Complex Geometry

Complex geometry is the branch of mathematics concerned with geometric structures and spaces defined using the complex numbers. It focuses on objects such as complex manifolds, complex algebraic varieties, functions of several complex variables, and holomorphic structures including holomorphic vector bundles and coherent sheaves. Because the theory draws simultaneously from algebraic geometry, differential geometry, and complex analysis, it forms a rich and highly interconnected area. This blending of methods frequently makes problems in complex geometry more tractable than analogous problems in purely real or smooth settings. The extra structure provided by complex analyticity allows the use of global analysis, classification techniques, and powerful analytic tools that often have no counterparts in the real category.

Background and conceptual foundations

At its core, complex geometry studies spaces that in some sense resemble the complex plane. The complex plane carries features that extend naturally to higher-dimensional complex spaces. These include an intrinsic notion of orientation and the rigid behaviour of holomorphic functions, whereby the existence of a single complex derivative implies differentiability to all orders. Such rigidity manifests in results such as the identity theorem and Liouville-type theorems, which inform the structure of complex manifolds and projective varieties.
One of the fundamental observations underpinning the subject is that complex manifolds admit no non-trivial partitions of unity by holomorphic functions. This distinguishes them strongly from smooth real manifolds and highlights the limitations and strengths of holomorphic analysis. Nonetheless, every complex manifold is naturally a smooth manifold of twice the real dimension because the complex plane is isomorphic to the real plane as a real vector space.
Complex geometry departs significantly in flavour from real geometry: meromorphic singularities, for example, are sharply constrained and therefore easier to analyse than typical singularities of real-valued functions. This enables the rigorous study of singular analytic and algebraic varieties. In turn, such structures provide a setting where techniques of geometry, algebra, and analysis intersect fruitfully.

Historical development and major advances

Throughout the twentieth century, complex geometry evolved into a central field shaped by developments in both algebraic and differential geometry. Deep global analytical results have marked its progress. Notable examples include the resolution of the Calabi conjecture by Shing-Tung Yau, which produced Ricci-flat Kähler metrics on compact Kähler manifolds and thereby established the mathematical existence of Calabi–Yau manifolds. Other achievements include the Hitchin–Kobayashi correspondence, relating stability of holomorphic vector bundles to Hermitian–Einstein metrics, and the non-abelian Hodge correspondence linking Higgs bundles with fundamental group representations.
These results have influenced algebraic geometry considerably. For instance, the classification of Fano manifolds has advanced through the theory of K-stability, an analytic notion that interacts with birational geometry. More broadly, the minimal model programme draws on both algebraic and complex techniques, relying on the interplay between singularities, curvature properties, and canonical models.
The impact of physics has also been profound. Complex geometry is indispensable in modern theoretical physics, particularly in conformal field theory, string theory, and mirror symmetry. The geometry of Calabi–Yau manifolds, for instance, models compact dimensions in string theory. Insights from physical dualities have suggested conjectures within geometry, including the Strominger–Yau–Zaslow (SYZ) conjecture predicting that mirror Calabi–Yau manifolds admit dual Lagrangian fibrations.
Conversely, developments in symplectic geometry such as Gromov–Witten theory have greatly enriched enumerative questions about complex varieties, linking counts of curves to intersection-theoretic invariants.

Complex manifolds and analytic varieties

A central object in the field is the complex manifold. A complex manifold of complex dimension n is a Hausdorff, second-countable topological space locally homeomorphic to open subsets of Cn\mathbb{C}^nCn, with the additional condition that coordinate changes are biholomorphic. Because biholomorphisms are smooth with smooth inverses, every complex manifold carries a natural smooth structure of real dimension 2n2n2n.
Complex geometry also studies singular spaces. A complex analytic variety arises as the common zero set of finitely many holomorphic functions, defined locally inside an open subset of Cn\mathbb{C}^nCn. Points where the Jacobian matrix fails to have full rank are singular. Projective analytic varieties are defined similarly using holomorphic data inside complex projective space. These notions connect analytic geometry with algebraic geometry: by results such as Serre’s GAGA theorem, projective analytic varieties correspond precisely to projective algebraic varieties, and holomorphic objects coincide with algebraic ones in this setting.
This equivalence places complex geometry closer to algebraic geometry than to differential geometry, despite its heavy use of analytical tools. The classical behaviour of singularities in one complex variable generalises in powerful ways, allowing explicit study of analytic and algebraic singularities that would be unwieldy in real geometry.

Intersections with other branches of mathematics

Complex geometry overlaps substantially with several major areas:

• Algebraic geometry: Analytic methods support the study of algebraic varieties, leading to Hodge theory, cohomology theorems, and insights that contributed to the formulation of the Weil conjectures and Grothendieck’s standard conjectures. Deformation theory for complex manifolds inspired analogous theories for schemes.
• Symplectic geometry: Kähler manifolds are symplectic, and many symplectic invariants have complex geometric interpretations, particularly in enumerative questions.
• Representation theory: Generalised flag varieties and their holomorphic line bundles yield the Borel–Weil–Bott theorem, linking geometry with highest-weight representations.
• Gauge theory: Holomorphic vector bundles serve as natural domains for equations such as the Yang–Mills equations, producing correspondences between geometric stability and analytical solutions.
• Riemannian geometry: Complex structures give rise to special metric geometries, including Calabi–Yau and hyperkähler manifolds. These spaces often serve as sources of exotic Riemannian phenomena.

Major themes and classification problems

A recurring theme in complex geometry is classification. The aim is to understand complex manifolds and varieties through invariants such as their curvature, canonical bundle, Hodge structure, and singularities. The minimal model programme seeks to simplify the birational classification of varieties by producing models with good canonical properties.
Another significant theme concerns the geometry of holomorphic objects. Holomorphic vector bundles, coherent sheaves, and moduli spaces of such objects are central topics, influenced by developments in both algebra and analysis.
Complex geometry also lies at the heart of major open problems. The Hodge conjecture, one of the Millennium Prize Problems, concerns the relationship between algebraic cycles and Hodge classes on projective complex manifolds. Its resolution would unify analytic and algebraic aspects of cohomology in a profound way.