We construct four-dimensional symplectic orbifolds admitting Hamiltonian circle actions with isolated fixed points, but not admitting any Hamiltonian action of a two-torus. One example is linear, and one example is compact

Author(s) S. F. Singer

J. Talvacchia

Janet Talvacchia

N. Watson

S. F. Singer

Stephanie Frank Singer

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Nontoric Hamiltonian Circle Actions on Four-Dimensional Symplectic Orbifolds

Singer, S. F.

Talvacchia, Janet

Watson, N.

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Swarthmore CollegeWorksMathematics & Statistics Faculty Works Mathematics & Statistics3-1-1999Nontoric Hamiltonian Circle Actions On Four-Dimensional Symplectic OrbifoldsS. F. SingerJanet TalvacchiaSwarthmore College, jtalvac1@swarthmore.eduN. WatsonFollow this and additional works at: http://works.swarthmore.edu/fac-math-statPart of the Mathematics CommonsThis Article is brought to you for free and open access by the Mathematics & Statistics at Works. It has been accepted for inclusion in Mathematics &Statistics Faculty Works by an authorized administrator of Works. For more information, please contact myworks@swarthmore.edu.Recommended CitationS. F. Singer, Janet Talvacchia, and N. Watson. (1999). "Nontoric Hamiltonian Circle Actions On Four-Dimensional SymplecticOrbifolds". Proceedings Of The American Mathematical Society. Volume 127, Issue 3. 937-940.http://works.swarthmore.edu/fac-math-stat/85Nontoric Hamiltonian Circle Actions on Four-Dimensional Symplectic OrbifoldsAuthor(s): S. F. Singer, J. Talvacchia and N. WatsonSource: Proceedings of the American Mathematical Society, Vol. 127, No. 3 (Mar., 1999), pp.937-940Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/119030 .Accessed: 15/01/2015 16:35Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp .JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org. .American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access toProceedings of the American Mathematical Society.http://www.jstor.org This content downloaded from 130.58.65.20 on Thu, 15 Jan 2015 16:35:06 PMAll use subject to JSTOR Terms and ConditionsPROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 3, March 1999, Pages 937-940 S 0002-9939(99)04767-X NONTORIC HAMILTONIAN CIRCLE ACTIONS ON FOUR-DIMENSIONAL SYMPLECTIC ORBIFOLDS S. F. SINGER, J. TALVACCHIA, AND N. WATSON (Communicated by Peter Li) ABSTRACT. We construct four-dimensional symplectic orbifolds admitting Hamiltonian circle actions with isolated fixed points, but not admitting any Hamiltonian action of a two-torus. One example is linear, and one example is compact. In this note we construct four-dimensional symplectic orbifolds admitting Hamil- tonian circle actions with isolated fixed points, but not admitting any Hamiltonian action of a two-torus. This work is part of our ongoing attempt to generalize to orbifolds the classification of four-dimensional compact symplectic manifolds with Hamiltonian circle actions due to Ahara and Hattori, Audin and Karshon ([AH], [Au], [K]) and is part of a larger program to understand torus actions of complex- ity one on orbifolds. The complexity of a Hamiltonian torus action on an orbifold is defined to be the difference between half of the dimension of the orbifold and the dimension of the torus. The complexity zero case (i.e., n-torus actions on 2n-dimensional orbifolds) has been analyzed by Lerman and Tolman [LT]. One of Karshon's theorems states that if a Hamiltonian circle action on a compact four-dimensional manifold has isolated fixed points, then the action can be extended to the action of a two-torus. It follows that if a compact symplectic four-dimensional manifold has a Hamiltonian circle action with isolated fixed points, then it is a toric variety. One