RIEMANN-ROCH FOR PUNCTURED CURVES VIA ANALYTIC PERTURBATION THEORY [SKETCH]

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1 RIEMANN-ROCH FOR PUNCTURED CURVES VIA ANALYTIC PERTURBATION THEORY [SKETCH] CHRIS GERIG Abstract. In [Tau96], Taubes proved the Rieann-Roch theore for copact Rieann surfaces, as a by-product of taking clever perturbations of the Cauchy-Rieann operator in order to define his Groov invariant for pseudoholoorphic curves. We will do the sae for noncopact surfaces, that is, we will recover the forula for the Fredhol index of a Cauchy-Rieann operator that is asyptotic to nondegenerate asyptotic operators. Contents 1. Setup 1 2. L-flat approxiation 3 3. Spectral flow for the perturbed asyptotic operator 3 4. Localization arguent 6 5. Appendix 6 References 7 1. Setup Suppose that C is a connected Rieann surface with punctures, and that E C is a holoorphic line 1 bundle with fixed trivialization τ near the punctures. Copactify C so that the neighborhood of the punctures are odeled on [0, ) S 1, and for each puncture denote the corresponding circle by γ. Take a Cauchy-Rieann type operator with its asyptotic operators D : Γ(E) Γ(T 0,1 C E) A γ : Γ(γ E) Γ(γ E) for all punctures. With respect to τ : γ E = S 1 C the asyptotic operator takes the for A γ = i t + A γ (t) where A γ (t) is a sooth loop of syetric atrices. Assue all asyptotic operators are nondegenerate (0 is not an eigenvalue) so that D is a Fredhol operator. 1 The technique also works for higher rank bundles: take deterinants and use the Splitting Lea. 1

2 2 CHRIS GERIG Exaple 1.1. In practice, C is a pseudoholoorphic curve in a 4-diensional syplectic cobordis between contact 3-anifolds, with punctures asyptotically approaching nondegenerate Reeb orbits. Here, E is its noral bundle and D is the (noral) deforation operator. Theore 1.2 (Schwarz [Sch95]). The Fredhol index of D is given by ind(d) = χ(c) + 2c 1 (E, τ) + γ CZ τ (A γ ) where CZ τ (A γ ) is the Conley-Zehnder index and c 1 (E, τ) is the relative 1st Chern class. Here is an outline of a novel proof of the index forula, using analytic perturbation theory as in [Kat95]: (1) Construct an L-flat approxiation for all A γ, i.e. a suitably nice asyptotic operator which A γ is hootopic to, such that ind(d) doesn t vary. (2) Prove ind(d) = ind(d + B) for soe B Γ(T 0,1 C E 2 ) whose winding nuber along the ends of C with respect to τ satisfies wind τ (B γ ) = CZ τ (A γ ). (3) Prove ind(d + rb) = #B 1 (0) by taking sufficiently large r R +, this being a concentration principle. We put these steps together, noting that by definition, to get ind(d) = c 1 (T 0,1 C, τ) + 2c 1 (E, τ) + γ #B 1 (0) = c 1 (T 0,1 C E 2, τ) + wind τ (B) wind τ (B γ ) = χ(c) + 2c 1 (E, τ) + γ CZ τ (A γ ) The proof of (1) is given in [Tau10, Appendix]. The proof of (3) is a regurgitation of [Tau96, 7], because the arguent is local in nature. A new contribution is (2), which did not arise in [Tau96] because there were no punctures (hence no asyptotic constraints). Reark 1.3. The Rieann-Roch forula for the case of closed Rieann surfaces follows by additivity of the Fredhol index with respect to gluing the punctured surfaces at their cylindrical ends. The Rieann-Roch forula can be recasted in the following for: the index equals twice the degree of the bundle plus the Euler characteristic of the surface. When we introduce punctures, the definition of the degree (as a relative 1st Chern class) requires soe choice of trivialization of the bundle along the ends of the surface, and a different choice ight change the degree. To get an invariant, a boundary correction ter is needed to copensate this is the Conley-Zehnder index. Reark 1.4. In using the Conley-Zehnder index, we will suppose the circles γ are nondegenerate (possibly ultiply covered) Reeb orbits. Acknowledgeents. The ain idea stes fro a chat with Cliff Taubes on how to extend his transversality results fro closed syplectic anifolds to syplectic cobordiss, and builds off of joint work with Chris Wendl on closed syplectic anifolds [GW17]. I appreciate the and their inspiration. Subsequently, a detailed version of this result (with a different style) now appears in Chris Wendl s book on Syplectic Field Theory [Wen, 5].

