I'm reading a paper in which the author use $(p,q)$-forms and currents on a complex analytic space.


My question is how to define $(p,q)$-current on complex space? Does it have similar properties like closeness, positivity and cohomology as in the smooth case? Could we define intersection theory or wedge product in this case?



How to define a current on a complex analytic space
The definition can be found in for example Section 3.3 of T. Bloom, M. Herrera: De Rham Cohomology of an Analytic Space (Inventiones math. 7, 275-296 (1969)) and Section 4.2 of M. Herrera, D. Lieberman: Residues and Principal Values on Complex Spaces (Math. Ann. 194, 259-294 (1971)) (in the more general setting of semianalytic sets).


If $M \subseteq U$, where $M$ is an analytic subset of an open set $U \subseteq \mathbb{C}^n$, and $\mathcal{E}\_U$ denotes the sheaf of smooth forms on $U$, then one may first define $N\_{M,U} \subseteq \mathcal{E}\_U$ as the sheaf of smooth forms on $U$ whose pullback to the regular part of $M$ vanishes. Then one defines the sheaf of smooth forms on $M$ as $\mathcal{E}\_M := \mathcal{E}\_U/N\_{M,U}$. This construction then globalizes to define smooth forms on an analytic space $\mathcal{E}\_X$ which on a local model $M \subseteq U$ coincides with $\mathcal{E}\_M$ defined above.


Then, one equips the space $\Gamma\_c(U,\mathcal{E}\_U)$ of sections with compact support with the usual appropriate (locally convex inductive) topology induced by uniform convergence of the coefficients of the form and of their derivatives. Since the kernel of the projection $\Gamma\_c(U,\mathcal{E}\_U) \to \Gamma\_c(M,\mathcal{E}\_M)$ is closed, one may equip $\Gamma\_c(M,\mathcal{E}\_M)$ with the quotient Hausdorff topology. Finally, for a general analytic space $X$, one equips $\Gamma\_c(X,\mathcal{E}\_X)$ with the appropriate topology which induces the given topology on the local models.


Regarding the question of what properties these currents have, this seems to be way broad question to answer here, but would have to be checked on a case by case basis.


One word of caution is that the definition in Bloom-Herrera is actually stated in two ways, claimed to be equivalent to each other, where the other definition is that $N\_{M,U}$ is alternatively defined as the space of smooth forms $\alpha$ on $U$ such that for any smooth map $g : W \to U$, with $W$ a manifold and $g(W) \subseteq M$, $g^\*\alpha = 0$. As far as I am aware, the equivalence of these two definitions is only addressed recently in M. Andersson, H. Samuelsson Kalm: A note on smooth forms on analytic spaces (Math. Scand. 127, 521–526 (2021)). With this alternative definition, it is straight-forward that a smooth map $f : N \to M$ of analytic spaces induces a pullback map $f^\*$ of smooth forms, and thus a push-forward of smooth proper maps of currents, while with the simpler definition of smooth forms I used above, this is not so immediate.




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The definition in the local models also has a more concrete version, unless I am mistaken about some topological subtlety:
Using the notation above, the push-forward of a current $T' := i\_\*T$ on $M$ under the inclusion $i : M \to U$ yields a current $T'$ on $U$ which vanishes on the ideal of test-forms in $N\_{M,U}$, and conversely, any current $T'$ with this property should be the push-forward of a unique current on $M$, and one could thus take this as the definition.


The global case should then be possible to describe through a set of compatible currents in the local models.


