Abstract
We show that direct summands (or more generally, pure images) of klt type singularities are of klt type. As a consequence, we give a different proof of a recent result of Braun, Greb, Langlois and Moraga that reductive quotients of klt type singularities are of klt type.
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Notes
This means there exists some effective ℚdivisor \(D\) on \(Y\) such that the pair \((Y,D)\) is klt. In some literature this is also called potentially klt. More generally we say a pair \((X,\Delta )\) (where \(\Delta \) is an effective ℚdivisor on \(X\)) is of klt (resp. plt, resp. lc) type if there exists some effective ℚdivisor \(D\) such that \((X,\Delta +D)\) is klt (resp. plt, resp. lc).
Thanks to Javier CarvajalRojas for suggesting this refinement.
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL 60607, USA (slyu@uic.edu)
The author thanks Zhiyuan Chen for pointing this out to me. The author does not know if a general pair \((X,\Delta )\) being formally of klt (resp. plt) type implies \((X,\Delta )\) being of klt (resp. plt) type, even when \(X\) is excellent and admits a dualizing complex.
References
Birkar, C., Cascini, P., Hacon, C.D., McKernan, J.: Existence of minimal models for varieties of log general type. J. Am. Math. Soc. 23(2), 405–468 (2010)
Boutot, J.F.: Singularités rationnelles et quotients par les groupes réductifs. Invent. Math. 88(1), 65–68 (1987)
Braun, L., Moraga, J.: Iteration of Cox rings of klt singularities. J. Topol. (2021, in press). arXiv:2103.13524
Braun, L., Greb, D., Langlois, K., Moraga, J.: Reductive quotients of klt singularities. Invent. Math. (2021, in press). arXiv:2111.02812
Chakraborty, S., Gurjar, R.V., Miyanishi, M.: Pure subrings of commutative rings. Nagoya Math. J. 221(1), 33–68 (2016)
Fedder, R.: \(F\)Purity and rational singularity. Trans. Am. Math. Soc. 278(2), 461–480 (1983)
Fujino, O., Gongyo, Y.: On canonical bundle formulas and subadjunctions. Mich. Math. J. 61(2), 255–264 (2012)
Godfrey, C., Murayama, T.: Pure subrings of Du Bois singularities are Du Bois singularities (2022). arXiv:2208.14429
Gongyo, Y., Okawa, S., Sannai, A., Takagi, S.: Characterization of varieties of Fano type via singularities of Cox rings. J. Algebraic Geom. 24(1), 159–182 (2015)
Grothendieck, A.: Éléments de géométrie algébrique: IV. étude locale des schémas et des morphismes de schémas, Troisième partie. Publ. Math. IHÉS 28, 5–255 (1966). With a collaboration of Jean Dieudonné
Hashimoto, M.: A pure subalgebra of a finitely generated algebra is finitely generated. Proc. Am. Math. Soc. 133(8), 2233–2235 (2005)
Hochster, M., Huneke, C.: Applications of the existence of big CohenMacaulay algebras. Adv. Math. 113(1), 45–117 (1995)
Hochster, M., Roberts, J.L.: Rings of invariants of reductive groups acting on regular rings are CohenMacaulay. Adv. Math. 13, 115–175 (1974)
Hochster, M., Roberts, J.L.: The purity of the Frobenius and local cohomology. Adv. Math. 21(2), 117–172 (1976)
Kollár, J.: Singularities of pairs. In: Algebraic Geometry, Santa Cruz, 1995, pp. 221–287 (1997)
Kollár, J.: Singularities of the Minimal Model Program. Cambridge Tracts in Mathematics, vol. 200. Cambridge University Press, Cambridge (2013). With a collaboration of Sándor Kovács
