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r The Schwarzschild Solution of Einsten s Equations M3920 Riemannian Geometry Fall 03 Haydar Oguz Erdin Conventions and Preliminary De nitions Mg will denote the pair consisting of the manifold M and the metric tensor g on M g will have signature l111 will be always torsion free and be compatible with the metric Whenever an upper index and a lower index is repeated it is understood that we have a sum over all the range the repeated indeces belongEinstein s summation convention A vector will be called timelike null or Spacelike if it s length is negative zero or positive respectively A curve will be called a timelike curve a null curve or a spacelike curve if its tangent vector at every point on the curve is timelike null or spacelike respectively 3 will denote real numbers will denote the set of smooth covariant k tensor elds and the Lie derivative of a tensor eld 139 with respect to a vector eld X will be denoted by x139 1Vacuum Schwarzschild Solution of Einstein s Equations In general relativity spacetime is a connected smooth 4manifold M on which a Lorentz metric gab is de ned such that the curvature of M is related to the matter distribution by Einstein s equations 1 1 1 Rab 81rTab EgabT Here Rab is the Ricci tensor and Tab is called the stressenergy or energymomentum tensor which describes the matter distribution in spacetime T is the trace of Tab Shortly after Einstein introduced the equations 11 Karl Schwarzschild found a solution ie a metric which models the spacetime outside a static spherically symmetric massive body In this case Tab and T are zero so that gab is a solution of Einstein s vacuum equation 1 2 Rab 0 391 hissolution models totagoodsapproximationftheloealgeemetry of spacetime in the solar system First we have to make the notions above precise De nition A spacetime is stationary if there exists a one parameter group of isometries 4 whose orbits are timelike curves A spacetime is static if it is both stationary and there exits Ia spacelike hypersurface E which is orthogonal to the orbits of the isometry A spacetime is spherically symmetric if its isometry group contains a subgroup isomorphic to the group 303 and the orbits of this subgroup are two dimensional spheres For physical interpretation of these de nitons see Wald ch61 De nition A vector eld X e T M is called a Killing eld if its one param eter group of diffeomorphismsor equivalently its ow are isometries The metric g is called invariant under the ow I of X if Vp e M 3 94419 9 In terms of this de nition X is a Killing eld if g is invariant under the ow of X and a stationary spacetime is a spacetime which admits a timelike Killing vector eld We ll show the following 39 Lemma 11 X is a Killing vector eld gt xg 0 This is a particular case of a more general theoremcf Lee ch18 for the follow ing We say that a tensor eld 139 is invariant under a ow 0 if 1393 9 7000 7399 Lemma 12 Let M be a smoothmanifold X a vector eld and 0 be the ow of X Then for any T E TquotM d n It Eltt00t mm 9 xT Mo 12 Proof The change of variables tto 3 yields 1 d d Ett00t mm Elao9 o79p d Els09to 0To9op a d 0 0 3003T9a9t01 9 EXT0eo p39 E Proposition 13 Let M be a smooth manifold X e T M and let 739 E 139quot M Then 139 is invariant under the ow 9 of X ltgt Lx r 0 Proof Let p E M Then 931704 Tp EXT 936 t lim90 t OOt 39 Gonvetsely assume xr nG39 39 Goasider 4T 6 c w TkTpM de ned by Tt 9339 7905 Lemma 11 gives d T39t 3 7902 9 x7 op 0 Vt E 6 6 Since 70 139 this shows that 92 Tom 7r Cl A 39 Lemma 11 follows immediately from this proposition For any X Y Z 6 T M we have Exng Z 9VYX 9Y VzX since Vg 0 Hence for a Killing vector eld X Lxg 0 implies 14 gmx 906 m o This equation is called Killing s equation Using this we have the following important lemma Lemma 14 Let X e T M be a Killing eld and let 7 be a geodesic with tangent 7 Then the angle gX7 is constant along 7 Proof 7 9X391 V1 9X YI 9V 7 7I 9X V Y39V 3 But ti39y 0 since 397 is a geodesic and gVX 1 0 because of 14 Hence 39y gX39y O which implies gX39y is constant along 397 E In physical terms Lemma 14 will give us laws like conservation of energy or conservation of angular momentum Using these symmetries in the above de nitions a coordinate patch can be con structed such that the metric is given by cf Hawking Appendix B 15 d32 1 3dt2 1 31dr2 r2d02 sin20d 2 As can be seen from the metric the topology covered by this patch is 32 x 0 2m U 2m 00 x 5392 0 and 5 are the usual spherical coordinates on 32 s 3 is the timelike Killing vector eld and the surfaces obtained by xing t and r are the orbits of the 803 group I is de ned intrinsically as 7 where A is the area of these orbits Here r need not represent the radii of these spheres from a center because a sphere need not have a center for example R x 3 Indeed we ll see that the points r 0 correspond to a singularity in our case m represents the total mass of the system This can be seen by comparing this expression with the Newtonian limitcf JampMh uh m m k g amp EMnmam 15 Then what is left is the at Lorentz metric In general relativity particles whose motions are determined only by gravity follow geodesics Particles without mass such as photons follow null geodesics and particles with mass follow timelike geodesics It is crucial to one s understanding of the geometry of a given metric to know what these geodesics look like Although we ll solve the geodesic equation for the metric 15 explicitly only for radial ingoing null geodesics for r gt 2m as a reference here are the Christoffel symbols and the full geodesic equation cf Ohanian pages 283287 2m m2 Flrritl r39 17 rt lt1 2m m2 tr quot1 T 17 2m r50 T1 T 2m r 39 20 1 44 rsm r 1 r29 Pgr 1 1 3 sin00080 1 FF i P34 9 cotO All other such gammas are zero The geodesic equations are 2 16 tquot 1 f r t39 o m 2 m 17 lt1 