aa r X i v : . [ m a t h . S G ] A ug Hamiltonian S -actions on complete intersections Nicholas LindsayAugust 7, 2020
Abstract
We assume that M is a closed symplectic manifold, diffeomorphic to a completeintersection of complex dimension 4 m , having a Hamiltonian S -action with finite fixedpoint set. We show that M is diffeomorphic to a CP m , a quadric Q ⊂ CP m +1 or anintersection of 2 quadrics Q ∩ Q ⊂ CP m +2 .We also prove the statement under a weaker assumption on the fixed point set, namelyif components of the fixed point set are isolated points or submanifolds with real dimensionnot divisible by 4. Smooth, complete intersections over C having infinite automorphism groups have been clas-sified by Benoist [2][Theorem 3.1]. When the complex dimension is at least 3, they are justprojective spaces and quadrics. Here, we prove a symplectic version of a partial case of thatresult. Namely: Theorem 1.1.
Let ( M, ω ) be a closed symplectic manifold with dim R ( M ) = 8 k , diffeomorphicto a complete intersection. Suppose that M has a Hamiltonian S -action with finite fixed pointset. Then, M is diffeomorphic to a complex projective space, a quadric or an intersection of quadrics. Our motivation for this result comes in two parts:1. It recovers a part of the algebraic result [2][Theorem 3.1] using purely topological meth-ods.2. It excludes the existence of Hamiltonian circle actions on a large class of Fano varieties.Exotic Hamiltonian torus actions on K¨ahler manifolds are known to exist, in varioussenses [4, 5, 8, 10]. Conjecturally, such actions do not occur on Fano varieties. In [3] it wasshown that smooth complete intersections with real dimension 6 have a smooth circle actionif and only if the holomorphic automorphism group is infinite.As was pointed out in [3], It is also possible to say something about smooth S -actionsin higher dimensions. Let X be a spin, complete intersection with even complex dimension,then ˆ A ( X ) = 0 ⇐⇒ X is Fano . A -genus rules out the existence of any smooth circle action. We alsorecall that there are many spin, Fano, complete intersections of even complex dimension. Forexample, any Fano, complete intersection of an odd number of even degree hypersurfaces inan odd dimensional projective space. Hence, our result applies to large variety of complexprojective manifolds with vanishing ˆ A -genus.In the final section, we note that a more general version of the signature formula givesthe following statement. We note that although this statement supersedes Theorem 1.1, wehave decided to preserve the original proof of Theorem 1.1, since it includes all of the mainfeatures of the argument and is less technichal. Theorem 1.2.
Suppose that Let ( M, ω ) be a closed symplectic manifold with dim R ( M ) = 8 k ,diffeomorphic to a complete intersection. Suppose that M has a Hamiltonian S -action whosefixed point set consists of either isolated fixed points or submanifolds whose real dimension isnot divisible by . Then, M is diffeomorphic to a complex projective space, a quadric or anintersection of quadrics. Theorem 1.2 has the following immediate corollary:
Corollary 1.3.
Suppose that Let ( M, ω ) be a closed symplectic manifold with dim R ( M ) =8 k , diffeomorphic to a complete intersection. Suppose that M has a Hamiltonian S -actionsuch that each component of the fixed point set has real dimension at most . Then, M isdiffeomorphic to a complex projective space, a quadric or an intersection of quadrics. Acknowledgements.
I would like to thank Volker Puppe for pointing me to the reference[3], which initiated my interest in this question.
We proceed first by showing that the intersection form on the middle cohomology groups ofmanifolds as in Theorem 1.1 is positive definite.
Proposition 2.1.
Suppose that M is a closed symplectic manifold of (real) dimension k with a Hamiltonian S -action with finite fixed point set. Suppose that b i ( M ) = b i ( CP k ) for i = 4 k . Then the intersection form on H k ( M, Z ) is positive definite.Proof. By [6, Theorem 1.1], the signature of M is equal to P j ∈ Z b j ( M ) − P j ∈ Z b j +2 ( M ) = b k ( M ). Since the signature of the intersection form is equal the rank of the middle coho-mology group, it is positive definite.By the Lefcshetz theorem, for a complete intersection X of real dimension 8 k , b i ( X ) = b i ( CP k ) for i = 4 k . Hence, by the above, if X has a Hamiltonian S -action with finite fixedpoint set, the intersection form on the middle cohomology is positive definite. Now, the proofof Theorem 1.1 follows from the following theorem of Libgober and Wood [7]][Page 638]. Theorem 2.2. [7] Suppose that X is a complete intersection with real dimension k , andsuppose that intersection form on H k ( X, Z ) is positive definite. Then, X is diffeomorphicto a projective space, a quadric or an intersection of two quadrics.
2e note that unfortunately, there is one unsettled case in each dimension: we didn’texclude Hamiltonian S -actions on symplectic manifolds diffeomorphic to the intersections oftwo quadrics. Such varieties do not have an algebraic torus action due to [2][Theorem 3.1].We leave it as a question: Question 2.3.
Does (the underlying smooth manifold) of an intersection of quadrics withcomplex dimension at least have a symplectic form exhibiting a Hamiltonian S -action withisolated fixed points? We finish by making one remark about the case that M is diffeomorphic to a completeintersection with dim R ( M ) = 8 k + 4, having a Hamiltonian S -action with finite fixed pointset. In this case, [6, Theorem 1.1] gives that the signature of M is 2 − b k +2 , i.e. the intersectionform is “almost negative definite”. It follows from [7][Theorem A], that if 3 ≤ b k +2 thenthere is a unimodular summand of the intersection form on H k +2 with rank at most 5. Thisdoes not occur in the algebraic case [2][Theorem 3.1]. In this section we prove an extension of Theorem 1.1, to allow certain classes of fixed subman-ifolds. Although this logically supersedes Theorem 1.1, we have decided to preserve the proofof Theorem 1.1 in its original form, since it contains all of the main features of the argumentwithout being overly technichal. Here we present a much more general result, which usesa combination of two well-known localisation theorems. First it is nessecary to recall onedefinition:
Definition 3.1.
