FANO多様体の諸問題 (複素幾何学の諸問題)

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1 $\bullet$ $\bullet$ FANO CONTENTS Fano MMP MMP Fano Fano Novelli-Occhetta Casagrande -BCHM Fano -VMRT VMRT Hwang $\tau:\mathcal{u}--*c$ Fano Fano (MMP) Fano ( ), 1 Kebekus, Mok-Hwang (VMRT ), [BCHM] Fano Fano Fano - - 1

2 107. Fano -VMRT - Fano $*$, $***$, $**$, ( ) $*$ ( ) 2. FANO Fano Fano (Fano ) $\mathbb{r}$- $D$ $(X, D)$ (klt pair) $K_{X}+D$ $\mathbb{r}$-cartier (discrepancy) $-1$ 2 $(X, D)$ $-(K_{X}+D)$ $(X, D)$ Fano Fano 21. $D=0$ 2.2. MMP. [BCHM] C. Birkar, P. Cascini, C. Hacon, and J. McKernan, Existence of minimal models for varieties of $log$ geneml type. J. Amer. Math. Soc. 23 (2010), no. 2, $(X, D)$ (MMP) 3 MMP $t$ MMP($MMP$ with scale) $(X, D)$ $A$ $\mathbb{q}$- $(X, D+A)$ $K_{X}+D+A$ $(K_{X}+D)$-MMP $(X, D)$ $(X, D+A)$ ($K_{X}+D+A$ ) MMP 4 2 Ja nos Kolla r, ( ) 3 4 Shokurov BCHM MMP

3 $\lambda_{1}$ $:= $\lambda_{2}$ $:= $\epsilon$ 108 Fano $(X_{1}, D_{1}):=(X, D),$ $A_{1}:=A$ $K_{X}+D$ $(X, D)$ $(K_{X}+D)$-MMP $0<\lambda_{1}\leq 1$ $\}$ \inf\{t\geq 0 K_{X_{1}}+D_{1}+tA_{1}$ $(K_{X_{1}}+D_{1})\cdot R_{1}<0,$ $(K_{X_{1}}+D_{1}+\lambda_{1}A_{1})\cdot R_{1}=0$ $R_{1}$ $R_{1}$ 5 $(K_{X}+D)$-MMP $f_{1}:x_{1}arrow Y_{1}$ $fi$ $fi$ $(K_{X}+D)$-MMP $fi$ $X_{2}:=Y_{1}$, $X_{2}$ $X_{1}$ $D_{2},$ $A_{2}$ $D_{1},$ $A_{1}$ $X_{2}$ $(K_{X_{1}}+D_{1}+\lambda_{1}A_{1})\cdot R_{1}=0$ $(X_{2}, D_{2}+\lambda_{1}A_{2})$ $K_{X_{2}}+D_{2}+\lambda_{1}A_{2}$ $X,$ $D,$ $A$ $(X_{2}, D_{2})$ $X_{2},$ $D_{2)}\lambda_{1}A_{2}$ $\}$ \inf\{t\geq 0 K_{X_{2}}+D_{2}+tA_{2}$ $\lambda_{2}\leq\lambda_{1}$ $(K_{X_{2}}+D_{2})\cdot R_{2}<0,$ $(K_{X_{2}}+D_{2}+\lambda_{2}A_{2})\cdot R_{2}=0$ $X_{1}-*\cdots--*$ $R_{2}$ $1\geq\lambda_{1}\geq\lambda_{2}\geq\ldots$ $D_{i},$ $A_{i}$ $X_{i}-*X_{i+1}-*\cdots$, $D_{1}$ $A_{1}$ $X_{i}$ $(X_{i}, D_{i}+\lambda_{i}A_{i})$ $K_{X_{i}}+D_{i}+\lambda_{i}A_{i}$ $(X_{i}, D_{i})$ $(K_{x_{:}}+D_{i})$. $<0$ $A_{i}$ $(K_{X}+D)$-MMP. $>0$ 1 (BCHM). $D$ (big) 6 $MMP$ 2.3. MMP. $(X, D)$ $K_{X}+$ $\mathbb{q}$- $D$ (pseudo-effective)7 $(K_{X}+D)$ MMP 2. $K_{X}+D$ $(K_{X}+D)-MMP$ 7 $Y$ MMP $H$ $A$ $H$ $\mathbb{q}$ $\mathbb{q}$- $A$ $(X, D+A)$ $K_{X}+D+A$ $(X, D+\epsilon A)$ $K_{X}+D+\epsilon A$ $D+\epsilon A$ 1 $D$ 1 $A$ $(K_{X}+D+\epsilon A)$-MMP $K_{X}+D+\epsilon A$ $(X, D+\epsilon A)$ $A$ $A$ MMP $(K_{X}+D+\epsilon A)$-MMP $(K_{X}+D)$-MMP $(X, D+\epsilon A)$ $\square$ $(X, D)$ [BCHM] 5 $D$ $\mathbb{q}arrow$ 6 $\mathbb{r}$- $\mathbb{r}$- $\mathbb{r}$ $\mathbb{q}$ 7

