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Let $2g+m\ge 3$. A holomorphic family of (non-singular, compact) Riemann surfaces of type $(g,m)$ is a triple $(X,Y,\pi,s_1,\ldots,s_m)$, where $X,Y$ are complex manifolds of (complex) dimension $n+1, …

Suppose $C$ is a (non-singular) compact Riemann surface of genus $g$ and with $n$ (distinct) marked points $p_1,\ldots,p_n$. If we assume the stability condition ($2-2g-n<0$), then it is proved in "Ge …

Suppose $C$ is a compact Riemann surface and $X$ is a compact Kähler manifold. Suppose $f:C\to X$ is a stable holomorphic map. Then, the deformations of $f$ are controlled by the complex $L^\bullet = …

I'm trying to learn about the deformation theory of compact complex spaces. I'm familiar with the case of compact complex manifolds from the paper "On the Locally Complete Families of Complex Analytic …

Suppose $(X,\omega)$ is a compact symplectic manifold and $J$ is an $\omega$-compatible almost complex structure on $X$ (the symplectic structure seems to be irrelevant for this question actually). Su …

While considering a "plumbed family of complex curves" (i.e. a complex $1$-parameter family of smooth curves degenerating to a nodal curve), I encountered an unexpected holomorphic tubular neighborhoo …

Suppose $C$ is a complex projective curve (or a compact $1$-dimensional connected reduced complex space). If $C$ is smooth, then its module of differentials $\Omega^1_C$ is locally free. If $C$ is a s …

I am looking for a reference which develops the theory of the cotangent complex for complex analytic spaces. I need this to justify some computations I did assuming some formal properties which hold i …