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1 [ ,12,16,17,18] [ ,4,5,6,7,8,9,10] K K K P 2 Winkelmann P 2 D D D C2 X X U p V p ϕ p U p U q ϕ q ϕ 1 p : ϕ p (U p U q ) ϕ q (U p U q ) X 1

2 R Φ : R G,G A G Φ Φ R Φ 1 (A) G A Φ 1 (A) R R Φ 1 (A) Φ 1 (A) (R, Φ, G) R G R R R R G σ R σ (R 1, Φ 1, G 1 ) D 1 (R 2, Φ 2, G 2 ) D 2 φ D 2 f f φ Φ σ = Φ 1 (A) R δ 1, δ 2 G-R, R S R S R Ψ S G S R R σ S R Φ(A) R Φ(A) G A R A R I inding point R A 2

3 C N G Σ Σ C N M(Σ) Sing(Σ) Σ M(Σ) Σ Σ M(Σ) Σ Σ C N f f M(Σ) Σ Σ Σ(M) Σ M(Σ) Σ Sing(Σ) Σ Sing(Σ) Σ Σ Σ = {z 2 xy = 0} σ (0, 0) R Σ Ψ : R Σ s.t. Σ in N, Ψ on R. R Rde R σ Σ Σ Σ N Σ R X X 3

4 X,Y C(X, Y ) X Y Y Z F F Z p F Hol(X, Y ) X Y Y Y d Y C(X, Y ) F Hol(X, Y ) F C(X, Y ) f f F F C(X, Y ) F C F X,Y d X,d Y D(X, Y ) C(X, Y ) C(X, Y ) Hol(X, Y ) D(X, Y ) Y X,Y F Hol(X, Y ) Φ : X F Y, T φ : X T Y φ(, t) F φ : T F such as φ := φ(, t) Douady X Y Hol(X, Y ) F Hol(X, Y ) X,Y,F X Φ : X F Y f 0 F Φ(, f 0 ) : X Y f F 4

5 f : X Y f : X X f Y p f : X X f onto, p (O(X)) = O(X f ). p f f : X f Y X f f 1 (y), y Y X,Y F Hol(X, Y ) s.t. f F F f : X X Y, f F p : X X onto f : X Y X π : X X Y f 0 Hol(X, Y ), f = f 0 π Hol( X, Y ) F(resp. F) f 0 (resp. f 0 ) Hol(X, Y )(resp.hol( X, Y )) f f π F F X,Y Hol(X, Y ) F x 0 X,f f(x 0 ) F Y x 0 X, y 0 Y F 0 = {f F; f(x 0 ) = y 0 } dimf dmy. X,Y f rankf = max x X {dim x X dim x f 1 (f(x))}, f 1 (f(x)) X k = 0, 1,, dimx Hol(X, Y, k) = {f Hol(X, Y ), rankf = k} Sur(X, Y ) Hol(X, Y ) surjective F in(x, Y ) X Sur(X, Y ) = Hol(X, Y, m), m = dimy. X,Y X F in(x, Y ), Sur(X, Y ), Hol(X, Y, k), 0 k m = dmx Hol(X, Y ) X Aut(X) Hol(X, X) 5

6 X,Y Y taut K X,L Y F K,L = {f Hol(X, Y ); f(k) L } Y Hol(X, Y ) Hol(X, Y ) F K,L Hol(X, Y ) X Y (taut x 0 X Hol(X, Y ) Y : f f(x 0 ) H.Cartan X Aut(X) n(n+2) x X, K inx {f Aut(X) : f(x) K} x X Aut(X) isotropy( G X x X g(x) = x G x Aut(X) Lie aut(x) X Aut(X) aut(x) Aut(X) X Aut(X) too large to be a Lie group as in the case of X = C n n 2. G G x, y (x, y) xy, x x 1 G 6

7 X X K i, K i K i+1 X = K i X Aut(X) X g 2 X 84(g 1). g 1 X o X f : X X, f(o) = o df o : T o X T o X o f df o 1, 2) df o T o X f X 3) det(df o ) = 1 f X ρ Hol(X, X) hol. retraction ρ 2 = ρ ρ(x) X hol. retracton ρ(x) ρ ρ X hol. retraction ρ(x) X x ρ(x) X x ρ(x) X taut {f Hol(X, X),f k := f f f} compactly div. X hol. ret. ρ {f k } h Hol(X, X) h = γ ρ γ Aut(ρ(X)). {f k } ρ f ρ(x) Aut(ρ(X)). 7