27x3 W 1 3gt12r39gt2 r1 331 0392 rsin201 32 2 0 18 0 gr O cosBsinOMJI2 0 and 19 4quot gr qi39 2coto 39939 o where f indicates g 1 aphotoninitially4neving4iirectlytotheEcenter ofthe sun for which 0 In other words we re assuming initially 0 0 0 and 0 Then 1 8 and 19 imply that 9 139 0 and 394 139 0 VT so that 9 325 and 0 VT The equations for t and r then become 2m 2m2 110 t 1e r 1 1 rt 0 and 2 m 111 1 9 393 W 1 2gt139E2r392 0 Solving these equations directly is dif cult but we can use the conserved quan tities from lemma 14 Let 39y be any geodesic then the timelike Killing eld amp gives using lemma 14 I 13912 quotE Itt quottgt quot1 nz where E gt 0 is a constant called energy Similarly the rotational symmetry gives 113 L rza39 where L is a constant called angular momentum In general relativity these two equations represent the conservation of energy and angular momentum For our geodesic L0 and since it is a null geodesic g7 7 0 or more explicitly A39B39 It co AAAAA AAAA AAAAAAAAAAMA O AAAnnnaagaAAAAA xx Figure 11cf Misner page 825 n1r 7 t391Egrw0 We ll show that curves whose t and r coordinates satisfy 112 and 114 also satisfy 110 and 111 Multiplying by 1 321 and using energy conservation the above equation becOmes 39 115 E2 r392 0 Hence 1quot le This implies that r 0 Putting this in 111 We obtain equation 1 14 Hence it satis es 141 Also differentiating 1 12 gives 2m 2m 0 7 2 lr39t39 lquot Ty 39 which is precisely the geodesic equation 110 Thus solutions of 114 that satisfy E142 with r E L 0 and 0 1239 are radial ingoing null geodesics Now 114 gives dt 2m 1 116 2 ll r Integrating this we have 1 17 t in constant where r r 2mln n 1 is the ReggeWheeler tortoise coordinate and 1 refers to outgoing or ingoing radial null geodesics A similar analysis for r lt 2m shows that in the tr plane our ingoing geodesic looks like the dotted line in gure 11 as a reference a timelike geodesic is also Here as can be seen from the graph t gt 00 as particles approach r2m That this behaviour is due to the coordinate system chosen and not of the spacetime will be seen in the next section Let s interpret our results so far in physical terms First of all the t coordinate is not the time measured by a clock put near the mass m It is the time measured by an observer far away in the asymtotically at part of the spacetime The surface r2m is called an event horizon We ll see that it acts as a oneway membrane Anything with r coordinate bigger than 2m can pass this point but nothing inside 139 lt 2m can go outside A region of spacetime inside an event horizon is called a black hole Note that for r e 0 2m 2 118 9 1 5 gt 0 and 119 gr 1 3331 lt 0 Hence r is a timelike t is a spacelike coordinate inside a black hole and the metric is time dependent shown that inside the black hole the metric is time de endent rd r choice of coordinate systemsT Hecall that for photons dr 2m 120 3 i1 r Hence for an observer using tr6 coordinates a photon at r2m is standing still The gure 12 from Ohanian page 314 shows the path of an ingoing photon and the shape of various light cones Since everything with nonzero mass is slower than a photon the trajectories of the particles with positive mass must lie strictly inside the lightcones The reversal of the orientation of cones inside the black hole represents the timelike character of r in the black hole Hence it is clear from the gure that nothing can come out of a black hole Ht 15 ms pun O Figure 12cf Ohanian page 314 7 As r 2m the fact that t 00 can be interpreted physically as follows Assume an astronaut sends signals with a xed period according to her clock to an observer far away in the asymtotically at region Then as she approaches r2m the far away observer will measure an ever increasing period Moreover according to the observer she will never reach the event horizon This is one manifestation of the time dilation phenomenon In general relativity the arclength parameter for timelike geodesics is interpreted as the time measured by the clock which moves along the geodesic and is called the proper time For null geodesics there does not exist such an interpretation Once inside the black hole it can be shown thatcf Ohanian page 315 amp 316 the proper time to reach r0 is nite For a black hole with the mass of our sun it is approximately 10395 seconds 2KruskalSzekeres Coordinates for the Swarzschild Solution We now come to a very important requirement of general relativity on the pair Mg First we have the following de nition De nition The pair M g is called an extension of M 39 9 if there exists an isometric imbedding V M gt M The requirement is that the model M 9 should not be extendable This is a requirement to ensure that M 9 contains all the nonsingular points of spacetime quotthat re coordinate patchquottOfc l39iswextendible con de the following 39 transformation r t 1 12 r4m h I utr 2m e cos 4m vtr 27 112e394msinhZ for r gt 2m q W 1 grayzernmsmng II I39 12 r4m t vtr 1 2m 6 cosh4m for r lt 2m and 0 4 is mapped to 0 4 Motivation for nding such a transformation is given in Appendix A For now we ll focus on the image of the tr plane and the form of the metric under the transformation above We ll show that it is an extension A straightforward calculation shows that the metric becomes 3 39 r 21 ds2 32m e dv2 dug r2 102 sin20d 2 where r is de ned implicitly by quot yaL2 22 2m 1e u 1 consider rst In Since cosh4 gt sinh Vt E Ru gt lvl Hence in the a ar92thatesi9n labala 139199212951 90th9 Figure 21 When rA a constant we have u constantcosha v constantsinhZ n Hence u2 v2 A2 so that rconstant lines are mapped to hyperbolas with asymptotes v in in region I on the right in gure 21 If tB a oonstantthen oashz so that tconstant lines are mapped to straight lines through the origin A similar analysis with lt1 will give gure 3922 It can be shown that r de ned by 2 2 