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Theta Characteristics of Riemann Surface 417 Now, Since Sˆ Is Not Identically Zero, It Defines a Meromorphic Section of the Holomor- =∼ ∗ Phic Tangent Bundle TX

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Theta Characteristics of Riemann Surface 417 Now, Since Sˆ Is Not Identically Zero, It Defines a Meromorphic Section of the Holomor- =∼ ∗ Phic Tangent Bundle TX J. Aust. Math. Soc. 76 (2004), 415–423 A NOTE ON THE THETA CHARACTERISTICS OF A COMPACT RIEMANN SURFACE INDRANIL BISWAS (Received 13 January 2002; revised 5 May 2003) Communicated by K. Wysocki Abstract Let X be a compact connected Riemann surface and ¾ a square root of the holomorphic cotangent bundle of X. Sending any line bundle L over X of order two to the image of dim H 0.X;¾ ⊗ L/ − dim H 0.X;¾/ = Z in Z 2 defines a quadratic form on the space of all order two line bundles. We give a topological interpretation of this quadratic form in terms of index of vector fields on X. 2000 Mathematics subject classification: primary 14F10, 57R25, 57R15. 1. The Arf function for a theta characteristic Let X be a compact connected Riemann surface of genus g. The holomorphic cotangent bundle of X will be denoted by K X .Let¾ be a holomorphic line bundle ⊗2 over X such that ¾ is holomorphically isomorphic to K X . A line bundle with this property is known as a theta characteristic of X. Since the degree of K X is even, X has a theta characteristic. There are exactly 22g theta characteristics of X,whereg is the genus of X. Indeed, if L is a holomorphic line bundle over X of order two, that is, L⊗2 is isomorphic to the trivial line bundle, then ¾ ⊗ L is also a theta characteristic, provided ¾ is one. It is easy to see that this action of the order two line bundles on the theta characteristics is free and transitive. In other words, the collection of all theta characteristics of X,which we will denote by S.X/, is an affine space for the collection of line bundle of order two. Note that the collection of all line bundle of order two, which we will denote by = Z J2.X/, is a vector space over Z 2 of dimension 2g. c 2004 Australian Mathematical Society 1446-8107/04 $A2:00 + 0:00 415 416 Indranil Biswas [2] On J2.X/ there is a bilinear form known as the Weil pairing (see [Mu1, page 183]). The Weil pairing  = Z (1.1) 2 : J2.X/ ⊗ J2.X/ → Z 2 = Z is antisymmetric (hence symmetric as the field is Z 2 ). Note that in [Mu1], the  = Z ∈ Z= Z image of 2 is identified with 1 by sending 1 and −1to0∈ Z 2 and 1 2 respectively. We recall a topological description of the pairing Â2. 1 = Z . / . ; Z= Z/ The Z 2 vector space J2 X is identified with H X 2 . With this identifi-  1 = Z/ . / cation, 2 is simply the cup product on H .X; Z 2 . It is easy to see that J2 X is . /; / ³ . / → identified with Hom H1.X; Z 1 . Indeed, using the natural projection 1 X / . ; Z/; / H1.X; Z an element in Hom H1 X 1 gives a character of order two of the fundamental group ³1.X/. A character of ³1.X/ gives a flat line bundle. Since the above character is of order two, the holomorphic line bundle defined by the corre- sponding flat line bundle is also of order two. By the above isomorphism of J2.X/ . ∼ 1 /; / . ; Z= Z/ with Hom H1.X; Z 1 = H X 2 , the cup product 1 1 2 = Z/ ⊗ . ; Z= Z/ → . ; Z= Z/ = Z= Z H .X; Z 2 H X 2 H X 2 2 translates to the Weil pairing Â2 defined in (1.1). Take ¾ ∈ S.X/, a theta characteristic. Define ! = Z (1.2) ¾ : J2.X/ → Z 2 7→ 0 0 = Z by L dim H .X;¾ ⊗ L/ − dim H .X;¾/ ∈ Z 2 . The bilinear form associated with the quadratic form !¾ in (1.2) coincides with the Weil pairing in (1.1). In other words, for any pair L1; L2 ∈ J2.X/ the identity (1.3) !¾ .L1 ⊗ L2/ − !¾ .L1/ − !¾ .L2/ = Â2.L1; L2/ is valid [Mu2, page 182, (?)]. A function on J2.X/ satisfying the identity (1.3)is known as an Arf function [Na, page 93]. In particular, !¾ is an Arf function. Any Arf function is of the form !¾ for some ¾ ∈ S.X/ [Na, page 100, Theorem 10.1]. We will give an alternative description of !