Let ( M, ω ) be a closed symplectic manifold with a Hamiltonian S -action.Let F be a connected component of the fixed point set M S . Denote by λ F the number of(strictly) negative weights along F , counted with multiplicity. We remark that equivalently λ F is half the Morse-Bott index of F , with respect to theHamiltonian, as is well known. We recall two localisation theorems. Theorem 3.2. [9][Page 9] Let ( M, ω ) be a closed symplectic manifold with a Hamiltonian S -action. Then each of the components of the fixed point set M S are symplectic submanifolds.For each k ∈ Z , b k ( M ) = X F ⊂ M S b k − λ F ( F ) where the sum runs over each connected component of the fixed point set. The following is an immediate consequence of the localisation formula of the signaturefor smooth circle actions, in the form proven by Atiyah and Singer [1].
Theorem 3.3.
Let ( M, ω ) be a closed symplectic manifold with a Hamiltonian S -action,such that dim R ( M ) is divisible by . Suppose that the fixed point set consists of isolated pointand submanifolds having real dimension not divisible by . Then the signature σ ( M ) of M isequal to σ ( M ) = X p ( − λ p , where p runs over the isolated fixed points. roof. This is the localisation of the signature formula for a smooth circle action, in theformula contained in [1]. Smooth submanifolds with real dimension not divisible by 4 do notcontribute to the localisation sum (see [6][Theorem 4.2]).
Proof of Theorem 1.2.
The proof is morally the same as the proof of Theorem 1.1, we start bygiving a brief overview of the argument. Combining Theorem 3.2, Theorem 3.3 and the factthat b i ( M ) = b i ( CP k ) for i = 4 k (due to the Lefcshetz Theorem) proves that the signaturesatisfies σ ( M ) = b k ( M ). Then, as before, the result follows from Theorem 2.2.We give additional details. Let N be a connected, fixed submanifold whose dimensionis not divisible by 4, we see that dim R ( N ) = 2 mod 4, since N is symplectic. Also, byTheorem 3.2 and the fact that M only has at most one Betti number which is at least 2 (bythe Lefcshetz Theorem), we see that N has at most one Betti number which is at least 2.By applying Poincare duality to N and dim R ( N ) = 2 mod 4, every integer which is aneven-degree Betti number of N appears in at least two degrees, hence all the Betti numbersof N are at most 1 by the above. Combining this with the fact that N is symplectic, sothat all even Betti numbers are non-zero, we obtain that b k ( N ) = b k ( CP m ) for all k and m = dim R ( N )2 . Below we will use the fact that m is odd.By Theorem 3.2 we may calculate the contribution of N to the Betti number sequenceof M . It contributes +1 in degrees 2 λ F , λ F + 2 , . . . , λ f + 2 m , and 0 to every other degree.We note that the contribution to the Betti number sequence of M is exactly the same as asequence of isolated fixed points p , . . . , p m such that λ p i = λ N + i .On the other hand, by Theorem, since the dimension of N is not divisible by 4 thecontribution of N to the signature formula in Theorem 3.3 is 0. A sequence of isolated fixedpoints p , . . . , p m such that λ p i = λ N + i , contributes 0 to the signature formula of Theorem3.3, by the fact that m is odd.Hence, we may formally replace each fixed submanifold with an appropriate sequence ofisolated fixed points, without changing the contribution to the Betti number sequence of M or the signature of M (the question whether a manifold with that fixed point data existsor not being irrelevant, it is a formal trick for the purpose of calculation). Hence, we mayformally apply Theorem 3.2 and Theorem 3.3, as if the fixed point set was isolated, obtaining σ ( M ) = P j ∈ Z b j ( M ) − P j ∈ Z b j +2 ( M ) = b k ( M ). In particular, the intersection form of M is positive definite, as required. References [1] M.F.Atiyah and I.M.Singer. The index of elliptic operators: Ann. of . Math. 87 (1968)546-604.[2] Olivier Benoist, Separation et propriete de Deligne-Mumford des champs de modulesdintersections completes lisse., J. Lond. Math. Soc., II. Ser. 87 (2013), no. 1, 138156(French).[3] A. Dessai and M. Wiemeler Complete Intersections with S -action, TransformationGroups, Vol. 22 (2017) pp. 295-320[4] O. Goertsches, P. Konstantis, L. Zoller. Symplectic and K¨ahler structures on biquotients.arXiv:1812.09689 (2019). 45] O. Goertsches, P. Konstantis, L. Zoller. Realization of GKM fibrations and new examplesof Hamiltonian non-K¨ahler actions. arXiv:2003.11298 (2020).[6] J.D.S. Jones , J.H. Rawnsley. Hamitonian circle actions and the Signature. Journal ofGeometry and Physics Volume 23, Issues 34, November 1997, Pages 301-307[7] A. S. Libgober and J. W. Wood, On the topological structure of even-dimensional com-plete intersections, Trans. Amer. Math. Soc. 267 (1981), no.2, 637660[8] N. Lindsay, D. Panov. Symplectic and K¨ahler structures on CP -bundles over CP2