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Let $M$ be a complex manifold with a hermitian metric (volumes and distances will be wrt this metric). Let $X\subset M$ be a complex analytic subspace of $M$ and $Y\subset X$ an analytic set containing $X_\mathrm{sing}$. Set $n=\dim X_\mathrm{reg}$.

Define $W_p$ as the space of holomorphic $p-$vectorfields $\xi\in\Omega_p(X\setminus Y)$ such that:

  1. $$\int_{X\setminus Y}\langle \xi, \phi_\epsilon\rangle dV_{X\setminus Y}\xrightarrow[\epsilon\to0]{}0$$
    with $\phi_\epsilon\in \mathcal{D}^{p,n}(M)$ supported in $Y_\epsilon=\{x\in X\ :\ d(x,Y)\leq \epsilon\}$;
  2. $$\int_{X\setminus Y}\xi, \overline{\partial}\langle\psi_\epsilon\rangle dV_{X\setminus Y}\xrightarrow[\epsilon\to0]{}0$$ with $\psi_\epsilon\in\mathcal{D}^{p,n-1}(M)$ supported in $Y_\epsilon$.

What can we say about $W_p$?

Remarks

  • The two conditions can be reformulated like this:
    $$\int_{Y_\epsilon}\langle \xi, dg_1\wedge\ldots\wedge dg_{n+p}\rangle dV\to0$$ as $\epsilon\to0$, for every $g_1,\ldots, g_{n+p}\in\mathcal{C}^\infty(M)$;
    $$\int_{bY_\epsilon}\langle \xi\llcorner\nu^{0,1}, dg_1\wedge\ldots\wedge dg_{n+p-1}\rangle dV'\to0$$
    as $\epsilon\to0$, for every $g_1,\ldots, g_{n+p}\in\mathcal{C}^\infty(M)$, with $\nu^{0,1}$ the $(0,1)$ component of the normal covector of $bY_\epsilon$ and $dV'$ the volume of $bY_\epsilon$.
  • On complex curves, the problem is purely local (as $Y$ has to be discrete); we can do all the computations and notice that $W_0$ consists of the $(0,1)-$vector fields which are holomorphic on the normalization (or better on a resolution of singularities) and $W_1$ consists of the $(1,1)-$vector fields which are, on the resolution of singularities, sections of $\mathcal{O}(|E|-E)$, with $E$ the exceptional divisor.
  • The main case I'm interested in is $M=\mathbb{CP}^m$, $Y=X_\mathrm{sing}$.
  • Morally, the two conditions relate the growth of $\xi$ near the singular set with the vanishing of differential forms due to the singularity, so it seemed to me a good idea to try and pull back the problem on the desingularization, but then I need to control, in some way, the vanishing of the maps induced by the desingularization morphism on the spaces of differential forms...

Thanks in advance.

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