%!PS-Adobe-2.0 %%Creator: dvips(k) 2023.1 (TeX Live 2023) Copyright 2023 Radical Eye Software %%Title: arXiv:2310.17608v1 [math.DG] 26 Oct 2023 %%CreationDate: Fri Oct 27 02:11:12 2023 %%Pages: 11 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentFonts: EUFM10 %%DocumentPaperSizes: Letter %%EndComments %%BeginDefaults %%ViewingOrientation: 1 0 0 1 %%EndDefaults %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -R2 -P pk -z numerical-criterion.dvi -o %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2023.10.27:0211 %%BeginProcSet: tex.pro 0 0 %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /dir 0 def/dyy{/dir 0 def}B/dyt{/dir 1 def}B/dty{/dir 2 def}B/dtt{/dir 3 def}B/p{dir 2 eq{-90 rotate show 90 rotate}{dir 3 eq{-90 rotate show 90 rotate}{show}ifelse}ifelse}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{/Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT)(LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{ BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat {BDot}imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B /M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M} B/g{0 M}B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{ 0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: l3backend-dvips.pro 0 0 %% %% This is file `l3backend-dvips.pro', %% generated with the docstrip utility. %% %% The original source files were: %% %% l3backend-header.dtx (with options: `header,dvips') %% %% Copyright (C) 2019-2023 The LaTeX Project %% %% It may be distributed and/or modified under the conditions of %% the LaTeX Project Public License (LPPL), either version 1.3c of %% this license or (at your option) any later version. 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Fs(\()p Fq(v)t Fs(\))27 b Fp(\021)h(N)2217 847 y Fn(T)2258 856 y Fj(0)2297 832 y Fs(\()p Fq(v)t Fs(\))k(the)g(w)m(eigh)m(t)h(p)s(olytop)s(e)e(of)h(an) h(elemen)m(t)-9 949 y Fq(v)i Fs(in)30 b(a)i(represen)m(tation)g(of)f Fq(G)p Fs(.)43 b(In)32 b(general)f Fp(N)15 b Fs(\()p Fq(v)t Fs(\))30 b(is)h(naturally)f(a)h(closed)h(in)m(terv)-5 b(al)30 b(in)h Fm(R)5 b Fs(.)49 b(A)31 b(non-trivial)-9 1070 y(irreducible)23 b(represen)m(tation)j(of)e Fq(G)h Fs(is)f(of)h(the)g(form)f(Sym)2064 1028 y Fn(k)2123 1070 y Fm(C)2189 1029 y Fo(2)2259 1070 y Fs(for)h(some)f(p)s(ositiv)m(e)h (in)m(teger)f Fq(k)s Fs(.)41 b(It)25 b(is)g(easy)h(to)-9 1186 y(see)i(that)f(to)g(study)i(Problem)1113 1186 y SDict begin H.S