4 $\mathbb{c}$ Fano 3(BCHM). Fano $(X, D)$ $B$ $B$ $\rangle$ $MMP$ B-MMP, Fano 3 Y. Hu and S. Keel, Mori dream spaces and $GIT$. Dedicated to William Fulton on the occasion of his 60th birthday. Michigan Math. J. 48 (2000), $B$ $A$ 3 3 $A$ B-MMP B-MMP $(K_{X}+D)$-MMP $B$ $B$ $m$ $m(k_{x}+$ $D)+B$ $H\in -m(k_{x}+$ $D)+B $ $A$ $H$ $\mathbb{q}$ $(X, D+A)$ $m\gg O$ $K+D+A$ $\frac{1}{m}b$ $K_{X}+D+ \frac{1}{m}a$ $A$ $\mathbb{q}$- 1 $A$ B-MMP 2 $\mathbb{q}$- $(K_{X}+D+ \frac{1}{m}a)$-mmp, $(Kx+D+ \frac{1}{m}a)-mmp$ $(Kx+D)-MMP$ $\square$ 2.5. Fano Fano $\pi:\mathcal{x}arrow S$ $\pi$ Fano Fano 9 Fano $\epsilon$ Fano $(X, D)$ $-1+\epsilon$ $(X, D)$ $\epsilon$-jll $D=0$ $(X, 0)$ 1- $\epsilon$ 1(BAB $***$ ). - Fano Alexeev ( V. Alexeev and S. Mori, Bounding Singular Surfaces of General Type 10) Borisov 11 3 toric Fano 15 3 (X ). $8[BCHM]$ Fano $(X, D)$ 9F. Campana, Connexite rationnelle des varietes de Fano, Ann. Sci. \ Ecole Norm. Sup. (4) 25 (1992), no. 5, J. Koll\ ar, Y. Miyaoka, and S. Mori, Rational connectedness and boundedness of Fano manifolds, J. Differential Geom. 36 (1992), no. 3, $1_{ //wwwmath princeton.edu $/kollar/$ 11 Alexeev

5 110 Fano J. McKernan and Y. Prokhorov, Threefold thresholds, Manuscripta Math. 114 (2004), no. 3, C. Birkar, Ascending chain condition for $log$ canonical thresholds and termination of $log$ fiips, Duke Math. J. 136 (2007), no. 1, $d$ 4. 2 (1) $\mathbb{q}$- $d$ $MMP$ (2) $d$ $\epsilon$ $BAB$ Picard 1, $\mathbb{q}$- - Fano $d+1$ C. Birkar and V. V. Shokurov, $Mld svs$ thresholds and flips, J. Reine Angew. Math. 638 (2010), ( 18 ). BAB 2( BAB $***$ ). $no$ $d$ 5. 3 (1) $\mathbb{q}$- $d$ $MMP$ (2) (minimal $d$ $log$ discrepancy $(mld)$) 12 (3) $d$ $\mathbb{q}$ $BAB$ Picard 1, - Fano f $d+1$ BAB Batyrev 3 $(**)$. $-mk_{x}$ Cartier Fano 13 $m$ J. McKernan, Boundedness of $log$ terminal Fano pairs of bounded index, $arxiv:math/ $ 3. Fano 14 Kollar 12 statement $13_{X}$ - $14_{n}$ $n-1$ $S$ $P^{1}\cross S-*X$

6 $\mathcal{h}$ $\rho \mathcal{u}\mathcal{k}\downarrowarrow^{\mu}$ $\mathcal{h}$ 111 J. Koll\ ar, Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 32. Springer-Verlag, Berlin, viii$+320$ pp 15 $\mathcal{o}_{x}$- $x\in X$ $\mathcal{f}^{x}:=\mathcal{f}_{x}\otimes_{0_{x,x}}k(x)$ 16 $n$ ( ) 1:1 ( ) RatCurve$n(X)$ 17 Univ $(X)arrow RatCurve^{n}(X)$ $\rho:\mathcal{u}arrow \mathcal{k}$ $\mathbb{p}^{1}$- $RatCurve^{n}(X)$ Univ $(X)arrow RatCurve^{n}(X)$ $\mu:\mathcal{u}arrow X$ $\mathcal{k}:={\rm Im}\mu$ Locus 18 (3.1) Locus $\mathcal{k}\subset X$ $\mu$ $x\in X$ $\mu^{-1}(x)$ (locally unsplit) ( ) $x\in X$ $\mu^{-1}(x)$ Univ $(X)arrow RatCurve^{n}(X)$ IP 1, $\mathbb{p}^{1}\cross Hom($ $X)_{red}arrow Hom(\mathbb{P}^{1}, X)_{red}$ $\mathbb{p}^{1}$ Aut $F:\mathbb{P}^{1}\cross Hom(\mathbb{P}^{1}, X)_{red}arrow X$ 20 (P., $[f )$ 19 ($p,$ $[f )\in \mathbb{p}^{1}\cross Hom(\mathbb{P}^{1}, X)_{red}$ $T_{P^{1}}^{p}\cross H^{0}(\mathbb{P}^{1}, f^{*}t_{x})arrow T_{X}^{f(p)}$ Zariski $\mathbb{p}^{1}\cross Hom(\mathbb{P}^{1}, X)arrow X$ $H^{0}(\mathbb{P}^{1}, T_{1P^{1}})arrow T_{1P^{1}}^{p}$ $T_{X}^{f(p)}$ $H^{0}(\mathbb{P}^{1}, T_{1P^{1}})$ $T_{X}^{f(p)}$ $H^{0}(\mathbb{P}^{1}, f^{*}t_{x})arrow$ $H^{0}(\mathbb{P}^{1}, f^{*}t_{x})arrow T_{X}^{f(p)}\simeq(f^{*}T_{X})^{p}$ $f^{*}t_{x}\simeq\oplus_{i=1}^{n}o_{p^{1}}(a_{i})$ $a_{i}$ $f^{*}t_{x}$ $f$ $f$ $f$ 21 $\grave$ $f^{*}t_{x}$ $\iota_{\llcorner}^{\vee}$ self-contained 16 $n$ 17 (normalization) $18\mathcal{K}$. $S$ Locus $S$ 19 $Hom(P^{1}, X)$ $P^{1}\cross \mathcal{h}arrow X$ $P^{1}$ 20 O 21 $f^{*}t_{x}$ $f:p^{1}arrow X$ $P^{1}\cross Hom(P^{1}, X)_{red}arrow X$

