A NEW INTERFACE BETWEEN ANALYSIS AND ALGEBRAIC GEOMETRY. by Yum-Tong Siu of Harvard University. June 29, 2009

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1 A NEW INTERFACE BETWEEN ANALYSIS AND ALGEBRAIC GEOMETRY by Yum-Tong Siu of Harvard University June 29, 2009 National Center for Theoretical Sciences National Tsing Hua University, Taiwan 1

2 ABSTRACT Multiplier ideal sheaves identify the jet directions where estimates for partial differential equations fail. They were first introduced by Joseph J. Kohn to study the complex Neumann problem for weakly pseudoconvex domains and by Alan M. Nadel to study the existence of Kaehler-Einstein metrics for Fano manifolds. The technique of multiplier ideal sheaves injects in a new way methods of algebraic geometry into problems of analysis. It also opens new channels of applying analysis to problems in algebraic geometry, leading to the solution or partial solution of a number of longstanding open problems in algebraic geometry such as the Fujita conjecture and other effective problems, the deformational invariance of plurigenera, the finite generation of the canonical ring, and the abundance conjecture. 2

3 Two Perspectives for Multiplier Ideal Sheaves Multiplier ideal sheaves arise from two different perspectives. Defined from Estimates: As the set of all jet directions where estimates for partial differential equations fail. Underlying space preserved. Defined from Instability: as the ideal sheaf of the subspace into which the original analytic entity degenerates when estimates in an evolution are kept by rescaling. Estimates preserved. 3

4 Historic Origins of Multiplier Ideal Sheaves Multiplier ideal sheaves were introduced, as measure of failure of estimates, by Joseph J. Kohn to study the complex Neumann problem for weakly pseudoconvex domains. Joseph J. Kohn, Subellipticity of the -Neumann problem on pseudoconvex domains: sufficient conditions. Acta Math. 142 (1979), Multiplier ideal sheaves from viewpoint of instability were introduced by Alan M. Nadel to study the existence of Kähler-Einstein metrics for Fano manifolds. A. Nadel, Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature. Ann. of Math. 132 (1990),

5 Two Kinds of Multiplier Ideal Sheaves Dynamic multiplier ideal sheaves, which are defined by a sequence or a family of inequalities. These are the ones originally introduced by Joe Kohn and Alan Nadel. Static multiplier ideal sheaf, which is defined by a single inequality. It can be defined algebraic-geometrically and is the kind which is used by algebraic geometers. It is used in many applications to algebraic geometry. 5

6 Estimates of Partial Differential Equations and Multiplier Ideal Sheaves A Priori Estimates. Consider the equation Lu = f by using weak solution. Use test function g to solve Lu = f. (g, Lu) = (g, f) (L g, u) Suffices to have linear functional L g (g, f) bounded, i.e., to have the a priori estimate. or more generously (g, f) C f L g g, g C L g g. 6

7 Intuitive Interpretation: g C L g g = injectivity of L = surjectivity of L = solvability of Lu = f. Method of Multiplier Ideal Sheaf. The multiplier ideal sheaf I measures the extent and location of the failure of the a priori estimate. It is the sheaf of germs of functions F satisfying F g C F L g g. The zero-set of I and its vanishing order give respectively the location and the extent of the failure of the a priori estimate. The multiplier ideal sheaf I is defined by a family of inequalities, which is parametrized by the test function g. Each g gives one inequality. 7

8 The above formulation deals with the existence of solution for Lu = f. For regularity problems, instead of g C L g g, we need g C L g for a stronger norm than. Then the multiplier ideal sheaf I for the regularity problem is the sheaf of germs of functions F satisfying F g C F L g g. g 8

9 Applied to Get Global Conditions for Solving PDE Since the zero-set and the vanishing order of the multiplier ideal sheaf identify the local position and the extent of the failure of estimates, it corresponds to iteration of many levels of microlocal analysis. Unlike micro-local analysis which treats all the directions the same way, the multiplier ideal sheaf technique distinguishes certain directions and jet directions. This enables the formulation of global conditions (as opposed to pointwise conditions such as ellipticity) to solve PDE in problems of analysis and complex geometry. 9

10 Multiplier Ideal Sheaves from the Viewpoint of Instability In many geometric applications Lu = f can be written as the limit of L ν u ν = f (as ν ), where a priori estimates are available for each L ν u ν = f. An Ascoli-Arzela argument is sought for the limiting case Lu = f, which requires a uniform bound for a stronger norm for the convergence of a subsequence in a weaker norm. For simplicity one considers the case for R with L 2 norm as the weaker norm and L 2 1 (the L 2 norm for derivatives up to order 1) as the stronger norm. A change of scale x λx has different effects on the two norms h 2 dx λ h 2 dx, R R h 2 1 dx λ h 2 dx. R R 10

