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A complex analytic space is a locally ringed space locally isomorphic to the common zero set of finitely many holomorphic functions on an open subset of ${\mathbb{C}}^n$, equipped with the quotient sheaf of holomorphic functions by a coherent ideal sheaf. This sheaf is part of the structure: the same underlying set can carry different analytic structures.

The cusp $y^2=x^3$ gives a reduced singular example. Its local holomorphic functions are represented by ambient holomorphic functions modulo the equation $y^2-x^3$. A complex manifold is the special smooth local model, with no singular equations needed after choosing coordinates.

General complex analytic spaces can also be nonreduced. The quotient $\mathbb{C}\{z\}/(z^2)$ has underlying set consisting of one point, but retains a nonzero nilpotent class of $z$. This distinguishes an analytic space from only a zero set. The notion extends complex geometry to singularities and infinitesimal structure that ordinary manifold charts cannot describe.

Complex geometry studies spaces locally described by complex coordinates with holomorphic transition maps. A complex $n$-dimensional complex manifold has real dimension $2n$, but its structure is more restrictive than a smooth atlas: the transition maps satisfy the Cauchy-Riemann equations. The complex projective line is a compact example, described by two complex charts related by $w=1/z$. Hermitian metrics and Kähler manifolds connect this analytic structure to Riemannian geometry, while complex algebraic varieties connect it to algebraic geometry.

Hans Grauert developed methods in complex analysis and complex geometry. His 1960 paper on analytic sheaf theory studies how sheaf cohomology behaves in families of complex spaces.^[ref]

A central theme is cohomology and base change: under appropriate properness, coherence and constancy hypotheses, fibrewise cohomology groups assemble into a locally free sheaf, and restriction to a fibre recovers the expected cohomology. This turns dimensions of solution spaces into geometric information about a family. The original archive also records a correction to the paper.^[ref]

Jean-Pierre Demailly developed analytic methods in complex geometry and algebraic geometry. His author bibliography includes extensive treatments of Hodge theory, positivity and the analysis of complex manifolds.^[ref]

His Complex Analytic and Differential Geometry by Jean-Pierre Demailly connects local analytic tools with global geometric objects: Cauchy integral formulas and the Dolbeault operator lead into sheaf cohomology, divisors and line bundles. His geometric lectures also derive the Ricci form from the logarithmic determinant of a Kähler metric, making curvature computable in complex coordinates.^[ref][ref]

Kähler geometry studies complex manifolds with a Hermitian metric $g$ whose associated real two-form $\omega (X,Y)=g(JX,Y)$ is closed. Here $J$ is the complex structure. Equivalently, the Levi-Civita connection preserves $J$. Thus the same metric simultaneously supports complex geometry, Riemannian geometry and symplectic geometry.^[ref]

On ${\mathbb{C}}^n$, the Euclidean metric gives $\omega ={\sum }_{j}d{x}_j\wedge d{y}_j$, which is closed and nondegenerate. Locally one can express a Kähler form as $\omega =i\partial \bar{\partial }\phi$ for a real Kähler potential $\phi$, using a stated normalization. On Complex projective space, $\phi =\log (1+{\sum }_j|z_j{|}^2)$ gives the Fubini-Study metric in an affine chart. The potentials need not glue as one global function; their forms glue. This is how locally elementary differentiation can define nontrivial global geometry.

A geometric Kähler potential is locally a real smooth function $\phi$ with $\omega =i\partial \bar{\partial }\phi$ for the Kähler form. Its complex Hessian must be positive definite to define a Kähler metric. Conventions sometimes absorb a factor of $1/2$ into $\phi$. Potentials differing by the real part of a holomorphic function determine the same form. This topic concerns ordinary complex geometry; the existing undisambiguated Kähler potential entry concerns superfields.

Marco Gualtieri works in differential geometry and mathematical physics, with a focus on generalized geometry. His own research page identifies that focus rather than presenting him only as an instructor.^[ref] The geometric viewpoint concerns structures carried by a smooth manifold, where tangent directions and covectors can be studied together. It connects the study of complex geometry and symplectic geometry with mathematical descriptions of physical fields.

Homepage: www.math.toronto.edu/mgualt/

Instructor for the 2012 Topology I: smooth manifolds (MAT1300HF) course; the course information explicitly identifies his role as instructor.