7 $ _{-\dot{\hat{\supset}}}^{-}--$ $([f, 112 Fano $x\in$ Locus (3.2) $\dim X+\deg \mathcal{k}-3\leq\dim \mathcal{k}=\dim$ Locus $\mathcal{k}+\dim$ Locus $\mathcal{k}_{x}-2$ (3.3). $\deg \mathcal{k}\leq\dim$ Locus $\mathcal{k}_{x}+1$ $\deg \mathcal{k}$ $(-K_{X}\cdot C)$ 22 $\mathcal{k}_{x}=\rho(\mu^{-1}(x))$ $Hom$ $\mathcal{u}arrow$ Locus $\mu^{-1}(x)$ $\dim \mathcal{k}_{x}$ $\mu _{\rho^{-1}(\mathcal{k}_{x})}$ (generically finite) $\mu _{\rho^{-1}(\mathcal{k}_{x})}$ bend and break 23 $\mathcal{k}_{x}$ Locus $\dim\mu^{-1}(x)=\dim$ Locus (3.2) $\mathcal{k}_{x}-1$ $\mathcal{u}arrow X$ $\dim$ Locus $\mathcal{k}=n$ $f:\mathbb{p}^{1}arrow Carrow X$ $Carrow X$ $f^{*}t_{x}\simeq\oplus_{i=1}^{n}o_{p^{1}}(a_{i})$ $a_{i}$ $Hom(\mathbb{P}^{1}, X)$ $[f]$ $H^{0}(\mathbb{P}^{1}, f^{*}t_{x})$ $[f]$ $[f]$ $Hom(\mathbb{P}^{1}, X)$ $[C]$ $H^{0}(\mathbb{P}^{1}, f^{*}t_{x})/h^{0}(\mathbb{p}^{1}, \sim 1)$ $[C]$ $Hom(\mathbb{P}^{1}, X, 0\mapsto x)$ (3.2) $\mathcal{k}_{x}$ $[f]$ ( ) $[C]$ $Hom(\mathbb{P}^{1}, X, 0\mapsto x)\cross \mathbb{p}^{1}arrow$ $\mathcal{k}_{x}$ Locus y)(y\neq 0)$ $\dim$ Locus $\mathcal{k}_{x}=\#\{i a_{i}\geq 1\}$ $\deg \mathcal{k}=\sum_{i=1}^{n}a_{i}$ (3.2) $0\leq p\leq n-1$ $\sum_{i=1}^{n}a_{i}=\#\{i a_{i}\geq 1\}+1$ Aut $\mathbb{p}^{1}$ $f^{*}t_{x}\simeq O_{P^{1}}(2)\oplus O_{1P^{1}}(1)^{\oplus p}\oplus O_{P^{1}}^{\oplus n-p-1}$ $f$ (standard rational curve) 24 $P$ $f$ 4. FANO Fano 3 Picard 1 Fano $F(X)$ $:= \max\{m\in N (-K_{X})/m\in$ Pic $X\}$ Fano Fano 1 3 Fano $g(x)=$ $22C$ 23 24Kolla r 25Max-Planck Golyshev

8 $\mathbb{q}$ Mazur 26? $(^{***})$ : $\mathbb{z}/m\mathbb{z}(1\leq m\leq 12, m\neq 11),$ $\mathbb{z}/2\mathbb{z}\oplus \mathbb{z}/2m\mathbb{z}(1\leq m\leq 4)$. ( ) 4 5 $(**)$. Picard 1 4 Fano Fano 6 $(*\sim**)$. Fano 4 : H.Sato, Toward the classification of higher-dimensional toric Fano varieties, Tohoku Math. J. 52 (2000), (221), (222), (223) 4 Fano Picard 29 3 Calabi-Yau 3 Calabi-Yau 4 Fano Calabi-Yau Picard $B_{5}$ 3 Fano 5 del Pezzo $U_{22}$ 4.2. Fano (geography) Fano S. Mukai, Problems on chmcterization of the complex projective space, Birational Geometry of Albgebraic Varieties Open Problems, The 23th international symposium, division of mathematics, the Taniguchi fundation, August 22 August 27, Katata, 1988 $26B$. Mazur, Modular curves and the Eisenstein ideal, IHES Publ. Math. 47 (1977), B. Mazur, Rational isogenies of prime degree, Invent. Math. 44 (1978), $27F(X)=1$ 28O. Kuchle, On Fano 4-fold of index 1 and homogeneous vector bundles over Grassmannians, Math. Z. 218 (1995), no. 4, Grassmann $29_{4.4}$ Casagrande 30 $F(X)\geq 2$ 4 Fano 3 Calabi-Yau?

9 114 Fano Picard Fano 7( ). $(F(X)-1)\rho(X)\leq\dim X$ $X\simeq(\mathbb{P}^{F(X)-1})^{\rho(X)}$ S. Mukai, Biregular classification of Fano 3-folds and Fano manifolds of coindex 3, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), no. 9, $F(X)=\dim X-3$ Fano $ -K_{X}/F(X) $ 31 Fano Fano Fano Fano $i(x):= \min\{(-kx\cdot C) C$ $\}$ Fano $(-K_{X}/F(X))\cdot C\geq 1$ $i(x)\geq F(X)$ $(-K_{X})\cdot C\geq F(X)$, 8( ). $(i(x)-1)\rho(x)\leq\dim X$ $X\simeq(\mathbb{P}^{i(X)-1})^{\rho(X)}$ - 32 ( ) - -Shepherd-Barron 6. $n$ Fano $i(x)\geq n+1$ $i(x)=n$ 33 9 $(**)$. $n\geq 4$ $n$ Fano $i(x)=n-1$ ) del Pezzo $F(X)$ $i(x)$ 36 $a>b>0)$ $F(X)=1,$ $i(x)=b+1$ $X=\mathbb{P}^{a}\cross \mathbb{p}^{b}(a+1,$ $b+1$ 31 Mella Ambro $32_{F(X)=\dim X+1}$ $F(X)=\dim X$ 33 Hwang-Mok $34F(X)=n-1$ Fano 35 $i(x)=n-1$ $n=4$ $\rho(x)\leq 2,$ $n\geq 5$ $\rho(x)=1$ $n\geq 5$ ( 6 ) 11 36