11 Can make an appropriate ν-dependent change of scale λ ν to make the stronger norm uniformly bounded in ν. Scaling done in a manifold X separately for ever smaller coordinate charts is equivalent to estimating F Dh 2 instead of X X Dh 2, where F is a smooth function on X (and Dh is the first-order differentiation of h). F is the multiplier and its absolute value describes the local rescalings of infinitesimally small coordinate charts. 11

12 Two Illustrations for Multiplier Ideal Sheaves from Viewpoint of Instability Will illustrate this viewpoint with two examples. One is Yang-Mills fields for stable vector bundles. The other is how Nadel s multiplier ideal sheaves arise. 12

13 Method of Multiplier Ideal Sheaves for Yang- Mills Fields Setting: X compact complex manifold with Kähler metric g i j. Holomorphic vector bundle V over X. Seek a metric h α β along the fibers of V such that g i j Ω α βi j = c h α β, where i,j and c is a constant. Ω α βi j = curvature of h α β For simplicity we take the constant c to be 1 in our illustration. 13

14 Equation ( ) g i j Ω α βi j = h α β i,j is elliptic along the directions of X. Unknown h α β is matrix-valued. Usual theory of elliptic equations with scalar unknown does not apply. Can consider unknown h α β as a scalar unknown over V. In that case there is no ellipticity along the fibers directions of V. Multiplier ideal sheaf, as a destabilizer, may occur when one approximates ( ) by equations elliptic also along the fiber directions of V. 14

15 Equation ( ) g i j Ω α βi j = h α β i,j persists at the limit. The destabilizer W is spanned by all nonzero eigenvectors of the limit h α β. All the curvature of V (after contraction by g i j) is concentrated on W whose rank < that of V, giving ( ) c 1 (W ) rank W > c 1(V ) rank V. Stability condition precisely stipulates the inequality direction < in ( ) for all proper subbundles W (subsheaves with torsion-free quotients) of V. References: K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Comm. Pure Appl. Math. 39 (1986), suppl., S257 S293. ibid. 42 (1989), S. K. Donaldson, Infinite determinants, stable bundles and curvature. Duke Math. J. 54 (1987), S. K. Donaldson, Infinite determinants, stable bundles and curvature. Duke Math. J. 54 (1987), Ben Weinkove, A complex Frobenius theorem, multiplier ideal sheaves and Hermitian-Einstein metrics on stable bundles. (English summary) Trans. Amer. Math. Soc. 359 (2007), no. 4,

16 How Nadel s Multiplier Ideal Sheaf Arises Setting: X Fano manifold (compact complex manifold with anticanonical line bundle K X positive). Problem: Construct Kähler-Einstein metrics for X (and find condition for its existence). Alan Michael Nadel, Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature. Ann. of Math. 132 (1990), Use Kähler metric g i j in the anti-canonical class K X. Solve complex Monge-Ampère equation det ( g i j + i jϕ ) = e ϕ F det ( ) g i j with where R i j g i j = i j log F, R i j = i j det ( ) g i j is the Ricci curvature of g i j. 16

17 Taking log of complex Monge-Ampère equation det ( g i j + i jϕ ) = e ϕ F det ( ) g i j yields ) R i j (g = i j g i j + ( R i j g i j) Ri j = g i j, where g i j = g i j + i jϕ, R i j = Ricci curvature of g i j. Solve det ( ) g i j + i jϕ t = e tϕ t F det ( ) g i j by continuity method with parameter 0 t 1, starting with t = 0. 17

18 Second, third-order a priori estimates for det ( ) g i j + i jϕ t = e tϕ t F det ( ) g i j in the continuity method hold, as given in Shing-Tung Yau, On the Ricci curvature of a compact Khler manifold and the complex Monge-Ampre equation. I. Comm. Pure Appl. Math. 31 (1978), The main trouble is with zeroth order estimate. 18

19 As closed positive (1, 1)-form in the class K X, some subsequence of g i j + i jϕ t converges weakly. Let ϕ t be the average of ϕ t over X with respect to g i j. By compactness of Green s kernel, a subsequence of ϕ t ϕ t converges in L 1. Zeroth order estimate holds unless the number ϕ t as t 1. Note that sup ϕ t is dominated by ϕ t plus O(1) by using the Green s kernel and its lower bound. 19

