Building on Fujita-Griffiths method of computing metrics on Hodge bundles, we show that the direct image of an adjoint semi-ample line bundle by a projective submersion has a continuous metric with Griffiths semi-positive curvature. This shows that for every holomorphic semi-ample vector bundle $E$ on a complex manifold, and every positive integer $k$, the vector bundle $S^kE\otimes\det E$ has a continuous metric with Griffiths semi-positive curvature. If $E$ is ample on a projective manifold, the metric can be made smooth and Griffiths positive.
Motivated by our previous work on Hodge-index type inequalities, we give a form of mixed Hodge-Riemann bilinear relation by using the notion of $m$-positivity, whose proof is an adaptation of the works of Timorin and Dinh-Nguyên. This mixed Hodge-Riemann bilinear relation holds with respect to mixed polarizations in which some satisfy particular positivity condition, but could be degenerate along some directions. In particular, it applies to fibrations of compact Kähler manifolds.
We prove the weak positivity of the kernels of Kodaira-Spencer- type maps for pure Hodge module extensions of generically defined variations of Hodge structure.
The main result of this paper concerns the positivity of the Hodge bundles of abelian varieties over global function fields. As applications, we obtain some partial results on the Tate--Shafarevich group and the Tate conjecture of surfaces over finite fields.
This is a survey of vanishing and positivity theorems for Hodge modules, and their recent applications to birational and complex geometry, expanding on my lecture at the 2015 AMS Summer Institute.
It is well known that positivity properties of the curvature of a vector bundle have implications on the algebro-geometric properties of the bundle, such as numerical positivity, vanishing of higher cohomology leading to existence of global sections etc. It is also well known that bundles arising in Hodge theory tend to have positivity properties. From these considerations several issues arise: (i) In general for bundles that are semi-positive but not strictly positive; what further natural conditions lead to the existence of sections of its symmetric powers? (ii) In Hodge theory the Hodge met…
We discuss the variations of mixed Hodge structure for cohomology with compact support of quasi-projective simple normal crossing pairs. We show that they are graded polarizable admissible variations of mixed Hodge structure. Then we prove a generalization of the Fujita--Kawamata semi-positivity theorem.
These are the lecture notes based on earlier papers with some additional new results. New and simple proofs are given for local freeness theorem and the semipositivity theorem. A decomposition theorem for higher direct images of dualizing sheaves of Kollár is extended to the sheaves of differential forms of arbitrary degrees in the case of a well prepared birational model. We will also prove the log versions of some of the results, i.e., the case where we allow horizontal boundary components.
The purpose of this paper is to study certain notions of metric positivity for the lowest nonzero piece in the Hodge filtration of a Hodge module. We show that the Hodge metric satisfies the minimal extension property. In particular, this singular Hermitian metric has semi-positive curvature.
We show that the dualizing sheaves of reduced simple normal crossings pairs have a canonical weight filtration in a compatible way with the one on the corresponding mixed Hodge modules by calculating the extension classes between the dualizing sheaves of smooth varieties. Using the weight spectral sequence of mixed Hodge modules, we then reduce the semi-positivity theorem for the higher direct images of dualizing sheaves to the smooth case where the assertion is well known. This may be used to simplify some constructions in a recent paper of Y. Kawamata. We also give a simple proof of the semi…
In this note, we show how the classical Hodge index theorem implies the Hodge index conjecture of Beilinson for height pairing of homologically trivial codimension two cycles over function field of characteristic 0. Such an index conjecture has been used in our paper on Gross-Schoen cycles to deduce the Bogomolov conjecture and a lower bound for Hodge class (or Faltings height) from some conjectures about metrized graphs which have just been recently proved by Zubeyir Cinkir.
In this paper, we investigate higher direct images of log canonical divisors. After we reformulate Kollár's torsion-free theorem, we treat the relationship between higher direct images of log canonical divisors and the canonical extensions of Hodge filtration of gradedly polarized variations of mixed Hodge structures. As a corollary, we obtain a logarithmic version of Fujita-Kawamata's semi-positivity theorem. By this semi-positivity theorem, we generalize Kawamata's positivity theorem and apply it to the study of a log canonical bundle formula. The final section is an appendix, which is a res…
Let f : X --> Y be a holomorphic map of complex manifolds, which is proper, Kahler, and surjective with connected fibers, and which is smooth over Y-Z the complement of an analytic subset Z. Let E be a Nakano semi-positive vector bundle on X, and consider direct image sheaves F = R^qf_*(K_{X/Y} \otimes E) for q \geq 0. In our previous paper, we discussed the Nakano semi-positivity of F with respect to the so-called Hodge metric, when the map f is smooth. In this paper, we discuss the extension of the induced metric on the tautological line bundle O(1) on the projective space bundle P(F) ``over…
We study how the geometry of a projective variety $X$ is reflected in the positivity properties of the diagonal $Δ_X$ considered as a cycle on $X \times X$. We analyze when the diagonal is big, when it is nef, and when it is rigid. In each case, we give several implications for the geometric properties of $X$. For example, when the diagonal is big, we prove that the Hodge groups $H^{k,0}(X)$ vanish for $k>0$. We also classify varieties of low dimension where the diagonal is nef and big.
