## Duke Mathematical Journal

- Duke Math. J.
- Volume 166, Number 1 (2017), 177-204.

### K-stability for Fano manifolds with torus action of complexity $1$

#### Abstract

We consider Fano manifolds admitting an algebraic torus action with general orbit of codimension $1$. Using a recent result of Datar and Székelyhidi, we effectively determine the existence of Kähler–Ricci solitons for those manifolds via the notion of *equivariant* K-stability. This allows us to give new examples of Kähler–Einstein Fano threefolds and Fano threefolds admitting a nontrivial Kähler–Ricci soliton.

#### Article information

**Source**

Duke Math. J., Volume 166, Number 1 (2017), 177-204.

**Dates**

Received: 28 July 2015

Revised: 4 March 2016

First available in Project Euclid: 26 October 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.dmj/1477494164

**Digital Object Identifier**

doi:10.1215/00127094-3714864

**Mathematical Reviews number (MathSciNet)**

MR3592691

**Zentralblatt MATH identifier**

1360.32020

**Subjects**

Primary: 32Q20: Kähler-Einstein manifolds [See also 53Cxx]

Secondary: 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 14J45: Fano varieties

**Keywords**

K-stability Kähler–Einstein metric $T$-varieties torus action Fano varieties

#### Citation

Ilten, Nathan; Süß, Hendrik. K-stability for Fano manifolds with torus action of complexity $1$. Duke Math. J. 166 (2017), no. 1, 177--204. doi:10.1215/00127094-3714864. https://projecteuclid.org/euclid.dmj/1477494164