20 Here is the key point how multiplier ideal sheaf arises Multiply det ( ) g i j + i jϕ t = e tϕ t F det ( ) g i j by e t ϕ t to get From e t ϕ t det ( g i j + i jϕ t ) = e t(ϕ t ϕ t ) F det ( g i j). X it follows that lim t det ( g i j + i jϕ t ) = X X det ( ) g i j e t(ϕ t ϕ t ) F det ( g i j) = along that subsequence t 1 when ϕ t as t 1. 20

21 From viewpoint of instability, the rescaling to preserve the estimate is to put in the function germ f such that ( ) lim f 2 e t(ϕ t ϕ t ) <. t We do the rescaling or put in the multiplier always in the crucial estimate of the PDE. That is why here the multiplier is defined by ( ). This sheaf I of germs of multipliers f, which is defined by a converging sequence of plurisubharmonic functions ψ ν := t ν (ϕ tν ϕ tν ) and the inequality lim sup f 2 e ψ ν < ν is Nadel s multiplier ideal sheaf. This multiplier ideal sheaf is dynamic in the sense that it is defined by a sequence of inequalities. 21

22 The multiplier ideal sheaf I (or destabilizing subsheaf) collapses the manifold X into the subspace with structure sheaf O X /I. The Fano manifold X (with anti-canonical line bnndle K X positive) has certain vanishing Hodge groups (by Kodaira s vanishing theorem) H p (X, O X ) = 0 for p 1. When X collapses into the destabilizing subspace defined by I, the destabilizing subspace inherits the property of certain vanishing Hodge groups from X. H p (X, O X /I ) = 0 for p 1. 22

23 Nadel proved this inheritance by showing H p (X, I) = 0 for p 1, which comes from the general Nadel vanishing theorem H p (X, I (L + K X )) = 0 for p 1 when L = K X. We can also consider a general abstract metric e ψ ν for L, instead of from the continuity method of the Monge-Ampère equation. When the every term in the sequence ψ ν is equal to just one ψ, H p (X, I ψ (L + K X )) = 0 for p 1. 23

24 Formulated separately: Let L be a holomorphic line bundle over a compact complex algebraic manifold X with metric e ψ whose curvature current 1 ψ is strictly positive. Then H p (X, I ψ (L + K X )) = 0 for p 1, where I ψ is the sheaf of germs of holomorphic functions f on X such that f 2 e ψ is locally integrable. In the algebraic setting, this result is known as the Vanishing Theorem of Kawamata-Viehweg-Nadel. Yujiro Kawamata, A generalization of Kodaira-Ramanujam s vanishing theorem. Math. Ann. 261 (1982), Eckart Viehweg, Vanishing theorems. J. Reine Angew. Math. 335 (1982), 1 8. The multiplier ideal sheaf I ψ, which is defined by a single inequality f 2 e ψ <, is a static multiplier ideal sheaf. This kind of static multiplier ideal sheaves is the kind used by algebraic geometers. They can be defined algebraic-geometrically by using a resolution of singularity when e ψ is algebraically defined. 24

25 Static multiplier ideal sheaves are useful in effective results in algebraic geometry, such as Fujita conjecture type problems, effective Matsusaka s big theorem, and the effective Nullstellensatz. However, in more powerful applications such as the analytic proof of the finite generation of the canonical ring, the abundance conjecture analytic construction of rational curves in Fano manifolds (not yet complete) the more versatile dyanamic multiplier ideal sheaves have to be used to terminate some infinite processes. 25

26 Remarks on Stability If one defines stability for a Fano manifold X as one for which a destabilizing subsheaf does not arise in the continuity method for the complex Monge- Ampère equation, it is a tautology to say that a Fano manifold admits a Kähler-Einstein metric if and only if it is stable. The challenge is to find easily verifiable conditions to rule out destabilizing subsheaves (to which the Fano manifold collapses). 26

27 A good guide is the stability condition for Hermitian- Einstein metrics for vector bundles (Yang-Mills fields). For Yang-Mills fields destabilizing subsheaves are over the same base manifold. For a vector bundle, one can find the maximum destabilizing subsheaf and inductively define the Harder- Narasimhan filtration. 27

28 Analogously: For a Fano manifold, one should try to define the maximum destabilizing ideal sheaf (which defines a subspace into which the Fano manifold collapses). One should look for subspaces inside the Fano manifold X. From viewpoint of multiplier ideal sheaves we should not just consider objects defined over all of X, for example, the space of all Kähler metrics on X. Global objects defined over X should be used only in the context of defining subspaces inside X as dynamic limits. 28

