Advances in Mathematics最新文献

{"title":"Morse properties in convex projective geometry","authors":"Mitul Islam ,&nbsp;Theodore Weisman","doi":"10.1016/j.aim.2025.110430","DOIUrl":"10.1016/j.aim.2025.110430","url":null,"abstract":"<div><div>We study properties of “hyperbolic directions” in groups acting cocompactly on properly convex domains in real projective space, from three different perspectives simultaneously: the (coarse) metric geometry of the Hilbert metric, the projective geometry of the boundary of the domain, and the singular value gaps of projective automorphisms. We describe the relationship between different definitions of “Morse” and “regular” quasi-geodesics arising in these three different contexts. This generalizes several results of Benoist and Guichard to the non-Gromov hyperbolic setting.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110430"},"PeriodicalIF":1.5,"publicationDate":"2025-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144654949","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Geometry and periods of G2-moduli spaces","authors":"Thibault Langlais","doi":"10.1016/j.aim.2025.110435","DOIUrl":"10.1016/j.aim.2025.110435","url":null,"abstract":"<div><div>This paper is concerned with the geometry of the moduli space <span><math><mi>M</mi></math></span> of torsion-free <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-structures on a compact <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-manifold <em>M</em>, equipped with the volume-normalised <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-metric <span><math><mi>G</mi></math></span>. When <span><math><msup><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>, this metric is known to be of Hessian type and to admit a global potential. Here we give a new description of the geometry of <span><math><mi>M</mi></math></span>, based on the observation that there is a natural way to immerse the moduli space into a homogeneous space <span><math><mi>D</mi></math></span> diffeomorphic to <span><math><mi>G</mi><mi>L</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mo>(</mo><mo>{</mo><mo>±</mo><mn>1</mn><mo>}</mo><mo>×</mo><mi>O</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></math></span>, where <span><math><mi>n</mi><mo>=</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>(</mo><mi>M</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span>. We point out that the formal properties of this immersion <span><math><mi>Φ</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>D</mi></math></span> are very similar to those of the period map defined on the moduli spaces of Calabi–Yau threefolds. With a view to understand the curvatures of <span><math><mi>G</mi></math></span>, we also derive a new formula for the fourth derivative of the potential and relate it to the second fundamental form of <span><math><mi>Φ</mi><mo>(</mo><mi>M</mi><mo>)</mo><mo>⊂</mo><mi>D</mi></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110435"},"PeriodicalIF":1.5,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144654840","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Virtual localization revisited","authors":"Dhyan Aranha ,&nbsp;Adeel A. Khan ,&nbsp;Alexei Latyntsev ,&nbsp;Hyeonjun Park ,&nbsp;Charanya Ravi","doi":"10.1016/j.aim.2025.110434","DOIUrl":"10.1016/j.aim.2025.110434","url":null,"abstract":"<div><div>Let <em>T</em> be a split torus acting on an algebraic scheme <em>X</em> with fixed locus <em>Z</em>. Edidin and Graham showed that on localized <em>T</em>-equivariant Chow groups, (a) push-forward <span><math><msub><mrow><mi>i</mi></mrow><mrow><mo>⁎</mo></mrow></msub></math></span> along <span><math><mi>i</mi><mo>:</mo><mi>Z</mi><mo>→</mo><mi>X</mi></math></span> is an isomorphism, and (b) when <em>X</em> is smooth the inverse <span><math><msup><mrow><mo>(</mo><msub><mrow><mi>i</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> can be described via Gysin pullback <span><math><msup><mrow><mi>i</mi></mrow><mrow><mo>!</mo></mrow></msup></math></span> and cap product with <span><math><mi>e</mi><msup><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>, the inverse of the Euler class of the normal bundle <em>N</em>. In this paper we show that (b) still holds when <em>X</em> is a quasi-smooth derived scheme (or Deligne–Mumford stack), using virtual versions of the operations <span><math><msup><mrow><mi>i</mi></mrow><mrow><mo>!