#### DOCUMENTA MATHEMATICA,
Vol. Extra Volume: Alexander S. Merkurjev's Sixtieth Birthday (2015), 367-385

** Nikita A. Karpenko
**
Minimal Canonical Dimensions
of Quadratic Forms

Canonical dimension of a smooth complete connected variety is the minimal
dimension of image of its rational endomorphism. The $i$-th canonical
dimension
of a non-degenerate quadratic form is the canonical dimension
of its $i$-th
orthogonal grassmannian. The maximum of a canonical dimension
for quadratic
forms of a fixed dimension is known to be equal to the
dimension of the
corresponding grassmannian. This article is about the
minima of the canonical
dimensions of an anisotropic quadratic form.
We conjecture that they equal the
canonical dimensions of an excellent
anisotropic quadratic form of the same
dimension and we prove it in a
wide range of cases.

2010 Mathematics Subject Classification: 14L17; 14C25

Keywords and Phrases: Algebraic groups, quadratic forms,
projective homogeneous varieties,
Chow groups and motives.

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