29 Application of Algebraic Techniques to Analysis by Multiplier Ideal Sheaves Key Technique: The non-integrability of the jets of failure of estimates (in an appropriate sense) yields the desired estimates. The derivation of estimates from the non-integrability of failure of estimates comes from integration by parts and other conventional techniques of PDE estimates. Non-integrability of failure of estimates can be formulated in terms of algebra and algebraic geometry. Algebraic techniques can be applied to get easily verifiable formulation of non-integrability of failure of estimates. We illustrate this application by using Kohn s original application of multiplier ideal sheaves to the complex Neumann problem. 29

30 Setting for the Complex Neumann Problem Setting: Bounded domain Ω in C n with smooth weakly pseudoconvex boundary defined by r < 0, dr is nowhere zero on the boundary Ω of Ω. Weakly pseudoconvex boundary 1 r T (1,0) Ω 0. Question of Regularity: u = f on Ω f is closed (0, 1) form smooth on Ω u holomorphic functions on Ω = u smooth function on Ω. The solution u perpendicular to all holomorphic functions always exists and is known as the Kohn solution. Its interior regularity is known. The problem is about its regularity up to the boundary. 30

31 Subelliptic Estimates Regularity for the complex Neumann problem holds if one has the following subelliptic estimate. For P Ω open neighborhood U of P in C n and positive numbers ɛ and C satisfying g 2 ɛ C ( g 2 + g 2 + g 2) for all (0, 1)-form g supported on U Ω which is in the domain of and. ɛ is the L 2 norm on Ω involving derivatives up to order ɛ in the boundary tangential direction of Ω. is the usual L 2 norm on Ω without involving any derivatives, and is the actual adjoint of with respect to. 31

32 g 2 is added to the right-hand side, because f = 0 enables us to limit ourselves to -closed test functions g. g 2 is added to the right-hand side, because solvability is always possible. Only regularity is in question. The Sobolev ε-norm is used to take care of the error from the commutator when differentiation is applied to both sides of the -equaiton the error from the cut-off function when the equation is localized at a boundary point. Tangential Sobolev norm ɛ is used (instead of the usual Sobolev norm ɛ ), because we would like to preserve in differentiation the condition for the actual adjoint of which requires the vanishing of normal component. 32

33 Kohn s Subelliptic Multipliers For the regularity question of the complex Neumann problem a scalar Kohn multiplier F is a smooth function germ at the point P Ω such that for some positive numbers ɛ F and C F F g 2 ɛ F C F ( g 2 + g 2 + g 2) for test (0, 1)-form g in the domain of and. I P is the ideal of all such multipliers. A vector Kohn multiplier means a smooth germ of (1, 0)-form θ at the point P Ω such that for some positive numbers ɛ θ and C θ θ g 2 ɛ θ C θ ( g 2 + g 2 + g 2) for test (0, 1)-form g in the domain of and. A P is the module of all such vector-multipliers. 33

34 Kohn s Algorithm (A) Initial Membership. (i) r I P. (ii) j r belongs to A P for every 1 j n 1 if r = z n at P for some local coordinate system (z 1,, z n ), where j means z j. (B) Generation of New Members. (i) If f I P, then f A P. (ii) If θ 1,, θ n 1 A P, then the coefficient of is in I P. (C) Real Radical Property. θ 1 θ n 1 r If g I P and f m g, then f I P. 34

35 Key Points of Kohn s Algorithm Allow certain differential operators to lower the vanishing order of multiplier ideals. Two limitations on differentiation to reduce vanishing order: Only (1, 0)-differentiation is allowed. Only determinants of coefficients of (1, 0)-differentials can be used (from Cramer s rule). Root-taking can be used to reduce vanishing order. 35

36 Algebraic Condition for the Regularity of the Complex Neumann Problem Algebraic Condition Turns Out to be: There is a finite bound on the (normalized) touching order to Ω of any local holomorphic curve in C n. Motivation for Algebraic Condition. If Ω is Leviflat (which means locally parametrized by complex submanifolds of real codimension 1), then regularity clearly fails. The reason is the lack of sufficient control over derivatives along the directions of the complex submanifolds in Ω. Even with the weaker assumption of a local complex curve inside Ω, regularity still fails. There is still trouble even for the very weak assumption of a local complex curve in C n touching Ω to an infinite order. These observations motivate the algebraic condition for regularity to be formulated in terms of a finite bound on the (normalized) touching order to Ω of any local holomorphic curve in C n. The precise definition is given below. 36

37 Global Condition of Finite Type for the Global Regularity Problem The type m at a point P of the boundary of weakly pseudoconvex Ω is the supremum of the normalized touching order ord 0 (r ϕ) ord 0 ϕ, to Ω, of all local holomorphic curves ϕ : C n with ϕ(0) = P, where is the open unit 1-disk and ord 0 is the vanishing order at the origin 0. John P. D Angelo, Finite type conditions for real hypersurfaces. J. Differential Geom. 14 (1979), Kohn s Conjecture: Kohn s algorithm terminates for smooth weakly pseudoconvex domains of finite type (with effectiveness involving type and order of subellipticity). Joseph J. Kohn, Subellipticity of the -Neumann problem on pseudoconvex domains: sufficient conditions. Acta Math. 142 (1979),