3 RIEMANN-ROCH FOR PUNCTURED CURVES 3 2. L-flat approxiation The point here is to reduce everything WLOG to the case that our Fredhol operator D has asyptotic operators with nice explicit descriptions, so that we can do hands-on coputations in the subsequent section. As explained in [Tau10, Appendix], there is a hootopy through nondegenerate operators (with fixed Conley-Zehnder index) fro a given asyptotic operator A γ to an L-flat asyptotic operator, specified below. The hootopy extends to a hootopy of Fredhol operators of constant index fro D to a Cauchy-Rieann type operator with L-flat asyptotics, and we abuse notation by denoting the resulting operator D with asyptotic operators {A γ }. We fix a positive real nuber L greater than the syplectic action of γ. We also fix the trivialization τ so that the Conley-Zehnder indices are 1 for ebedded elliptic and ebedded negative hyperbolic orbits, and 0 for ebedded positive hyperbolic orbits. For ultiply covered orbits we can pull back the asyptotic operator of the underlying ebedded orbit, so we assue γ is ebedded. The result of how an L-flat A γ (t) acts on η(t) Γ(γ E) is given as follows (see [Tau10, Lea 2.3]): (1) If γ is an ebedded elliptic orbit (irrational rotation nuber θ) then η θ η (2) If γ is an ebedded positive hyperbolic orbit (rotation nuber 0) then for soe sufficiently sall positive constant ε. η εi η (3) If γ is an ebedded negative hyperbolic orbit (rotation nuber 1) then for soe sufficiently sall positive constant ε η 1 2 η + εieit η 3. Spectral flow for the perturbed asyptotic operator This section prescribes the asyptotic liits (with respect to the fixed trivialization τ) of the desired section B Γ(T 0,1 Σ E E). All asyptotic operators are assued to be L-flat fro now on. The nonzero coplex nubers C will be identified as a subset of GL 2 (R) via a + bi ( ) a b b a. The perturbed operator D r := D + rb will reain Fredhol of constant index for all r R if and only if each perturbed asyptotic operator A γ,r := A γ + rb γ (t) reains nondegenerate. In other words, A γ,r ust not have any spectral flow as a function of r. Theore 3.1. A γ,r has no spectral flow if and only if wind τ (B γ ) = CZ τ (A γ ), with the additional constraint that B γ (0) / ir when γ is a positive hyperbolic orbit or an even cover of a negative hyperbolic orbit.

4 4 CHRIS GERIG Proof. As the focus is now on a single asyptotic operator, the subscript γ will be dropped. The Reeb orbit γ is an -fold cover of soe ebedded orbit α, and is paraetrized by t R/2πZ. Note that the perturbed asyptotic operators act on coplex-valued sections η(t) L 2 1 (R/2πZ, C), and the study of A γ reduces to the study of A α by pullback via R/2πZ R/2πZ. Suppose α is elliptic with rotation nuber 0 < θ < 1. Then CZ τ (γ) = 2 θ + 1 and ( ) θ 0 A(t) η(t) = η(t) = θ η(t). 0 θ We ay assue that B satisfies wind τ (B γ ) = n Z and takes the for B(t) η(t) = B 0 e int/ η(t), B 0 C. We now look for solutions to A r η = 0. This first-order coplex differential equation is i dη dt + θη + rb 0e int/ η = 0. Use the Fourier expansion η(t) = k Z a ke ikt/ and copare odes to obtain the recurrence relation (θ k )a k + rb 0 ā n k = 0. Swapping k n k gives another recurrence relation (concerning the sae coefficients a k and a n k ), and these two relations cobine to give the constraint r 2 = 1 B 0 2 (θ k )(θ n k ). If r 2 > 0 then there is spectral flow, and if r 2 < 0 then there is no spectral flow. Indexing over l {0,..., 1}, suppose l < θ < l+1, so that CZ τ (γ) = 2l + 1. If n 2l + 2 then there is spectral flow (choosing k = l +1) because (θ k n k ) and (θ ) are both negative. If n 2l then there is spectral flow (choosing k = l) because (θ k n k ) and (θ ) are both positive. If n = 2l + 1 then there is no spectral flow because (θ k n k ) and (θ ) are of opposite sign for any k (if k l then n k l+1 n k, and if k l + 1 then l ). We re done. Suppose α is positive hyperbolic with rotation nuber 0. Then CZ τ (γ) = 0 and ( ) 0 ε A(t) η(t) = η(t) = εi η(t) ε 0 for ε R + sall. Again we ay assue that B satisfies wind τ (B γ ) = n Z and takes the for B(t) η(t) = B 0 e int/ η(t), B 0 C. The first-order coplex differential equation to solve is now i dη dt + (rb 0e int/ + εi) η = 0.

5 RIEMANN-ROCH FOR PUNCTURED CURVES 5 Use the Fourier expansion η(t) = k Z a ke ikt/ to obtain the recurrence relation k a k + rb 0 ā n k + εiā k = 0. There is spectral flow when n 0 (see Appendix 5), but when n = 0 the recurrence relation contradicts itself (unless B 0 is purely coplex). Indeed, k a k + (rb 0 + εi)ā k = 0 which by swapping k k gives another recurrence relation (concerning the sae coefficients a k and a k ), and these two relations cobine to give the constraint B 0 2 r 2 + ( B 0 B 0 )εir + ε 2 = k2 2. This is never satisfied unless B 0 = b 0 i (where b 0 R ) and k = 0, in which case r = ε b 0 (hence η(t) = 1 is in the kernel). We re done. Suppose α is negative hyperbolic with rotation nuber 1. Then CZ τ (γ) = and A(t) η(t) = 1 η(t) + εieit η(t) 2 for ε R + sall. Again we ay assue that B satisfies wind τ (B γ ) = n Z and takes the for B(t) η(t) = B 0 e int/ η(t), B 0 C. The first-order coplex differential equation to solve is now i dη dt η + (rb 0e int/ + εie it ) η = 0. Use the Fourier expansion η(t) = k Z a ke ikt/ to obtain the recurrence relation ( 1 2 k )a k + rb 0 ā n k + εiā k = 0. There is spectral flow when n (see Appendix 5), but when n = the recurrence relation contradicts itself (unless B 0 is purely coplex and is even). Indeed, ( 1 2 k )a k + (rb 0 + εi)ā k = 0 which by swapping k k gives another recurrence relation (concerning the sae coefficients a k and a k ), and these two relations cobine to give the constraint B 0 2 r 2 + ( B 0 B 0 )εir + ε 2 = ( 1 2 k )2. This is never satisfied unless B 0 = b 0 i (where b 0 R ) and k = 2 with even, in which case r = ε b 0 (hence η(t) = e it/2 is in the kernel). We re done.

6 6 CHRIS GERIG 4. Localization arguent Using hootopy theory (i.e. obstruction theory), we can extend the choices of B γ given in Theore 3.1 to a global B Γ(T 0,1 C E 2 ) having nondegenerate zeros. Note that the zeros are bounded away fro the punctures due to the existence of the trivialization τ. Theore 4.1. For r >> 0 and B as above, ind(d + rb) = #B 1 (0). Proof. Using a sequence of cutoff functions to invoke a noncopact version of Stokes theore, there is a Bochner-Weitzenböck forula for Dη + rb η 2 in ters of the quantities r 2 C B η 2 and rr C B η2 (here, R eans the real part). The sae localization arguent as in [Tau96, 7] shows that the positive zeros of B contribute to the kernel of D + rb for sufficiently large r R +, while the negative zeros of B contribute to the cokernel. Roughly speaking, the support of any sequence η r Ker(D + rb) ust concentrate near the (positive) zeros of B. The result follows. Reark 4.2. This proof hinges on the asyptotic behavior of the perturbation object B. If B had copact support or exponentially decayed to zero at the punctures of C, we could have a sequence {(η r, r)} Γ(E) R + with η r Ker(D + rb) and r such that sup C (η r ) runs away towards the punctures. 5. Appendix We supply the issing coputation in Section 3 but only in the case that γ is an ebedded positive hyperbolic orbit and the perturbation is B γ (t) = e it (so n = = B 0 = 1), because the other cases are handled in the sae way. Solving for the kernels of the perturbed asyptotic operators A γ,r is equivalent to solving the following 1st order coplex differential equations for sooth functions η : R/2πZ C {0} defined on the circle, i η t + (reit + εi) η = 0 where ε R + is sufficiently sall and fixed and r R + is a nonzero positive real paraeter. We would like to find the set of all such r for which Ker A γ,r, as well as di R Ker A γ,r, though it suffices to prove the existence of one such r. There are at least three ways to do this. The result is a unique solution to A γ,r η = 0 which occurs around r ε. Two ethods are given in a MathOverflow post: athoverflow.net/questions/280825/solution-to-at-least-one-ode-in-a-faily-of-odes 5.1. Alternate solution: When γ is a (cover of a) hyperbolic orbit, the perturbed asyptotic operator A γ,r has trivial kernel when r = 0 and r = r >> 0 (use the Bochner- Weitzenböck forula to see this). The Conley-Zehnder index is a Z-valued invariant of hootopy classes of such operators, and the net spectral flow of r [0, r ] A γ,r is CZ τ (A γ,r ) CZ τ (A γ ) = wind τ (B γ ) CZ τ (γ) Thus when wind τ (B γ ) CZ τ (γ) there ust exist at least one r > 0 with Ker A γ,r 0.

7 RIEMANN-ROCH FOR PUNCTURED CURVES Previous attept: My original attept was to Fourier expand η(t) = k Z a ke ikt and obtain the recurrence relation ka k + rā 1 k + εiā k = 0 I need a specific collection {a k } C which solves this (for soe r > 0), but I get stuck. What I get at the least is r = εi a 0 a 1 (and subsequently a 0 a 1 ir ). The coefficients a k will blow up as k if not chosen carefully. The recurrence relation cobines with another relation (obtained by replacing k 1 k) to yield a k = 1 ) (iεa k 1 (k 1)ā 1 k = a ( 1 i(k 1) ) (5.1) a k 1 + ā 1 k r a 0 ε ā k = 1 ) ( k(k 1) (iεa k 1 (r 2 + k(k 1))ā 1 k = a k 1 + i ε a2 ) 0 (5.2) iε ε a 2 ā 1 k 1 This is too coplicated for e to predict the next coefficient: We are iteratively adding and conjugating coefficients, flipping their real and iaginary parts via ultiplication by i, and putting sall ε in both nuerators and denoinators. I cannot parse whether these are helpful or harful, and I have no indication what the values a 0 and a 1 should be to have sufficient decay in the nors of the coefficients. References [GW17] C. Gerig and C. Wendl, Generic transversality for unbranched covers of closed pseudoholoorphic curves, Co. Pure Appl. Math. 70 (2017), no. 3, [Kat95] T. Kato, Perturbation theory for linear operators, Classics in Matheatics, Springer-Verlag, Berlin, Reprint of the 1980 edition. [Sch95] M. Schwarz, Cohoology operations fro S 1 -cobordiss in Floer hoology, Ph.D. Thesis, ETH Zürich, [Tau96] C. Taubes, Counting pseudo-holoorphic subanifolds in diension 4, J. Differential Geo. 44 (1996), no. 4, [Tau10], Ebedded contact hoology and Seiberg-Witten Floer cohoology I, Geo. Topol. 14 (2010), no. 5, [Wen] C. Wendl, Lectures on syplectic field theory. Preprint arxiv: Matheatics Departent, Evans Hall, University of California, Berkeley CA 94720, USA E-ail address: cgerig@berkeley.edu