Kollár, J., Mori, S.: Birational Geometry of Algebraic Varieties. Cambridge Tracts in Mathematics, vol. 134. Cambridge University Press, Cambridge (1998). With the collaboration of, Clemens, C. H. and Corti, A., Translated from the 1998 Japanese original
Lyu, S., Murayama, T.: The relative minimal model program for excellent algebraic spaces and analytic spaces in equal characteristic zero (2022). arXiv:2209.08732v1
Murayama, T.: Relative vanishing theorems for \(\mathbf{Q}\)schemes (2022). arXiv:2101.10397v2
Schoutens, H.: Logterminal singularities and vanishing theorems via nonstandard tight closure. J. Algebraic Geom. 14(2), 357–390 (2005)
Schwede, K., Smith, K.E.: Globally \(F\)regular and log Fano varieties. Adv. Math. 224(3), 863–894 (2010)
Takagi, S., Yamaguchi, T.: On the behavior of adjoint ideals under pure morphisms (2023). arXiv:2312.17537
Temkin, M.: Desingularization of quasiexcellent schemes in characteristic zero. Adv. Math. 219(2), 488–522 (2008)
Zhuang, Z.: On boundedness of singularities and minimal log discrepancies of Kollár components. J. Algebraic Geom. 33(3), 521–565 (2024)
Acknowledgements
The author is partially supported by the NSF Grants DMS2240926, DMS2234736, a Clay research fellowship, as well as a Sloan fellowship. He would like to thank Lukas Braun, Javier CarvajalRojas, Shiji Lyu, Linquan Ma, Joaquín Moraga, Karl Schwede, Kevin Tucker and Chenyang Xu for helpful discussions and comments. He also wants to thank the anonymous referee for careful reading of the manuscript and several helpful suggestions.
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Appendix: Pure images of klt type excellent schemes
Appendix: Pure images of klt type excellent schemes
Shiji Lyu^{Footnote 4}
In this appendix, we explain how to extend the previous results to morphisms between excellent schemes admitting dualizing complexes, and slightly further.
1.1 A.1 Main theorems for nonfinitetype schemes
In this subsection we prove Theorems 1.1 and 2.10 for excellent schemes admitting dualizing complexes.
Let \(X\) be a Noetherian excellent scheme of equal characteristic zero that admits a dualizing complex. We say \((X,\Delta )\) is of klt (resp. plt) type if Zariski locally on \(X\) there exists a ℚdivisor \(D\geq 0\) such that \((X,\Delta +D)\) is klt (resp. plt).
Since log resolutions exist [23, Theorem 2.3.6], being klt or plt can be detected using a single log resolution, thus \((X,\Delta )\) is of klt (resp. plt) type if and only if for all \(x\in X\), \((\mathbf{Spec}(\mathcal{O}_{X,x}),\Delta _{\mathbf{Spec}( \mathcal{O}_{X,x})})\) is of klt (resp. plt) type.
We say that a morphism of Noetherian schemes \(f:Y\to X\) is pure if for all \(x\in X\), there exists \(y\in Y\) such that \(f(y)=x\) and \(\mathcal{O}_{X,x}\to \mathcal{O}_{Y,y}\) is pure. If \(Y=\mathbf{Spec}(B)\) and \(X=\mathbf{Spec}(A)\) are affine, \(f\) is pure if and only if \(A\to B\) is pure, see [12, Lemma 2.2].
Here is our extension of Theorems 1.1 and 2.10.
Theorem A.1
Let \(f:Y\to X\) be a pure morphism between Noetherian schemes of equal characteristic zero. Assume that both \(X\) and \(Y\) are excellent and admit dualizing complexes. Then the followings hold.

