By Patrick Shanahan (auth.)

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If < , > is the for w h i c h a partial ~(D * ) = has (_ l ) k to s h o w directly that whose x [ei} extension is an = - self-adjoint, operator a~A then formally V that to might be Since , is an e l l i p t i c <~a,b> for 9 a multiplication calculation product = , V by this index need ~(D) and is so. is s e l f - a d j o i n t its not be differential * this operator suggests in fact it is not We the omit index zero, that restrict To get V The operator manifold X V+ . : c~(Zh+(T)) is c a l l e d the D i r a c , c'(A-(T)) operator .

In the Then 2 xI n series is a p o l y n o m i a l - / the c o h o m o l o g y . x2 tanh x 2 [X] -x~ I 2 I11 ( i - e xl) ( i - e -x~) the we have l -E-) Since 42 When n = 6 , a similar computation xi tan~ x 1 yields - 1 + 13-Pl l x. In g e n e r a l , a sum the p o l y n o m i a l Lj (Pl ..... c a n be w r i t t e n l t a n h x. of p o l y n o m i a l s PJ ) with as rational j=o coefficients. L o For small j one has = 1 1 L1 - ~Pl L2 = A (7P2- P~) 1 L 3 - 945(62P3 L4 - 1 4 , 117 5 (The d e n o m i n a t o r s this way~ form 14,175 34-52-7 = Combining index + 2p~) - 71P3PI become very to the p o i n t , (see H i r z e b r u c h _ 19p2 + 22P2p2 impressive of c o u r s e , [i, p.

Interpreting meromorphic classical dim H°(X, D) functions inequality as the d i m e n s i o n of the space of subordinate of Riemann~ to the divisor identifying D gives the dim HI(x, D) the index of speciality then gives the equation of Roch Chern [i, §i0] for details) (see as 31 6. The Hodge O p e r a t o r Let X dimension be a compact oriented r i e m a n n i a n m a n i f o l d of n , and let T = TX be the c o t a n g e n t bundle. The r i e m a n n i a n structure and the o r i e n t a t i o n may be used to define a linear t r a n s f o r m a t i o n , .