of our orbifolds is compact, proving that Karshon's result does not generalize to orbifolds. Our result is slightly stronger: not only do the circle actions fail to extend, but the underlying spaces admit no Hamiltonian two-torus action whatsoever. In the first section we consider a compact orbifold that is the quotient of a non- singular four-dimensional toric variety by a finite group. Specifically, we consider CP1 x CP1 modulo an action of the group Z/4Z. The natural Hamiltonian action of the two-torus on CP1 x CP1 does not descend to the quotient; however, the action of one circle subgroup does descend, giving the required circle action. In the second section we consider the vector orbi-space C2/r where r is a finite subgroup of U(2) that is not abelian. Here the diagonal action of S1 on C2 descends to the orbifold but, due to the nature of the group r, there is no possible two-torus Received by the editors July 8, 1997. 1991 Mathematics Subject Classification. Primary 58Fxx. The second author was supported in part by a fellowship from the American Association of University Women and NSF grant DMS 9304580. (?)1999 American Mathematical Society 937 This content downloaded from 130.58.65.20 on Thu, 15 Jan 2015 16:35:06 PMAll use subject to JSTOR Terms and Conditions938 S. F. SINGER, J. TALVACCHIA, AND N. WATSON action on the orbifold. Unlike symplectic manifolds, where the only obstructions to large torus actions are global, symplectic orbifolds can have local obstructions. 1. A COMPACT EXAMPLE: CP' x CP//r Consider the real four-dimensional manifold CP1 x CP1. We denote an arbitrary point on this manifold by (x, y), with x, y E CP1. This manifold has a standard symplectic form given by i ( dxAdt dyAdy '=2 +t 1X)2 1+ I 012)2 Let r denote the group of symplectomorphisms generated by (x, y) a-4 (-y, x). Note that r = Z/4Z. Then CP1 x CPl/r is a symplectic orbifold. We denote a point of this orbifold by [x, y], the r-equivalence class of (x, y). The only singular points of this orbifold are [0, 0], [oo, oo] and [0, oc], and these have cyclic orbifold structure groups of orders 4, 4 and 2, respectively. We can define a Hamiltonian circle action on CP1 x CPl/r. The function 0 ([X ______=___ $([xy]) = 4 ((lJ + J1F ) (1 + Jy2)) is well-defined on CP1 x CPl/r, and its Hamiltonian flow is given by [eit/2x, eit/2y]. Notice that the minimum period of this flow is 27r. The fixed points are [0, 0], [oc, oc] and [0, oc]. The image of 0 is [0, 2]. The function 0 is a Morse function. One can apply the results of [LT], section 4.1, to see that the associated Morse polynomial is 1+ X2 + X4, that this polynomial must equal the Poincare polynomial, and hence that dim H2 = 1 i.e., the second cohomology is one-dimensional. It remains to show that this orbifold admits no effective Hamiltonian action of a two-torus. We argue by contradiction. Suppose : cP1 x CPl/r - R2 is the Hamiltonian for an effective two-torus action. Let A be the image of A. Then A is a convex polytope. By applying Bott-Morse theory to a generic one- dimensional projection of the moment map AP and comparing with the Poincare polynomial, we know that the number of vertices of A is equal to dim H2 + 2 = 3. Hence A is a triangle. Because CP1 x CPl/r has only three singular points, of orders 4, 4 and 2, it follows from the normal form theorem for group actions on orbifolds, in section 3.5 of [LT], that these points must correspond to the vertices of the triangle. Without loss of generality (by choosing an automorphism of the two- torus) we may assume that the edge between the two vertices of order 4 is parallel to the vector (1, 0). Let (a, b) and (c, d) be primitive integer vectors parallel to the other two sides. Because the order of an isolated singular point in a toric variety is given by the determinant of the matrix of primitive integer vectors parallel to edges emanating from the corresponding vertex of the associated polytope, we find that b d ?