Restricted spaces of holomorphic sections vanishing along subvarieties

Coman, Dan; Marinescu, George; Nguyên, Viêt-Anh

doi:10.1007/s00209-025-03706-w

Abstract LetXbe a compact normal complex space of dimensionnandLbe a holomorphic line bundle onX. Suppose that$$\Sigma =(\Sigma _1,\ldots ,\Sigma _\ell )$$

Σ = (Σ_{1}, \dots, Σ_{ℓ})

is an$$\ell $$

ℓ

-tuple of distinct irreducible proper analytic subsets ofX,$$\tau =(\tau _1,\ldots ,\tau _\ell )$$

τ = (τ_{1}, \dots, τ_{ℓ})

is an$$\ell $$

ℓ

-tuple of positive real numbers, and let$$H^0_0(X,L^p)$$

H_{0}^{0} (X, L^{p})

be the space of holomorphic sections of$$L^p:=L^{\otimes p}$$

L^{p} : = L^{\otimes p}

that vanish to order at least$$\tau _jp$$

τ_{j} p

along$$\Sigma _j$$

Σ_{j}

,$$1\le j\le \ell $$

1 \leq j \leq ℓ

. If$$Y\subset X$$

Y \subset X

is an irreducible analytic subset of dimensionm, we consider the space$$H^0_0 (X|Y, L^p)$$

H_{0}^{0} (X | Y, L^{p})

of holomorphic sections of$$L^p|_Y$$

L^{p} |_{Y}

that extend to global holomorphic sections in$$H^0_0(X,L^p)$$

H_{0}^{0} (X, L^{p})

. Assuming that the triplet$$(L,\Sigma ,\tau )$$

(L, Σ, τ)

is big in the sense that$$\dim H^0_0(X,L^p)\sim p^n$$

dim H_{0}^{0} (X, L^{p}) \sim p^{n}

, we give a general condition onYto ensure that$$\dim H^0_0(X|Y,L^p)\sim p^m$$

dim H_{0}^{0} (X | Y, L^{p}) \sim p^{m}

. WhenLis endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces$$H^0_0(X|Y,L^p)$$

H_{0}^{0} (X | Y, L^{p})

converge to a certain equilibrium current onY. We apply this to the study of the equidistribution of zeros inYof random holomorphic sections in$$H^0_0(X|Y,L^p)$$

H_{0}^{0} (X | Y, L^{p})

as$$p\rightarrow \infty $$

p \to \infty

.

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