10 $\overline{\mathcal{k}}_{1}$ $(**)$. Fano $i(x)=$ $\min$ $\{(-K_{X}\cdot l_{i}) [l_{i}]\in R_{i},$ $l_{i}$ $\}$ $R_{i}:X$ 3 4 T. Tsukioka, On the minimal length of extremal rays for Fano 4-folds, arxiv: Picard $(**)$. Picard 1 Fano $i(x)=1$? $F(X)$ $i(x)$ $F(X)=i(X)$? $F(X)=1$ $-K_{X}$ $F(X)=1$? $\dim X=3$ Fano $\dim$ $\llcorner\hat$ t $X\leq$ [NO] C. Novelli and G. Occhetta, Rational curves and bounds on the Picard number of Fano manifolds, Geometriae Dedicata, 147, no. 1, 207-2I7 4, $(*\sim**)$ $6$. 8? 4.3. Novelli-Occhetta [NO] Picard Cartier $S$ $N_{1}(X)$ $N_{1}(S, X)$ RatCurve$n(X)$ $:={\rm Im}(N_{1}(S)arrow N_{1}(X))$ : $S$ Chow -S $(X)$ RatCurv$e^{}$ $\mathcal{k}_{1}$ $\pi_{1}:x-*y_{1}$ $\mathcal{k}_{1}$ $Y_{1}$ $\overline{\mathcal{k}}_{1}$ $Y_{1}$ 39 $\mathcal{k}_{2}$ $\mathcal{k}_{1}$ $\mathcal{k}_{1},$ $\mathcal{k}_{2}$ $\pi_{2}:x--*$ 37Kolla r 113. $38J$. Koll\ ar, Y. Miyaoka, and S. Mori, Rational connectedness and boundedness of Fano manifolds, J. Differential Geom. 36 (1992), no. 3, $39X-*Y_{1}$ $\mathcal{k}_{1}$

11 $\Gamma_{j_{0}}$ $\overline{\mathcal{k}}_{1}$ $\overline{\mathcal{k}}_{1}$ $\overline{\mathcal{k}}_{1}$ $\mathcal{k}_{1},$ $\mathcal{k}_{m}$ 116 Fano $\overline{\mathcal{k}}_{2}$ $Y_{2}$ $\mathcal{k}_{1},$ $\mathcal{k}_{2}$ $\mathcal{k}_{3}$ $\mathcal{k}_{1},$ $\mathcal{k}_{m}$ $\ldots,$ $\overline{\mathcal{k}}_{2}$ 2 $\overline{\mathcal{k}}_{m}$ (4.1) $\dim X\geq(\sum_{i=1}^{m}\deg \mathcal{k}_{i})-m$ 40 $\deg \mathcal{k}_{i}\geq i(x)$ $\dim X\geq m(i(x)-1)$ $\mathcal{k}_{i}$ 41 (quasiunsplit), $\rho(x)=m$ 42 $\overline{\mathcal{k}}_{i}$ $\mathcal{k}_{i}$ [NO] $i(x) \geq\frac{\dim X+3}{3}$ $\mathcal{k}_{i}$ ( ) $\deg \mathcal{k}_{i\text{ }}\geq 2i(X)$ i $i(x) \geq\frac{\dim X+3}{3}$ (41) $m=1$ 2 $\overline{\mathcal{k}}_{1}$ $\overline{\mathcal{k}}_{1}$ $k$ 2 2 $\overline{\mathcal{k}}_{1}$ $k$ $x\in X$ $\Gamma_{1},$ $\Gamma_{k}$ $x\in\gamma_{1},$ $\Gamma_{1}\cap\Gamma_{2}\neq\emptyset,$ $\Gamma_{k-1}\cap\Gamma_{k}\neq\emptyset$ $\overline{\mathcal{k}}_{1}$ $\ldots,$ $\ldots,$ $\Gamma_{j}$ $\Gamma_{j}$ $\Gamma_{j}$ $i$ $j_{0}$ $\Gamma_{1}\cup\cdots\cup\Gamma_{k}$ $\Gamma$j $\Gamma_{j_{0}}$ $\mathcal{k}_{1}$ ( ). $j_{0}\geq 2$ $\Gamma_{j_{0}-1}$ $x_{1}$ $io=2$ $x_{1}=x,$ $j_{0}\geq 3$ $\Gamma_{j_{0}-2}\cap\Gamma_{j_{\text{ }}-1}$ $Y$ $(\mathcal{k}_{1})_{x_{1}}$ Locus $\Gamma_{j_{0}}$ (3.3) $\dim Y\geq\deg \mathcal{k}_{1}-1$ $i(x) \geq\frac{\dim X+3}{3},$ $\deg \mathcal{k}_{1}\geq 2i(X)$ $\Gamma_{j_{0}}$ $\dim Y>\dim X-i(X)$ $\Gamma $ $Y$ $j_{0}$ $x_{1}$ 40 (3.3) [ACO]M. Andreatta, E. Chierici, and G. Occhetta, Generalized Mukai conjecture for special Fano varieties, Cent. Eur.. Math. 2 (2004), no. 2, Lemma $J$ 54 $X\simeq(P^{i(X)-1})^{m}$ 41 [NO] Lemma44 $A$ G. Occhetta, characterizalion of products of projective spaces, Canad. Math. Bull. 49 (2006), no. 2, $\mathcal{k}_{i}$ 42 [ACO] Lennna 44 $\ldots,$