38 Kohn s conjecture solved for real-analytic case without effectiveness. K. Diederich and J. E. Fornaess, Pseudoconvex domains with realanalytic boundary. Ann. of Math. 107 (1978), Effectiveness of Kohn s algorithm is given for special domains in Yum-Tong Siu, Effective Termination of Kohn s Algorithm for Subelliptic Multipliers, arxiv: with a sketch of how to extend the proof to the effective real-analytic case and the effective smooth case. We are going to explain the algebraic condition of finite type in terms of the non-integrability of the failure of estimates. 38

39 RELATION BETWEEN KOHN ALGORITHM AND FROBENIUS INTEGRABILITY THEOREM Usual Frobenius Theorem (for R n ) U R n open subset. x V x T R n = R n distribution of (n k)- dimensional subspaces of T R n. V x integrable (i.e. V x = tangent space of (n k)-fold U) [V x, V x ] V x x U dω j = k l=1 ω l η j,l where ω 1,, ω k are 1-forms defining V x, η j,1,, η j,k some other 1-forms. 39

40 Frobenius Theorem Over Artinian Subschemes For integrability over an Artinian subscheme, we consider integrability not over U but only over some multiple point, for example, the ringed space (0, O C n /I ) with (m C n,0) N I for some integer N 1, where m C n,0 is the maximum ideal of the structure sheaf O C n of C n at 0. The ringed space (0, O C n /I ) is an Artinian subscheme. 40

41 Partial integrability requiring integrable curves was studied in C. Carathéodory, Untersuchungen über die Grundlagen der Thermodynamik. Math. Ann. 67 (1909), Wei-Liang Chow, Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung. Math. Ann. 117, (1939) The partial integrability of distribution of linear subspaces was studied by Carathéodory in his differential geometric and PDF description of the second law of thermodynamics in which certain points can be joined by integrable curves and how adiabatic and diabatic processes are related to connectedness by integrable curves. Carathéodory dealt with the codimension-one case and Wei-Liang Chow with the higher-codimension case. Kohn s algorithm involves multiple-points (Artinian subschemes or jets) instead of curves. Another feature in Kohn s algorithm is that only multiple-points with complex structure (J-invariant) are considered. 41

42 Carnot-Carathéodory Metric Nowadays Carathéodory s work is used in the context of the Carnot-Carathéodory metric. The French engineer Carnot introduced the Carnot cycle giving theoretically the optimal efficiency of a heat engine from the Second Law of Thermodynamics. Carathéodory gave the mathematical formulation of the reachability of one state from another in view of the second law of thermodynamics. Reachability is given by a curve whose tangent belongs to a given distribution of linear subspaces. For a distribution of linear subspaces V, the Carnot- Carathéodory metric is the infimum of all piecewise smooth curves whose tangents are in V at its smooth points. When the bracket-generating condition is satisfied (i.e., iterated Lie brackets of V -valued tangent vector fields generate full tangent spaces), the Carnot- Carathéodory metric is finite. Sub-Riemannian geometry deals with such metrics. 42

43 Interpretation in Cauchy-Riemann Geometry. Real hypersuface M = {r = 0} U C n weakly pseudoconvex. Distribution T R M JT R M Usual Frobenius integrability means Leviflat. Integrability over an Artinian subscheme (which is holomorphic) means local holomorphic curve touching M to high order at one point. Finite type roughly means limit on the order of the Artinian subscheme of integrability. 43

44 Kohn s Algorithm as Differential Condition for Frobenius Theorem. The condition of Kohn s algorithm generating the unit in I p is the differential condition for the Frobenius theorem over Artinian subschemes with a bound on order. Frobenius condition: exterior differentiation of ω 1,, ω k would not get anything new, i.e., already spent by ω 1,, ω k. Kohn s algorithm is the opposite: want new entities in the admissible differentiation process until the unit ideal 1 is generated. Recall Key Technique. Pointwise condition (like ellipticity or subellipticity) for symbols may fail at every point. As long as they do not fail in an integrable manner, one can still get estimates. Idea of non-integrability of failure of pointwise condition of symbol (à la Frobenius) should be applicable to general partial differential equations. Theory not yet developed. 44