</mo></mrow></msup></math></span> and <span><math><mo>(</mo><mo>−</mo><mo>)</mo><mo>∩</mo><mi>e</mi><msup><mrow><mo>(</mo><mi>N</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>. As a corollary we prove the virtual localization formula <span><math><msup><mrow><mo>[</mo><mi>X</mi><mo>]</mo></mrow><mrow><mi>vir</mi></mrow></msup><mo>=</mo><msub><mrow><mi>i</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><msup><mrow><mo>[</mo><mi>Z</mi><mo>]</mo></mrow><mrow><mi>vir</mi></mrow></msup><mo>∩</mo><mi>e</mi><msup><mrow><mo>(</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>vir</mi></mrow></msup><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> of Graber–Pandharipande without global resolution hypotheses and over arbitrary base fields. We include an appendix on fixed loci of group actions on (derived) stacks which should be of independent interest.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110434"},"PeriodicalIF":1.5,"publicationDate":"2025-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144633177","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Exact structures and purity","authors":"Kevin Schlegel","doi":"10.1016/j.aim.2025.110433","DOIUrl":"10.1016/j.aim.2025.110433","url":null,"abstract":"<div><div>We relate the theory of purity of a locally finitely presented category with products to the study of exact structures on the full subcategory of finitely presented objects. Properties in the context of purity are translated to properties about exact structures. We specialize to the case of a module category over an Artin algebra and show that generic modules are in one to one correspondence with particular maximal exact structures.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110433"},"PeriodicalIF":1.5,"publicationDate":"2025-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596904","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Blow-ups and the quantum spectrum of surfaces","authors":"Ádám Gyenge ,&nbsp;Szilárd Szabó","doi":"10.1016/j.aim.2025.110432","DOIUrl":"10.1016/j.aim.2025.110432","url":null,"abstract":"<div><div>We investigate the behavior of the spectrum of the quantum (or Dubrovin) connection of smooth projective surfaces under blow-ups. Our main result is that for small values of the parameters, the quantum spectrum of such a surface is asymptotically the union of the quantum spectrum of a minimal model of the surface and a finite number of additional points located “close to infinity”, that correspond bijectively to the exceptional divisors. This proves a conjecture of Kontsevich in the surface case.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110432"},"PeriodicalIF":1.5,"publicationDate":"2025-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144587582","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Rees algebra and analytic spread of a divisorial filtration","authors":"Steven Dale Cutkosky","doi":"10.1016/j.aim.2025.110428","DOIUrl":"10.1016/j.aim.2025.110428","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In this paper we investigate some properties of Rees algebras of divisorial filtrations and their analytic spread. A classical theorem of McAdam shows that the analytic spread of an ideal &lt;em&gt;I&lt;/em&gt; in a formally equidimensional local ring is equal to the dimension of the ring if and only if the maximal ideal is an associated prime of &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mover&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mo&gt;‾&lt;/mo&gt;&lt;/mover&gt;&lt;/math&gt;&lt;/span&gt; for some &lt;em&gt;n&lt;/em&gt;. We show in Theorem 1.5 that McAdam's theorem holds for &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-divisorial filtrations in an equidimensional local ring which is essentially of finite type over an excellent local ring of dimension less than or equal to 3. This generalizes an earlier result for &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-divisorial filtrations in an equicharacteristic zero excellent local domain by the author. This theorem does not hold for more general filtrations.&lt;/div&gt;&lt;div&gt;We consider the question of the asymptotic behavior of the function &lt;span&gt;&lt;math&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;↦&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; for a &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-divisorial filtration &lt;span&gt;&lt;math&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;-primary ideals on a &lt;em&gt;d&lt;/em&gt;-dimensional normal excellent local ring. It is known from earlier work of the author that the multiplicity&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;!&lt;/mo&gt;&lt;munder&gt;&lt;mi&gt;lim&lt;/mi&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; can be irrational. We show in Lemma 4.1 that the limsup of the first difference function&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;munder&gt;&lt;mrow&gt;&lt;mrow&gt;&lt;mi&gt;lim&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mrow&gt;&lt;mi&gt;sup&lt;/mi&gt;&lt;/mrow&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;→&lt;/mo&gt;&lt;mo&gt;∞&lt;/mo&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;I&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; is always finite for a &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;-divisorial filtration. We then give an example in Section 4 showing that this limsup may not exist as a limit.