Let $H$ be a finite-dimensional complex [vector space](/page/Vector%20Space), and let $p_{\min},p_{\max}\in\mathbb Z$ with $p_{\min}\le p_{\max}$. Let $F^\bullet$ be a decreasing filtration of $H$ indexed by the integers $p_{\min},\dots,p_{\max}+1$, meaning

\begin{align*} H=F^{p_{\min}}\supseteq F^{p_{\min}+1}\supseteq \cdots \supseteq F^{p_{\max}}\supseteq F^{p_{\max}+1}=0. \end{align*}

For each $p\in\{p_{\min},\dots,p_{\max}\}$, define the graded piece $\operatorname{Gr}_F^pH:=F^p/F^{p+1}$. Let $\operatorname{Flag}(H,(\dim_{\mathbb C}F^p)_p)$ denote the complex flag variety parametrizing decreasing filtrations $E^\bullet$ of $H$ with $\dim_{\mathbb C}E^p=\dim_{\mathbb C}F^p$ for every $p$. Then the tangent space at $F^\bullet$ has the canonical quotient description

\begin{align*} T_{F^\bullet}\operatorname{Flag}(H,(\dim_{\mathbb C}F^p)_p)\cong \operatorname{End}_{\mathbb C}(H)\big/\{A\in \operatorname{End}_{\mathbb C}(H):A(F^p)\subseteq F^p\text{ for every }p\}. \end{align*}

Equivalently, after choosing a complex-linear splitting of the filtration $H=\bigoplus_p G^p$ with $F^p=\bigoplus_{r\ge p}G^r$, this tangent space is identified with

\begin{align*} T_{F^\bullet}\operatorname{Flag}(H,(\dim_{\mathbb C}F^p)_p)\cong \bigoplus_p\operatorname{Hom}_{\mathbb C}(\operatorname{Gr}_F^pH,H/F^p). \end{align*}

If $Q:H\times H\to\mathbb C$ is a [bilinear form](/page/Bilinear%20Form) and $\operatorname{Per}\subseteq\operatorname{Flag}(H,(\dim_{\mathbb C}F^p)_p)$ is the locally closed subvariety cut out near $F^\bullet$ by algebraic polarization relations expressed in terms of $Q$ and the filtration, then the Zariski tangent space $T_{F^\bullet}\operatorname{Per}$ is the linear subspace of $T_{F^\bullet}\operatorname{Flag}(H,(\dim_{\mathbb C}F^p)_p)$ cut out by the differentials at $F^\bullet$ of those polarization relations. On the smooth locus of $\operatorname{Per}$, this Zariski tangent space is the ordinary complex tangent space.