end 1113 1186 a 0 TeXcolorgray Fs(1)p 0 TeXcolorgray 1162 1122 a SDict begin H.R end 1162 1122 a 1162 1186 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (problem.1) cvn H.B /ANN pdfmark end 1162 1186 a 27 w Fs(without)e(loss)g(of) g(generalit)m(y)g(w)m(e)h(ma)m(y)f(alw)m(a)m(ys)h(assume)g Fq(V)49 b Fs(is)27 b(irreducible)-9 1302 y(so)32 b(is)g(of)h(the)g (form)e Fq(V)49 b Fs(=)28 b(Sym)1115 1260 y Fn(n)1178 1302 y Fm(C)1244 1261 y Fo(2)1290 1302 y Fs(.)-9 1410 y SDict begin H.S end -9 1410 a -9 1410 a SDict begin 14 H.A end -9 1410 a -9 1410 a SDict begin [/View [/XYZ H.V]/Dest (subsection.2.1) cvn /DEST pdfmark end -9 1410 a 97 x Fs(2.1.)48 b Fv(Irreducible)43 b(case.)49 b Fs(Here)39 b(w)m(e)h(pro)m(v)m(e)f(P)m(art)g(\(1\))f(of)g(Theorem)2628 1507 y SDict begin H.S end 2628 1507 a 0 TeXcolorgray Fs(1.3)p 0 TeXcolorgray 2752 1443 a SDict begin H.R end 2752 1443 a 2752 1507 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.1.3) cvn H.B /ANN pdfmark end 2752 1507 a Fs(.)61 b(As)39 b(w)m(e)h(assume)f Fq(W)52 b Fs(is)38 b(irre-)-9 1624 y(ducible,)31 b(it)g(follo)m(ws)g(that)g Fq(W)41 b Fs(=)28 b(Sym)1404 1582 y Fn(m)1487 1624 y Fm(C)1553 1583 y Fo(2)1630 1624 y Fs(for)k(some)f(p)s(ositiv)m(e)h(in)m(teger)f Fq(m)p Fs(.)44 b(Supp)s(ose)33 b(\()p Fq(v)t(;)17 b(w)s Fs(\))26 b Fp(2)i Fq(V)43 b Fp(\010)21 b Fq(W)-9 1740 y Fs(is)32 b(n)m(umerically)e(semistable)i(but)h(not)f(semistable,)g(w) m(e)h(will)e(dra)m(w)i(a)f(con)m(tradiction.)43 b(W)-8 b(e)33 b(can)f(write)1171 2009 y Fq(v)f Fs(=)1395 1880 y Fn(\014)1435 1889 y Fj(1)1361 1914 y Fg(X)1353 2124 y Fn(i)p Fo(=)p Fn(\013)1477 2133 y Fj(1)1528 2009 y Fq(c)1570 2024 y Fn(i)1599 2009 y Fq(e)1644 1968 y Fn(i)1644 2033 y Fo(1)1683 2009 y Fq(e)1728 1968 y Fn(n)p Fk(\000)p Fn(i)1728 2033 y Fo(2)1854 2009 y Fq(;)115 b(w)30 b Fs(=)2242 1880 y Fn(\014)2282 1889 y Fj(2)2207 1914 y Fg(X)2199 2124 y Fn(i)p Fo(=)p Fn(\013)2323 2133 y Fj(2)2375 2009 y Fq(d)2426 2024 y Fn(i)2454 2009 y Fq(e)2499 1968 y Fn(i)2499 2033 y Fo(1)2538 2009 y Fq(e)2583 1968 y Fn(m)p Fk(\000)p Fn(i)2583 2033 y Fo(2)-9 2277 y Fs(for)41 b(some)h(0)h Fp(\024)h Fq(\013)678 2292 y Fo(1)761 2277 y Fp(\024)g Fq(\014)937 2292 y Fo(1)1020 2277 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y(for)32 b(some)i(p)s(ositiv)m(e)e (in)m(teger)i Fq(p)f Fs(and)g Fq(k)s Fs(\()p Fq(t)p Fs(\))d(=)f(Id)23 b(+)f Fq(\017)p Fs(\()p Fq(t)p Fs(\))34 b(with)f Fq(\017)p Fs(\(0\))c(=)g(0.)46 b(W)-8 b(e)33 