¾ using the notion of index of a vector field on X. 2. Vector fields and Arf function We continue with the notation of the preceding section. Take ¾ ∈ S.X/.Takea meromorphic section s of the line bundle ¾ which is not identically zero. Therefore, s ⊗ s is a meromorphic section of K X .Inotherwords,sˆ := s ⊗ s is a meromorphic one form of X which is not identically zero. [3] Theta characteristics of Riemann surface 417 Now, since sˆ is not identically zero, it defines a meromorphic section of the holomor- =∼ ∗ phic tangent bundle TX. Indeed, since TX K X , we have a (unique) meromorphic section − of TX defined by the condition that the evaluation −.sˆ/ is the constant function 1 on X. In particular, the zeros (respectively, poles) of sˆ becomes poles (respectively, zeros) of −.LetC ⊂ X be the union of all the zeros and poles of −. Since X is a compact connected oriented smooth manifold of (real) dimension 1 ∼ = Z/ . ; Z= Z/ two, using Poincar´e duality we have H .X; Z 2 = H1 X 2 . Using this ! 1 = Z/ isomorphism, the quadratic form ¾ on J2.X/ = H .X; Z 2 defined in (1.2) would = Z/ be considered as a quadratic form on H1.X; Z 2 . ∈ = Z/ Take any homology class c H1.X; Z 2 .Let be a smooth oriented loop ∞ 1 2 | 2 + 2 = } \ (that is, a C immersion of the circle S ={.x; y/ ∈ Ê x y 1 )inX C representing the homology class c. Since the set C of poles and zeros of the section − = Z is finite, such a loop exists. Since c is a homology class with coefficients in Z 2 , if we reverse the orientation of then also it represents c. .−; / ∈ − \ Let index Z be the index of the vector field on X C for the oriented loop . We recall the definition of index. If the vector field − rotates n times clockwise with respect to the tangent vectors of the curve (recall that is an immersion), then index.−; / = n + 1. For any p ∈ S1, the quotient of the (real) + nonzero tangent space T .p/ X −{0} by the multiplication action of Ê can be identified 1 2 | 2 + 2 = } . ; / ∈ 1 with S ={.x; y/ ∈ Ê x y 1 by sending 1 0 S to the tangent vector 0.p/ along the loop . Using this identification, − defines a map from S1 to S1.The above integer n is the degree of this map. .−; / = Z Henceforth, by index we will always mean the image in Z 2 of the above constructed number. Note that each pole or zero of − is of even order. Therefore, although is a loop in X \ C, the dependence of index.−; / on factors through the image of in 0 = Z/ H1.X; Z 2 . In other words, if a loop represents a homology class in the kernel = Z/ → . ; Z= Z/ of the natural homomorphism H1.X \ C; Z 2 H1 X 2 (induced by the \ .−; 0 = Z inclusion map of X C in X), then index / = 0 ∈ Z 2 . If s and s0 are two meromorphic sections of ¾, then we have a one-parameter family ¾ ½ 7→ 0 of meromorphic sections of defined by s½ := ½s + .1 − ½/s ,where½ ∈ C. 0 So, we have s1 = s and s0 = s . Note that index of a vector field along a loop is a topological invariant. In particular, it does not change under continuous deformations of the vector field. Let − 0 be the meromorphic vector field constructed using s0.Nowin view of the above remark that the dependence of index.−; /on factors through the 0 = Z/ .−; / = .− ; / image of in H1.X; Z 2 it follows immediately that index index . .−; / ∈ = Z ¾ Consequently, index Z 2 depends only on and c. In other words, we are justified in using the notation index.¾; c/ in place of index.−; /. Our aim here is to prove the following theorem. 418 Indranil Biswas [4] = Z ! . / .¾; / THEOREM 2.1. The two elements in Z 2 , namely ¾ c and index c , coincide. PROOF. Let XT be a holomorphic family of compact connected Riemann surfaces with a theta characteristic parameterized by a complex manifold T .Inotherwords, XT is a complex manifold equipped with a holomorphic submersion ³ : XT → T and a holomorphic line bundle ¾T over XT such that for any point t ∈ T ,thefiber −1 Xt := ³ .t/ is a compact connected Riemann surface and the restriction of the line bundle ¾T to Xt is a theta characteristic of Xt . The restriction of ¾T to Xt will be denoted by ¾t . A basic theorem due to Atiyah and Mumford says that if the parameter space T 0 = Z is connected then the image of dim H .Xt ;¾t / in Z 2 is independent of t (see [At, page 28, Theorem 1], [Mu2, page 184, Theorem]). In other words, the parity of 0 dim H .Xt ;¾t / remains constant over T , provided T is connected. Now suppose that family XT of Riemann surfaces are equipped with a choice of = Z ∈ a first homology class with coefficients in Z 2 . In other words, for each t T we = Z/ have t ∈ H1.Xt ; Z 2 with the property that for every contractible open subset U of T , the homology class ∼ −1 0 0 = Z/ .³ . /; Z= Z/; t ∈ H1.Xt ; Z 2 = H1 U 2 where t 0 ∈ U, is independent of t0.
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