Semi-positivity from Higgs bundles

Abstract

We prove a generalization of the Fujita-Kawamata-Zuo semi-positivity Theorem [Fuj78, Kaw81, Zuo00] for filtered regular meromorphic Higgs bundles and tame harmonic bundles. Our approach gives a new proof in the cases already considered by these authors. We give also an application to the geometry of smooth quasi-projective complex varieties admitting a semisimple complex local system with infinite monodromy.

Canonical bundle formula and degenerating families of volume forms

“…It is a consequence of deep results in the theory of variation of Hodge structure by Cattani, Kaplan and Schmid [CKS86] (cf. also [Ks85] and some more related references listed in [FF17], [Br17]). Here the asymptotics is given by a plurisubharmonic weight with vanishing Lelong numbers in the special case of the unipotent monodromies condition, which amounts to generalizing the logarithmic singularity log |z| 2 β .…”

“…In contrast to the above numerous previous results when the base dimension dim Y is equal to 1, the only previous result to our knowledge in the general case of dim Y ≥ 2 is the asymptotics of the L 2 metric for the direct image f * (K X/Y ) given in the Kawamata semipositivity theorem (see Theorem 5.1) [Ka00] (cf. [Ka81]), [FF17], [Br17] under the unipotent monodromies condition. It is a consequence of deep results in the theory of variation of Hodge structure by Cattani, Kaplan and Schmid [CKS86] (cf.…”

Nearby cycles and semipositivity in positive characteristic

“…But even the last part of this theorem was not known in characteristic zero. Already this part implies essentially all known semipositivity results (see below) for Higgs bundles or complex polarized variations of Hodge structures due to Fujita [Fu], Kawamata [Kw], Zuo [Zu], Fujino-Fujisawa [FF,Theorem 5.21], Brunebarbe [Br1,Theorems 1.8 and 4.5], [Br2,Theorem 1.2] and many others. Note that almost all the proofs of such results are analytic and use Hodge theory.…”

“…The following corollary is a direct analogue of [Br2,Theorem 1.2] in positive characteristic. In fact, it implies its generalization from polystable to the semistable case.…”

Hyperbolicity of bases of log Calabi-Yau families

“…Then N q = (B −1 ⊗K q )∩F q . By the work of Zuo [Zuo00] (see also [PW16,Bru17,FF17,Bru18] for various generalizations) on the negativity of kernels of Kodaira-Spencer maps of Hodge bundles, K * q is weakly positive in the sense of Viehweg 2 , cf. [VZ02, Lemma 4.4.(v)].…”