b(can)h(express)i(the)d(action)g(of) -9 3587 y Fq(\017)p Fs(\()p Fq(t)p Fs(\))g(as)f(follo)m(ws:)1436 3719 y SDict begin H.S end 1436 3719 a 1436 3719 a SDict begin 14 H.A end 1436 3719 a 1436 3719 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.2) cvn /DEST pdfmark end 1436 3719 a Fq(\017)p Fs(\()p Fq(t)p Fs(\))p Fq(:e)1658 3734 y Fo(2)1726 3719 y Fs(=)27 b Fq(p)p Fs(\()p Fq(t)p Fs(\))p Fq(e)2034 3734 y Fo(1)2096 3719 y Fs(+)22 b Fq(q)t Fs(\()p Fq(t)p Fs(\))p Fq(e)2397 3734 y Fo(2)2437 3719 y Fq(;)1245 b Fs(\(2.2\))-9 3871 y(where)37 b Fq(p)p Fs(\()p Fq(t)p Fs(\))d(=)h Fq(O)s Fs(\()p Fq(t)p Fs(\))g(and)i Fq(q)t Fs(\()p Fq(t)p Fs(\))d(=)g Fq(O)s Fs(\()p Fq(t)p Fs(\).)54 b(Since)37 b Fq(p)d(>)g Fs(0,)j(the)g(leading)e(order)h(term) g(of)g Fq(g)t Fs(\()p Fq(t)p Fs(\))p Fq(:v)k Fs(as)c Fq(t)f Fp(!)f Fs(0)-9 3997 y(arises)e(from)f Fq(g)t Fs(\()p Fq(t)p Fs(\))p Fq(:)p Fs(\()p Fq(e)759 3950 y Fn(\014)799 3959 y Fj(1)759 4022 y Fo(1)837 3997 y Fq(e)882 3950 y Fn(n)p Fk(\000)p Fn(\014)1020 3959 y Fj(1)882 4022 y Fo(2)1058 3997 y Fs(\).)43 b(As)33 b Fp(N)15 b Fs(\()p Fq(v)t Fs(\))27 b Fp(\032)h(N)15 b Fs(\()p Fq(w)s Fs(\))31 b(and)i Fq(g)t Fs(\()p Fq(t)p Fs(\))p Fq(:)p Fs([\()p Fq(v)t(;)17 b(w)s Fs(\)])30 b(con)m(v)m(erges)35 b(to)d Fm(P)p Fs(\()p Fq(V)45 b Fp(\010)22 b(f)p Fs(0)p Fp(g)p Fs(\),)31 b Fq(p)p Fs(\()p Fq(t)p Fs(\))-9 4114 y(can)e(not)h(b)s(e)f (iden)m(tically)f(zero.)43 b(Therefore)30 b(in)f(\()1765 4114 y SDict begin H.S end 1765 4114 a 0 TeXcolorgray Fs(2.2)p 0 TeXcolorgray 1889 4049 a SDict begin H.R end 1889 4049 a 1889 4114 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (equation.2.2) cvn H.B /ANN pdfmark end 1889 4114 a Fs(\))h(w)m(e)g(can)g (write)f Fq(p)p Fs(\()p Fq(t)p Fs(\))f(=)g Fq(ct)2888 4072 y Fn(\034)2947 4114 y Fs(+)16 b Fq(o)p Fs(\()p Fq(t)3159 4072 y Fn(\034)3202 4114 y Fs(\))29 b(for)g Fq(c)f Fp(6)p Fs(=)f(0.)42 b(Note)-9 4230 y(that)392 4438 y(\()p Fq(\025)p Fs(\()p Fq(t)p Fs(\)\))636 4397 y Fk(\000)p Fn(p)730 4438 y Fq(:)p Fs(\()p Fq(e)840 4391 y Fn(\014)880 4400 y Fj(1)840 4463 y Fo(1)919 4438 y Fs(\()p Fq(e)1002 4453 y Fo(2)1063 4438 y Fs(+)22 b Fq(t)1196 4397 y Fn(\034)1240 4438 y Fq(e)1285 4453 y Fo(1)1324 4438 y Fs(\))1362 4397 y Fn(n)p Fk(\000)p Fn(\014)1500 4406 y Fj(1)1538 4438 y Fs(\))28 b(=)f Fq(t)1742 4397 y Fo(\()p Fn(n)p Fk(\000)p Fo(2)p Fn(\014)1942 4406 y Fj(1)1977 4397 y Fo(\))p Fn(p)2061 4309 y(n)p Fk(\000)p Fn(\014)2199 4318 y Fj(1)2075 4343 y