is positive and analytic Note that r gt 0 implies u2 v2 gt 1 Note also that r2m line does not belong to the tr plane and hencequot the diagonals v 2111 are not in the image of the transformations although from the metric we see that the only problematic points in the uv plane are the points in the image of the r0 line I 39 A a 1 g2 3 39 3 I 439 o II l I Iquot N l ix l gt 5 lt 5 2 39 W l Figure 22cf Ohanian page 318 9 Now consider the new pair M g with the new coordinate patch vu0 where the metric g is 21 If we can show that the transformations above are isometric imbeddings then M 9 will be an extension of M g with only possible singularities at r0 since 939 is well behaved on the whole uv plane except at the origin 39 To see this it suf ces only to consider the pushforward obtained from ltIgt1 the case gt11 is similar Let Q 1 be the matrix of the pushforward of 1 Then to show that it is an isometry we have to show that 23 9 1T939 1 I where f represents the matrix of the map f For r gt 2m 80 81 39 a cosM Asinhhtm zsz39nh Acosh t m Where B 1quot 28 599514 3985 Let C Then the right hand l sidle of equation I 2 3 becomes GA2 0 2 gt 0 r f which is indeed What about the u lt v part of the uv plane The metric behaves nicely in that region and indeed there exists a set of maps from the tr plane to this part of the uv plane cf Misner page 833 I utr 421 112e394mcoshz III vtr 4 112e394msmhf for r gt 2m utr 1 ign Pzer msinh I IV 739 12 r4m t t q c v r 1 2m 6 cosh4m for r lt 2m Doing a similar analysis as in the case of ltIgt 1we obtain gure 23 cf Misner page 834 39 Several remarks about this maximal Schwarzschild geometry are in order First a spacetime is called maximal if every geodesic is either de ned on all of SR or else ends only at a singularity of the spacetime In general relativity only maximalspacetimes are considered39since physically it makes no sense for stable particles to vanish or to appear out of nothing In maximal spacetimes particles are created or annihilated only at singularities The spacetime on the right in gure 23 is maximal and two tr patches are required to cover the whole uv plane A ItcanheshoumthattheKmskalextensionisthemniqueanalytiaand locallyinexw tendible extension of the Schwarzchild solution The singularity at r0 is physical 10 J u SChgtfzsh39ld KruskalSzckcrcs 3 Figure 23cf Misner page 834 since as we will show at Appendix B as r r0 the scalar R dRabc diverges By considering the new pair M 9 with 9 given by 21 we obtained an extension singularity isfor points whose r coordinate is zero The topology covered by this patch is the part of the uv plane in gure 23 between the hyperbolas v2 u2 1 where each point represents a two dimensional sphere Note that since the uv part of the metric is of the form 32m3 132 25 r d112 du2 null geodesics follow d245 lines and hence timelike geodesics must have higher slopes This means that the two asymtotically at universes regions I and III in gure 23 can not communicate with each other On the other hand as Ohanian page320 remarks we can see from the gure that if astronauts from both uni verses jump into the black holeregion 11 then they can meetembrace and die together For reference gure 24 shows and compares some timelike and null geodesics in tr and vu planescf Misner page 385 L 43339 M at t 00 AAAA AAAAAAAALLA AALALAAAAAAAA Figure 24cf Misner page 835 KruskalSzekc res 11 From gure 24 we see that r0 corresponds to hyperpolas v l l 112 These are spacelike surfaces To better understand the physical meaning of the two regions I and III note that for time 12 gt 1 or v lt 1 the regions have two separate singular ities while at times 1 t1 the two asymptotically at spaces are joined together Such a region is called a wormholeor EinsteinRosen bridge or a Schwarzschild throat The regions I and III can be interpreted as either two different universes or as two far away regions of the same universe This ambiguity arises because Einstein s eld equations x only the local nature of the universe whereas the in terpretation above depends on the global topology To better understand this x v0 0 g and embed the remaining Schwarzschild metric dr2 ds2 5 1 gm r2d 2 r on a curved surface in at 923A quick way of doing this is to write ds2 dz2 dr2 r2d 2 where r 6 z are cylindirical coordinates and then solve the differential equation 32 1le 1 to obtain the surface One obtains gure 25aThe regions u gt 0 and u lt 0 corresponds to regions I and III In gure 25b 39we see that they can be the far away regions of the same universe 2 R u n u n F N l P 5 Figure 25cf Misner page 837 Appendix AzKruskal Coordinates Since geodesics have a close relationship to the geometry of the metric it is natural to use their af ne parameters for constructing coordinates Recall that the radial null geodesics of tr0 coordinates satisfy t in constant where n r 2mlnlf39 1 De ne the coordinates T R by A l T t n R t ran which are known as EddingtonFinkelstein coordinates The metric becomes A3 132 1 3deR where r is now de ned implicitly by r R T A4 r 2mln 1 r Using A4 we can write A3 in the form A 5 ds2 me ir w Calculating the af ne parameter along null geodesics of this metric gives A 6 U 3 A7 V V ez The metric becomes 3 57 ds2 2m re dUdV Finally setting A8 v U V A9 21 V gives the form that we have been using 32m3efv7r r 32 dv2 duz 1 2d6l2 sin20d 2 Appendix B Curvature Tensor Coef cients of 14 We ll calculate the coef cients of the curvature tensor of the more general metric B1 d32 frdt2 lnrdr2 1392d02 sin29d 2 by rst de ning B 2 9 RjHdk A 039 where Rjk is found using a basis 6 such that a e and then using Cartan s equations B 3 do w A aj i 39 i I For an explicit explanation of this method see Misner section 146 Now choose the following orthonormal basis lforms 0 0 dt 1 dr a2 rd0 03 rsin9d Then 0 I do mdr A dt f 1 o 0 A0 2f xT dal o 1 d02 drAd0 