mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Fujita's conjecture is a problem in the theories of

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 ...

and

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...

s, unsolved . It is named after Takao Fujita, who formulated it in 1985.

Statement

In complex geometry, the conjecture states that for a

positive Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posi ...

holomorphic line bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a com ...

''L'' on a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

complex manifold ''M'', the line bundle ''K''_''M'' ⊗ ''L''^⊗''m'' (where ''K''_''M'' is a

canonical line bundle In mathematics, the canonical bundle of a 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 the complex numbers, it ...

of ''M'') is *

spanned by sections In mathematics, a sheaf of ''O''-modules or simply an ''O''-module over a ringed space (''X'', ''O'') is a sheaf ''F'' such that, for any open subset ''U'' of ''X'', ''F''(''U'') is an ''O''(''U'')-module and the restriction maps ''F''(''U'') � ...

when ''m'' ≥ ''n'' + 1 ; *

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 ...

when ''m'' ≥ ''n'' + 2, where ''n'' is the

complex dimension In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex dimension of an algebraic variety, algebraic variety. These are spaces in which the local neighborhoods of points (or of non-singular points in the ...

of ''M''. Note that for large ''m'' the line bundle ''K''_''M'' ⊗ ''L''^⊗''m'' is very ample by the standard

Serre's vanishing theorem In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the exist ...

(and its complex analytic variant). Fujita conjecture provides an explicit bound on ''m'', which is optimal for

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

Known cases

For surfaces the Fujita conjecture follows from

Reider's theorem In algebraic geometry, Reider's theorem gives conditions for a line bundle on a projective surface to be very ample. Statement Let ''D'' be a nef divisor on a smooth projective surface ''X''. Denote by ''K'X'' the canonical divisor In mathemat ...

. For three-dimensional algebraic varieties, Ein and Lazarsfeld in 1993 proved the first part of the Fujita conjecture, i.e. that ''m''≥4 implies global generation.

References

*. *. *. *. *{{citation , last = Smith , first = Karen E. , doi = 10.1007/s002080000094 , issue = 2 , journal = Mathematische Annalen , mr = 1764238 , pages = 285–293 , title = A tight closure proof of Fujita's freeness conjecture for very ample line bundles , volume = 317 , year = 2000, url = https://deepblue.lib.umich.edu/bitstream/2027.42/41935/1/208-317-2-285_03170285.pdf , hdl = 2027.42/41935 , s2cid = 55051810 , hdl-access = free .

External links

supporting facts to fujita conjecture
Algebraic geometry Complex manifolds Conjectures Unsolved problems in geometry

Statement

Known cases

See also

References

External links