We study the degeneration of semipositive smooth hermitian line bundles on open complex manifolds, assuming that the metric extends well away from a codimension two analytic subset of the boundary. Using terminology introduced by R. Hain, we show that under these assumptions the so-called height jump divisors are always effective. This result is of particular interest in the context of biextension line bundles on Griffiths intermediate jacobian fibrations of polarized variations of Hodge structure of weight -1, pulled back along normal function sections. In the case of the normal function on M…
We show that, for any fixed weight, there is a natural system of Hodge sheaves, whose Higgs field has no poles, arising from a flat projective family of varieties parametrized by a regular complex base scheme, extending the analogous classical result for smooth projective families due to Griffiths. As an application, based on positivity of direct image sheaves, we establish a criterion for base spaces of rational Gorenstein families to be of general type. A key component of our arguments is centered around the construction of derived categorical objects generalizing relative logarithmic forms …
In this paper, we provide a Hodge-theoretic interpretation of Laurent phenomenon for general skew-symmetric quantum cluster algebras, using Donaldson-Thomas theory for a quiver with potential. It turns out that the positivity conjecture reduces to the certain statement on purity of monodromic mixed Hodge structures on the cohomology with the coefficients in the sheaf of vanishing cycles on the moduli of stable framed representations. As an application, we show that the positivity conjecture (and actually a stronger result on Lefschetz property) holds if either initial or mutated quantum seed i…
For an embedding of sufficiently high degree of a smooth projective variety X into projective space, we use residues to define a filtered holonomic D-module (M, F) on the dual projective space. This gives a concrete description of the intermediate extension to a Hodge module on P of the variation of Hodge structure on the middle-dimensional cohomology of the hyperplane sections of X. We also establish many results about the sheaves F_k M, such as positivity, vanishing theorems, or reflexivity.
Using the harmonic theory developed by Takegoshi for representation of relative cohomology and the framework of computation of curvature of direct images bundles by Berndtsson, we prove that the higher direct images by a smooth morphism of the relative canonical bundle twisted by a semi-positive vector bundle are locally free and semi-positively curved, when endowed with a suitable Hodge type metric.
We give new characterizations of plurisubharmonic functions and Griffiths positivity of holomorphic vector bundles with singular Finsler metrics. As applications, we present a different method to prove plurisubharmonic variation of generalized Bergman kernel metrics and Griffiths positivity of the direct images of twisted relative canonical bundles associated to holomorphic families of Kähler manifolds.
We first provide details for the proof of Fujita's second theorem for Kähler fibre spaces over a curve, asserting that the direct image $V$ of the relative dualizing sheaf splits as the direct sum $ V = A \oplus Q$, where $A$ is ample and $Q$ is unitary flat. Our main result then answers in the negative the question posed by Fujita whether $V$ is semiample. In fact, $V$ is semiample if and only if $Q$ is associated to a representation of the fundamental group of $B$ having finite image. Our examples are based on hypergeometric integrals.
We prove an analogue for Hodge modules of Pink's theorem on the degeneration of l-adic sheaves (Math. Ann. 292). Let j be the open immersion of a Shimura variety M into its Baily-Borel compactification. Its boundary has a natural stratification into locally closed subsets, each of which is itself a Shimura variety (up to taking the quotient by the action of a finite group). Let i be the inclusion of an individual such stratum M'. Saito's formalism gives a functor i^* j_* from the bounded derived category of Hodge modules on M to that of Hodge modules on M'. Our result gives a formula for the e…
Given a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds and a relative ample line bundle $L\to \mathcal{X}$, the higher direct images $R^{n-p}f_*Ω^p_{\mathcal{X}/S}(L)$ carry a natural hermitian metric. Using the explicit formula for the curvature tensor of these direct images, we prove that the determinant of the twisted Hodge filtration $F^p_L=\oplus_{i\geq p}R^{n-i}Ω^i_{\mathcal{X}/S}(L)$ is (semi-) positive on the base $S$ if $L$ itself is (semi-) positive on $\mathcal{X}$.