01 A02 rxI z ale3 sinOdr A dd rcosOdO A d 1 01 A a3 lcattle2 A 03 r h T Then B3 implies o 1 f o w w or 1 2 1 2 w w a l 2 r h wa w1 1 03 1 3 r h wg w3 1cot003 all others being zero Hence 0 r o I39 o dwl zf dra 2f da fl f 1 1 o aAa2f l WW f f39 a a 2mgt2m21 1A 2 A01 A 0 l l 1 dw2 39 0102 d0102 1 r hH r22 1 1I 1 1 2 BO 102 cot0 dw Ba1 a3 a2 A 03 r2 h Then equation B4 implies Q A01 A00 S20 f 0 A02 2fhr 9 2 f 0 A03 2fhr Q Bala2 Q Ball03 1 1 2 2 3 AO39 All others being zero So R901 Av f f R0303 2 fhr 112 B R l3 quot Bv These imply 39 r f39 2 ROIOI Aquot2f 2fW IH 1f39y 2W7 2f R0202 t r 30303 30202 R1212 B 51 1 1 R2323 21 Then f39 ROD Afhr3 R11 A ZB f39 39 1 1 R22 fhr Br21 h f 1 1 R33 fhr Br2l h R2A 4B 2 1 l fhr r2 h For the original Schwarzschild metric we then have 2 RabcdRabcd 48 139 NOTES ON HODGE THEORY ON KAHLER MANIFOLDS BENNI GOETZ 1 INTRODUCTION Differentiable manifolds are familiar objects de ned as topological spaces locally diffeomorphic wit R A natural extension of the concept is to e ne a complex manifold to be a topological space locally diffeomorphic with C The transition functions of coordinate charts determine the local properties of the manifold In the case of a real differentiable manifold we required the local diffeomorphisms to admit smooth transition functions In the case of complex manifolds we ask that the transition functions be holomorphic In the course of analyzing the structure of a complex manifold we notice that the complex coordinate maps de ne an action of the imaginary numbers on each tangent space The complex structure of the complex manifold induces endomorphisms on the real tangent spaces of the underlying real differentiable manifold For each point z E M we write the induced endormorphism on Tch as J75 lf J75 varies smoothly along the manifold we can de ne J to be a l ltensor eld on M which acts as J75 at each point z E M This tensor eld has the crucial property that J 71 and is called an almost complex structure Given a complex manifold we get an almost complex structure on the underlying real manifold as above We ask the converse question given a real manifold does it admit an almost complex structure J and is it the underlying manifold for a complex manifold that induces J The answer in general is no but easily checked conditions allow us to determine when this is the case Every real manifold admits a Riemannian metric and a unique connection which is compatible with this metric Given a manifold with an almost complex structure we can ask how the extra structure of the J tensor eld interacts with a metric and connection Exploring the dependencies leads us to the de nition of a Kahler metric and Kahler manifolds A Kahler metric on a complex manifold is roughly analogous to a Riemannian metric on a real manifold a Kahler holonomy group is a subgroup of the unitary group the 77 complex version77 of the orthogonal holonomy group of a Riemannian metric Kahler metrics are pretty restrictive and impose a rigid local structure on the manifold The rigidity of a Kahler metric gives us extra identities in Hodge theory and gives us two useful decompositions of the comohomology of a Kahler manifold the Hodge and Lefschetz decompositions Both of these depend on the class of harmonic forms the kernel of the LaplaceBeltrami operator These introductory notes were written for my own bene t and to satisfy a class requirement The theorems proofs and explanations are taken liberally from Bryant 1 Grif ths amp Harris 2 Joyce 3 and Kobayashi amp Nomizu Some paragraphs and sections are lifted verbatim Neither the math nor the presentation are new Date December 4 2003 2 BENNI GOETZ 2T ALMOST COMPLEX STRUCTURES ON DIFFERENTIABLE MANIFOLDS 2T1T Coordinates And Cotangent Space On CC Let 21 T T T 2 be a coordi nate system on CnT Then we can write 21 II V71 yi for each i The cotangent space to a point in C is spanned by dzidyi but we7ll nd it more convenient to work with the following change of basis dzi dzi V71 dyi d i dzi 7 V71 dyiT The dual basis for the tangent space becomes 6 1 E E E 1 E E 1 77 ixili 77 7 V717 T l l 62 2 an 6 62 2 611 6 In terms of these bases7 the formula for the total differential of a complexvalued function is df 7 842 7 2832142 We write the rst term as 8f and the second as For a function f C A C the CauchyRiemann equations are equivalent to g 07 as you can check directly from This is equivalent to the statement that 5f 0T 2T2 Manifolds And Their Tangent Spaces An ndimensional real manifold is a topological space that is locally homeomorphic with RnT We de ne complex manifolds by analogy De nition 21 An ndimensional complex manifold is a 2ndimensional di er ential manifold together with an open cover Ua and coordinate maps 45 Ua A C such that the maps 4150 o abgl are holomorphic on 453Ua U3 C C Note the important fact that an n dimensional complex manifold M is simulta neously a 2ndimensional real manifoldT De nition 22 An almost complex structure is a 17 1tensor eld on a di eren tial manifold where J is an endomorphism oprM for each p E M and J2 71 An almost complex manifold is a di erential manifold together with an almost complex structure A short argument shows that an almost complex manifold is orientable and must have even dimensionT Every complex manifold M has a natural almost complex structure Suppose p E M and 21 T T T 2 is a local complex coordinate system for p7 where 2139 11 V71 yiT Then ail 321 is a basis for TpM7 and we de ne a a a a Jltazgt 6y Jltaygt an We re left with an obvious question when is a manifold M with an almost complex structure J actually a complex manifold The answer lies in a 12tensor on M called the torsion of J7 also called the Nijnhuis tensorT We de ne the torsion by NX Y 7 2JX JY 7 my 7 JX JY 7 JJX YT A theorem by Newlander and Nirenberg shows that J actually de nes a complex structure