&lt;/div&gt;&lt;div&gt;In the final section, we","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110428"},"PeriodicalIF":1.5,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144581248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Floer cohomology and higher mutations","authors":"Soham Chanda","doi":"10.1016/j.aim.2025.110425","DOIUrl":"10.1016/j.aim.2025.110425","url":null,"abstract":"<div><div>We extend the construction of higher mutation as introduced in <span><span>[42]</span></span> to local higher mutation, which is applicable to a larger class of monotone Lagrangians. For two-dimensional Lagrangians, local higher mutation is the same as performing a Lagrangian anti-surgery in the sense of <span><span>[24]</span></span> followed by a Lagrangian surgery. We prove that up to a change of local systems, the Lagrangian intersection Floer cohomology of a pair of Lagrangians is invariant under local mutation. This result generalizes the wall-crossing formula in <span><span>[42]</span></span>. For two-dimensional Lagrangians, this result agrees with the invariance result in <span><span>[43]</span></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110425"},"PeriodicalIF":1.5,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144581249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Equidistribution of polynomial sequences in function fields, with applications","authors":"Thái Hoàng Lê ,&nbsp;Yu-Ru Liu ,&nbsp;Trevor D. Wooley","doi":"10.1016/j.aim.2025.110424","DOIUrl":"10.1016/j.aim.2025.110424","url":null,"abstract":"<div><div>We prove a function field analog of Weyl's classical theorem on equidistribution of polynomial sequences. Our result covers the case in which the degree of the polynomial is greater than or equal to the characteristic of the field, which is a natural barrier when applying the Weyl differencing process to function fields. We also discuss applications to van der Corput, intersective and Glasner sets in function fields.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110424"},"PeriodicalIF":1.5,"publicationDate":"2025-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144581250","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Three-dimensional positively curved generalized Ricci solitons with SO(3)-symmetries","authors":"Fabio Podestà ,&nbsp;Alberto Raffero","doi":"10.1016/j.aim.2025.110426","DOIUrl":"10.1016/j.aim.2025.110426","url":null,"abstract":"<div><div>We prove the existence of a one-parameter family of pairwise non-isometric, complete, positively curved, steady generalized Ricci solitons of gradient type on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> that are invariant under the natural cohomogeneity one action of SO(3). In the context of generalized Ricci flow, this result represents the analogue of Bryant's construction of the complete rotationally invariant steady soliton for the Ricci flow.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110426"},"PeriodicalIF":1.5,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570738","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Spectral flow of Callias operators, odd K-cowaist, and positive scalar curvature","authors":"Pengshuai Shi","doi":"10.1016/j.aim.2025.110429","DOIUrl":"10.1016/j.aim.2025.110429","url":null,"abstract":"<div><div>On a complete Riemannian manifold <em>M</em>, we study the spectral flow of a family of Callias operators. We derive a codimension zero formula when the dimension of <em>M</em> is odd and a codimension one formula when the dimension of <em>M</em> is even. These can be seen as analogues of Gromov–Lawson's relative index theorem and classical Callias index theorem, respectively. Secondly, we introduce an intrinsic definition of K-cowaist on odd-dimensional manifolds, making use of the odd Chern character of a smooth map from the manifold to a unitary group. It behaves just like the usual K-cowaist on even-dimensional manifolds. We then apply the notion of odd K-cowaist and the tool of spectral flow to investigate problems related to positive scalar curvature on spin manifolds. In particular, we prove infinite odd K-cowaist to be an obstruction to the existence of PSC metrics. We obtain quantitative scalar curvature estimates on complete non-compact manifolds and scalar-mean curvature estimates on compact manifolds with boundary. They extend several previous results optimally, which unfolds a major advantage of our method via spectral flow and odd K-cowaist.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"479 ","pages":"Article 110429"},"PeriodicalIF":1.5,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}