8 X taut f Hol(X, X) f(x) X f x 0 X {f k } x 0 X f Hol(X, X) k s.t. f k hol. retraction, {f k } hol. retraction f hol.retraction Denjoy-Wolff f Hol(D, D), D if and only if {f k } X in C n with C 3 boundary f Hol(X, X) X if and only if {f k } P n,h 1,, H n+p f Hol(C, P n H i ) [ n p ]. p = n + 1 f P n 2n+1 i=1 H i P n T : C n P m m + 2 T P m n, m 1. Y: XA codim 2 f : Hol(X A, Y ) Hol(X, Y ) Y f H(, Y ) Hol(, Y ) X A X Y Z Y Z h Hol(X A, Y ) Hol(X, Z) Y A 8

9 X A Y h Hol(X A, Y ) Hol(X, Y ) X d X 0 A X X A X A Y Z X A Hol(X A, Y ) Hol(X, Z) {f n } Hol(X A, Y ) Hol(X, Z) X Y f Mer(X, Y ) X Y Hol(X, Y ) X,Y Y F Hol(X, Y ) Hol(X, Y ) F Φ : X F X Y (x, f) (x, f(x)) evaluation map dimf dimy. Z B Cartier Y = Z B Z X A Hol(X A, Y, n) n Hol(X A, Y, n) F onto map f X Y dominant G f X Y Y onto map under the natural projection X Y Y 9

10 f onto map X,Y f dominant x X Sin(X) s.t. f regular df : T x X T f(x) Y onto X,Y g(y ) 2 {f : X Y }, f onto X,Y Y Dom(X, Y ), Dom(X, Y ) X Y onto Lang X,Y Y Dom(X, Y ) Y A X B Y {f : f Hol(X, Y ), f(a) = B} X with a Cartier divisor A,Z with a Cartier divisor B, s.t. Y = Z B Z Dom(X A, Y ). X with a Cartier divisor A s.t. X A X Bim(X A) A X,B Z Hol(X A, Z B) dimz 1. X f : X X onto X section X A X 10

11 Y Y X Y K Y ample f : X A Y maximal rank X Y X A X Y smooth variety with a very ample K Y s.t. dimy dimx f : X A Y maximal rank X Y X X X mod X p, q, p q d X (p, q) 0 X = {p X; q X, s.t.p qwithd X (p, q) = 0} X. X X X Remark X C 2 X X X X X X taut mod D F Hol(D, X) mod F = {f j } F Hol(D, X) D conpact diverge to. taut space mod mod. D Σ D Σ D inclusion D taut. C n taut. C n C 1 taut. 11

12 taut. C 1 Example 5.2.9) C n taut taut. X = {g = 0, g O(X)} hyp. mod taut mod? C n taut X taut D Hol(D, X) Aut(X-E) Aut(X)? ( R R R X R = R X t (t R ) F = {f j (z), f j O(D)} D f O F f j O (D) f j M M 12

13 X A X Y C n Y S, S C n f Hol(X A, Y ) A f(x A) S. X s.t. X C n X S where S X X mod S taut mod S. X X S = {g = 0, whereg O X taut mod S. M = C m,a C m S C m M mod S f Hol(, M) f(0 : M), f( ) S, f(0 : M) S ρ 1, s.t. f( ) S =. M E f Hol(M E, X) X f E U(p) X 0 f(u E) X 0 f Hol(M, X) M E X X K 13

14 f Hol(M E, X) s.t. f(m E) X X f Hol(M, X) M M,E X X f Hol(M E, X) Hol(M, X) M M,E f : M E X X f Hol(M, X) X E 14

9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x

9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x 2009 9 6 16 7 1 7.1 1 1 1 9 2 1 f(x, y) = xy sin x cos y x y cos y y x sin x d (x, y) = y cos y (x sin x) = y cos y(sin x + x cos x) x dx d (x, y) = x sin x (y cos y) = x sin x(cos y y sin y) y dy 1 sin

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