By I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov
"... To sum up, this booklet is helping to benefit algebraic geometry very quickly, its concrete type is pleasant for college kids and divulges the wonderful thing about mathematics." --Acta Scientiarum Mathematicarum
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Additional resources for Algebraic geometry 01 Algebraic curves, algebraic manifolds and schemes
1. Let Z be a Cm-hermitian line bundle on X such that nef and CQ is ample. z)(x) Proof: } zis vertically - d+l &(""' 2 inf- h(x,z)(x) (d 1)deg(C&) X€X(Q) + Let us begin with the following lemma. 2. 2 we may assume C has a small section s. We set Y = div(s). Then, for any x E (XQ \ Y)(Q), - Let us start the proof of the theorem. Let c be a real number with 0 < c < 1. Let A be a hermitian line bundle on Spec@) given by 71 = (Ospec(z), cl . I). First, let us consider the left inequality. Let X be a rational number with 58 SHU KAWAGUCHI, ATSUSHI MORIWAKI AND KAZUHIKO YAMAKI Then, by an easy calculation, G ((6(c) - ~4(a*( 2 1 ) ) ~ >~ 0, ' ) where 7r : X -+ Spec(Z) is the natural morphism.
D e g ( ( ~ ~ c ) d oN(nd-') ) + + + oN(nd-'). )= O(Ndnd)log(O(Ndnd))= oN(ndf ') (rk(r2) - r k ( r l ) ) ( N n c 4 + N + n ) log = O(Ndndf') + oN(nd). 2 Combining these estimates, we obtain + - ( N n i)d+l deg(6 ( z ) d f l ) O(Ndndf') oN(ndf l ) . (d l ) ! Step 4: We will bound x s u p ( r l ) from below. We equip P(V3) c Vl with the induced norm. ) + (rk(r1) - rk(r3)) ( N n + N + n ) log (:-C4 ) . 3 that + - (Nn z)~+' deg(21(z)d+') O(Ndndfl) oN(ndf l ) . (d l ) ! For rk(r3), we have a similar estimate as for r k ( r l ) and rk(F2): For any n >> 1, N >> 1, and 0 5 i N - 1, we have xsUp(rs) = + + + < + ( N n i)d d e g ( ( ~ C ) d ) O(Nd-lnd) d!
Let Txly = Ker(TX -+ f * ( T y ) ) be the relative tangent sheaf. We fix a Kahler metric hf on Tx(c)ly(c) such that hf is invariant under the complex conjugation. Then, the arithmetic Todd class td(Txly) E C H ~ ( Xis) defined, although the coherent sheaf Txly is not in general a vector bundle (cf. 11). Let R(T) be the Gillet-Soul6 power series: A- epZO A + + where [(s) is the Riemann zeta function. Set R(T1, . . ,T,) = R(T1) . . R(T,) E R[[Tl,. _, T,]]. 1. 1 (Arithmetic Riemann-Roch Theorem).
Algebraic geometry 01 Algebraic curves, algebraic manifolds and schemes by I. R. Shafarevich (editor), V.I. Danilov, V.V. Shokurov