Fg(X)2086 4553 y Fn(j)t Fo(=0)2250 4438 y Fq(t)2285 4397 y Fo(\()p Fn(\034)8 b Fk(\000)p Fo(2)p Fn(p)p Fo(\))p Fn(j)2541 4298 y Fg(\022)2614 4371 y Fq(n)23 b Fp(\000)f Fq(\014)2849 4386 y Fo(1)2728 4506 y Fq(j)2889 4298 y Fg(\023)2962 4438 y Fq(e)3007 4391 y Fn(\014)3047 4400 y Fj(1)3081 4391 y Fo(+)p Fn(j)3007 4463 y Fo(1)3173 4438 y Fq(e)3218 4391 y Fn(n)p Fk(\000)p Fn(\014)3356 4400 y Fj(1)3390 4391 y Fk(\000)p Fn(j)3218 4463 y Fo(2)3481 4438 y Fq(:)-9 4695 y Fs(Giv)m(en)j(that)g Fq(n)8 b Fp(\000)g Fs(2)p Fq(\014)725 4710 y Fo(1)793 4695 y Fp(\025)28 b Fq(m)8 b Fp(\000)g Fs(2)p Fq(\014)1180 4710 y Fo(2)1245 4695 y Fs(and)26 b Fq(g)t Fs(\()p Fq(t)p Fs(\))p Fq(:)p Fs([\()p Fq(v)t(;)17 b(w)s Fs(\)])24 b(con)m(v)m(erges)k(to)d Fm(P)p Fs(\()p Fq(V)31 b Fp(\010)8 b(f)p Fs(0)p Fp(g)p Fs(\))25 b(as)h Fq(t)i Fp(!)f Fs(0,)g(the)f(follo)m(wing)-9 4811 y(conditions)31 b(m)m(ust)i(b)s(e)g(satis\014ed:)480 4983 y SDict begin H.S end 480 4983 a 480 4983 a SDict begin 14 H.A end 480 4983 a 480 4983 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.3) cvn /DEST pdfmark end 480 4983 a Fq(pn)23 b Fp(\000)f Fq(\034)11 b Fs(\()p Fq(n)23 b Fp(\000)g Fq(\014)1036 4998 y Fo(1)1075 4983 y Fs(\))28 b Fq(>)f Fs(\(2)p Fq(\014)1386 4998 y Fo(2)1448 4983 y Fp(\000)22 b Fq(m)p Fs(\))p Fq(p)33 b Fs(and)g Fq(pn)22 b Fp(\000)h Fq(\034)11 b Fs(\()p Fq(n)22 b Fp(\000)h Fq(\014)2497 4998 y Fo(1)2537 4983 y Fs(\))k Fq(>)h(pm)22 b Fp(\000)h Fq(\034)11 b Fs(\()p Fq(m)23 b Fp(\000)f Fq(\014)3315 4998 y Fo(2)3355 4983 y Fs(\))p Fq(:)289 b Fs(\(2.3\))-9 5155 y(Let)32 b Fq(\015)216 5170 y Fo(1)283 5155 y Fs(=)c(2)p Fq(\014)491 5170 y Fo(1)552 5155 y Fp(\000)23 b Fq(n)28 b Fp(\024)g Fq(\015)894 5170 y Fo(2)960 5155 y Fs(=)g(2)p Fq(\014)1168 5170 y Fo(2)1229 5155 y Fp(\000)23 b Fq(m)p Fs(,)33 b(then)g(\()1734 5155 y SDict begin H.S end 1734 5155 a 0 TeXcolorgray Fs(2.3)p 0 TeXcolorgray 1859 5091 a SDict begin H.R end 1859 5091 a 1859 5155 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (equation.2.3) cvn H.B /ANN pdfmark end 1859 5155 a Fs(\))f(is)g(equiv)-5 b(alen)m(t)33 b(to)657 5355 y Fq(n)22 b Fp(\000)869 5288 y Fq(\034)p 847 5332 98 4 v 847 5423 a Fs(2)p Fq(p)954 5355 y Fs(\()p Fq(n)h Fp(\000)f Fq(\015)1223 5370 y Fo(1)1262 5355 y Fs(\))28 b Fq(>)f(\015)1482 5370 y Fo(2)1554 5355 y Fs(and)33 b Fq(n)22 b Fp(\000)1956 5288 y Fq(\034)p 1934 5332 V 1934 5423 a Fs(2)p Fq(p)2041 5355 y Fs(\()p Fq(n)h Fp(\000)f Fq(\015)2310 5370 y Fo(1)2349 