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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?

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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.

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.

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  • $\begingroup$ Sorry for my confusion. Is it clear that the definition does not depend on the choice of local embeddings $M\hookrightarrow U$? $\endgroup$
    – Z. M
    Jun 13, 2022 at 20:23
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    $\begingroup$ I believe it is quite straightforward: If $F : U \to U'$ is a holomorphic map, which induces an isomorphism of local embeddings $M \stackrel{\cong}{\to} M'$, then the independence should follow from the fact that $F^*$ maps $N_{M',U'}$ to $N_{M,U}$, and it is continuous with respect to the semi-norms, basically by the chain rule. $\endgroup$
    – Richard Lärkäng
    Jun 14, 2022 at 7:27
  • $\begingroup$ Do these forms recover the Betti cohomology? I see that you are restricting to the regular locus, while Kashiwara's definition of D-modules imposes that the module is supported on the whole subvariety, not necessarily on its regular locus. Seemingly the definition you give does not extend to de Rham complex of D-modules. Hartshorne's definition of algebraic de Rham complex also depends on the embedding. That is why I am suspicious. $\endgroup$
    – Z. M
    Jun 14, 2022 at 9:10
  • $\begingroup$ I'm not sure what you mean by Betti cohomology. If you mean $H^k(M,\mathbb{C})$, then as is well-known, if $M$ is smooth, then this may be represented by cohomology of the de Rham complex of global sections of smooth forms or of currents, by the Poincaré lemma and de Rhams theorem. However, neither the sheaf of smooth forms or currents are locally exact on a singular space in general, and would thus not represent $H^k(M,\mathbb{C})$. $\endgroup$
    – Richard Lärkäng
    Jun 14, 2022 at 9:23
  • $\begingroup$ The sheaves are not restricted to the regular locus. Only the definition of $N_{M,U}$ is given in terms of the regular locus, which is reasonable if you want a torsion-free sheaf, since then a form should vanish if it vanishes generically on $M$, i.e., on the regular locus. $\endgroup$
    – Richard Lärkäng
    Jun 14, 2022 at 9:25

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