In the mathematical fields of

differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

and

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

, the Frankel conjecture was a problem posed by

Theodore Frankel Theodore Frankel (June 17, 1929 – August 5, 2017) was a mathematician who introduced the Andreotti–Frankel theorem and the Frankel conjecture. Frankel received his Ph.D. from the University of California, Berkeley in 1955. His doctoral adv ...

in 1961. It was resolved in 1979 by

Shigefumi Mori is a Japanese mathematician, known for his work in algebraic geometry, particularly in relation to the classification of three-folds. He won the Fields Medal in 1990. Career Mori completed his Ph.D. titled "The Endomorphism Rings of Some Abelian ...

, and by

Yum-Tong Siu Yum-Tong Siu (; born May 6, 1943) is a Chinese mathematician. He is the William Elwood Byerly Professor of Mathematics at Harvard University. Siu is a prominent figure in the study of functions of several complex variables. His research interes ...

and

Shing-Tung Yau Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and professor emeritus at Harvard University. Until 2022, Yau was the William Caspar ...

. In its differential-geometric formulation, as proved by both Mori and by Siu and Yau, the result states that if a closed

Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnol ...

has positive bisectional curvature, then it must be

biholomorphic In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose inverse is also holomorphic. Formal definition ...

complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...

. In this way, it can be viewed as an analogue of the

sphere theorem In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows. I ...

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

, which (in a weak form) states that if a closed and

simply-connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoint ...

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

has positive curvature operator, then it must be diffeomorphic to a

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

. This formulation was extended by

Ngaiming Mok Ngaiming Mok (; born 1956) is a Hong Kong mathematician specializing in complex differential geometry and algebraic geometry. He is currently a professor at the University of Hong Kong. After graduating from St. Paul's Co-educational College in H ...

to the following statement: In its algebro-geometric formulation, as proved by Mori but not by Siu and Yau, the result states that if is an irreducible and nonsingular

projective variety In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, th ...

, defined over an

algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...

, which has

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 ...

tangent bundle, then must be isomorphic to the

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 ...

defined over . This version is known as the Hartshorne conjecture, after

Robin Hartshorne __NOTOC__ Robin Cope Hartshorne ( ; born March 15, 1938) is an American mathematician who is known for his work in algebraic geometry. Career Hartshorne was a Putnam Fellow in Fall 1958 while he was an undergraduate at Harvard University (under ...

References

* Theodore Frankel. ''Manifolds with positive curvature.'' Pacific J. Math. 11 (1961), 165–174. * Robin Hartshorne
''Ample subvarieties of algebraic varieties.''
Notes written in collaboration with C. Musili. Lecture Notes in Mathematics, Vol. 156 (1970). Springer-Verlag, Berlin-New York. xiv+256 pp. * Shoshichi Kobayashi and Takushiro Ochiai. ''Characterizations of complex projective spaces and hyperquadrics.'' J. Math. Kyoto Univ. 13 (1973), 31–47. * Ngaiming Mok. ''The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature.'' J. Differential Geom. 27 (1988), no. 2, 179–214. * Shigefumi Mori. ''Projective manifolds with ample tangent bundles.'' Ann. of Math. (2) 110 (1979), no. 3, 593–606. * Yum Tong Siu and Shing Tung Yau
''Compact Kähler manifolds of positive bisectional curvature.''
Invent. Math. 59 (1980), no. 2, 189–204. {{closed access Differential geometry Algebraic geometry Conjectures