on M iff the torsion is 0T NOTES ON HODGE THEORY ON KA39HLER MANIFOLDS 3 We turn now to the tangent spaces of an almost complex manifold For real man ifolds we have one commonly encountered tangent space but for complex manifolds we have three i Since an almost complex manifold M is an evendimensional real manifold we can choose coordinates 11 y1i i In yn around a point p 6 Mi Then 3 3 is a basis for the usual tangent space TpM and we can write 6 E W R l This is the real tangent space socalled because it is a real vector space and when we want to distinguish it from the other tangent spaces we will write it as TpRMi Working with complex manifolds it will sometimes be convenient to have tangent spaces that are complex vector spaces To obtain them we simply complexify the real tangent spaces so TEM TIBM ER C With a change 39 R 3 3 3 3 39 of bas1s for T1 M from 311 m to 371 37 we can wr1te a a C 7 if TPM Clazi ayZl 6 6 Notice that if the underlying real manifold of M is 2ndimensional then the underlying real vector space of TEM is 4ndimensiona iii Since J2 71 J is nondegenerate and its eigenvalues are 7H Thus we can split TEM Tplgt0M EB TEAM where Tplgt0M is the F eigenspace and T19gt1Mis the 7H eigenspacei A proposition tells us that Tplgt0M Xi JX and TEAM XJr JX for X a real tangent vector 239e X E TpRMi Thus we can also write our new tangent spaces as 6 6 1gt0 7 0gt1 7 T1 M aZi Tp M Ca ii Tplgt0M is called the holomorphic tangent space and TEAM is called the antiholomomhic tangent space C 39 The spaces Tplgt0M and T19gt1M are isomorphic and the isomorphism between them is called conjugation Thus TEAM is sometimes written Tg OMi We therefore know that both the holomorphic and antiholomorphic tangent spaces are 2ndimensional as real vector spaces and thus we have an Rlinear isomorphism between TPRM and T1gt0M 17 We call a map f M A N between almost complex manifolds a holomorphic map if it maps the holomorphic tangent space at p E M into the holomorphic tangent space at E N ie f TgtOM C T 2N for all p 6 Mi If f is just a map from C to C then the holomorphic tangent space at any point 20 is C Bilbo and 8 8 fdz id TE 62 df 4 BENNI GOETZ so mapping the source holomorphic tangent space into the target holomorphic tan gent space is equivalent to E 0 which is the usual condition that f be holomor phici 23 Differential Forms And Cohomology We want to understand differential forms on complex manifolds Welll be looking at complexvalued forms on complex manifolds and they will take vectors from the complexi ed tangent space TfMi Since we can split the tangent space as TEM TzlpM EB T291 our complex differ ential forms will re ect this structure First de Rham cohomologyi De nition 23 If M is a di ferential manifold let A M denote the complex valued nforms on M and let Z M denote the complexvalued closed nforms on M Then the nth de Rham cohomology group is Z M dAn M lf Hg2 M R denotes the usual realvalued de Rham cohomology then Hg2 HER M R C Now we assume that M is a complex manifold The holomorphic and antiholo morphic tangent spaces give us a decomposition of the complex nformsi We write TEX for the complex cotangent bundlei With some linear algebra A TEYM Tgt ltMgt ea T2gtlltMgt EB Aijgt M qT2gt1Mi pq HiRltMgt This gives us a decomposition at each individual tangent space We de ne AMM as e A M 14152 e p T3gt M quot T2gt1M for all 2 e M so A M EB AWM pqn If 45 E ApgtqM we say 45 is of type p q and then dab e APTgt ltMgt quot T2gt1ltMgt A TERM 6 Ap1gtqM GB Apgtq1Mi We de ne operators 8 ApgtqM A Ap1gtqM 3 ApgtqM A Apgtq1M by projection so d a a If 45 ZIpq abudzI d2 where I and J are multiindices then writing out 8415 and 8415 explicitly aqs Z B Wdzi A dz A d2 82 IJi gqs Z 8 d2 A dz A d2 IJj NOTES ON HODGE THEORY ON KA39HLER MANIFOLDS 5 Note that since partial derivatives commute7 32 0 Thus we have a 3cohomology theory Let Zg qM be the 3 closed pqforms De nition 24 The p97 Dolbeault cohomology group is the quotient Zg q M ngM 9417117 1 M 3 METRICS AND KAHLER MANIFOLDS 31 Hermitian Metrics Let M be a complex manifold of dimension n De nition 31 A hermitian metric on M is a positive de nite hermitian inner product at each point of M which also varies smoothly over M An inner product h on a complex vector space V is hermitian if it is linear in the rst term7 and conjugate linear in the second7 so haz y a hzy Another way ofkoking at this is to say that h V697 A C is linear on each factor of V697 where V is the conjugate vector space If 21 2 is a local coordinate system around 2 6 M7 and h is a hermitian metric on M7 we can write it as h Zhijdzi dzj where hijz z is a smooth function7 2 is a hermitian inner product7 and hij hji By the GramSchmidt process7 at each 2 E M we can nd an orthonormal basis 012 on T0 for the inner product With this orthonormal basis we can rewrite the metric as dsQ thzwzi dgj 280139 Jw A hermitian metric is complexvalued Looking at its real and imaginary parts splits the metric into a symmetric and antisymmetric part Restricting our metric to the holomorphic tangent space which is isomorphic to the underlying real tangent space7 we get a Riemannian metric and a special differential form on the underlying manifold Explicity7 if 1gon is a unitary coframe for ds2 and if we write goi ai J where ai i are real differential forms7 then d82 291139 V 1 5139 04139 i V 1 5i 291139 804139 Jr z39 85139 V71 2701139 i i 804139 The rst term is a symmetric7 nondegenerate bilinear form7 so Reds2 is a Rie mannian metric on the holomorphic tangent space The second term is an alternat ing bilinear form7 ie a real differential 2form We can consider these as bilinear forms on the underlying real manifold7 using the isomorphism between TlR and Tzl o We call w 7Imds2 the associated l7 lform of the metric In terms 5 BENNI GOETZ of the above 1 W ai i i ai Z 01139 