=2, b = ?4 and d = ?4. This content downloaded from 130.58.65.20 on Thu, 15 Jan 2015 16:35:06 PMAll use subject to JSTOR Terms and ConditionsNONTORIC HAMILTONIAN CIRCLE ACTIONS 939 But these three equations cannot be simultaneously satisfied by integers, as a | must be divisible by 4 if a and c are integers. Contradiction! Hence CP1 x CPl/r admits no effective action of a two-torus. We remark that this orbifold has a Kaehler structure inherited from CP1 x CP1. While Karshon has shown that any compact symplectic four-dimensional manifold with a Hamiltonian circle action (with no assumptions on the fixed-point sets) must be Kaehler [K] and Tolman has constructed a six-dimensional, non-Kaehler compact symplectic manifold with a Hamiltonian action of a two-torus [T], it is unknown whether every compact symplectic four-dimensional orbifold with a Hamiltonian circle action must be Kaehler. 2. A VECTOR ORBI-SPACE EXAMPLE: C2/r Let r be a finite subgroup of U(2) that is not abelian. (For example, let r be a dihedral group.) Consider C2 with coordinates (z1, Z2) and symplectic form W =(dz A dz1 + dz2 A d2). 2 The diagonal Hamiltonian action of Si on (C2, W) given by (e, (z, Z2)) e (&0zie0 i z2) with moment map q(Z1, Z2) = 4(tzi12 + Z2 12) descends to C2/r. There is one singular point of the orbifold at [(0,0)] which is also an isolated fixed point of the S1 action. The orbifold structure group of [(0 0)] is r. By Lemma 6.2 in [LT], however, there can be no faithful Hamiltonian two- torus action on C2/r. The image of the moment map (D for any such two-torus action would have to be a simplicial cone containing D([(O, 0)]) as a vertex, and the orbifold structure group at [(O 0)] would have to be a quotient of lattices, and hence abelian. But r is not abelian; hence the vector orbi-space admits no Hamiltonian two-torus action. ACKNOWLEDGEMENTS We thank S. Tolman for suggesting this problem and A. Knutson, E. Lerman and S. Tolman for many helpful conversations. REFERENCES [AH] Ahara, K. and A. Hattori, 4 dimensional symplectic S1-manifolds admitting moment map, J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 38 (1991), 251-298. MR 93b:58048 [At] Atiyah, M. F., Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15. MR 83e:53037 [Au] Audin, M., Hamiltoniens periodiques sur les varieties symplectiques compactes de dimension 4, Geometrie symplectique et mechanique, Proceedings 1988, C. Albert ed., Springer Lecture Notes in Math. 1416 (1990). MR 91f:57013 [GS] Guillemin, V. and S. Sternberg, Convexity properties of the moment mapping I, Invent. Math. 67 (1982), 491-513. MR 83m:58037 [K] Karshon, Y., Periodic Hamiltonian flows on four-dimensional manifolds, Trans. AMS, sub- mitted. (Available electronically at dg-ga/9510004.) This content downloaded from 130.58.65.20 on Thu, 15 Jan 2015 16:35:06 PMAll use subject to JSTOR Terms and Conditions940 S. F. SINGER, J. TALVACCHIA, AND N. WATSON [LT] Lerman, E. and S. Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. AMS. 349 (1997), 4201-4230. (Available electronically at dg-ga/9511008.) MR 98a:57043 [T] Tolman, S., Examples of Non-Kaehler Hamiltonian Torus Actions, Invent. Math., to appear. DEPARTMENT OF MATHEMATICS, HAVERFORD COLLEGE, HAVERFORD, PENNSYLVANIA 19041 E-mail address: ssinger~haverford. edu DEPARTMENT OF MATHEMATICS, SWARTHMORE COLLEGE, SWARTHMORE, PENNSYLVANIA 19081 E-mail address: jtalvaclswarthmore.edu DEPARTMENT OF MATHEMATICS, THE WHARTON SCHOOL, UNIVERSITY OF PENNSYLVANIA, PHILADELPHIA, PENNSYLVANIA 19104 This content downloaded from 130.58.65.20 on Thu, 15 Jan 2015 16:35:06 PMAll use subject to JSTOR Terms and Conditions

