Details
| Original language | English |
|---|---|
| Pages (from-to) | 439-455 |
| Number of pages | 17 |
| Journal | MOSC MATH J |
| Volume | 3 |
| Issue number | 2 |
| Publication status | Published - 2003 |
Abstract
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In: MOSC MATH J, Vol. 3, No. 2, 2003, p. 439-455.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Indices of 1-forms on an isolated complete intersection singularity
AU - Ebeling, Wolfgang
AU - Gusein-Zade, Sabir M.
PY - 2003
Y1 - 2003
N2 - There are some generalizations of the classical Eisenbud-Levine-Khimshashvili formula for the index of a singular point of an analytic vector field on Rn for vector fields on singular varieties. We offer an alternative approach based on the study of indices of 1-forms instead of vector fields. When the variety under consideration is a real isolated complete intersection singularity (icis), we define an index of a (real) 1-form on it. In the complex setting we define an index of a holomorphic 1-form on a complex icis and express it as the dimension of a certain algebra. In the real setting, for an icis V=f−1(0), f:(Cn,0)→(Ck,0), f is real, we define a complex analytic family of quadratic forms parameterized by the points ϵ of the image (Ck,0) of the map f, which become real for real ϵ and in this case their signatures defer from the "real" index by χ(Vϵ)−1, where χ(Vϵ) is the Euler characteristic of the corresponding smoothing Vϵ=f−1(ϵ)∩Bδ of the icis V.
AB - There are some generalizations of the classical Eisenbud-Levine-Khimshashvili formula for the index of a singular point of an analytic vector field on Rn for vector fields on singular varieties. We offer an alternative approach based on the study of indices of 1-forms instead of vector fields. When the variety under consideration is a real isolated complete intersection singularity (icis), we define an index of a (real) 1-form on it. In the complex setting we define an index of a holomorphic 1-form on a complex icis and express it as the dimension of a certain algebra. In the real setting, for an icis V=f−1(0), f:(Cn,0)→(Ck,0), f is real, we define a complex analytic family of quadratic forms parameterized by the points ϵ of the image (Ck,0) of the map f, which become real for real ϵ and in this case their signatures defer from the "real" index by χ(Vϵ)−1, where χ(Vϵ) is the Euler characteristic of the corresponding smoothing Vϵ=f−1(ϵ)∩Bδ of the icis V.
UR - https://arxiv.org/abs/math/0105242
U2 - 10.17323/1609-4514-2003-3-2-439-455
DO - 10.17323/1609-4514-2003-3-2-439-455
M3 - Article
VL - 3
SP - 439
EP - 455
JO - MOSC MATH J
JF - MOSC MATH J
SN - 1609-3321
IS - 2
ER -