5355 y Fs(\))28 b Fq(>)f(m)c Fp(\000)2758 5288 y Fq(\034)p 2736 5332 V 2736 5423 a Fs(2)p Fq(p)2843 5355 y Fs(\()p Fq(m)g Fp(\000)f Fq(\015)3139 5370 y Fo(2)3178 5355 y Fs(\))p Fq(:)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 4 4 TeXDict begin HPSdict begin 4 3 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -18 -200 a SDict begin H.S end -18 -200 a -18 -200 a SDict begin H.R end -18 -200 a -18 -200 a SDict begin [/View [/XYZ H.V]/Dest (page.4) cvn /DEST pdfmark end -18 -200 a 0 TeXcolorgray -9 -126 a Fi(4)781 b(SEAN)25 b(TIMOTHY)g(P)-6 b(A)n(UL,)25 b(SONG)g(SUN,)g(AND)f(JUNSHENG) h(ZHANG)p 0 TeXcolorgray -9 74 a Fs(So)32 b(in)g(particular)f(w)m(e)i (m)m(ust)g(ha)m(v)m(e)h Fq(m)28 b(>)f(n)h(>)g(\015)1757 89 y Fo(2)1796 74 y Fs(,)33 b Fq(m)22 b Fp(\000)h Fq(\015)2114 89 y Fo(2)2175 74 y Fp(\000)f Fq(n)h Fs(+)f Fq(\015)2504 89 y Fo(1)2571 74 y Fq(>)27 b Fs(0,)32 b(and)1287 294 y SDict begin H.S end 1287 294 a 1287 294 a SDict begin 14 H.A end 1287 294 a 1287 294 a SDict begin [/View [/XYZ H.V]/Dest (equation.2.4) cvn /DEST pdfmark end 1287 294 a 1297 227 a Fq(n)22 b Fp(\000)h Fq(\015)1528 242 y Fo(2)p 1297 271 271 4 v 1297 362 a Fq(n)f Fp(\000)h Fq(\015)1528 377 y Fo(1)1599 294 y Fp(\000)1920 227 y Fq(m)f Fp(\000)h Fq(n)p 1709 271 688 4 v 1709 362 a(m)f Fp(\000)h Fq(\015)1967 377 y Fo(2)2028 362 y Fp(\000)g Fq(n)f Fs(+)g Fq(\015)2357 377 y Fo(1)2434 294 y Fq(>)27 b Fs(0)p Fq(:)1096 b Fs(\(2.4\))-9 526 y(Consider)33 b(the)g(quadratic)f(function)842 710 y Fq(Q)p Fs(\()p Fq(x)p Fs(\))c(=)g(\()p Fq(n)22 b Fp(\000)h Fq(x)p Fs(\)\()p Fq(m)g Fp(\000)f Fq(x)h Fp(\000)g Fq(n)f Fs(+)g Fq(\015)2145 725 y Fo(1)2184 710 y Fs(\))g Fp(\000)h Fs(\()p Fq(n)f Fp(\000)h Fq(\015)2613 725 y Fo(1)2652 710 y Fs(\)\()p Fq(m)f Fp(\000)h Fq(n)p Fs(\))p Fq(:)-9 893 y Fs(It)37 b(has)h(ro)s(ots)f(at)g Fq(x)f Fs(=)f Fq(m)j Fs(or)f Fq(x)f Fs(=)g Fq(\015)1359 908 y Fo(1)1398 893 y Fs(.)58 b(Since)37 b Fq(\015)1793 908 y Fo(2)1868 893 y Fp(2)f Fs([)p Fq(\015)2048 908 y Fo(1)2087 893 y Fq(;)17 b(m)p Fs(],)39 b(w)m(e)f(ha)m(v)m(e)h Fq(Q)p Fs(\()p Fq(\015)2853 908 y Fo(2)2892 893 y Fs(\))d Fp(\024)g Fs(0.)58 b(This)37 b(con)m(tradicts)-9 1009 y(\()29 1009 y SDict begin H.S end 29 1009 a 0 TeXcolorgray Fs(2.4)p 0 TeXcolorgray 153 945 a SDict begin H.R end 153 945 a 153 1009 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (equation.2.4) cvn H.B /ANN pdfmark end 153 1009 a Fs(\).)-9 1083 y SDict begin H.S end -9 1083 a -9 1083 a SDict begin 14 H.A end -9 1083 a -9 1083 a SDict begin [/View [/XYZ H.V]/Dest (theorem.2.1) cvn /DEST pdfmark end -9 1083 a 0 TeXcolorgray 116 x Fl(R)-5 b(emark)36 b Fs(2.1)p Fl(.)p 0 TeXcolorgray 35 w Fs(As)24 b(one)g(can)g(observ)m(e)i(the)e(pro)s(of) f(hinges)h(on)g(the)g(sp)s(eci\014c)g(situation)f(here.)41 b(The)25 b(argumen)m(t)-9 1315 y(fails)30 b(either)j(when)g Fq(W)47 b Fs(is)32 b(reducible)g(or)g(when)i Fq(G)e Fs(has)h(a)g (higher)f(rank.)-9 1408 y SDict begin H.S end -9 1408 a -9 1408 a SDict begin 14 H.A end -9 1408 a -9 1408 a SDict begin [/View [/XYZ H.V]/Dest (subsection.2.2) cvn /DEST pdfmark end -9 1408 a 96 x Fs(2.2.)48 b Fv(Reducible)42 b(case.)50 b Fs(Here)39 b(w)m(e)g(giv)m(e)f(the)g(example)g(claimed)e (in)h(P)m(art)h(\(2\))g(of)f(Theorem)3556 1504 y SDict begin H.S end 3556 1504 a 0 TeXcolorgray Fs(1.3)p 0 TeXcolorgray 3680 1440 a SDict begin H.R end 3680 1440 a 3680 1504 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.1.3) cvn H.B /ANN pdfmark end 3680 1504 a Fs(.)60 b(Fix)-9 1622 y(an)34 b(in)m(teger)g Fq(n)e Fp(\025)f Fs(3.)49 b(Let)35 b Fq(V)53 b Fs(=)31 b(Sym)1356 1580 y Fn(n)1419 1622 y Fm(C)1485 1581 y Fo(2)1565 1622 y Fs(and)k Fq(W)44 b Fs(=)31 b(Sym)2187 1580 y Fn(n)2251 1622 y Fm(C)2317 1581 y Fo(2)2386 1622 y Fp(\010)24 b Fs(Sym)2674 1580 y Fn(n)p Fk(\000)p Fo(2)2827 1622 y Fm(C)2893 1581 y Fo(2)2939 1622 y Fs(.)49 b(Let)38 b(~)-52 b Fq(e)3237 1637 y Fo(1)3276 1622 y Fs(,)39 b(~)-52 b Fq(e)3384 1637 y Fo(2)3458 1622 y Fs(denote)35 b(the)-9 1738 y(standard)e(basis)f(of)g (the)h(second)h Fm(C)1297 1697 y Fo(2)1375 1738 y Fs(in)e(the)h (de\014nition)e(of)h Fq(W)14 b Fs(.)44 b(Then)33 b(w)m(e)h(c)m(ho)s (ose)1134 1922 y Fq(v)e Fs(=)27 b Fq(e)1361 1937 y Fo(1)1401 1922 y Fq(e)1446 1881 y Fn(n)p Fk(\000)p Fo(1)1446 1946 y(2)1611 1922 y Fp(2)h Fq(V)5 b(;)135 b(w)30 b Fs(=)e Fq(e)2178 1881 y Fn(n)2178 1946 y Fo(1)2247 1922 y Fs(+)e(~)-53 b Fq(e)2390 1881 y Fn(n)p Fk(\000)p Fo(2)2390 1946 y(2)2555 1922 y Fp(2)28 b Fq(W)m(:)-9 2105 y Fp(\017)36 b Fs(\()p Fq(v)t(;)17 b(w)s Fs(\))35 b(is)h(n)m(umerically)f(semistable.)54 b(This)37 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1066 y Fs(\))g(=)g(2)p Fq(k)d Fs(+)d(2)p Fq(:)296 b Fs(\(3.1\))-9 1230 y(This)33 b(is)g(the)g(desired)h(prop)s(ert)m(y)g(that)f(motiv)-5 b(ates)31 b(our)i(c)m(hoice)h(of)e Fq(w)2512 1245 y Fo(0)2585 1230 y Fs(and)h Fq(v)t Fs(.)45 b(Recall)31 b(that)i(w)m(e)h(ha)m(v)m(e) h(the)-9 