Nontoric Hamiltonian Circle Actions On Four-Dimensional Symplectic Orbifolds

Singer, Stephanie Frank

Talvacchia, J.

Haverford College: Haverford Scholarship

Haverford College Haverford Scholarship Faculty Publications Mathematics & Statistics 1999 Nontoric Hamiltonian circle actions on four-dimensional symplectic orbifolds Stephanie Frank Singer Haverford College J. Talvacchia N. Watson Follow this and additional works at: https://scholarship.haverford.edu/mathematics_facpubs Repository Citation Singer, S., J. Talvacchia, and N. Watson. "Nontoric Hamiltonian circle actions on four-dimensional symplectic orbifolds." Proceedings of the American Mathematical Society 127.3 (1999): 937-940. This Journal Article is brought to you for free and open access by the Mathematics & Statistics at Haverford Scholarship. It has been accepted for inclusion in Faculty Publications by an authorized administrator of Haverford Scholarship. For more information, please contact nmedeiro@haverford.edu. Nontoric Hamiltonian Circle Actions on Four-Dimensional Symplectic OrbifoldsAuthor(s): S. F. Singer, J. Talvacchia and N. WatsonSource: Proceedings of the American Mathematical Society, Vol. 127, No. 3 (Mar., 1999), pp.937-940Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/119030 .Accessed: 29/03/2013 13:09Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp .JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org. .American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access toProceedings of the American Mathematical Society.http://www.jstor.org This content downloaded from 165.82.168.47 on Fri, 29 Mar 2013 13:09:22 PMAll use subject to JSTOR Terms and ConditionsPROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 127, Number 3, March 1999, Pages 937-940 S 0002-9939(99)04767-X NONTORIC HAMILTONIAN CIRCLE ACTIONS ON FOUR-DIMENSIONAL SYMPLECTIC ORBIFOLDS S. F. SINGER, J. TALVACCHIA, AND N. WATSON (Communicated by Peter Li) ABSTRACT. We construct four-dimensional symplectic orbifolds admitting Hamiltonian circle actions with isolated fixed points, but not admitting any Hamiltonian action of a two-torus. One example is linear, and one example is compact. In this note we construct four-dimensional symplectic orbifolds admitting Hamil- tonian circle actions with isolated fixed points, but not admitting any Hamiltonian action of a two-torus. This work is part of our ongoing attempt to generalize to orbifolds the classification of four-dimensional compact symplectic manifolds with Hamiltonian circle actions due to Ahara and Hattori, Audin and Karshon ([AH], [Au], [K]) and is part of a larger program to understand torus actions of complex- ity one on orbifolds. The complexity of a Hamiltonian torus action on an orbifold is defined to be the difference between half of the dimension of the orbifold and the dimension of the torus. The complexity zero case (i.e., n-torus actions on 2n-dimensional orbifolds) has been analyzed by Lerman and Tolman [LT]. One of Karshon's theorems states that if a Hamiltonian circle action on a compact four-dimensional manifold has isolated fixed points, then the action can be extended to the action of a two-torus. It follows that if a compact symplectic four-dimensional manifold has a Hamiltonian circle action with isolated fixed points, then it is a toric variety. One of our orbifolds is compact, proving that Karshon's result does not generalize to orbifolds. Our result is slightly stronger: not only do the circle actions fail to extend, but the underlying spaces admit no Hamiltonian two-torus action whatsoever. In the first section we consider a compact orbifold that is the quotient of a non- singular four-dimensional toric variety by a finite group. Specifically, we consider CP1 x CP1 modulo an action of the group Z/4Z. The natural Hamiltonian action of the two-torus on CP1 x CP1 does not descend to the quotient; however, the action of one circle subgroup does descend, giving the required circle action. In the second section we consider the vector orbi-space C2/r where r is a finite subgroup of U(2) that is not abelian. Here the diagonal action of S1 on C2 descends to the orbifold but, due to the nature of the group r, there is no possible two-torus Received by the editors July 8, 1997. 