45 ILLUSTRATION: Algebraic Formulation for Special Domains. Special domain Ω in C n+1 defined by N Re w + F j (z 1,, z n ) 2 < 0, j=1 where F j (z 1,, z n ) is a holomorphic function defined on some open neighborhood of the closure Ω which depends only on the first n variables z = (z 1,, z n ). The (n + 1)-ball given by {F j } = {z j } after the higher-dimensional analog of the biholomorphism between the disk and the upper half-plane. Finite Type Condition of Special Domain. Assume 0 Ω. Let m C n,0 be the maximum ideal of O C n,0 at 0. Finite type at 0 means that N (m C n,0) p j=1 O C n,0f j for some positive integer p (which is explicitly related to the type). For special domains Kohn s algorithm of generating new multipliers by differentiation is as follows, giving also effectiveness. 45

46 Effective Subellipticity for Special Domain of Finite Type. F 1,, F N holomorphic function germs on C n at the origin with the origin as the only common zero. The multiplicity dim C O C n,0 / N j=1 O C n,0f j is the number used for effective statements. Effective quantities are explicit functions of it. Select n C-linear combinations g 1,, g n of F 1,, F N. Form the Jacobian determinant of g 1,, g n with respect to z 1,, z n. Such a Jacobian determinant is a multiplier in Kohn s algorithm and so are elements in the ideal they generate. 46

47 In contrast, we use the term pre-multiplier for the original given function germs F 1,, F N, because they are not multipliers and only the Jacobian determinants of their C-linear combinations are multipliers. Pre-multipliers do not form an ideal. Take the ideal I generated by all such Jacobian determinants. Take a positive number q and a number of function germs ϕ 1,, ϕ l whose q-th power belong to the ideal I. The function germs ϕ 1,, ϕ l are also multipliers in Kohn s algorithm because of the Real Radical Property in Kohn s algorithm. Add ϕ j to F 1,, F N to enlarge F 1,, F N. Repeat the procedure with the set F 1,, F N replaced by its enlargement. Repeat until I becomes the unit ideal. Effective subellipticity means the procedure stops with an effective number of steps and the q at each step is effective. 47

48 Rôles of Analysis in Algebraic Geometry Historic Use of Singularities for Constructing Holomorphic Forms on Riemann Surfaces by Using Electrotatic Potentials Let X be a compact Riemann surface. For a point charge at P and an opposite equal charge Q nearby we can construct an electrostatic potential u for the pair P, Q as a dipole by minimizing the Dirichlet integral of the square of the gradient of u with log singularities log z P and log z Q (up to some normalizing factor) at P and Q respectively. Constants of integration from loops pose the problem of multi-valuedness for the electrostatic potential, which we deal with by differentiation. The (1, 0)-derivative u is a meromorphic 1-form 1 whose principal parts are z P 0 and 1 z Q (up to some normalizing factor) If X has positive genus g, g C-linearly independent holomorphic 1-forms on X are constructed by the necklace trick as illustrated by the following picture. 48

49 Cancelation of principal parts by adjacent principal parts of opposite sign for dipoles arranged as pearls on a necklace around one of the necks of the Riemann surface. Two adjacent pearls form one dipole. 49

50 Technique of Curvature to Metric to Section The dipole formed from the two points P, Q of the Riemann surface X can be regarded as the difference δ P δ Q of two Dirac deltas, which is the curvature current of a metric of the line bundle associated to the divisor P Q. The above argument goes the following route. Curvature Metric Section Near the dipole the metric (up to normalization) is actually mainly e ϕ with ϕ(z) = log z P log z Q which is the electrostatic potential. The main point is the need to use differentiation to get rid of the ambiguity from the constants of integration along loops. This differentiation in the (1, 0)-direction adds the canonical line bundle K X as a summand when the section is produced. The necklace is used to make sure that the final result is not identically zero. It is a condition for a nonvanishing statement. 50

51 The Rôle Played by Adding the Canonical Line Bundle In the application of analysis to algebraic geometric problems the rôle played by the addition of the canonical line bundle is extremely important. It is a key point in the following. Kodaira s vanishing theorem with K X added to a positive line bundle L to get vanishing for L + K X. The process of multivalued sections of L metric of L sections for the adjoint bundle L + K X makes pluricanonical extension possible, giving the deformational invariance of plurigenera. Pluricanonical extension makes possible the induction on dimension in the proof of the finite generation of the canonical ring (minimal model program), in both the algebraic and analytic versions of the proof. Pluricanonical extension also makes possible the proof of the abundance conjecrure. 51

