On the Donaldson polynomials of elliptic surfaces
On the Donaldson polynomials of elliptic surfaces
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DOI：
10.1007/bf01459803
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发表时间：
1994-05
影响因子：
1.4
通讯作者：
P. Lisca
中科院分区：
文献类型：
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作者：
P. Lisca
设q ′是tEIP 1上的光滑单连通极小椭圆曲面，其几何亏格为n-1，重数为p和q.由于n ′ q的同胚类型只依赖于n，所以设qvn是它的交形式。设f表示一般纤维的2-同调类，k表示唯一本原类，使得f= pqk。唐纳森SU（2）不变量？c（Vn ′ 4）是H2（Vn）上的齐次多项式，H2（Vn）上的复值多项式代数可以等同于H2（Vn）上的对称多线性型空间; C）被赋予合适的对称乘积（见下文），其同构于Sym*（H2（Vn; C））。因此，用k表示上面定义的同调类和它的Poincar 6对偶qv。和k可以被认为是Sym*（H2（V.;（E））。”（《易经》）c（Imn '~）是qv和k中的多项式，所以我们可以写为
Let~'q be a smooth simply connected minimal elliptic surface over tEIP 1 with geometric genus n-1 and two multiple fibers of multiplicities p and q. The homeomorphism type of~'q depends only on n, so let qvn be its intersection form. Let fdenote the 2-homology class of a generic fiber and k the unique primitive class such that f= pqk. The Donaldson SU (2)-invariants? c (/~'4) are homogeneous polynomials on H2 (V,), and the algebra of complex-valued polynomials on H2 (Vn) can be identified with the space of symmetric multilinear forms on H2 (V.; C) endowed with a suitable symmetric product (see below), which is isomorphic to Sym*(H2 (Vn; C)). Hence, denoting by k both the homology class defined above and its Poincar6 dual, qv. and k can be considered as elements of Sym*(H2 (V.;(E)). By [FM] the? c (Imn'~)'s are polynomials in qv, and k, so we may write