1991 Mathematics Subject Classification. Primary 58Fxx. The second author was supported in part by a fellowship from the American Association of University Women and NSF grant DMS 9304580. (?)1999 American Mathematical Society 937 This content downloaded from 165.82.168.47 on Fri, 29 Mar 2013 13:09:22 PMAll use subject to JSTOR Terms and Conditions938 S. F. SINGER, J. TALVACCHIA, AND N. WATSON action on the orbifold. Unlike symplectic manifolds, where the only obstructions to large torus actions are global, symplectic orbifolds can have local obstructions. 1. A COMPACT EXAMPLE: CP' x CP//r Consider the real four-dimensional manifold CP1 x CP1. We denote an arbitrary point on this manifold by (x, y), with x, y E CP1. This manifold has a standard symplectic form given by i ( dxAdt dyAdy '=2 +t 1X)2 1+ I 012)2 Let r denote the group of symplectomorphisms generated by (x, y) a-4 (-y, x). Note that r = Z/4Z. Then CP1 x CPl/r is a symplectic orbifold. We denote a point of this orbifold by [x, y], the r-equivalence class of (x, y). The only singular points of this orbifold are [0, 0], [oo, oo] and [0, oc], and these have cyclic orbifold structure groups of orders 4, 4 and 2, respectively. We can define a Hamiltonian circle action on CP1 x CPl/r. The function 0 ([X ______=___ $([xy]) = 4 ((lJ + J1F ) (1 + Jy2)) is well-defined on CP1 x CPl/r, and its Hamiltonian flow is given by [eit/2x, eit/2y]. Notice that the minimum period of this flow is 27r. The fixed points are [0, 0], [oc, oc] and [0, oc]. The image of 0 is [0, 2]. The function 0 is a Morse function. One can apply the results of [LT], section 4.1, to see that the associated Morse polynomial is 1+ X2 + X4, that this polynomial must equal the Poincare polynomial, and hence that dim H2 = 1 i.e., the second cohomology is one-dimensional. It remains to show that this orbifold admits no effective Hamiltonian action of a two-torus. We argue by contradiction. Suppose : cP1 x CPl/r - R2 is the Hamiltonian for an effective two-torus action. Let A be the image of A. Then A is a convex polytope. By applying Bott-Morse theory to a generic one- dimensional projection of the moment map AP and comparing with the Poincare polynomial, we know that the number of vertices of A is equal to dim H2 + 2 = 3. Hence A is a triangle. Because CP1 x CPl/r has only three singular points, of orders 4, 4 and 2, it follows from the normal form theorem for group actions on orbifolds, in section 3.5 of [LT], that these points must correspond to the vertices of the triangle. Without loss of generality (by choosing an automorphism of the two- torus) we may assume that the edge between the two vertices of order 4 is parallel to the vector (1, 0). Let (a, b) and (c, d) be primitive integer vectors parallel to the other two sides. Because the order of an isolated singular point in a toric variety is given by the determinant of the matrix of primitive integer vectors parallel to edges emanating from the corresponding vertex of the associated polytope, we find that b d ?