Abstract 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 for smooth maps and their functorial properties.
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NC-smooth algebroid thickenings for families of vector bundles and quiver representations
In his work on deformation quantization of algebraic varieties Kontsevich introduced the notion of algebroid as a certain generalization of a sheaf of algebras. We construct algebroids which are given locally by NC-smooth thickenings in the sense of Kapranov, over two classes of smooth varieties: the bases of miniversal families of vector bundles on projective curves, and the bases of miniversal families of quiver representations.
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- Award ID(s):
- 1700642
- PAR ID:
- 10093481
- Date Published:
- Journal Name:
- Compositio Mathematica
- Volume:
- 155
- Issue:
- 4
- ISSN:
- 0010-437X
- Page Range / eLocation ID:
- 681 to 710
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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