12 $\Gamma_{j_{0}}$ 117 $\Gamma $ $N_{1}(Y, X)$ $W$ $\Gamma $ $\dim$ Locus $\mathcal{w}_{y}\geq\dim Y+(-K_{X}\cdot\Gamma )-1>\dim X-i(X)+i(X)-1$ Locus Wy $Y$ $\mathcal{w}$ Locus, $\mathcal{w}_{y}=x$ $\mathcal{w}$ $\mathcal{k}_{1}$ 13 $(**)$. $i(x)= \frac{n+2}{3}$, $\frac{n+1}{3}$ 43 $\rho(x)\geq 2$? 4.4. Casagrande BCHM -. C. Casagrande, On the Picard number of divisors in Fano manifolds, arxiv: [BCHM] Fano ( 16). Picard ( 4 ) Fano Picard ( 12) 4 Fano Picard $S$ $\rho(x)-\rho(s)\leq$ codim $N_{1}(S, X)$ Fano Fano (1) $i(x)=2$ 1 45 Fano $Z$ $\mathbb{p}^{1}$ (2) $D$ $\rho(d)$, ) $i(x)$ 1 $Z$ - Fano $N_{1}(D, X)=N_{1}(X)$ $\rho(x)\leq$ $N^{1}(X)arrow N^{1}(D)$ ($i(x)>1$ ) $D$ 3 D-MMP $K_{X}$-MMP Fano MMP $Y$ $Yarrow Z$ MMP $N_{1}(D, X)$ $D$ $Y$ $D_{Y}$ $D_{Y}$ $D_{Y}arrow Z$ $\rho(y)-\rho(z)=1$ codim $N_{1}(D_{Y}, Y)\leq 1$ $X_{i}$ MMP (X ) $X_{i}arrow Y_{i}$ $R_{i}$ $X_{i}--*X_{i+1}$ $X_{i}arrow Y_{i}$ 46 Fano $X_{i}arrow Y_{i}$ $D$ $D_{i},$ $D_{i+1}$ $R_{i}\subset N_{1}(D_{i}, X_{i})$ $X_{i},$ $X_{i+1}$ codim $N_{1}(D_{i}, X_{i})=$ codim $N_{1}(D_{i+1}, X_{i+1})$, codim codim $N_{1}(D_{i+1}, X_{i+1})$ codim $N_{1}(D_{Y}, Y)\leq 1$ $N_{1}(D_{i}, X_{i})-1=$ 43 $n=5$ Occhetta $45Y$ 8 46 (2) Picard Picard

13 118 Fano codim $N_{1}(D, X)$ $R_{i}\not\subset N_{1}(D_{i}, X_{i})$ $R_{i}\not\subset N_{1}(D_{i}, X_{i})$ $X_{i}arrow Y_{i}$ $D_{i}$ $l$ $X_{i}arrow Y_{i}$ 1 $D$ Sing $X_{i}\subset D_{i}$ $X_{i}-*X$ $l$ $X_{i}$ Sing $l$ $-K_{X_{1}}\cdot l\leq 1$ $X_{i}$ 47 Sing $l$ $X-*X_{i}$ MMP $-K_{X}$ ( 48 ). Fano $l$ $X_{i}$ Sing $X_{i}arrow Y_{i}$ $Y_{i}$ Sing 2 $\not\subset N_{1}(D_{i}, X_{i})$ $X_{i}arrow Y_{i}$ Fano $i(x)$ 1 MMP $\not\subset$ $N_{1}(D_{i}, X_{i})$ $X_{i}arrow Y_{i}$ $-K_{X}$ 1 codim $N_{1}(D, X)=$ codim $N_{1}(D_{Y}, Y)\leq 1$. codim $N_{1}(D_{Y}, Y)=1$ $D_{Y}$ $Yarrow Z$ $Yarrow Z$ 1 $i(x)\geq 2$ $\square$ $X=Y$ $Xarrow Z$ $\mathbb{p}^{1}$- 5. FANO -VMRT VMRT Kebekus Hwang Mok $\mathbb{p}(t_{x}^{*})$ 49 $\mathbb{p}(t_{x})$ Grothendieck $\mathbb{p}^{*}(t_{x})$ $V$ 1 $\mathbb{p}_{*}(v)$ $RatCurve^{n}(X)$ $\tau:\mathcal{u}-*\mathbb{p}(t_{x}^{*})$ : $\alpha\in \mathcal{u}$ $\rho(\alpha)$ $x:=\mu(\alpha)$ $\mathbb{p}(t_{x}^{*})$ $C\subset \mathbb{p}(t_{x}^{*})$ $\pi:c\subset \mathbb{p}(t_{x}^{*})arrow X$ (Variety of Minimal Rational Tangents), VMRT $\tau$ $\mathcal{k}_{x}$ bend and break $x\in X$ $p$ $p$ $f:\mathbb{p}^{1}arrow X$ $f^{*}t_{x}\simeq O_{P^{1}}(2)\oplus O_{P^{1}}(1)^{\oplus p}\oplus O_{P^{1}}^{\oplus n-p-1}$ $\tau$ $p$ $\mu^{-1}(x)arrow \mathcal{k}_{x}$ $\mathcal{k}_{x}-*c_{x}$ Kebekus $p$ $\mu^{-1}(x)-*c_{x}$ 47S. Mori, Flip theorem and the existence of minimal models for 3-folds, J. Am. Soc. Sci., 1 (1988) Kolla r $49_{T_{X}}$ $T_{X}^{*}$