1346 y(w)m(eigh)m(t)d(decomp)s(osition)f Fq(W)42 b Fs(=)27 b Fp(\010)1256 1362 y Fn(\033)r Fk(2)p Fn(Lie)p Fo(\()p Fn(T)1518 1371 y Fj(0)1554 1362 y Fo(\))1581 1343 y Fe(\003)1622 1346 y Fq(W)1714 1361 y Fn(\033)1761 1346 y Fs(.)43 b(W)-8 b(e)33 b(de\014ne)1543 1538 y SDict begin H.S end 1543 1538 a 1543 1538 a SDict begin 14 H.A end 1543 1538 a 1543 1538 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.2) cvn /DEST pdfmark end 1543 1538 a Fq(W)1649 1497 y Fk(0)1699 1538 y Fs(=)1913 1444 y Fg(M)1803 1660 y Fn(l)1824 1669 y Fj(3)1858 1660 y Fo(\()p Fn(\033)r Fo(\))p Fk(\024)p Fo(2)p Fn(k)r Fk(\000)p Fo(7)2191 1538 y Fq(W)2283 1553 y Fn(\033)2330 1538 y Fq(:)1352 b Fs(\(3.2\))-9 1715 y SDict begin H.S end -9 1715 a -9 1715 a SDict begin 14 H.A end -9 1715 a -9 1715 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.1) cvn /DEST pdfmark end -9 1715 a 0 TeXcolorgray 116 x Fv(Prop)s(osition)45 b(3.1.)p 0 TeXcolorgray 46 w Fl(Supp)-5 b(ose)41 b Fq(k)k Fp(\025)d Fs(12)p Fl(,)i(then)e(the)h(p)-5 b(air)42 b Fs(\()p Fq(v)t(;)17 b(w)s Fs(\))41 b Fl(is)h(numeric)-5 b(al)5 b(ly)42 b(semistable)f(but)i(not)-9 1947 y(semistable,)33 b(wher)-5 b(e)34 b Fq(w)d Fs(=)c Fq(w)1035 1962 y Fo(0)1096 1947 y Fs(+)22 b Fq(w)1267 1906 y Fk(0)1325 1947 y Fl(for)35 b Fq(w)1554 1906 y Fk(0)1611 1947 y Fl(a)g(generic)f(ve)-5 b(ctor)35 b(in)f Fq(W)2549 1906 y Fk(0)2572 1947 y Fl(.)90 2125 y Fs(It)k(is)f(easy)i(to)f(see)h(that)e(\()p Fq(v)t(;)17 b(w)s Fs(\))37 b(is)g(not)h(semistable.)58 b(F)-8 b(or)37 b(this)h(w)m(e)h(tak)m(e)f Fq(\025)p Fs(\()p Fq(t)p Fs(\))f(=)f(diag)q (\()p Fq(t)3429 2084 y Fk(\000)p Fn(\026)3530 2125 y Fq(;)17 b(t)3609 2084 y Fk(\000)p Fn(\026)3711 2125 y Fq(;)g(t)3790 2084 y Fo(2)p Fn(\026)3872 2125 y Fs(\))-9 2241 y(for)33 b Fq(\026)d Fp(\025)h Fs(1.)48 b(Notice)33 b(that)h(the)h(comp)s(osition)c(of)j(the)g(action)g(of)f Fq(l)j Fs(man)m(y)e Fq(E)2791 2256 y Fo(23)2866 2241 y Fs('s)h(on)f Fq(w)3176 2200 y Fk(0)3229 2241 y Fp(2)d Fq(W)3432 2200 y Fk(0)3489 2241 y Fs(lies)i(in)g(the)-9 2374 y(direct)f(sum)640 2279 y Fg(M)473 2495 y Fn(l)494 2504 y Fj(3)529 2495 y Fo(\()p Fn(\033)r Fo(\))p Fk(\024)p Fo(2)p Fn(k)r Fk(\000)p Fo(7+3)p Fn(l)974 2374 y Fq(W)1066 2389 y Fn(\033)1146 2374 y Fs(for)g Fq(l)e Fs(=)d(1)p Fq(;)17 b Fs(2)p Fq(;)g Fs(3.)42 b(Then)34 b(b)m(y)h(\()2191 2374 y SDict begin H.S end 2191 2374 a 0 TeXcolorgray Fs(3.1)p 0 TeXcolorgray 2315 2310 a SDict begin H.R end 2315 2310 a 2315 2374 