We consider a holomorphic family $f:\mathcal{X} \to S$ of compact complex manifolds and a line bundle $\mathcal{L}\to \mathcal{X}$. Given that $\mathcal{L}^{-1}$ carries a singular hermitian metric that has Poincaré type singularities along a relative snc divisor $\mathcal{D}$, the direct image $f_*(K_{\mathcal{X}/S}\otimes \mathcal{D} \otimes \mathcal{L})$ carries a smooth hermitian metric. In case $\mathcal{L}$ is relatively positive, we give an explicit formula for its curvature. The result applies to families of log-canonically polarized pairs. Moreover we show that it improves the general…
Our main goal in this article is to establish a quantitative version of the positivity properties of twisted relative pluricanonical bundles and their direct images. The notion of "singular Hermitian metric" on vector bundles (together with an appropriate definition of positivity of the associated curvature) plays a major role here, and its properties are studied via a version of Ohsawa-Takegoshi extension theorem. Part of this article is based on the joint work of the first named author with Bo Berndtsson, and it can be seen as an expanded and updated version of it.
Faltings initiated in 2005 a p-adic analogue of the (complex) Simpson correspondence whose construction has been taken up by various authors, according to several approaches. Following the one we initiated previously, we present an overview of a new monograph developing new features of the p-adic Simpson correspondence, inspired by our construction of the relative Hodge-Tate spectral sequence. First, we address the connection to Hodge-Tate local systems. Second, we establish the functoriality of the p-adic Simpson correspondence by proper direct image. Along the way, we expand the scope of our…
This is an expository article. In the first part we recall the definition and a few results concerning singular Hermitian metrics on torsion-free coherent sheaves. They offer the perfect platform for the study of properties of direct images of twisted pluricanonical bundles which we will survey in the second part.
We give a new proof of a theorem by Pareschi, Popa and Schnell that the direct image of the canonical bundle of a smooth projective variety along a morphism to an abelian variety admits a Chen-Jiang decomposition, without using the theory of Hodge modules.
We prove that the $L^2$ metric on the direct image of an adjoint positive line bundle by a locally trivial submersion between projective manifolds is Nakano positive, under the assumption that the typical fiber has zero first Betti number. As a consequence, we get that the symmetric powers of an ample vector bundle tensorized by its determinant are Nakano positive, in particular Griffiths positive. This in turn gives vanishing theorems and an analytic characterization of ample vector bundles.
We exhibit a direct correspondence between the potential defining the H^{1,1} small quantum module structure on the cohomology of a Calabi-Yau manifold and the asymptotic data of the A-model variation of Hodge structure. This is done in the abstract context of polarized variations of Hodge structure and Frobenius modules.
In this paper we explain how non-abelian Hodge theory allows one to compute the $L^2$ cohomology or middle perversity higher direct images of harmonic bundles and twistor D-modules in a purely algebraic manner. Our main result is a new algebraic description for the fiberwise $L^2$ cohomology of a tame harmonic bundle or the corre- sponding flat bundle or tame polystable parabolic Higgs bundle. Specifically we give a formula for the Dolbeault version of the $L^2$ pushforward in terms of a modification of the Dolbeault complex of a Higgs bundle which takes into account the monodromy weight filtr…
In this paper, we prove that the statement: ``The (Generalized) Hodge Conjecture holds for codimension-two cycles on a smooth projective variety $X$" is a birationally invariant statement, that is, if the statement is true for $X$, it is also true for all smooth varieties $X'$ which are birationally equivalent to $X$. We also prove the analogous result for 1-cycles. As direct corollaries, the Hodge Conjecture holds for smooth rational projective manifolds with dimension less than or equal to five, and, the Generalized Hodge Conjecture holds for smooth rational projective manifolds with dimensi…
We prove the Hodge symmetry type result on the Dolbeault cohomology of Oeljeklaus-Toma manifolds with values in the direct sum of holomorphic line bundles. Consequently, we show the vanishing and non-vanishing of Dolbeault cohomology of Oeljeklaus-Toma manifolds with values in holomorphic line bundles.
In this paper, we study pushforwards of log pluricanonical bundles on projective log canonical pairs $(Y,Δ)$ over the complex numbers, partially answering a Fujita-type conjecture due to Popa and Schnell in the log canonical setting. We show two effective global generation results. First, when $Y$ surjects onto a projective variety, we show a quadratic bound for generic generation for twists by big and nef line bundles. Second, when $Y$ is fibered over a smooth projective variety, we show a linear bound for twists by ample line bundles. These results additionally give effective non-vanishing s…
We give equivalent descriptions for the augmented and diminished base loci of vector bundles in characteristic zero. We show that these base loci behave well under pullback, tensor product, and direct sum. Pathological behavior is observed on some nonsplit exact sequences.
Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complex numbers. Fix $n\geq 2$, and an integer $d$. A pair $(E,ϕ)$ over $X$ consists of an algebraic vector bundle $E$ of rank $n$ and degree $d$ over $X$ and a section $ϕ$. There is a concept of stability for pairs which depends on a real parameter $τ$. Let $M_τ(n,d)$ be the moduli space of $τ$-semistable pairs of rank $n$ and degree $d$ over $X$. We prove that the cohomology groups of $M_τ(n,d)$ are Hodge structures isomorphic to direct summands of tensor products of the Hodge structure $H^1(X)$. This implies a similar result f…
According to the decomposition and relative hard Lefschetz theorems, given a projective map of complex quasi projective algebraic varieties and a relatively ample line bundle, the rational intersection cohomology groups of the domain of the map split into various direct summands. While the summands are canonical, the splitting is certainly not, as the choice of the line bundle yields at least three different splittings by means of three mechanisms in a triangulated category introduced by Deligne. It is known that these three choices yield splittings of mixed Hodge structures. In this paper, we…
Given an n-dimensional variety Z with rational singularities, we conjecture that for a resolution of singularities whose reduced exceptional divisor E has simple normal crossings, the (n-1)-th higher direct image of the sheaf of differential forms with log poles along E vanishes. We prove this when Z has isolated singularities and when it is a toric variety. We deduce that for a divisor D with isolated rational singularities on a smooth complex n-dimensional variety X, the generation level of Saito's Hodge filtration on the localization of the structure sheaf along D is at most n-3.
This paper is a sequel to \cite{Berndtsson}. In that paper we studied the vector bundle associated to the direct image of the relative canonical bundle of a smooth Kähler morphism, twisted with a semipositive line bundle. We proved that the curvature of a such vector bundles is always semipositive (in the sense of Nakano). Here we adress the question if the curvature is strictly positive when the Kodaira-Spencer class does not vanish. We prove that this is so provided the twisting line bundle is stricty positive along fibers, but not in general.
The transcendental Hodge lattice of a projective manifold $M$ is the smallest Hodge substructure in $p$-th cohomology which contains all holomorphic $p$-forms. We prove that the direct sum of all transcendental Hodge lattices has a natural algebraic structure, and compute this algebra explicitly for a hyperkahler manifold. As an application, we obtain a theorem about dimension of a compact torus $T$ admitting a symplectic embedding to a hyperkahler manifold $M$. If $M$ is generic in a $d$-dimensional family of deformations, then $\dim T\geq 2^{[(d+1)/2]}$.
This is a translation of a final paper. It contains proofs of some well-known Positivstellensätze. The approach in this paper differs in two respects from presentations in literature. We use the Stone-Weierstrass Theorem directly to prove an inductive property, which makes the Kadison-Dubois Theorem superfluous in our paper, and we also consider homogeneous and projective Positivstellensätze. This leads us to a homogeneous version of Schmüdgen's theorem and a projective version of Putinar's theorem.
We show that the direct image of the filtered logarithmic de Rham complex is a direct sum of filtered logarithmic complexes with coefficients in variations of Hodge structures, using a generalization of the decomposition theorem of Beilinson, Bernstein and Deligne to the case of filtered $D$-modules. The advantage of using the logarithmic complexes is that we have the strictness of the Hodge filtration by Deligne after taking the cohomology group in the projective case. As a corollary, we get the total infinitesimal invariant of a (higher) cycle in a direct sum of the cohomology of filtered lo…
Let $n=2g+2$ be a positive even integer, $f(x)$ a degree $n$ complex polynomial without multiple roots and $C_f: y^2=f(x)$ the corresponding genus $g$ hyperelliptic curve over the field $\C$ of complex numbers. Let a $(g-1)$-dimensional complex abelian variety $P$ be a Prym variety of $C_f$ that corresponds to a unramified double cover of $C_f$. Suppose that there exists a subfield $K$ of $\C$ such that $f(x)$ lies in $K[x]$, is irreducible over $K$ and its Galois group is the full symmetric group. Assuming that $g>2$, we prove that $End(P)$ is either the ring of integers $Z$ or the direct sum…
Let $\Mg$ denote the moduli space of compact Riemann surfaces of genus $g$. Mumford had proved that, for each fixed genus $g$, there are isomorphisms asserting that certain higher $DET$ bundles over $\Mg$ are certain fixed (genus-independent) tensor powers of the Hodge line bundle on $\Mg$. Here we obtain a coherent, genus-independent description of the Mumford isomorphisms over certain infinite-dimensional ``universal'' parameter spaces of compact Riemann surfaces (with or without marked points). We work with an inductive limit of Teichmüller spaces comprising complex structures on a certain …