52 Effectiveness from Taking Roots in the Metric Defined by Multivalued Holomorphic Sections Another application from the process of multivalued sections of L metric of L sections for the adjoint bundle L + K X is that the use of multivalued sections to define a metric allows the taking of roots (explained in more details below). This process of constructing sections of the adjoint bundle L+K X, coupled with the theorem of Hirzebruch- Riemann-Roch, makes it possible to get effective results in algebraic geometry such as the following. The Fujita Conjecture: Γ (X, ml + K X ) free (or very ample) for L > 0 and m m n with n = dim C X. Effective Matsusaka s Big Theorem. Effective Nullstellensatz. 52

53 Going from Riemann Surfaces to Higher Dimension For the technique of taking the (1, 0)-differential of an electrostatic potential on a Riemann surface, if we allow the holomorphic 1-form to have value in a holomorphic line bundle, for example in the line bundle associated to P, we do not need to use the necklace trick and can just consider the Dirac delta δ P as the curvature of the line bundle associated to P. Note that without using the necklace one has to guarantee nonvanishing in another way. For this lecture we are going to suppress this aspect of making sure that the final result is not identically zero. 53

54 When X is a compact complex manifold of complex dimension n > 1, we replace δ P by a closed positive (1, 1)-current Θ. Such Θ can be constructed by Θ = 1 log j s j 2, for some multi-valued holomorphic sections s j of some line bundle L. Multi-valued holomorphic section s of L means s N is a usual holomorphic section of NL for some positive integer N. 54

55 For higher dimension, the condition for a local function to be holomorphic is an over-determined system of PDE. On the other hand, the analog of electrostatic potential corresponds only to solving a just determined PDE like the Laplace equation. Technically to get a holomorphic section of L + K X from Θ is more complicated than in the case of a Riemann surface. It involves the Bochner-Kodaira technique of completion of squares for symbols. = i,j gīj j ī + Curvature Term analogous to ax 2 + bx + c = a ( x + b ) 2 b2 4ac. 2a 4a Adding K X as volume form by using L 2 estimates corresponds to taking the (1, 0)-derivative of the electrostatic potential in the case of a Riemann surface. 55

56 Analysis transforms the problem of holomorphic sections of a line bundle L to a problem for the absolute value L of the line bundle, because metrics and curvature depend only on the absolute values of the transition functions. We can take roots of L but not L. Kodaira s embedding theorem gives an ineffective number m for the very ampleness of ml for a positive line bundle L. Ineffectivity comes from the assumption on holomorphic injectivity radius in Kodaira s proof. Effective results with techniques of multiplier ideal sheaves come from the technique of root-taking for L. 56

57 Examples of Effective Results in Algebraic Geometry Freeness in the Fujita Conjecture. For any positive line bundle L over a compact complex manifold X of complex dimension n, Γ (X, ml + K X ) generate ml + K X if m ( n) + 1. Conjectured result with m only n + 1. Urban Angehrn and Yum-Tong Siu, Effective freeness and point separation for adjoint bundles. Invent. Math. 122 (1995), no. 2, Effective Matsusaka Big Theorem. B be a numerically effective line bundle over X. Then ml B is very ample for m no less than ( ) 2 max(n 2,0) ( ) C n L n 1 2 max(n 2,0) KX 1 + Ln 1 KX, L n where C n = 2 n 1+2n 1 ( n k=1 ) (k2 ) (n k 1)(n k) 2 max(k 2,0) 2 and K X = ( 2n ( ) ) 3n 1 n + 2n + 1 L + B + 2KX. Yum-Tong Siu, A new bound for the effective Matsusaka big theorem. Special issue for S. S. Chern. Houston J. Math. 28 (2002),

58 Deformational Invariance of Plurigenera Conjecture for the Kähler Case. Let π : X be a holomorphic family of compact Kähler manifolds over the open unit 1-disk with fiber X t. Then for any positive integer m the complex dimension of Γ (X t, m K Xt ) is independent of t for t. Proved only for algebraic X t. Kähler case still open. m = 1 is the deformational invariance of the Hodge number. For higher m by semi-continuity of the dimension of fiber cohomology the problem is pluricanonical extension. It is handled by the key technique is induction on m by adding one canonical line bundle in the process of multivalued sections of L metric of L sections for the adjoint bundle L + K X 58

59 Trouble with the Kähler case is that, in one step of the use of the analog for multiplier ideals sheaves of Theorem A of Cartan-Oka of global generation, an ample line bundle A is needed so that pluricanonical extension for mk is first done with mk+a and then A is removed by dividing by N and let N. Yum-Tong Siu, Invariance of plurigenera, Invent. Math. 134 (1998), Yum-Tong Siu, Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semipositively twisted plurigenera for manifolds not necessarily of general type. In: Complex Geometry: Collection of Papers Dedicated to Professor Hans Grauert, Springer-Verlag 2002, pp