14 119 S. Kebekus, Families of singular rational curves, J. Algebraic Geom. 11 (2002), no. 2, VMRT $C_{x}\subset \mathbb{p}_{*}(t_{x}^{x})$ 14 $(**)$. Picard 1 $S$ Fano Picard 1 50 $x\in X$ VMRT $C_{x}\subset \mathbb{p}_{*}(t_{x}^{x})$ $s\in S^{51}$ $C_{s} \subset \mathbb{p}_{*}(t_{s}^{s})$ VMRT ( ) $X\simeq S$. Mok 14 N. Mok, Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of.minimal mtional tangents, AMS $/IP$ Studies in Advanced Mathematics, 42, Hong-Hwang long simple root 14 J. Hong and J. Hwang, Characterization of the mtional homogeneous space associated to a long simple root by its variety of minimal rational tangents, Algebraic geometry in East Asia-Hanoi 2005, Advanced Studies of Pure Mathematics, 50, Shepherd-Barron 8. VMRT $\mathbb{p}_{*}(t_{x}^{x})$ 52 VMRT $\mathbb{p}_{*}(t_{x}^{x})$ K. Cho, Y. Miyaoka, N. I. Shepherd-Barron, Chamcterizations of projective space and applications to cornplex symplectic $\gamma nanifolds$, Higher dimensional birational geometry (Kyoto, 1997), 1-88, Adv. Stud. Pure Math., 35, Math. Soc. Japan, Tokyo, 2002 S. Kebekus, the rojective $Cha7^{\cdot}acter\cdot izing$ $p$ $space$ after $Cho$, Miyaoka and Shepherd-Barron, Complex geometry (G\"ottingen, 2000), , Springer, Berlin, 2002 VMRT 53 VMRT 9. Picard 1 Fano $\mathbb{p}_{*}(t_{x}^{x})$ VMRT Y. Miyaoka, Numerical characterisations of hyperquadrics, Adv. Stud. Pure Math. 42, Math. Soc. Japan, Tokyo (2004) VMRT $50_{S}$ $51S$ 52 6

15 $\mathcal{e}$ slope $\mathbb{p}^{2}$ 120 Fano 5.2. Hwang Hwang VMRT Hwang 15 $(**)$. Picard 1 Fano $n$ 8, 9 $P=n-2,$ $n-1$ Reid $F(X)=1$ Peternell Wis niewski $F(X)\geq\dim X-2$ Hwang 5 6 J. Hwang, Stability of tangent bundles of low-dimensional Fano manifolds with Picard number 1, Math. Ann. 312 (1998), no. 4, Picard 1 Fano Picard 1 $\mu(\mathcal{e}):=\frac{c_{1}(\mathcal{e})\cdot C}{rank\mathcal{E}}$ 51 $P$, $\mu(t_{x})=l+\underline{2}$ T $r=1$ $r$ Wahl $T_{X}$ $r\geq 2$ Zariski $\mathcal{f} c\subset T_{X} c$ $\mathcal{f} _{C}=\oplus_{i=1}^{r}O_{P^{1}}(a_{i})(a_{1}\geq a_{2}\geq\cdots\geq a_{r})$ $\mu(\mathcal{f})>\mu(t_{x})>0$ $a_{i}$ 2 $a_{1}>0$ $a_{i}$ 1 $T_{X} c$ $a_{1}$ 2 $r<n$ $\mu(\mathcal{f})=\sigma\underline{a_{l}}\leq 1$ Picard 1 $p=0$ $T_{X}$ $T_{X}$ $\mathcal{f} c$ $O_{P^{1}}(1)$ $T_{X} c$ $O_{P^{1}}(2)$ $\mathcal{f} c\subset T_{X} c$ $n=6$ $P=0,4,5$ $p=3$ $\deg \mathcal{k}=5$ $F(X)=1$ 5 $F(X)=1$ Reid, $F(X)=5$ Peternell $p=1$ $T_{X} c$ $a_{1}=1,$ $a_{i}\leq 0(i\geq 2)$ Wi\ {s}niewski $\tau_{x}$ $\mu(\mathcal{f})>0$ $a_{i}=0(i\geq 2)$ $\mu(\mathcal{f})=\frac{1}{r}\leq\frac{1}{2}$. $\mu(\mathcal{f})=\mu(t_{x})$ $n>4$ $\mu(\mathcal{f})<1$ $\mu(t_{x})=\frac{3}{6}$ $T_{X}$ $p=2$ 53 $n=6$ $\mu(\mathcal{f})=\mu(t_{x})$ $T_{X}$ $\mathcal{f} _{C}$ enumeration $\mu(\mathcal{f})=1$ $\mathbb{p}^{2}$ distribution 54 $\mathcal{f} $ $\mathcal{f} <n$ rank Picard 1 16 $(**)$. Picard 1 $n$ Fano $X\simeq \mathbb{p}^{n}$ $(-K_{X})^{n}\leq(n+1)^{n}$ $53VMRT$ fundamental (saturated)