=2, b = ?4 and d = ?4. This content downloaded from 165.82.168.47 on Fri, 29 Mar 2013 13:09:22 PMAll use subject to JSTOR Terms and ConditionsNONTORIC HAMILTONIAN CIRCLE ACTIONS 939 But these three equations cannot be simultaneously satisfied by integers, as a | must be divisible by 4 if a and c are integers. Contradiction! Hence CP1 x CPl/r admits no effective action of a two-torus. We remark that this orbifold has a Kaehler structure inherited from CP1 x CP1. While Karshon has shown that any compact symplectic four-dimensional manifold with a Hamiltonian circle action (with no assumptions on the fixed-point sets) must be Kaehler [K] and Tolman has constructed a six-dimensional, non-Kaehler compact symplectic manifold with a Hamiltonian action of a two-torus [T], it is unknown whether every compact symplectic four-dimensional orbifold with a Hamiltonian circle action must be Kaehler. 2. A VECTOR ORBI-SPACE EXAMPLE: C2/r Let r be a finite subgroup of U(2) that is not abelian. (For example, let r be a dihedral group.) Consider C2 with coordinates (z1, Z2) and symplectic form W =(dz A dz1 + dz2 A d2). 2 The diagonal Hamiltonian action of Si on (C2, W) given by (e, (z, Z2)) e (&0zie0 i z2) with moment map q(Z1, Z2) = 4(tzi12 + Z2 12) descends to C2/r. There is one singular point of the orbifold at [(0,0)] which is also an isolated fixed point of the S1 action. The orbifold structure group of [(0 0)] is r. By Lemma 6.2 in [LT], however, there can be no faithful Hamiltonian two- torus action on C2/r. The image of the moment map (D for any such two-torus action would have to be a simplicial cone containing D([(O, 0)]) as a vertex, and the orbifold structure group at [(O 0)] would have to be a quotient of lattices, and hence abelian. But r is not abelian; hence the vector orbi-space admits no Hamiltonian two-torus action. ACKNOWLEDGEMENTS We thank S. Tolman for suggesting this problem and A. Knutson, E. Lerman and S. Tolman for many helpful conversations. REFERENCES [AH] Ahara, K. and A. Hattori, 4 dimensional symplectic S1-manifolds admitting moment map, J. Fac. Sci. Univ. Tokyo Sect. IA, Math. 38 (1991), 251-298. MR 93b:58048 [At] Atiyah, M. F., Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), 1-15. MR 83e:53037 [Au] Audin, M., Hamiltoniens periodiques sur les varieties symplectiques compactes de dimension 4, Geometrie symplectique et mechanique, Proceedings 1988, C. Albert ed., Springer Lecture Notes in Math. 1416 (1990). MR 91f:57013 [GS] Guillemin, V. and S. Sternberg, Convexity properties of the moment mapping I, Invent. Math. 67 (1982), 491-513. MR 83m:58037 [K] Karshon, Y., Periodic Hamiltonian flows on four-dimensional manifolds, Trans. AMS, sub- mitted. (Available electronically at dg-ga/9510004.) This content downloaded from 165.82.168.47 on Fri, 29 Mar 2013 13:09:22 PMAll use subject to JSTOR Terms and Conditions940 S. F. SINGER, J. TALVACCHIA, AND N. WATSON [LT] Lerman, E. and S. Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. AMS. 349 (1997), 4201-4230. (Available electronically at dg-ga/9511008.) MR 98a:57043 [T] Tolman, S., Examples of Non-Kaehler Hamiltonian Torus Actions, Invent. Math., to appear. DEPARTMENT OF MATHEMATICS, HAVERFORD COLLEGE, HAVERFORD, PENNSYLVANIA 19041 E-mail address: ssinger~haverford. edu DEPARTMENT OF MATHEMATICS, SWARTHMORE COLLEGE, SWARTHMORE, PENNSYLVANIA 19081 E-mail address: jtalvaclswarthmore.edu DEPARTMENT OF MATHEMATICS, THE WHARTON SCHOOL, UNIVERSITY OF PENNSYLVANIA, PHILADELPHIA, PENNSYLVANIA 19104 This content downloaded from 165.82.168.47 on Fri, 29 Mar 2013 13:09:22 PMAll use subject to JSTOR Terms and Conditions

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Nontoric Hamiltonian circle actions on four-dimensional symplectic orbifolds

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Nontoric Hamiltonian Circle Actions On Four-Dimensional Symplectic Orbifolds

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