Bi Ll 7 T 280139 8quot Conversely given an associated l lform w we can reconstruct the original her mitian metric Example 32 On C the standard hermitian metric is the usual one given by ds2 Eda d i We call a hermitian metric a Kahler metric if do 0 in which case w is called a Kahler form A complex manifold with a Kahler metric is called unsurprisingly a Kahler manifold Note that since w must be nondegenerate a Kahler manifold must also be symplectic The Kahler condition may seem somewhat random and it certainly doesn7t give much geometrical insight but it is very powerful and very common To give some better motivation and geometrical insight on it llll digress from the main development here and give some alternative formulations Welll start with the holonomy of a Riemannian manifo 32 Kiihler Metrics From The Holonomy Classi cation The fundamental theorem of Riemannian geometry is that on any Riemannian manifold M we have a canonical connection the LeviCivita connection This connection de nes the parallel transport of a vector 1 along a piecewisesmooth curve 7 01 A M If we x I E M and let 7 be a loop at I so 70 7l I then the parallel transport of v 6 T7 de nes an endomorphism on Tch which we denote P7 The composition of two endomorphisms R a P5 P73 is de ned by reparametrizing 76 so that its domain is 01 If we de ne 7 1 01 A M by 7 1t 71 7 t then one can check that P771 is an inverse for P7 Thus the collection of endomorphisms P7 is a group Since P7 is invertible P7 6 GLT M De nition 33 Let Mg be a Riemannian manifold and x I E M De ne a loop at z to be a piecewisesmooth curve 7 01 A M such that 70 7l z The LeviCivita connection de nes an invertible endomorphism R on Tch by parallel transport We de ne the holonomy group of g at z to be Hol g P7 7 is a loop based at I C GLT M If M is connected then there exists a path 739 between any two points Ly E M Taking P7 to be the isomorphism between Tch and TyM given by parallel translation it s clear that if 7 is a loop at y then 777 1 is a loop at 1 Thus if R E Holyg then 1377771 P a R 0 P771 6 Hol M and in particular P7 Holyg P771 Holy Thus up to conjugacy in GLnR n is the complex dimension the holonomy of a Riemannian manifold is independent of basepoint Therefore we can de ne Holg to be the conjugacy class of Holy g in GL n R for any I E There is a subgroup H010 9 of the general holonomy group Holg called the restricted holonomy group which is the holonomy group generated by nullhomotopic loops There is a surjective group homomorphism 45 7r1M A HolgHol0 9 so if M is simplyconnected Holog Holg From here on I will concentrate on NOTES ON HODGE THEORY ON KA39HLER MANIFOLDS 7 simplyconnected manifolds when talking about holonomy groups in order to avoid global topological issues and to concentrate on local properties of the metric It is a fact which I won t prove here that if M g is a simplyconnected Rie mannian manifold which can be written as a product M1 91 X M2gg then the holonomy group Holg Holgl gtlt Holgg So in studying Riemannian holonomy groups we can concentrate on those manifolds which are irreducible those that can7t be decomposed as a product Now in order to classify Riemannian manifolds by their holonomy groups we just need to introduce the notion of symmetric spaces De nition 34 A Riemannian manifold Mg is a Riemannian symmetric space if for every p E M there exists an involutive isometry sp M A M such that s2 is the identity and such that p is an isolated xed point of sp We say Mg is locally symmetric iffor every p E M there exists an open neighborhood Up and an involutive isometry sp Up A Up with xed point p M g is nonsymmetric if it is not locally symmetric Elie Cartan introduced and classi ed Riemannian symmetric spaces and their holonomies in 1925 Thus we re left to classify the nonsymmmetric Riemannian manifolds This was rst accomplished by Berger in 1955 who provided a list of the possible holonomies which was later re ned by Alekseevskii and also by Brown and Gray Theorem 35 Berger Suppose M is a simplyconnected manifold of dimension n and g is a Riemannian metric on M that is irreducible and nonsymmetric Then exactly one of the following seven cases holds lt1 Holltggt soltngt ii n 2mHolg Um Kahler iii n 2m Holg SUm Calabi Yau iv n 4m Holg Sp Hyperkahler v n 4mHolg SpmSpl Quaternionic Kahler vi n 7 Holg G2 vii n 8 Holg Spin7 Metrics g with Holg C SUm are called Kahler metrics This de nition is equivalent to our earlier de nition although showing this requires some work This formulation makes the Kahler condition a little less random Kahler manifolds are simply a particular class of Riemannian manifolds given by the holonomy clas si cation Note that Calabi Yau and hyperkahler metrics are also Kahler since Spm C SU2m C U2m 33 Other Forms Of The Kz39ihler Condition The holonomy classi cation gives us a natural way of seeing where Kahler manifolds come from and the de nition of a Kahler manifold as a hermitian manifold with a closed associated 11 form gives us a relatively easy way of checking if a metric is Kahler But there are additional equivalent conditions that are sometimes useful Another useful way of looking at Kahler metrics is to view them as metrics which locally look like the Euclidean metric up to order 2 De nition 36 A metric ds2 on M osculates1 to order k to the Euclidean metric on C iffor every 20 E M we can nd a holomorphic coordinate system 21 2n 1Oscillate comes from Latin for to kiss 8 BENNI GOETZ around 20 such that d32 25139139 Qijd2i din where gij vanishes up to order h In this terminology a Kahler metric is a metric which osculates to the Euclidean metric to order 2 The proof isnlt too dif cult but I won t go into it here it s covered in Grif ths amp Harris This is an extremely useful conditioni Anything that can be proven for C with the standard hermitian metric