16 121 J. Hwang, On the degrees of Fano four-folds of Picard number 1, J. Reine Angew. Math. 556 (2003), Hwang 4 55 Bogomolov 8 $(-K_{X})^{2}c_{2}(X)\geq$ $3(-K_{X})^{4}$, Riemann-Roch $\dim -K_{X} =\frac{(-k_{x})^{4}}{6}+\frac{(-k_{x})^{2}c_{2}(x)}{12}$ $\dim -K_{X} \geq\frac{19}{96}(-k_{x})^{4}$ $\dim -K_{X} $ $\dim C_{x}=0,1,2,3$ $\dim C_{x}=2,3$ 8, 9 $\dim$ C. $=1$ $F(X)=4,3,2$ $F(X)=1$, $ -K_{X} $ $\dim -K_{X} $ $ -K_{X} $ $F(X)=1$ $\dim -K_{X} \leq 120$ $\dim -K_{X} >120$ $k$ $k\geq 2$ $\frac{1}{6}(k^{3}+6k^{2}-k-6)$ 56 $\dim -K_{X} >120$ $x,$ $y$ 5 $ -K_{X} $ 3 35 $P_{*}(T_{X}^{x})\simeq \mathbb{p}^{3}$ $\deg \mathcal{k}=3$ 4 $ -K_{X} $ $D$ $\mathcal{k}_{x}$ $\mathbb{p}_{*}(t_{x}^{x})$ 4 25 $\dim -K_{X} >2(35+25)=120$ $\mathcal{k}_{x}$ $\dim S_{x}=2$ $\mathcal{k}_{x}$ $M_{C}$ $\bigcup_{y\in C}S_{y}$ $M_{C}$ 3 $H$ 57 $x,$ $y\in C$ $ -K_{X} $ Nadel ( 10) $ -K_{X} $ 4 $H$ $H$ $\square$ $F(X)=1$ Nadel Nadel Picard 1 Fano 10. $C\subset X$ $L$ $\nu:\mathbb{p}^{1}arrow C$ $\nu^{*}t_{x}=\sum_{i=1}^{n}$ $O_{P^{1}}$ $(a_{i})$ $d \leq\min\{a_{1}, \ldots, a_{n}\}$ $d$ 58 $ L $ $D$ $x_{1},$ $\ldots,$ $\sum_{i=1}^{m}(m_{x_{i}}(d)-m_{c}(d))\leq C\cdot L-d\cdot m_{c}(d)$, $x_{m}\in C$ $m_{x_{i}}(d);m_{c}(d)$ $D$ $x_{i}$ 59 K. Watanabe, Lengths of chains 555 $(-K_{X})^{5}\leq 9^{5}$ of minimal mtional curves on Fano manifolds, arxiv: , to appear in Journal of Algebra 56 3 VMRT fundamental $d=0$ $59C$ $m_{x_{i}}(d)\geq m_{c}(d)$.

17 122 $k_{i}:=m_{x_{i}}(d),$ Fano $l:=mc(d)$ $D$ l-jet $\nu^{*}(sym^{l}t_{x}^{*}\otimes L)$ $\nu^{*}(sym^{l}t_{x}^{*}\otimes L)$ $x_{i}$ ki l $\sum_{i=1}^{m}$(ki l). L dl. $\leq C$ $\square$. C ld $L$ Fano $(-K_{X})^{\dim X}$ 60 ( ) Ran Clemens $A$ ${\rm Res}$ Z. Ran, H. Clemens, New Method in Fano Geometry, Int Math Notices (2000) 2000 (10): (Campana-Peternell61 $**$). Fhno $T_{X}$ : $\mu$ 62 ( ) 63 J. Hwang, Rigidity of rational homogeneous spaces, International Congress of Mathematicians. Vol. II, , Eur. Math. Soc., Z\"urich, 2006 Hwang 4 Mok (3.1) N. Mok, On Fano manifolds with $nef$ tangent bundles admitting l-dimensional varieties of minimal mtional tangents, Trans. A.M.S. 354 (2002), Picard $>1$ Campana-Peternell Picard $>1$ 18 $(**)$. Picard 2 17 Picard 1 $\dim C_{x}\leq 3$ $\dim C_{x}=3,2$ 8, 9 $\dim C_{x}=0$ $\mu$ $\mu$ $\mathbb{p}^{1}$- Picard $\mathcal{u}$ 1 $\dim C_{x}=1$ Mok Mok (4 ), (31) $\mathbb{p}^{3}$ $X\simeq \mathbb{p}^{2},$ $X\simeq Q^{3},$ (5 )G2 $\square$ $61F$. Campana and T. Peternell, Projective manifolds whose tangent bundles are numerically effective, Math. Ann. Vol. 289, Number 1, $64Mok$ Hwang $65_{\mu}$ $P^{1}$ $\rho$ 2

18 $\mathcal{k}_{x}$ $\rangle$ $(*\sim**)$. 4, 5 $\mathbb{p}^{4}$ VMRT 20 $(*\sim**)$. $Z$ Fano 66 $Z$ VMRT Hwang 67 Fano $Z\subset \mathbb{p}^{n}$ 11. $H$ $\pi:xarrow \mathbb{p}^{n}$ $Z$ $Z$ $x\in X\backslash \pi^{-1}(h)$ ( $\pi(x)$ $Z$ $Z$ ), $Z$ 5.3. $\tau:\mathcal{u}-*c$ VMRT Hwang-Mok fundamental 68 J. Hwang, Geometry of minimal mtional curves on Fano manifolds, School on Vanishing Theorems and Effective Results in Algebraic Geometry (Thrieste, 2000), , ICTP Lect. Notes, 6, Abdus alam Int. Cent. Theoret. Phys., $S$ $rrieste$, $\tau:\mathcal{u}-*c$ $\mathcal{k}_{x}arrow C_{x}$ $x\in X$ $(***)$. Picard 1 Fano $?X$? VMRT 12 J. Hwang, N. Mok, Birationality of the tangent map for minimal mtional curves, Asian J. Math. 8 (2004), no. 1, Picard 1 $67_{j}$. Hwang, Equivalence problem for minimal rational curves with isotrivial varieties of minimal rational tangents, arxiv: $6S$ Hwang $69\dim C_{x}=0$ $7_{F}$. Zak

