View source for Algebraic variety, automorphism of an

An invertible morphism of an algebraic variety (or scheme) into itself. The group of all automorphisms of an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a0117401.png" />, which is usually denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a0117402.png" />, is an important invariant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a0117403.png" />. Studies of the action of the automorphism group of an algebraic variety on objects functorially connected with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a0117404.png" /> such as the [[Picard group|Picard group]]; the [[Chow ring|Chow ring]]; the [[K-functor|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a0117405.png" />-functor]], and the cohomology group are a tool used in the study of the varieties themselves. The automorphism group of an algebraic variety is important for the concept of forms (cf. [[Form|Form]]) of an algebraic variety. For complete algebraic varieties over the field of complex numbers, the automorphism group is identical with the group of biholomorphic automorphisms.

The structure of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a0117406.png" /> is known for a number of simple algebraic varieties. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a0117407.png" /> is projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a0117408.png" />-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a0117409.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174010.png" />, then any one of its automorphisms is a linear projective transformation and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174011.png" /> becomes identical with the projective linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174012.png" />. The automorphism group of an elliptic curve and, in general, of any Abelian variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174013.png" />, is an extension of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174014.png" /> of automorphisms which preserve the structure of the Abelian variety, by the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174015.png" /> of translations in the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174016.png" />, i.e. the sequence of groups

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is exact. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174018.png" /> is a smooth, complete algebraic curve of genus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174019.png" />, then the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174020.png" /> is finite; an estimate of its order as a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174021.png" /> is known (cf. [[Algebraic curve|Algebraic curve]]). For automorphisms of surfaces, see [[Algebraic surface|Algebraic surface]].

In the case of algebraic varieties with an ample canonical or anti-canonical invertible sheaf the automorphism group is an algebraic subgroup of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174022.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174023.png" />. The automorphism group of a smooth hypersurface of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174024.png" /> and degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174025.png" /> is finite [[#References|[1]]].

In the above examples, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174026.png" /> has the natural structure of an algebraic group, perhaps with an infinite number of connected components; this is also true in the general case [[#References|[2]]].

Families of automorphisms are considered in the modern approach to automorphism groups of algebraic varieties. A family of automorphisms of a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174027.png" /> with parameter scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174028.png" /> is a set of automorphisms of the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174029.png" /> that commute with projection onto the second factor; the set of families of automorphisms with parameter scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174030.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174031.png" />. One thus obtains a contravariant functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174032.png" />. If the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174033.png" /> is complete, then this functor is locally representable (cf. [[Representable functor|Representable functor]]) by an algebraic group scheme with at most a countable number of connected components [[#References|[3]]]. A. Grothendieck gave a proof of this fact for projective varieties, and this theorem has been extended to the case of proper flat schemes of morphisms. The scheme representing this functor is not necessarily reduced even if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174034.png" /> is a smooth projective surface; however, if the characteristic of the ground field is zero, or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174035.png" /> is a smooth curve or a smooth hypersurface, then the connected component of the unit of this scheme is a variety.

For incomplete varieties the automorphism functor is not always representable in the category of schemes. For an affine variety, the automorphism functor is representable in the category of inductive limits of schemes.

Apart from the simple case of the affine straight line, for the affine spaces only the automorphism group of the affine plane is known. It is a free product of two of its subgroups with as amalgamated subgroup their intersection, viz. the subgroup of linear affine transformations and the subgroup of triangular automorphisms, i.e. transformations of the form

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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174037.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174038.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174039.png" /> is an arbitrary polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011740/a01174040.png" /> [[#References|[4]]], [[#References|[5]]]. For the treatment of affine algebraic surfaces transitively acted upon by the automorphism group, see [[#References|[6]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Matsumura,   P. Monsky,   "On the automorphisms of hypersurfaces"  ''J. Math. Kyoto Univ.'' , '''3'''  (1964)  pp. 347–361  {{MR|0168559}} {{ZBL|0141.37401}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  T. Matsusaka,   "Polarized varieties, fields of moduli and generalized Kummer varieties of polarized Abelian varieties"  ''Amer. J. Math.'' , '''80'''  (1958)  pp. 45–82  {{MR|0094360}} {{ZBL|0085.15304}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Matsumura,   F. Oort,   "Representability of group functors and automorphisms of algebraic schemes"  ''Invent. Math.'' , '''4'''  (1967)  pp. 1–25  {{MR|0217090}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  W. Engel,   "Ganze Cremona-Transformationen von Primzahlgrad in der Ebene"  ''Math. Ann.'' , '''136'''  (1958)  pp. 319–325  {{MR|0103888}} {{ZBL|0082.36502}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  I.R. Shafarevich,   "On some infinite-dimensional groups"  ''Rend. di Mat. e Appl.'' , '''25'''  (1966)  pp. 208–212  {{MR|0485898}} {{ZBL|0149.39003}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M.Kh. Gizatullin,   "Quasihomogeneous affine surfaces"  ''Math. USSR-Izv.'' , '''5''' :  5  (1971)  pp. 1057–1082  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''35'''  (1971)  pp. 1047–1071  {{MR|0286791}} {{ZBL|0221.14023}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  L. Roth,   "Algebraic threefolds" , Springer  (1955)  {{MR|0076426}} {{ZBL|0066.14704}} </TD></TR></table>