Naslov
Flags in zero dimensional complete intersection algebras and indices of real vector fields.

Autori
Giraldo, L. ; Gómez-Mont, X. ; Mardešić, Pavao

Izvornik
Mathematische Zeitschrift (0025-5874) 260 (2008), 1; 77-91

Vrsta, podvrsta i kategorija rada
Radovi u časopisima, članak, znanstveni

Ključne riječi
GSV-index; vector field

Sažetak
This paper concludes the search for an algebraic formula for the real GSV-index. The case of an even-dimensional ambient space was solved in [X. Gómez-Mont and P. Mardešić, Funktsional. Anal. i Prilozhen. 33 (1999), no. 1, 1–13, 96 ; Let $X=\sum X_i\partial_i$ be a real analytic vector field with algebraic isolated singularity at the origin in $\Bbb R^n$, with $n$ odd, which is tangent to $V_0\coloneq f^{; ; ; -1}; ; ; (0)$, so $df(X)=fh$ for some function $h$. By results of [X. Gómez-Mont and P. Mardešić, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 5, 1523–1539] the GSV-index ${; ; ; \rm Ind}; ; ; _{; ; ; V_{; ; ; \pm, 0}; ; ; }; ; ; (X)$ and the signature $\sigma_{; ; ; {; ; ; \bf B}; ; ; , h, 0}; ; ; $ differ by constants $K_\pm$ depending only on $f$. This signature is the first of a whole series, related to a flag of ideals. Let ${; ; ; \bf A}; ; ; \colon={; ; ; \scr A}; ; ; /(f_i)$ be a real zero-dimensional complete intersection local algebra, and $f\in{; ; ; \bf A}; ; ; $ an element in the maximal ideal. The flag of ideals is $K_m={; ; ; \rm Ann}; ; ; _{; ; ; \bf A}; ; ; (f)\cap(f^{; ; ; m-1}; ; ; )$, and projection on the socle of ${; ; ; \bf A}; ; ; $induces a nondegenerate bilinear form on $K_m/K_{; ; ; m+1}; ; ; $, with signature $\sigma_{; ; ; {; ; ; \bf A}; ; ; , f, m}; ; ; $. The algebra ${; ; ; \bf B}; ; ; $ is${; ; ; \scr A}; ; ; /(X_i)$. To compute the constants $K_\pm$ it suffices to do this for one vector field. Using results on the Jantzen filtration from [J. C. Jantzen, Moduln mit einem höchsten Gewicht, Lecture Notes in Math., 750, Springer, Berlin, 1979 ; and [D. A. Vogan, Jr., Ann. of Math. (2) 120 (1984), no. 1, 141–187 ; the constants are related to the higher-order signatures of $f$ on its Jacobi algebra ${; ; ; \bf A}; ; ; \colon={; ; ; \scr A}; ; ; /(\partial_i f)$: $$ K_+=\sum_{; ; ; m\geq1}; ; ; \sigma_{; ; ; {; ; ; \bf A}; ; ; , f, m}; ; ; , \qquad K_-=\sum_{; ; ; m\geq1}; ; ; (-1)^m\sigma_{; ; ; {; ; ; \bf A}; ; ; , f, m}; ; ; . $$

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA