Algebraic formula for $L^2$ higher direct images in non-abelian Hodge theory beyond relative dimension one

Donagi, Pantev, and Simpson (arXiv:1612.06388) proved that for a semistable family of curves, the $L^2$ (middle-perversity) higher direct images of a tame harmonic bundle can be computed purely algebraically, as the higher direct images of a modified Dolbeault complex of the associated parabolic Higgs bundle, cut by monodromy weight filtrations; the parabolic structure is identified through the $V$ -filtration, and local freeness rests on the Sabbah–Mochizuki pure twistor $\mathcal{D}$ -module decomposition theorem. The paper explicitly records that no theorem is possible at present for fibers of dimension greater than one, and lists as open: an algebraic definition of the $L^2$ Dolbeault complex when several weight filtrations interact, a direct non-twistor proof of local freeness and strictness, and a global formula under alterations and toric semistable reduction; a uniform-in- $\lambda$ account is known only through an unpublished private note of Mochizuki acknowledged in the paper. The problem is to construct the algebraic theory and prove the analogue of the main theorem in relative dimension at least two.

Background

Non-abelian Hodge theory relates the topology of a complex algebraic variety to holomorphic linear-algebra data on it. A Higgs bundle on a smooth complex variety $X$ is a pair $(E,\varphi)$ with $E$ a holomorphic vector bundle and $\varphi:E\to E\otimes\Omega^1_X$ an $\mathcal{O}_X$ -linear Higgs field satisfying $\varphi\wedge\varphi=0$ ; a flat bundle is a vector bundle with a flat (integrable) connection. A harmonic bundle is a flat bundle equipped with a harmonic metric, that is, a Hermitian metric on the fibers satisfying a harmonic-map type equation for the flat connection - the non-abelian analogue of a harmonic function on a domain. On the complement $X-D$ of a simple normal crossing divisor $D$ (one locally defined by a product of coordinate functions), the non-abelian Hodge correspondence (NAHC) identifies tame harmonic bundles - those with at worst logarithmic growth of the Higgs field along $D$ and nilpotent residues - with tame polystable parabolic Higgs bundles on $(X,D)$ , where a parabolic structure is, along each irreducible component of $D$ , a decreasing filtration of the bundle indexed by real weights. For a local system underlying a harmonic bundle, the natural cohomology to push forward along a map is $L^2$ cohomology: cohomology represented by forms that are square-integrable, together with their derivatives, for a complete metric behaving like the Poincaré (hyperbolic-type) metric near $D$ . A theorem of Zucker for curves, extended within Hodge theory, identifies this $L^2$ cohomology with middle-perversity intersection cohomology, that is, with the cohomology of the intermediate (minimal) extension $j_*L$ of the local system across $D$ ; correspondingly $R^if_*(j_*L)$ is called the middle-perversity higher direct image. Donagi–Pantev–Simpson (arXiv:1612.06388, an unpublished 2016 preprint; v1 is the only version) asked for a purely algebraic counterpart of these analytic pushforwards. Their Main Theorem 3.6 concerns a semistable family of curves: $f:X\to Y$ from a smooth projective surface to a smooth projective curve, with reduced divisors $Q\subset Y$ and $D=D_V+D_H\subset X$ of simple normal crossings such that $D_V=f^{-1}(Q)$ has reduced normal-crossing fibers (this is the semistability of the family) and the horizontal divisor $D_H$ is étale over $Y$ . For a tame harmonic bundle on $X-D$ satisfying their Nilpotence Hypothesis (nilpotent Higgs residues and monodromy eigenvalues around components of $D$ that are roots of unity), with corresponding parabolic Higgs bundle $(E,\varphi)$ , they define a modification of the relative Dolbeault complex - the $L^2$ parabolic Dolbeault complex $\operatorname{DOL}^{par}_{L^2}(X/Y,E_{\alpha(a)})$ - in which the graded pieces of the parabolic filtration, in the normal directions to $D_H$ , are further cut by the monodromy weight filtration, the canonical filtration $W(N)$ associated to the nilpotent monodromy operator $N$ acting on each graded piece. Their theorem states that $F^i_a=R^if_*(\operatorname{DOL}^{par}_{L^2}(X/Y,E_{\alpha(a)}))$ are locally free, compatible with base change, fit together over the parabolic levels $a$ into a parabolic Higgs bundle on $(Y,Q)$ (the Higgs field induced by wedge product with $\varphi$ on relative forms, a Gauss–Manin-type construction - the Gauss–Manin connection is the canonical flat connection on the relative de Rham cohomology of a smooth family), and that this parabolic Higgs bundle is exactly the one associated by the NAHC to the middle-perversity higher direct image of the underlying local system on $Y-Q$ . The parabolic weights of the pushforward are identified through the Kashiwara–Malgrange $V$ -filtration - the filtration of a $\mathcal{D}$ -module along a function, encoding the exponents of its nearby cycles - computed via the graph embedding at singular points of $f$ , and the local freeness ultimately rests on strictness in the Sabbah–Mochizuki theory of pure twistor $\mathcal{D}$ -modules, the category that packages the whole $\lambda$ -line of flat ( $\lambda=1$ ) to Higgs ( $\lambda=0$ ) objects into one structure and whose pure objects satisfy a decomposition theorem. Section 7 of the same paper extends the result to families of curves over higher-dimensional bases, but explicitly states that for fibers of dimension greater than one the authors are not able to state a theorem, and the present problem records what remains. Adjacent directions have been developed in the later literature: twistor-side comparison theorems and Hard Lefschetz for pushforwards (Mochizuki), a non-twistor proof of Sabbah's decomposition theorem for proper maps (Wei–Yang), algebraic direct images for maps between curves (Alfaya–Biswas), and analytic positivity packages (Shentu–Zhao, Zhao); no resolution of the higher-relative-dimension case is known.

