In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, a Fano variety, introduced by
Gino Fano
Gino Fano (5 January 18718 November 1952) was an Italian mathematician, best known as the founder of finite geometry. He was born to a wealthy Jewish family in Mantua, in Italy and died in Verona, also in Italy.
Fano made various contributions ...
in , is a
complete variety
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety , such that for any variety the projection morphism
:X \times Y \to Y
is a closed map (i.e. maps closed sets onto closed sets). Thi ...
''X'' whose
anticanonical bundle In mathematics, the canonical bundle of a non-singular variety, non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''.
Over th ...
''K''
X* is
ample In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of a ...
. In this definition, one could assume that ''X'' is
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
over a field, but the
minimal model program
In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible. The subject has its or ...
has also led to the study of Fano varieties with various types of singularities, such as
terminal
Terminal may refer to:
Computing Hardware
* Terminal (electronics), a device for joining electrical circuits together
* Terminal (telecommunication), a device communicating over a line
* Computer terminal, a set of primary input and output dev ...
or
klt singularities. Recently techniques in differential geometry have been applied to the study of Fano varieties over the complex numbers, and success has been found in constructing
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ...
s of Fano varieties and proving the existence of
Kähler–Einstein metric
In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admits a Kähler–Einstein metric. The ...
s on them through the study of
K-stability of Fano varieties
In mathematics, and in particular algebraic geometry, K-stability is an algebro-geometric stability condition for projective algebraic varieties and complex manifolds. K-stability is of particular importance for the case of Fano varieties, where ...
.
Examples
* The fundamental example of Fano varieties are the
projective spaces
In mathematics, the concept of a projective space originated from the visual effect of perspective (graphical), perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean s ...
: the
anticanonical line bundle of P
''n'' over a field ''k'' is ''O''(''n''+1), which is
very ample In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of a ...
(over the complex numbers, its
curvature is ''n+1'' times the
Fubini–Study symplectic form).
* Let ''D'' be a smooth codimension-1 subvariety in P
n. The
adjunction formula In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedde ...
implies that ''K''
''D'' = (''K''
''X'' + ''D''),
''D'' = (−(''n''+1)''H'' + deg(''D'')H),
''D'', where ''H'' is the class of a hyperplane. The
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidea ...
''D'' is therefore Fano if and only if deg(''D'') < ''n''+1.
* More generally, a smooth
complete intersection
In mathematics, an algebraic variety ''V'' in projective space is a complete intersection if the ideal of ''V'' is generated by exactly ''codim V'' elements. That is, if ''V'' has dimension ''m'' and lies in projective space ''P'n'', there shou ...
of hypersurfaces in ''n''-dimensional projective space is Fano if and only if the sum of their degrees is at most ''n''.
*
Weighted projective space In algebraic geometry, a weighted projective space P(''a''0,...,''a'n'') is the projective variety Proj(''k'' 'x''0,...,''x'n'' associated to the graded ring ''k'' 'x''0,...,''x'n''where the variable ''x'k'' has degree ''a'k''.
Prop ...
P(''a''
0,...,''a''
''n'') is a singular (
klt) Fano variety. This is the projective scheme associated to a graded polynomial ring whose generators have degrees ''a''
0,...,''a''
''n''. If this is well formed, in the sense that no ''n'' of the numbers ''a'' have a common factor greater than 1, then any complete intersection of hypersurfaces such that the sum of their degrees is less than ''a''
0+...+''a''
''n'' is a Fano variety.
* Every projective variety in characteristic zero that is homogeneous under a linear algebraic group is Fano.
Some properties
The existence of some ample line bundle on ''X'' is equivalent to ''X'' being a
projective variety
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables w ...
, so a Fano variety is always projective. For a Fano variety ''X'' over the complex numbers, the
Kodaira vanishing theorem
In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices ''q'' > 0 are automatically zero. The implicat ...
implies that the
sheaf cohomology groups
of the
structure sheaf vanish for
. In particular, the Todd genus
automatically equals 1. The
cases of this vanishing statement also tell us that the
first Chern class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau ...
induces an isomorphism
.
By Yau's solution of the
Calabi conjecture
In the mathematical field of differential geometry, the Calabi conjecture was a conjecture about the existence of certain kinds of Riemannian metrics on certain complex manifolds, made by . It was proved by , who received the Fields Medal and Oswal ...
, a smooth complex variety admits Kähler metrics of positive
Ricci curvature if and only if it is Fano.
Myers' theorem
Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941. It asserts the following:
In the special case of ...
therefore tells us that the
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
of a Fano manifold is compact, and so can only be a finite covering. However, we have just seen that the Todd genus of a Fano manifold must equal 1. Since this would also apply to the manifold's universal cover, and since the Todd genus is multiplicative under finite covers, it follows that any Fano manifold is
simply connected.
A much easier fact is that every Fano variety has
Kodaira dimension In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''.
Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the ...
−∞.
Campana and
Kollár–
Miyaoka–
Mori
Mori is a Japanese and Italian surname, and also a Persian pet name for Morteza. It is also the name of two clans in Japan, and one clan in India.
Italian surname
*Barbara Mori, Uruguayan-Mexican actress
* Camilo Mori, Chilean painter
* Cesare ...
showed that a smooth Fano variety over an algebraically closed field is
rationally chain connected; that is, any two closed points can be connected by a chain of
rational curves.
Kollár–Miyaoka–Mori also showed that the smooth Fano varieties of a given dimension over an algebraically closed field of characteristic zero form a bounded family, meaning that they are classified by the points of finitely many algebraic varieties.
[J. Kollár. Rational Curves on Algebraic Varieties. Corollary V.2.15.] In particular, there are only finitely many deformation classes of Fano varieties of each dimension. In this sense, Fano varieties are much more special than other classes of varieties such as varieties of
general type In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical ring, canonical model of a projective variety ''X''.
Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the ...
.
Classification in small dimensions
The following discussion concerns smooth Fano varieties over the complex numbers.
A Fano curve is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the
projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
.
A Fano surface is also called a
del Pezzo surface
In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in some sense the opposite of surfaces of general ...
. Every del Pezzo surface is isomorphic to either P
1 × P
1 or to the projective plane blown up in at most 8 points, which must be in general position. As a result, they are all
rational
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
.
In dimension 3, there are smooth complex Fano varieties which are not rational, for example cubic 3-folds in P
4 (by
Clemens Clemens is both a Late Latin masculine given name and a surname meaning "merciful". Notable people with the name include:
Surname
* Adelaide Clemens (born 1989), Australian actress.
* Andrew Clemens (b. 1852 or 1857–1894), American folk artist
* ...
-
Griffiths
The surname
Griffiths is a surname with Welsh origins, as in Gruffydd ap Llywelyn Fawr. People called Griffiths recorded here include:
* Alan Griffiths (born 1952), Australian politician and businessman
* Alan Griffiths (cricketer) (born 1957), ...
) and quartic 3-folds in P
4 (by
Iskovskikh -
Manin). classified the smooth Fano 3-folds with second
Betti number
In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplici ...
1 into 17 classes, and classified the smooth ones with second Betti number at least 2, finding 88 deformation classes. A detailed summary of the classification of smooth Fano 3-folds is given in .
See also
*
Periodic table of shapes a project to classify all Fano varieties in three, four and five dimensions.
Notes
External links
Fanography- A tool to visually study the classification of threedimensional Fano varieties.
References
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Algebraic geometry
3-folds