and doesn7t depend on derivatives of order higher than 1 will also hold for any Kahler manifold There are two convenient equivalent descriptions of a Kahler metrici If h is a hermitian metric and V is the associated LeviCivita connection then i VJ 0 ii Vw 0 In other words J and w are parallel with respect to the LeviCivita connection 4 THE HODGE DECOMPOSITION AND THE HARD LEFSCHETZ THEOREM The Hodge Theorem for a compact complex manifold is a cornerstone of modern differential geometry It allows us to identify a unique representative for each co homology class with the special property that this unique representative is in the kernel of the Laplacian which we will de ne The extra structure of a Kahler man ifold gives us extra symmetries in the Hodge structure of the manifold and leads to the powerful Lefschetz decompositioni But before we tackle the Hodge theory I need to introduce the Hodge star operator and the Laplacian 41 The Laplacian Suppose we have an ndimensional compact complex mani fold M with a hermitian metric h The metric h induces a metric on every vector bundle 7r E A M in particular we have a metric on each ber of ApgtqM which we7ll denote 2 at the ber Eli The metric varies smoothly between bers so we can extend it to a global metric on ApgtqM by integration am Maz zzlt1gt2 where 39i39 is the volume form remember w is the associated 1 1form for h Notice that this integral is always de ned on a compact mani old If we venture away from compact manifolds we need to put restrictions on the forms such as considering only forms with compact support or more generally forms for which the integral is nite these are called LQforms We de ne the Hodge star operator to be the map 96 ApgtqM A An pm q such that 12 A MZ a27 2zlt1gt2 for all a E ApgtqMi This gives us a compact way of writing the metric as 15 a 965 M With the metric ApgtqM is a Hilbert space check or justify and we can de ne a formal adjoint for 3 which we denote 3 Apgtq1M A ApgtqM so that if a E ApgtqM and 6 Apgtq1M then 3175 173 A form is coolosed if 5 0i NOTES ON HODGE THEORY ON KA39HLER MANIFOLDS 9 We can now de ne the central operator in Hodge theory7 the Laplacian also called the LaplaceBeltrami operator De nition 41 The Laplacian is an operator A z A A A de ned by A 33 3 3 A form a is harmonic if Aa 0 and we denote the space of harmonic pqforms by wow ker A APgt4ltMgt e ApgtqltMgt39 Later7 to avoid confusion7 we may write this Laplacian as A5 and refer to the 3 Laplacian and 3 harmonic forms You can check that a form is harmonic iff it is 3 closed and 3 coclosed These forms are also the forms of minimal norm in the Dolbeault cohomology We are now ready to state the Hodge Theorem for an n dimensional compact complex manifold M Theorem 42 Hodge Theorem Suppose M is an ndimensional compact complex manifold Then i dim U CpgtqM lt 00 ii the orthogonal projection U C ApgtqM A U CpgtqM is wellde ned and there exists a unique operator G ApgtqM A ApgtqM called the Green s operator such that CU CpgtqM 0 G commutes with 3 and 3 and I9 AG One of the major implications of this theorem is that every Dolbeault cohomology class is represented by a unique harmonic form up to scaling7 which we show on page 12 Every harmonic form is automatically closed7 so we have an isomorphism between the space of p97 Dolbeault cohomology classes and the space of p97 harmonic forms 42 The Hodge Decomposition On A Kz39ihler Manifold The extra structure of a Kahler manifold tells us more about the harmonic forms on it For this section assume that M is an n dimensional Kahler manifold with metric ds2 and associated Kahler form w Welve de ned a bunch of operators on M already Now welll de ne some more Let d be the adjoint of d7 let 3 be the adjoint of 37 and let Ad dd dd and A3 33 33 be the respective Laplacians As a convenience we de ne L1 dc 7 3 7 3 4W and write its associated adjoint as d Note that ddc idcd We also have the projection operators Hp AM A ApgtqM HT EB HM A M A ARM pqr by type and degree The Hodge decomposition of a Kahler manifold follows from a basic identity involving the operator L z A A Ap1gtq1M7 which we de ne by L07 Mm 10 BENNI GOETZ A L Ap qM 7gt p71gtq 1M be its adjoint The basic identity we want to prove is A d 747 or equivalently L d 47rd 1711 call this the Kzihler identity By decomposition of type the identity is equivalent to 1L3 771 and A49 713 Since A d and d0 are real operators either of these implies the other Proving AB V71 3 is not intrinsically difficult but it is messy We7ll leave the dirty work to Griffiths amp Harris They prove the identity on C and then argue that since their proof didn t use any derivatives of the metric of order greater than one the proof carries over to Kahler manifolds This identity has major consequences including the Hodge and Lefschetz decompositions We lead off with two preliminary consequences The proofs aren7t important but theylre short so 1711 include them Proposition 43 L Ad 0 Proof Note that since Ad is selfadjoint this is equivalent to A Ad 0 By our assumption that M is Kahler dnAw dnAwinAdw d7 m Now we can compute L d L din Ldn 7 dLn dnAw 7dnAw 0 Taking the adjoint of both sides A d 0 Using the Kahler identity Add d d dA 7 47rdcd d Ad dAd 7 47rdcd d Ad dAd 47rd dw d Ad dAdquot dquot 47 Ad dAd d dA dd d dA Proposition 44 Ad 2A3 2A5 Proof From part of the Kahler identity A8 7 8A V71 5 We calculate V71 63 7 3 8 8A8 7 8A 7 A8 7 8A 6A6 7 8A6 0 NOTES ON HODGE THEORY ON KAHLER MANIFOLDS 11 So 33 733i From part of the Kahler identity7 Ad 6 66 6 6 66 6 66 66 66 66 66 66 66 66 33 33 33 33 A3 Agi We7ll be done if we can show that A3 Agi Examine V71 A5 3A3 7 3A A3 7 3A3 7 3A3 7 33A A33 7 3A3 7 3A3 7 3A A3 7 3A3 V71 33 33 V 71 A3 since A73 7x7l 3 D Note how both of these propositions rely on the Kahler identitiesi The second proposition tells us that7up to a factor of 27on a Kahler manifold the operators A01 A3 A5 all coincide7 and the corresponding harmonic forms are all the same Since A3 and Ag preserve type7 so does Ad iiei Add 173 0 This commutativity is crucial to the Hodge and Lefschetz decompositions De ne Z gtqltMgt dA Zg q U CpgtqM 0 E ApgtqM Adgo 0 go 6 ATM 1A0ng 0i ApgtqM and Ad commutes with Hp 7 if a E ATM7 HMM Since ATM pqT Ada Ad 030 a7 711 I I I 10 Adamo Adar711 I I I Ada07 where a Hagar Thus Ada 0 iff Ada 0 for each p7417 and 2 WWW 69 90730 pqr The complex differential forms are the complexi cation of the real forms7 which we can write as A A M R C Since the differential d is de ned on the real forms7 Ad can be considered as an operator on the real forms which is extended to the complex forms by complex linearityi In the tensor product described above7 conjugation just acts on the factor of C From this description it is clear that Ad commutes with 39 39 Since quot is an i omoiphi in between ApgtqM and AqgtpM7 we obtain 3 U Cp qM U39C 1gtPMi 12 BENNI GOETZ From the Hodge Theorem7 we know that if go 6 ApgtqM and dgo 07 then so 9W MGM W80 MGM dWGW 9W MGM since d and G commute U Cgo is also of type 107 q7 so 4 Hp qM E 39fp qM The Hodge Theorem for Ad also tells us that HRM 2 3f Putting this together with 27 3 and 47 we get Theorem 45 Hodge Decomposition On a compact Kahler manifold M we have HTltMgt e19 HPgt4ltMgt pqr HpgtqM H 1gtPM Since Ad 2A5 we also have 90 E U Cg qM7 which means HpgtqM Hg qM An immediate consequence of the Hodge theorem is a strong condition on the Betti numbers of a Kahler manifold Proposition 46 On a compact Kahler manifold M the odd Betti numbers must be even This gives us a topological obstruction to a complex manifold admitting a Kahler structure As an example7 start with the usual complex structure on C2 7 07 and take the quotient by the group action 2 gt gt 22 The quotient inherits the com plex structure7 and is a complex manifold Topologically this space is equivalent to 53 X 5391 and is called the Hopf surface By the Kiinneth formula7 H S3 X 5391 H 53 E H 53917 so H1S3 X 51 E C and therefore the manifold can t admit a Kahler structure 43 The Lefschetz Decomposition On A Kz39ihler Manifold The operators LA and their commutativity with Ad on a Kahler manifold give us another de composition of the cohomology of the manifold These operators generate a vector space which we can identify with 52 With this identi cation we get the fantastic fact that the cohomology is an slgrepresentationl Here s a blitzprimer2 on slg the Lie algebra for SL2 52 can be written as the vector space of traceless 2 X 2 complex matrices with commutator AB AB 7 BA The standard basis for 52 is given by Hg 5 H 8 H5 31 HX 2X my 72y XY H with relations 2blitzprimei Lightmngefast primer NOTES ON HODGE THEORY ON KA39HLER MANIFOLDS 13 A representation of52 is a complex vector space V and a Lie algebra homomorphism p 52 7gt EndV In this case V is called an 52module Unless there is some possible confusion llll omit the p7s for the action of 52 on V A submodule is a subspace of V xed by the action of 52 V is irreducible if V has no nontrivial submodules lrreducible representations of 52 have a particular structure given by the eigenspaces of H which are called weightspoces If v is in a weightspace for H with eigenvalue A then X1 and Y1 are also in weightspaces with eigenvalues A 2 and A 7 2 respectively We7ll do the calculation for X1 which is very similar to the case of Yo HXv HXv XHv 2Xv Xv A 2Xv Since V is nitedimensional H only has a nite number of eigenvalues This implies that X and Y are nilpotent We call X primitive if v is an eigenvector for H and X 1 0 1711 now list some important propositions without proof Proposition 47 va is primitive then V is generated by vYvY2v Since the nonzero elements of V of the form an all have different eigenvalues they are linearly independent If we write the A eigenspace of H as VA we have H05 VA XV VH2 YV VA727 and Proposition 48 All eigenvalues ofH are integers and we can write VVnGBVn2 BEBVLQEBVW Now for any 52module V not necessarily irreducible if PV v E V Xv 0 then we have the Lefsohetz decomposition of V VPVEBYPVEBY2PVGBm We also know that the maps Ym Vm 7 V7 and Xm Vim 7gt Vm are isomor phisms We are close to our goal We need to gure out how 52 acts on the cohomology ring of an n dimensional compact Kahler manifold to make it an 52module The relevant calculation conveniently omitted here is simple but tedious A L n 7 r on AT De ne h 7 rllT r0 Since AL n 7 r on ATM we know on AM A L 7 rllT hl 14 BENNI GOETZ We perform similar calculations 2n 2n h L 7 rllTgt L 7 L 7 rllTgt n7r2L7Ln7r on ATM 72L and hA 7 rllTgt A 7 A 7 rllTgt n7 r72A7An7r on ATM 2L The two calculations above extend linearly to A Pulling these calculations together and restating them on A for succinctness we have A L h h L 72L h A 2A Since the operators hLA commute with Ad they act on the harmonic forms 3 By the Hodge theorem H E 90 M so we can de ne an action of 52 on the cohomology of M by sending X 7gt A Y 7gt L H 7gt hi De ne the primitive cohomology as Pn kM ker Lk1 H L WM a Hnk2M ker A N Together with the facts above about representations of 52 we obtain Theorem 49 Hard Lefschetz Theorem The map Lk Hnik 7gt Hnk is an isomorphism and we have the Lefschetz decomposition HmM EB LkPm QWM k So the Kahler condition imposes some strict requirements on the topology of the manifol i REFERENCES 1 Bryant R Lie Groups And Symplectic Geometry In Geometry And Quantum Field Theory IASPark City Mathematics Series Volume 1 Providence American Mathematical Society 1995 2 Grif ths P Harris J Principles Of Algebraic Geometry New York John Wiley amp Sons 1978 3 Joyce D Compact Manifolds With Special Holonomy Oxford Oxford University Press 2000


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