(1)
Assume that \(Y\) is of klt type. Then \(X\) is also of klt type.

(2)
Let \(P\) be a prime divisor on \(X\), and let \(Q\) be the divisorial part of the schemetheoretic pullback \(f^{1}(P)\). If \((Y,Q)\) is of plt type, then \((X,P)\) is also of plt type.
We also remark that Corollary 2.9 can be extended as well.
Corollary A.2
Let \(f:Y\to X\) be an equidimensional morphism of finite type between Noetherian schemes of equal characteristic zero. Assume that \(X\) is normal and excellent, and that \(X\) admits dualizing complexes. Then the followings hold.

(1)
Assume that \(Y\) is of klt type. Then \(X\) is also of klt type.

(2)
Let \(P\) be a prime divisor on \(X\), and let \(Q\) be the divisorial part of the schemetheoretic pullback \(f^{1}(P)\). If \((Y,Q)\) is of plt type, then \((X,P)\) is also of plt type.
Proof
A finite type map preserves being excellent and admitting dualizing complexes. Thus it suffices to show \(f\) is pure.
By [10, Proposition 13.3.1], \(f\) factors locally as \(Y\xrightarrow{g}\mathbb{A}^{e}_{X}\to X\) where \(g\) is quasifinite and \(e=\dim Y\dim X\). Since \(\mathbb{A}^{e}_{X}\) is normal and of equal characteristic zero, \(g\) is pure, thus so is \(f\). □
We cannot readily extend Corollary 1.2, since we do not know if excellence and dualizing complexes can be descended from \(A\) to \(A^{G}\). However, see Corollary A.4 below.
We now turn to the proof of Theorem A.1. We will not give all the details in our case, since the argument will be completely parallel to the proof of Theorems 1.1 and 2.10. We will only indicate which parts of the argument need to be modified in our situation.
In the proof of Lemma 2.3, [17, Proposition 5.20] works with no problem in our case, and the reference [2] can be replaced by [19, Theorem C]. Proper birational map from a regular scheme satisfies GrauertRiemenschneider [19, Theorem A]. Thus one can apply the argument in [15, §11] to see that [17, Theorem 5.22 and Corollary 5.24] hold in our case.
In the proof of Lemma 2.4 (for klt type), [17, Lemma 6.2] works with no problem, and the required Bertini theorem is [18, Corollary 10.4]. We still need to prove Lemma 2.5. We follow the proof of [24, Lemma 4.7]. The existence of ℚfactorialization is [18, Corollary 22.3]. Small perturbation of a klt pair is klt [18, Lemma 6.9], and the log canonical model exists due to the finite generation of relative adjoint rings [18, Theorem 17.3].
The proof of Lemmas 2.6, 2.7 and 2.8 works verbatim. At this point, we have proved statement (1) of Theorem A.1.
Let us now consider statement (2). Again, we follow the proof of Theorem 2.10. The first step is to show \(Y\) of klt type, so \(X\) of klt type. It suffices to prove [17, Proposition 2.43] for every affine local excellent scheme \(X\) of equal characteristic zero that admits a dualizing complex. We can, for simplicity, assume \(H=0\). We use the same argument as in [17], but we write out much of the details since we do not have a Bertini theorem stated in [18] for the linear system of a Weil divisor. Since our \(X\) is local and excellent, by Hironaka’s resolution of singularities, there is a log resolution \(\pi :X'\to X\) that is an iterated blowup of regular centers disjoint from \(X\setminus Z\). In particular, there is a \(\pi \)ample \(\pi \)exceptional Cartier divisor \(H'(\leq 0)\) on \(X'\). Then \(aH'+\pi ^{1}_{*}m\Delta _{1}\) is \(\pi \)generated [18, Definition 4.1] for some positive integer \(a\). Some member \(D'\in aH'+\pi ^{1}_{*}m\Delta _{1}\) then satisfies \((X',\pi ^{1}_{*}\Delta +D')\) snc by [18, Theorem 10.1 and Remark 10.2], so \(D:=\pi _{*}D'\) is such that \(D\sim m\Delta _{1}\) and that \((X\setminus Z,(\Delta +D)_{X\setminus Z})\) is snc. We can then argue as in the second last paragraph of the proof of [17, Proposition 2.43].
We now need Lemma 2.4 for plt type. Since we know [17, Proposition 2.43] in our case, we can apply the same proof for the klt case, except for the Bertini theorem. [18, Corollary 10.4] holds for plt instead of klt by a similar proof, which is what we want. (Alternatively, one can use inversion of adjunction as noted below.)
Now we can follow the argument until the last paragraph. For the last step of the proof, inversion of adjunction holds in our case since resolutions exist and the connectedness theorem [17, Theorem 5.48] holds, the latter depending (only) on Kawamata–Viehweg vanishing [19, Theorem A]. We have now proved Theorem A.1.
1.2 A.2 Schemes formally of klt or plt type
For a general Noetherian scheme \(X\) of equal characteristic zero, we say \((X,\Delta )\) is formally of klt (resp. plt) type if for all \(x\in X\), the completion \(A=\mathcal{O}_{X,x}^{\wedge}\) is normal and \((\mathbf{Spec}(A),\Delta _{\mathbf{Spec}(A)})\) is of klt (resp. plt) type. Here, for a prime divisor \(P\) on \(X\), \(P_{\mathbf{Spec}(A)}\) is the divisorial part of \(P\times _{X} \mathbf{Spec}(A)\) and \(\Delta _{\mathbf{Spec}(A)}\) is defined by linearity. We note that \(\mathbf{Spec}(A)\) is excellent and admits a dualizing complex since \(A\) is complete.
If \(X\) is excellent and admits a dualizing complex, and \((X,\Delta )\) is of klt (resp. plt) type, then \((X,\Delta )\) is formally of klt (resp. plt) type (cf. [16, Proposition 2.15]). The converse holds when \(\Delta =0\) (resp. \(\Delta \) is a prime divisor) by our Theorem A.1.^{Footnote 5}
Now, if \(A\to B\) is a pure local map of Noetherian local rings, then by [6, Proposition 1.3(5)] it is clear that \(A^{\wedge}\to B^{\wedge}\) is pure. Let \(P\) is a prime divisor on \(\mathbf{Spec}(A)\) and let \(Q\) (resp. \(P^{\wedge}\), \(Q^{\wedge}\)) be the divisorial part of its schemetheoretic pullback to \(\mathbf{Spec}(B)\) (resp. \(\mathbf{Spec}(A^{\wedge})\), \(\mathbf{Spec}(B^{\wedge})\)). If \((\mathbf{Spec}(B^{\wedge}),Q^{\wedge})\) is of plt type, then \(Q^{\wedge}\) must be a prime divisor, thus so is \(P^{\wedge}\). Therefore Theorem A.1 gives
Theorem A.3
Let \(f:Y\to X\) be a pure morphism between Noetherian schemes of equal characteristic zero. Then the followings hold.

