In algebraic geometry, a **Todorov surface** is one of a class of surfaces of general type introduced by Todorov ( 1981 ) for which the conclusion of the Torelli theorem does not hold.

In mathematics, **complex geometry** is the study of complex manifolds, complex algebraic varieties, and functions of several complex variables. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis.

In algebra, **ring theory** is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings, as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities.

**Oscar Zariski** was a Russian-born American mathematician and one of the most influential algebraic geometers of the 20th century.

In relation with the history of mathematics, the **Italian school of algebraic geometry** refers to the work over half a century or more done internationally in birational geometry, particularly on algebraic surfaces. There were in the region of 30 to 40 leading mathematicians who made major contributions, about half of those being in fact Italian. The leadership fell to the group in Rome of Guido Castelnuovo, Federigo Enriques and Francesco Severi, who were involved in some of the deepest discoveries, as well as setting the style.

**David Bryant Mumford** is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science. He is currently a University Professor Emeritus in the Division of Applied Mathematics at Brown University.

In mathematics, the **Jacobian variety***J(C)* of a non-singular algebraic curve *C* of genus *g* is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of *C*, hence an abelian variety.

In mathematics, **Hodge theory**, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold *M* using partial differential equations. The key observation is that, given a Riemannian metric on *M*, every cohomology class has a canonical representative, a differential form which vanishes under the Laplacian operator of the metric. Such forms are called **harmonic**.

In mathematics, a complex analytic **K3 surface** is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface

**Kunihiko Kodaira** was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds, and as the founder of the Japanese school of algebraic geometers. He was awarded a Fields Medal in 1954, being the first Japanese national to receive this honour.

In mathematics, an **algebraic surface** is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two and so of dimension four as a smooth manifold.

In mathematics, the **canonical bundle** of a non-singular algebraic variety of dimension over a field is the line bundle , which is the *n*th exterior power of the cotangent bundle Ω on *V*.

**Shigefumi Mori** is a Japanese mathematician, known for his work in algebraic geometry, particularly in relation to the classification of three-folds.

In mathematics, **arithmetic geometry** is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.

In algebraic geometry, a **surface of general type** is an algebraic surface with Kodaira dimension 2. Because of Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 will actually be an algebraic surface, and in some sense most surfaces are in this class.

In mathematics, the **Torelli theorem**, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve *C* is determined by its Jacobian variety *J*(*C*), when the latter is given in the form of a principally polarized abelian variety. In other words, the complex torus *J*(*C*), with certain 'markings', is enough to recover *C*. The same statement holds over any algebraically closed field. From more precise information on the constructed isomorphism of the curves it follows that if the canonically principally polarized Jacobian varieties of curves of genus are *k*-isomorphic for *k* any perfect field, so are the curves.

In mathematics, the **Schottky problem,** named after Friedrich Schottky, is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties.

In mathematics, **real algebraic geometry** is the sub-branch of algebraic geometry studying real algebraic sets, i.e. real-number solutions to algebraic equations with real-number coefficients, and mappings between them.

**Fedor Alekseyevich Bogomolov** is a Russian and American mathematician, known for his research in algebraic geometry and number theory. Bogomolov worked at the Steklov Institute in Moscow before he became a professor at the Courant Institute in New York. He is most famous for his pioneering work on hyperkähler manifolds.

**David Archibald Cox** is a retired American mathematician, working in algebraic geometry.

- Morrison, David R. (1988), "On the moduli of Todorov surfaces",
*Algebraic geometry and commutative algebra*,**I**, Tokyo: Kinokuniya, pp. 313–355, MR 0977767 - Todorov, Andrei N. (1981), "A construction of surfaces with
*p*_{g}= 1,*q*= 0 and 2 ≤ (*K*^{2}) ≤ 8. Counterexamples of the global Torelli theorem.",*Invent. Math.*,**63**(2): 287–304, doi:10.1007/BF01393879, MR 0610540

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