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I have a hyperbolic 3-manifold $M$ that I'd like to see sit inside a complex algebraic variety $V$ as beautifully and snugly as possible. I don't want to specify attributes of "beauty" (e.g., embedding) and "snugness" (e.g., low-dimensional) in order not to constrain the possible answers too much. Are there any general results in this direction?

EDIT: After Will Sawin's comments and riccoli's answer that are very helpful in refining the question I'd like to instead ask: What is a complex algebraic variety of lowest possible dimension into which a given hyperbolic 3-manifold can be (1) isometrically $C^\infty$-embedded if the variety has a Kähler metric, or (2) $C^\infty$-embedded, or (3) mapped in a way that is injective on $\pi_1$'s? Is it possible to do these with an algebraic surface?

I have a hyperbolic 3-manifold $M$ that I'd like to see sit inside a complex algebraic variety $V$ as beautifully and snugly as possible. I don't want to specify attributes of "beauty" (e.g., embedding) and "snugness" (e.g., low-dimensional) in order not to constrain the possible answers too much. Are there any general results in this direction?

EDIT: After Will Sawin's comments and riccoli's answer that are very helpful in refining the question I'd like to instead ask: What is a complex algebraic variety of lowest possible dimension into which a given hyperbolic 3-manifold can be (1) isometrically $C^\infty$-embedded if the variety has a Kähler metric, or (2) $C^\infty$-embedded, or (3) mapped in a way that is injective on $\pi_i$'s? Is it possible to do these with an algebraic surface?

I have a hyperbolic 3-manifold $M$ that I'd like to see sit inside a complex algebraic variety $V$ as beautifully and snugly as possible. I don't want to specify attributes of "beauty" (e.g., embedding) and "snugness" (e.g., low-dimensional) in order not to constrain the possible answers too much. Are there any general results in this direction?

I have a hyperbolic 3-manifold $M$ that I'd like to see sit inside a complex algebraic variety $V$ as beautifully and snugly as possible. I don't want to specify attributes of "beauty" (e.g., embedding) and "snugness" (e.g., low-dimensional) in order not to constrain the possible answers too much. Are there any general results in this direction?

EDIT: After Will Sawin's comments and riccoli's answer that are very helpful in refining the question I'd like to instead ask: What is a complex algebraic variety of lowest possible dimension into which a given hyperbolic 3-manifold can be (1) isometrically $C^\infty$-embedded if the variety has a Kähler metric, or (2) $C^\infty$-embedded, or (3) mapped in a way that is injective on $\pi_1$'s? Is it possible to do these with an algebraic surface?