(1)
Assume that \(Y\) is formally of klt type. Then \(X\) is also formally of klt type.

(2)
Let \(P\) be a prime divisor on \(X\), and let \(Q\) be the divisorial part of the schemetheoretic pullback \(f^{1}(P)\). If \((Y,Q)\) is formally of plt type, then \((X,P)\) is also formally of plt type.
Now we are able to extend Corollary 1.2.
Corollary A.4
Let \(k\) be a field of characteristic zero, \(G\) a reductive \(k\)group, \(A\) a Noetherian \(k\)algebra that admits a \(k\)rational \(G\)action. Then the followings hold.

(1)
Assume that \(\mathbf{Spec}(A)\) is formally of klt type. Then \(\mathbf{Spec}(A^{G})\) is also formally of klt type.

(2)
Let \(P\) be a prime divisor on \(\mathbf{Spec}(A^{G})\), and let \(Q\) be the divisorial part of the schemetheoretic pullback \(P\times _{\mathbf{Spec}(A^{G})}\mathbf{Spec}(A)\). If \((\mathbf{Spec}(A),Q)\) is formally of plt type, then \((\mathbf{Spec}(A^{G}),P)\) is also formally of plt type.
Proof
The map \(A^{G}\to A\) is pure, in fact split, see for example [13, §10]. Thus \(A^{G}\) is Noetherian (cf. the proof of Lemma 2.7) and Theorem A.3 applies. □
Again, we have a version of Corollary 2.9. The proof is the same as Corollary A.2.
Corollary A.5
Let \(f:Y\to X\) be an equidimensional surjective morphism of finite type between Noetherian schemes of equal characteristic zero. Assume that \(X\) is normal. Then the followings hold.

(1)
Assume that \(Y\) is formally of klt type. Then \(X\) is also formally of klt type.

(2)
Let \(P\) be a prime divisor on \(X\), and let \(Q\) be the divisorial part of the schemetheoretic pullback \(f^{1}(P)\). If \((Y,Q)\) is formally of plt type, then \((X,P)\) is also formally of plt type.
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Zhuang, Z. Direct summands of klt singularities. Invent. math. 237, 1683–1695 (2024). https://doi.org/10.1007/s00222024012811
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DOI: https://doi.org/10.1007/s00222024012811