19 $\hat{\tau}_{c_{\pi(\alpha)}^{\alpha}}$ $\mathcal{u}$ $\prime \mathcal{r}$ $\mathcal{u}$ 124 Fano 12 $\rho:\mathcal{u}arrow \mathcal{k}$ $\tau$ $\tau$ $x\in X$ $p :=\dim C_{x}$ $C_{\pi(\alpha)}$ $\alpha$ $\hat{t}_{c_{\pi(\alpha)}}^{\alpha}$ $p +1$ $T_{X}^{\pi(\alpha)}$ $f$ $f^{*}t_{x}\simeq O_{P^{1}}(2)\oplus O_{P^{1}}(1)^{\oplus p}\oplus O_{P^{1}}^{\oplus n-1-p}$ 3 $*$ $\alpha$ f T $\hat{t}_{c_{\pi(\alpha)}}^{\alpha}$ 71 $p=p $ $O_{P^{1}}(2)\oplus O_{lP^{1}}(1)^{\oplus p}$ distribution $P$ $\alpha$ $\mathcal{p}^{\alpha}=(d\pi)^{-1}(\hat{t}_{c_{\pi(\alpha)}}^{\alpha})\subset T_{C}^{\alpha}$ $d\pi$ $P$ $\pi$ $\alpha$ subdistribution $\mathcal{p}$ $(\mathcal{p})^{\alpha}:=\{p\in \mathcal{p}^{\alpha} \lceil p, P^{\alpha}]=0\}$ Ch $S$ Jacobi Frobenius $(\mathcal{p})$ Ch $S$ $P$ Gauss $C_{x}-*G(p+1, T_{X}^{x})(\alpha\mapsto\hat{T}_{C_{x}}^{\alpha})$ Gauss $S$ 72 $\mathcal{r}$ $P$ $\tau:\mathcal{u}-*c$ distribution $Hom(\mathbb{P}^{1}, X)$ 3 $[f]$ $[f]$ $H^{0}(\mathbb{P}^{1}, f^{*}t_{x})$ $[C]$ $[C]$ $H^{0}(\mathbb{P}^{1},0_{P^{1}}(1)^{\oplus p})$ \mathcal{r}$ $\prime distribution $\mathcal{f} $ $\alpha$ $H^{0}(\mathbb{P}^{1}, O_{P^{1}}(1)^{\oplus p}\oplus O_{P^{1}}^{\oplus n-1-p})$ $V$ distribution $d\rho$ $\mathcal{u}$ $\mathcal{r}$ $P$ $\tau$ $\pi:carrow X$ $\mathcal{j}$ $\mathcal{f} $ $\mathcal{j}^{\alpha}=(d\pi)^{-1}(\mathbb{c}\alpha)$ $\alpha\in \mathcal{u}$ distribution73 $\alpha$ ( $)v+\mathcal{f} =\mathcal{j}$ (5.1) $P=V+\mathcal{F} +[\mathcal{f}, V]$ $(\mathcal{r})$ Ch Ch $(P)$ $\rho$ $\mathcal{f}\subset$ $\prime p$ $\mathcal{f} \subset$ $( \mathcal{r})$ $\mathcal{r}$ Ch Ch $(P)$ $\mathcal{g}$ Gauss $V$ $\mathcal{g}=$ Ch 74 (51) Ch $(\mathcal{p})\cap V$ $(\mathcal{p})=\mathcal{g}+\mathcal{f} +[\mathcal{f}, \mathcal{g}]$ $\mathcal{g}$ Gauss rank $k-1$ rank Ch $(\mathcal{p})=2k-1$ $T=\pi(S)$ $\dim T=k$ $\mathcal{f} $ $\pi$ $\alpha$ $T$ $T$ $T$ $T^{0}$ $\alpha\in C$ $S$ $\mathbb{p}(t_{t^{0}}^{*})$ $\mathcal{r}$ 72 $\mathbb{p}(t_{t^{\text{ }}}^{*})\cap S$ $\mathbb{p}(t_{t^{\text{ }}}^{*})$ N. Mok $73\mathbb{C}\alpha$ $T_{\lambda}^{\pi(\alpha)}$ $\alpha$ 1 74Gauss

20 $\overline{t}^{n}$ $\overline{t}$ $\overline{t}$ 125 $T$ $t$ $t$ Gauss $\overline{t}$ Gauss $\mathcal{k}^{t}\subset \mathcal{k}$ $\mathcal{u}^{t}$ $T$ $u$ $\mathcal{k}^{t}$ $\mathcal{k}^{t}$ $C^{T}$ VMRT $\overline{t}$ $\overline{t}$ $t\in T$ $\mathbb{p}_{*}(t_{t}^{t})$ $\mathbb{p}^{k}arrow\ovalbox{\tt\small REJECT} T$ 8 Sing $(\mathcal{p})$ Ch $S$ $U^{T}-*\mathcal{K}^{T}$ 8 $\overline{t}$ 12 $T \frac{t}{t}$ $C_{t}$ 21 Gauss $\square$ $t$ Cauchy Gauss Zak $S$ Cauchy $(\mathcal{f}$ Cauchy ) 22 $(**)$. Picard 1 Fano Cauchy??75 23 $(**)$. Picard 1 Fano $\dim$ $\geq 1$. $\ovalbox{\tt\small REJECT} X$ 8 $X^{0}$ Zariski $\mathbb{p}^{p+1}$ - $\varphi:x^{0}arrow T^{0}$ $X^{0}$ ) $X^{0}$ $\varphi$ $p:=\dim C_{x}\geq 1$ C. Araujo, Rational curves of minimal degree and chamcterizations of projective spaces, Math. Ann. 335 (2006), no. 4, $\mathbb{p}^{p+1}$ - Cauchy Andreatta-Wis niewski $T_{X}$ 76 Cauchy [HM] 12 Lazarsfeld 14. Picard 1 $G/P$ $f$ 77 $f:g/parrow X$ $X\simeq \mathbb{p}^{n_{l}}$ 75 $C_{x}=P_{*}(T_{X}^{x})$ $77_{Picard}$ 2 C. Lau, Holomorphic maps from rational homogeneous spaces onto projective manifolds, J. Algebraic Geom. 18 (2009), no. 2,

21 126 Fano 4 Fano VMRT N. Mok VMRT address: takagi@ms.u-tokyo.ac.jp