooth embedded submanifold of M of dimension 2j such that JM (TK) = TK. By ⊂ g Remark 2.2, the submanifold i=1 Z(ωi) satisfies these conditions. Note that JM induces an automorphism T J : ((TM) )/TK ((TM) )/TK M/K |K −→ |K of the quotient bundle over K. Since JM is an almost complex structure it follows that JM/K JM/K = Id . Therefore, ((TM) )/TK is a complex vector bundle on K of rank k j.◦ We would − ((T M)|K )/T K |K − like to replace K by the (complex) projectivized normal bundle P(((TM) K )/TK) which will be called the almost complex blow-up of M along K. | For notational convenience, the intersection g Z(ω ) will be denoted by . i=1 i Z We first projectivize (TM) to get a CPk−1Tbundle over ; this CPk−1 bundle will be denoted by . |Z Z B So parametrizes the space of all (real) two dimensional subspaces of (TM) Z preserved by JM (such twoB dimensional subspaces are precisely the complex lines in (TM) equipped| with the complex vector |Z bundle structure defined by JM ). Let π : B −→ Z be the natural projection. For notational convenience, the pulled back vector bundle π∗((TM) ) will be denoted by π∗TM. |Z Let T := kernel(dπ) T π ⊂ B be the relative tangent bundle, where dπ : T π∗T is the differential of π. The map π is almost B −→ Z holomorphic, and Tπ has the structure of a complex vector bundle. It is known that the complex vector bundle T is identified with the vector bundle HomC( , (π∗TM)/ ) = ((π∗TM)/ ) C ∗, where π L L L ⊗ L π∗TM L ⊂ is the tautological real vector bundle of rank two; note that both and (π∗TM)/ have structures of L L complex vector bundles given by JM , and the above tensor product and homomorphisms are both over C. Therefore, the pullback π∗TM splits as ((π∗TM)/ ) = (T ) . L⊕ L L⊕ π ⊗ L The image of the zero section of the complex line bundle is identified with , and the normal L −→ B B bundle of is identified with . A small deleted normal neighborhood UB of in can be identified with a deletedB ⊂ neighborhoodL U of L in M. Let U := U be the neighborhoodB L of in (it is no Z B B B B L S ANALMOSTCOMPLEXCASTELNUOVODEFRANCHISTHEOREM 3 longer a deleted neighborhood), where is again identified with the image of the zero section of . In B L the disjoint union (M ) U B, we may identify U with UB. The resulting topological space will be called the almost complex\ Z ⊔ blow-up of M along . Z Remark 2.3.

(1) Consider the foliation on the deleted neighborhood U of in M defined by . Let LU denote the leaf space for it. After identifying with the image ofZ the zero section ofFπ∗TM, we obtain B locally, from a neighborhood of , a map to the leaf space LU . (2) Note that the notion of an almostB complex blow-up above is well-defined up to the choice of an identification of U with UB. Therefore, the construction yields a well-defined almost complex manifold, up to an isomorphism.

3. Proof of Theorem 1.2

We will ﬁrst show that the leaves of in Section 2 are proper embedded submanifolds of V = M . F \ Z Suppose some leaf L0 does not satisfy the above property. Then there exists x V and a neighborhood U of x such that U L contains inﬁnitely many leaves (of ) accumulating at∈ x. Let σ be a smooth x x 0 F path starting at x transverseT to . Thus for every ǫ > 0 there exist F (1) distinct (local) leaves F , F U , such that (globally) F , F L , 1 2 ⊂ x F 1 2 ⊂ 0 (2) points y F , j = 1 , 2, T j ∈ j (3) a sub-path σ12 σ Ux joining y1 and y2, and (4) a path τ L⊂joining⊂ y and y such that 12 ⊂ 0 1 2

ωi < ǫ i . | Zσ12 | ∀ Since is in the kernel of each ω , F i ωi = 0, i . | Zτ12 | ∀

Setting ǫ smaller than the absolute value of any non-zero period of the ωi’s, it follows that

ωi =0 1 i g . Zτ12∪σ12 ≤ ≤ Hence

ωi = 0 1 i g . Zσ12 ∀ ≤ ≤ ′ Taking limits we obtain that ωi(σ (0)) = 0; but this contradicts the choice that σ is transverse to . Therefore, the leaves of are proper embedded submanifolds of V . F F Consequently, the leaf space D = V/ hFi for the foliation is a smooth 2-manifold. Let F q : V D −→ be the quotient map. Remark 3.1. The referee kindly pointed out to us the following considerably simpler proof of the fact that leaves of are closed in V : F By equation (1), we have ωi = fi,j ωj. Since ωi are closed, d(f ) ω = 0, i , j . i,j ∧ j ∀ 1 The functions fi,j deﬁne a map f : V CP . Since d(fi,j ) ωj = 0, it follows that f is constant on the leaves of the foliation deﬁned by−→ the forms ω , 1 ileqn∧. F i ≤ 4 INDRANIL BISWAS AND MAHAN MJ

Next, let ξ := TV/ F be the quotient complex line bundle on V . Then ξ carries a natural ﬂat partial connection along the leaves (cf. [5]); this is known as the Bott partial connection. It is straightforward to checkD that D D preserves the complex structure on ξ. Indeed, this follows immediately from the fact that ωi are of type (1 , 0). Hence ξ induces an almost complex structure on D. Since D is two–dimensional, this almost complex structure is integrable giving D the structure of a Riemann surface. Further, there exist closed integral (1 , 0) forms η , 1 i g, on D such that i ≤ ≤ ∗ ωi = f ηi .

3.1. Removing indeterminacy. Since Z(ωi) are mutually transverse, we can construct the almost complex blow-ups of M along successively along the µ–fold intersections, µ = 2 , ,g, to obtain M such that Z ··· c (1) there is an extension of the line bundle ξ to all of M, (2) there is a well-deﬁned smooth map c q : M D −→ extending q, and b c (3) the blown-up locus has the structure of complex analytic CP1–bundles as usual (due to transver- sality). The Riemann surface D is compact because M is so. Further, D has genus greater than one as g > 1. So any complex analytic map from CP1 to D must be constant. Therefore, q actually induces a smooth c map from M to D, and we may assume that the indeterminacy locus of q was empty to start oﬀ with, or equivalently that q extends to a smooth map q : M D. This furnishesb the required conclusion and completes the proof of Theorem 1.2. −→

4. Refinements and consequences

4.1. Stein factorization. We now proceed as in the proof of the classical Castelnuovo-de Franchis Theorem, [1, p. 24, Theorem 2.7], to deduce a posteriori that D is a Stein factorization of a map to a compact Riemann surface. Define h : V CPg−1 , m [ω (m): : ω (m)] . −→ 7−→ 1 ··· g Suppose ω (m) = 0. Then there exists a small neighborhood U(m) such that i 6 h(x) = [f (x): : f (x) : 1 : f (x): : f (x)] , x U(m) , 1,i ··· i−1,i i+1,i ··· g,i ∀ ∈ where f are defined in (1). Since ω ω = 0 and dω = 0 for all i , j, it follows that df ω = 0. l,j j ∧ i j j,i ∧ i Consequently, each fj,i is constant on the leaves of . Hence h has a (complex) one dimensional image, so the image of h is a Riemann surface C. F Further, h induces a holomorphic map h : D C. 1 −→ We note that D may be thought of as the Stein factorization of h, and h factors as h = h q. 1 ◦ 4.2. Further generalizations. Let M be a compact manifold, and let TM be a nonsingular foliation such that M has a flat transversely almost complex structure. ThisS ⊂ means that we have an automorphism J : TM/ TM/ S −→ S such that

(1) J J = IdTM/S , and (2) J ◦is ﬂat with− respect to the Bott partial connection on TM/ . S Let ω , 1 i g, be linearly independent smooth sections of (TM/ )∗ C such that i ≤ ≤ S ⊗ ANALMOSTCOMPLEXCASTELNUOVODEFRANCHISTHEOREM 5

∗ (1) each ωi is ﬂat with respect to the connection on (TM/ ) C induced by the Bott partial connection on TM/ , S ⊗ S (2) each ωi is of type (1 , 0), meaning the corresponding homomorphism TM/ M C is C– linear, S −→ × (3) each ωi is closed when considered as a complex 1–form on M using the composition ωi TM C (TM/ ) C M C ⊗ −→ S ⊗ −→ × (4) ω ω = 0 for all i , j, and i ∧ j (5) ωj are in general position. Theorem 1.2 can be generalized to this set–up.

Remark 4.1. The only real use we have made of the hypothesis in Theorem 1.2 that ωi’s are in general position is to ensure that has complex codimension at least two. This is what allows us to remove indeterminacies in the smoothZ category (as opposed to the algebraic or complex analytic categories, where complex codimension greater than one follows naturally). Any hypothesis that ensures that the indeterminacy locus has complex codimension greater than one would suﬃce.

Acknowledgements

We are extremely grateful to the referee for very detailed and helpful comments and particularly for Remark 3.1 that considerably simpliﬁes of the proof of the main theorem. Both authors acknowledge support of J. C. Bose Fellowship.

References [1] J. Amoros, M. Burger, K. Corlette, D. Kotschick and D. Toledo, Fundamental groups of compact Kähler manifolds. Mathematical Surveys and Monographs, 44. American Mathematical Society, Providence, RI, 1996. [2] G. Castelnuovo, Sulle superficie aventi il genere aritmetico negativo. Rend. Circ. Mat. Palermo 20 (1905), 55–60. [3] F. Catanese, Moduli and classification of irregular Kaehler manifolds (and algebraic varieties) with Albanese general type fibrations. Invent. Math. 104 (1991), 263–289. [4] M. De Franchis, Sulle superficie algebriche le quali contengono un fascio irrazionale di curve. Rend. Circ. Mat. Palermo 20 (1905), 49–54. [5] H. B. Lawson, The quantitative theory of foliations. Expository lectures from the CBMS Regional Conference held at Washington University, St. Louis, Mo., January 6–10, 1975. Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 27. American Mathematical Society, Providence, R. I., 1977.

School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India E-mail address: [email protected]