Problem Statement

Let $f:X\to Y$ be a morphism of smooth projective varieties of relative dimension at least two - the basic open case is a smooth projective variety of dimension at least three fibered over a smooth projective curve $Y$ - with a reduced divisor $Q\subset Y$ , and a reduced simple normal crossing divisor $D=D_V+D_H\subset X$ with $D_V=f^{-1}(Q)$ and $D_H$ horizontal, $f$ smooth away from $D_V$ . Let $(E,\varphi)$ be a tame polystable parabolic Higgs bundle on $(X,D)$ corresponding under the non-abelian Hodge correspondence to a tame harmonic bundle satisfying the Nilpotence Hypothesis of Donagi–Pantev–Simpson (nilpotent Higgs residues and monodromy eigenvalues that are roots of unity), and let $L$ be the underlying local system on $X-D$ . The problem, recorded as an explicit expectation in section 7.4 of the source paper, is to extend the algebraic description of the $L^2$ pushforward to this setting, specifically: (1) define algebraically - in terms of parabolic structures and monodromy weight filtrations - the $L^2$ parabolic Dolbeault complex $\operatorname{DOL}^{par}_{L^2}(X/Y,E_{\alpha(a)})$ in this generality, including the configuration where components of $D_H$ intersect each other and where crossings of $D_V$ touch $D_H$ , so that several monodromy weight filtrations interact (the source conjectures the answer is the complex of holomorphic forms square-integrable with their derivatives for the Poincaré metric along $D$ , known analytically only for variations of Hodge structure - local systems underlying families of polarized Hodge structures, such as the primitive cohomology of a family of projective varieties - by Jost–Yang–Zuo; a candidate algebraic complex is written down only when $D_H$ is smooth over $Y$ and meets only smooth points of $D_V$ ); and (2) prove the analogue of the source's Main Theorem 3.6: the sheaves $F^i_a=R^if_*(\operatorname{DOL}^{par}_{L^2}(X/Y,E_{\alpha(a)}))$ are locally free, their formation is compatible with base change, they fit together into a parabolic Higgs bundle with Gauss–Manin Higgs field on $(Y,Q)$ , and this parabolic Higgs bundle is the one associated by the non-abelian Hodge correspondence to the middle-perversity higher direct image $R^if_*(j_*L)$ ; equivalently, identify the parabolic weights through the $V$ -filtration for the graph embedding at normal-crossing points where $D_V$ meets $D_H$ . The source verifies this expectation only for relative dimension one (its main theorem) and for a one-point base (absolute cohomology, by analytic arguments); everything between - genuine relative dimension at least two - is open. The same paper also records related open questions in this direction: a direct algebraic or analytic proof of the local freeness and strictness of these higher direct images, which currently is obtained only through the Sabbah–Mochizuki pure twistor $\mathcal{D}$ -module decomposition theorem; a global algebraic formula for the higher direct images under de Jong-type alterations (an alteration is a proper, generically finite surjection from a smooth variety; Abramovich–Karu weak semistable reduction is the related procedure that replaces a family by one whose fibers are reduced with controlled toric singularities), which requires new calculations on those toric singularities in higher relative dimension; and a uniform-in- $\lambda$ presentation of the comparison statements, for which the only known argument is an unpublished private note of Mochizuki acknowledged in the source paper.

References

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