60 Finite Generation of Canonical Ring. Statement: m=1 Γ (X, mk X) is finitely generated for a compact complex algebraic manifold X. An analytic proof is first done with the case of general type. Then a fibration is used. In both the analytic and the algebraic-geometric proof, the key technique (of induction on dimension) is an adaptation of the pluricanonical extension technique to extend pluricanonical sections from a hypersurface Y to X with appropriate modifications, where Y contains the common zero-set of the pluricanonical sections of X. That is the reason why this result had to wait for so many years until the introduction of the pluricanonical technique from the deformational invariance of plurigenera made its solution possible. 60

61 The analytic proof actually uses the pluricanonical extension in the form of the related Skoda s result on ideal generation instead of its use. Moreover, in the analytic proof a dynamic multiplier ideal sheaf argument involving an infinite sum of multivalued holomorphic sections is used to terminate an infinite process of blow-ups. Yum-Tong Siu, A general non-vanishing theorem and an analytic proof of the finite generation of the canonical ring, arxiv:math/ Yum-Tong Siu, Additional explanatory notes on the analytic proof of the finite generation of the canonical ring, arxiv: Yum-Tong Siu, Finite generation of canonical ring by analytic method (arxiv: ), J. Sci. China 51 (2008), Yum-Tong Siu, Techniques for the analytic proof of the finite generation of the canonical ring (arxiv: ), Proceedings of the Conference on Current Developments in Mathematics, November Algebraic Proof: C. Birkar, P. Cascini, C. Hacon, and J. McKernan, Existence of minimal models for varieties of log general type, arxiv:math/

62 Statement Abundance Conjecture lim sup m log dim Γ (X, mk X + A) log m = lim sup m for an ample line bundle A on a compact complex algebraic manifold X. Analogous to PDE estimates of adding a positive number ε times a friction term and then letting ε 0 with the estimates independent of ε. Proof by the technique of deformational invariance of plurigenera for the case of non general type. Paper still being written up. log dim Γ (X, mk X ) log m 62

63 Generic Existence Versus Special Existence There are two kinds of constructions from the technique of multiplier ideal sheaves. One kind uses artificial singularities (to imitate electric charge in a Riemann surface) to construct holomorphic sections in order to obtain effective results. This kind of construction puts a singularity at a general point. It is a kind of generic existence. Another kind of construction is by instability. To be constructed is a destabilizing subspace. For example, a rational curve in a Fano manifold. Such a curve is a very special curve. It is a kind of special existence. 63

64 Construction of Rational Curves in Fano Manifolds Without Using Positive Characteristic. Problem. Let X be a Fano manifold, i.e., X is a compact complex manifold whose anticanonical line bundle K X is positive (admitting a metric e ϕ with ϕ smooth strictly plurisubharmonic). How to prove the existence of rational curves in X. A rational curve in X means a nonconstant holomorphic image of the Riemann sphere P 1. The existence of rational curves in Fano manifolds was proved by Mori in 1979 by using characteristic p > 0 and the method of bend-and-break. S. Mori, Projective manifolds with ample tangent bundles, Ann. of Math. 111 (1979), So far there is no analytic method (or any other method not using characteristic p > 0) which can prove the existence of rational curves in Fano manifolds, except the following. 64

65 Special case of positive bisection curvature (which becomes a trivial case after the proof of Hartshorne s conjecture. Yum-Tong Siu and Shing-Tung Yau, Compact Kähler manifolds of positive bisectional curvature. Invent. Math. 59 (1980), Method uses energy-minimizing harmonic map from the 2-sphere. Needs positivity in every direction of normal bundle from the bisectional curvature. 65

66 Thorn on the Side of Analysts Usually a technique is first obtained in analysis and then adapted to algebraic geometry. For example, Kodaira s vanishing theorem and Hodge theory. So far the technique of rational curves in Fano manifolds can only be done by algebraic geometric methods of characteristic p > 0. For the last thirty years this has been a thorn on the side of analysts. 66

67 One approach is to analytically construct rational curves of Fano manifolds by using destabilizing subspaces in complex Monge-Ampère equations of the type which magnify singularities (such as the one used for the construction of Kähler-Einstein metrics of positive Chern class). A destabilizing subspace inherits some properties of being Fano from the ambient Fano manifolds. When the dimension of a destabilizing subspace is 1, its support gives a rational curve. There is still on hurdle remaining in this analytic approach, which is the lower bound of positivity for the analog of the Ricci curvature for the destabilizing subspace. Yum-Tong Siu, Dynamic multiplier ideal sheaves and the construction of rational curves in Fano manifolds, arxiv: