The Non-abelian Hodge Correspondence for Non-Compact Curves

Transcription

1 1 Section 1 Setup The Non-abelian Hoge Corresponence for Non-Compact Curves Chris Elliott May 8, Setup In this talk I will escribe the non-abelian Hoge theory of a non-compact curve. This was worke out by Simpson in the paper Harmonic Bunles on Non-Compact Curves, an as such almost everything I say here can be foun in that paper in more etail. Let X be a smooth non-compact curve, with smooth completion j : X X, an S = X \ X a finite set of punctures. For simplicity, I ll assume we have only a single puncture S = {s}. To prouce a non-abelian Hoge corresponence in this setting, we ll nee to consier filtere analogues of flat an Higgs bunles, an impose tameness conitions on the behaviour of the Higgs fiel or connection with respect to this filtration. One way to think about objects like Higgs an flat bunles on non-compact varieties X is as singular objects living on a smooth completion X, with singularities supporte along the ivisor X \ X. The iea of tameness or regularity is to control how baly behave these singularities are allowe to be. In what follows, let U be a small contractible neighbourhoo of the puncture s in X. Let z be a local coorinate on U vanishing at s. Definition 1.1. A filtere vector bunle on X is a locally free sheaf E on X, equippe with a ecreasing filtration of j E by coherent subsheaves satisfying α R E α E α = β<α E β E α+1 = ze α (so in particular the filtration is etermine by α [0, 1).) One coul view this as a collection of extensions of the vector bunle E on X across the puncture s that form a ecreasing left continuous filtration. In this vein, a filtere Higgs or flat bunle is just a filtere vector bunle on X with a Higgs fiel θ or flat connection. For the moment we o not impose any conitions relating this aitional ata to the filtration. If we allowe multiple punctures the efinitions woul be extremely similar, we woul merely ask for a filtration on every j si, E, one for each puncture. Remark 1.2. I coul efine the notion of a filtere local system as the Betti analogue of a filtere flat bunle: wherever I escribe corresponences between flat an Higgs bunles in this setting there is always also a local systems analogue. For brevity, I will omit these etails. How can we use a filtration to control the local behaviour (e.g. growth rate ) of something like a connection or Higgs fiel near the puncture? To see this, we will consier the classical notion of regular singularities.

2 2 Section 2 Regularity an Tameness 2 Regularity an Tameness First, we consier the notion of tameness for a harmonic bunle on X as our main way of controlling growth rates of sections near the puncture. Tameness means that the Higgs fiel θ associate to a harmonic bunle has eigenvalues that have moerate growth near the puncture: λ c r ε > 0 where r = z enotes the istance from the puncture in our stanar local coorinate system: as when we escribe the notion of moerate growth above, this shoul hol in angular sectors about the puncture, ensuring that the λ are single-value functions. The reason the tameness/regularity hypothesis is neee is, in the course of the proof of this fact we nee to establish analytic estimates on the size of these eigenvalues. I ll say a little more about this shortly. This eigenvalue efinition of tameness is equivalent to the following notion: the ata of a harmonic bunle inclues a π 1 (X) equivariant map F : X GL(n, C)/U(n). We say the harmonic bunle is tame if whenever we choose locally a ray ρ extening out from the puncture, we can lift F ρ to a map ρ GL(n, C) whose image grows in norm at most polynomially in 1/r, where r is the istance out from the puncture along ρ. We ll nee to restrict to the tame case to prove a non-abelian Hoge theorem over X: in the course of the proof it will be necessary to establish analytic estimates on things like the curvature of our bunles in certain metrics so that the natural functors between harmonic bunles an filtere flat or Higgs bunles are well-efine. Moerate growth is neee to establish these estimates. So with this in min we can work out what restrictions on flat an Higgs bunles we nee to introuce to correspon to this moerate growth conition. This conition will be calle regularity. Regular singularities arise as a notion in the stuy of systems of linear ODEs. Let K enote the fiel of meromorphic functions holomorphic away from zero, i.e. K is the fiel of fractions of the stalk O = (O C ) 0. More concretely, K = C{{z}}[z 1 ] where C{{z}} enotes the ring of convergent power series at z = 0. Consier a ifferential equation n ( ) n i P = a i (x) i=o for a i K, a 0 0. This is equivalent to a system of linear first orer equations, which we can express in the form n u i(x) = a ij (x)u j (x), for i = 1,..., n, or equivalently j=0 u(x) = A(x)u(x) for A Mat n (K). Two such systems u(x) = A i(x)u(x), i = 1, 2 are calle equivalent if there exists a matrix T GL n (K) such that A 1 = T A 2 T 1 T T 1 which is simply a change of variable conition: the result of setting v(x) = T u(x). Classically, a system of ODEs was calle regular or Fuchsian if its solutions have a property calle moerate growth. To be precise, Definition 2.1. Consier a multivalue solution u Õ of the system u(x) = A(x)u(x). The solution u is sai to have moerate growth if for any sector of the form S = {z = (r, θ) : 0 < r < ε, θ 0 < θ < θ 1 }, there exists a constant c > 0 an an integer j N such that u(z) < c z j

3 3 Section 2 Regularity an Tameness for all z S. If the solutions of the system have moerate growth, we say the system is regular, or has regular singularity at 0. The following is a classical result on regularity. Theorem 2.2. The following are equivalent: 1. The system is regular. u(x) = A(x)u(x) (1) 2. The system (1) is equivalent to one of the form B(x) u(x) = x u(x) where B(z) is a matrix with holomorphic coefficients. 3. The system (1) is equivalent to one of the form where C is a matrix with constant coefficients. u(x) = C x u(x) Theorem 2.3 (Fuchs). The system u(x) = A(x)u(x) is equivalent to a single equation n i=0 a i(x) ( n i ) u = P u = 0. It is regular if an only if ai a 0 has a pole at 0 of orer at most i, for i = 1,..., n. More immeiately applicable to our purposes is the following reformulation. A system of linear ODEs of this form is equivalent to a vector bunle over C equippe with a meromorphic connection: Definition 2.4. A meromorphic connection on C at 0 is a finite imensional vector space M over K equippe with a C-linear map : M M such that (fm) = f m + f (m) f K, m M. Two such meromorphic connections (M 1, 1 ) an (M 2, 2 ) are isomorphic if there exists an isomorphism φ: M 1 M 2 such that φ 1 = 2 φ. In terms of meromorphic connections, this means we can make the following efinition: Definition 2.5. Let (M, ) be a meromorphic connection. We say the connection is regular (at 0) if there exists a basis e 1,..., e n of M over K such that e i = j b ij (z) e j b ij O. z Equivalently, there exists a finitely generate submoule L of M such that z L L, generating M over K. In view of the previous two theorems, we can see that this is equivalent to the regularity of the associate system of linear ifferential equations. Inee, the meromorphic connection correspons to the system u i = j b ij (z) u j z which, by 2.2, is regular precisely when we can choose b ij O, i.e. when there exists a basis of the above form for the associate meromorphic connection. This shoul motivate the following efinition of regularity as imposing moerate growth on flat sections:

4 4 Section 3 Resiues Definition 2.6. A connection on the filtere vector bunle E = E α is calle regular if on the filtere pieces it maps : E α E α Ω 1 (log s) X where Ω 1 z (log s) is the sheaf of ifferentials generate by X z, for z as usual a local coorinate vanishing at the puncture s: the sheaf of logarithmic ifferentials. This means in particular that the action of z z preserves the filtration. Similarly, a Higgs fiel θ on E is calle regular if it maps θ : E α E α Ω 1 (log s). X A regular filtere flat bunle is in particular a holonomic D-moule on X with a regular singularity at the point s. Locally, this is the same as a meromorphic connection with regular singularity, as escribe above (or rather, the algebraic analogue of the analytic efinition given above). These regular objects correspon to tame harmonic bunles on X, an thus we have a non-abelian Hoge theorem in this setting. Theorem 2.7 (Non-abelian Hoge corresponence for a non-compact curve). There is a natural equivalence of categories between stable regular filtere Higgs bunles of egree zero an stable regular filtere flat bunles of egree zero on the curve X. By the egree of a filtere vector bunle, we mean its algebraic egree eg(e, E α ) = eg(e 0 ) + rk(gr α (E s )). α [0,1) The corresponence is essentially the same as in the compact case, an the filtrations on the two sies are the same. One proves that both sies are equivalent to irreucible tame harmonic bunles on the puncture curve. What is interesting an new in this setting is the corresponence also neatly relates the growth behaviour of the Higgs fiel an the connection near the puncture s, in a way we will attempt to make precise. Before we o this, let me say a few wors about a tame harmonic bunle gives a filtration on the unerlying vector bunle, with respect to which the usual constructions of θ an will be regular. If E is a holomorphic vector bunle with a metric K, following Simpson we efine Ξ(E) to be the filtere vector bunle where the germs of Ξ(E) α at s are those sections of E in a puncture neighbourhoo of s satisfying the growth conition e K Cr α ε for all ε > 0. This is where tameness comes in: we use the analytic estimates I mentione earlier to ensure that this functor is well-efine: i.e. that the Ξ(E) s are coherent subsheaves of j E. 3 Resiues So far all we have seen is that in a suitably restricte setting where tameness is impose a non-abelian Hoge corresponence still hols. However, we can actually say something stronger. By efining a resiue that captures the limiting behaviour of the objects near the puncture s, we can actually show that the local behaviour of the Higgs, flat an harmonic bunles near s are relate in a rigi way. Let me give a efinition escribing what ata we have at a singularity, then unpack it to see what it actually implies. Definition 3.1. Let (E, E α, θ) be a regular filtere Higgs bunle. The resiue of E at the puncture s is the pair (res(e), res(θ)) consisting of the grae vector space res(e) = gr α (E) α [0,1) the associate grae of the filtere extension E of E to X given by pushing forwar, an the enomorphism res(θ) = zθ( z ). Here we are using the fact that E is regular to prouce an enomorphism of the grae vector space.

5 5 Section 3 Resiues Similarly, let (E, E α, ) be a regular filtere flat bunle. We can similarly efine its resiue to be the same grae vector space, but with enomorphism res( ) given by the action of z ( z ) on the associate grae. More explicitly, this is the resiue evaluation given by taking an evaluating z/z 1. In both cases we are essentially just killing the z z gr( ): gr(e α ) C z z part of the stalk of E Ω(log z). Let me try to explain how to quantify this resiue ata an how it relates to the local behaviour of the Higgs fiel or connection by means of a a simple example. Example 3.2. Suppose E is a line bunle on X with filtration E α. Then one can fin a point j [0, 1) such that for all ε > 0 E j+ε E j, i.e. the filtration jumps at j. Since E is a line bunle this point is unique. So res(e) = E j. If we have a regular Higgs fiel or flat connection on E, its resiue is an enomorphism of a one-imensional vector space, i.e. a number. Thus our resiue ata consists of two numbers: 1. a real number j [0, 1): the jump. 2. a complex number λ C: the eigenvalue. What o these numbers say about the Higgs fiel or connection? First of all, recall where the filtration E α came from, from the point of view of the harmonic bunle. A germ of E in a puncture neighbourhoo of s being in E α but not E α+ε meant that in the harmonic metric, it grew at least as fast as r α, but not as fast as r α+ε for all ε > 0. So the jump of our bunle shoul tell us about the growth rate of local sections of E in the harmonic metric. In a more complicate situation: a rank n vector bunle, we can escribe the ata we have as follows: the jumps in the filtration i.e. the points in [0, 1) at which gr(e) is supporte give a partition of gr(e) into vector spaces E j of sizes summing to n. This ecomposition reflects the possible polynomial growth rates of the sections of E in the harmonic metric. Each of these pieces amits a further ecomposition into generalise eigenspaces of the enomorphism res(θ) or res( ), so the pairs (j, λ) escribe a labelle partition P j,λ of n. However, there is still more information. On each such piece, the enomorphism acts like a iagonal matrix plus a nilpotent, e.g. λ λ λ This nilpotent part contains further information, which can be escribe in a canonical way by a weight filtration. But before we escribe this, let me escribe how the coarser part the jumps an eigenvalues of the resiue relate uner the non-abelian Hoge corresponence: essentially they are preserve, up to a permutation which one can explicitly compute. This is what I meant earlier when I mentione rigiity of the non-abelian Hoge corresponence. Example 3.3. Lets consier the example of the line bunle once again: so there is a single block P j,λ labelle by a real an a complex number in each of the Higgs an flat settings. The corresponence is: Higgs Flat j j 2b λ a + ib j + 2bi This is a reasonably simple computation on the growth rate of holomorphic sections of the line bunle in the stanar metric, carrie out in section 5 of Simpson s paper. I ll say a few wors about the weight filtration on the nilpotent pieces, an their preservation uner the corresponence.

6 6 Section 4 Some Remarks on the Higher-Dimensional Case Definition 3.4. On such a P j,λ, there exists a unique increasing exhaustive filtration W k, k Z the weight filtration such that the enomorphism res(θ) or res( ) lowers weights by two, an so that the weights of each Joran block are arrange symmetrically about the origin. Example 3.5. The crucial example is the harmonic rank 2 bunle over the puncture isc corresponing to the variation of Hoge structure W given by the stanar representation ( ) of SL 2. In this case we can choose a 0 1 basis such that the resiue enomorphism N = res(θ) has form. Notice this is C 0 0 invariant, so oes inee unerly a variation of Hoge structure. Here there are two non-trivial pieces of the weight filtration: W < 1 = 0 W 1 = ker(n) = W 0 W 1 = W. The reason this is crucial is, by taking symmetric powers of this VHS we can prouce nilpotent Joran blocks of arbitrary size. Then twisting by line bunles an taking irect sums allows us to prouce all possible resiue ata as arising from harmonic bunles. 4 Some Remarks on the Higher-Dimensional Case How oes this generalise to the case where X is a higher-imensional variety, so we have to consier singularities in the neighbourhoo of a ivisor, not just an isolate point? Let me begin to iscuss what kin of thing one can say, to be elaborate on in subsequent talks. Firstly, there is a natural extension of the concept of regular singularities for D-moules on higher imensional varieties, generalising the notion we have iscusse. A flat bunle on X is regular if for every morphism f : C X where C is a smooth algebraic curve, the pullback of the flat bunle uner f is regular, in the sense we have alreay consiere. This further generalises to all holonomic D-moules by a classification theorem: a composition series is given where the composition factors are simple D-moules, we say the D-moule is regular is these simple D-moules are minimal extensions of flat bunles on affine open subvarieties. Simpson prove that on a general Kähler manifol not necessarily compact there is an equivalence of categories between harmonic bunles an pure polarize twistor structures of egree zero. In the tame case, one can say something stronger: Theorem 4.1 (Tame harmonic bunles are equivalent to twistor D-moules). Let j : X X be a smooth completion of X, with complement N = X \ X a normal crosssings ivisor. Then we know flat connections (E, ) on X are equivalent to so-calle variations of pure polarizable egree zero twistor structures on X a generalisation of the notion of a variation of Hoge structure. But in fact, if is regular on N, then the corresponing variation of twistor structure extens to a twistor D-moule on X. Generalising this, Mochizuki prove that on a quasiprojective variety, there is an equivalence between polarizable twistor D-moules of weight zero an semisimple regular holonomic D X -moules: this is a generalisation of the above theorem, which consiere regular holonomic D-moules which restrict to flat connections on the complement of a normal crosssings ivisor. Twistor D-moules can be thought of as something like λ- connections, parameterise by λ P 1, an the functor here is simply restriction to λ = 1.