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (equation.3.1) cvn H.B /ANN pdfmark end 2315 2374 a Fs(\))e(as)g Fq(t)28 b Fp(!)f Fs(0)32 b(w)m(e)i(ha)m(v)m(e)976 2668 y Fp(k)p Fq(\025)p Fs(\()p Fq(t)p Fs(\))p Fq(:)17 b Fs(exp)q(\()o Fq(tE)1531 2683 y Fo(23)1607 2668 y Fs(\))p Fq(:w)s Fp(k)27 b(\030)h(j)p Fq(t)p Fp(j)2018 2627 y Fo(min)n Fk(f\000)p Fo(\(2)p Fn(k)r Fk(\000)p Fo(1\))p Fn(\026;)p Fo(3)p Fk(\000)p Fo(\(2)p Fn(k)r Fo(+2\))p Fn(\026)p Fk(g)2897 2668 y Fq(;)-9 2831 y Fs(and)1249 2954 y Fp(k)p Fq(\025)p Fs(\()p Fq(t)p Fs(\))p Fq(:)17 b Fs(exp\()p Fq(tE)1804 2969 y Fo(23)1879 2954 y Fs(\))p Fq(:v)t Fp(k)28 b(\030)g(j)p Fq(t)p Fp(j)2269 2913 y Fo(2)p Fk(\000)p Fo(\(2)p Fn(k)r Fo(+2\))p Fn(\026)2624 2954 y Fq(:)-9 3097 y Fs(It)k(follo)m(ws)f(that) i(\()p Fq(v)t(;)17 b(w)s Fs(\))31 b(is)h(not)h(semistable.)90 3213 y(The)f(remaining)c(part)i(of)g(this)g(section)h(is)f(dev)m(oted)i (to)e(the)h(pro)s(of)f(of)g(the)h(fact)f(that)g(\()p Fq(v)t(;)17 b(w)s Fs(\))30 b(is)g(n)m(umer-)-9 3330 y(ically)g (semistable)i(for)g Fq(w)943 3289 y Fk(0)998 3330 y Fs(generic.)44 b(W)-8 b(e)33 b(\014x)g Fq(k)e Fp(\025)d Fs(12.)43 b(F)-8 b(or)32 b Fp(f)p Fq(\013)q(;)17 b(\014)6 b(;)17 b(\015)5 b Fp(g)26 b Fs(=)h Fp(f)p Fs(1)p Fq(;)17 b Fs(2)p Fq(;)g 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1393 y Fg(\023\022)1726 1466 y Fs(2)p Fq(k)e Fp(\000)d Fq(i)1734 1602 y(j)28 b Fp(\000)23 b Fq(i)1935 1617 y Fo(1)1983 1393 y Fg(\023)2057 1534 y Fq(a)2108 1490 y Fn(i)2132 1499 y Fj(1)2108 1558 y Fo(11)2183 1534 y Fq(a)2234 1490 y Fn(i)p Fk(\000)p Fn(i)2337 1499 y Fj(1)2234 1558 y Fo(31)2376 1534 y Fq(a)2427 1486 y Fn(j)t Fk(\000)p Fn(i)2539 1495 y Fj(1)2427 1558 y Fo(13)2577 1534 y Fq(a)2628 1486 y Fo(2)p Fn(k)r Fk(\000)p Fn(i)p Fo(+)p Fn(j)t Fk(\000)p Fn(i)2948 1495 y Fj(1)2628 1558 y Fo(33)2986 1534 y Fq(e)3031 1486 y Fn(j)3031 1558 y Fo(1)3070 1534 y Fq(e)3115 1486 y Fo(2)p Fn(k)r Fk(\000)p Fn(j)3115 1558 y Fo(3)3303 1534 y Fp(\012)f Fq(e)3447 1493 y Fk(\003)3447 1558 y Fo(2)-9 1776 y Fs(This)32 b(is)g(similar)e(to)i Fv(Case)38 b(1)p Fs(.)90 2009 y Fv(Case)g(3.)44 b Fq(b)530 2024 y Fo(12)633 2009 y Fp(6)p Fs(=)27 b(0.)43 b(In)33 b(this)g(case)g(w)m (e)h(need)f(to)g(consider)f(\005)2317 2024 y Fo(231)2460 2009 y Fs(and)h(\005)2723 2024 y Fo(321)2833 2009 y Fs(.)43 b(W)-8 b(e)33 b(ha)m(v)m(e)413 2322 y(\005)486 2337 y Fo(231)596 2322 y Fs(\()p Fq(A:\030)777 2337 y Fn(i)805 2322 y Fs(\))28 b(=)f Fq(b)1015 2337 y Fo(12)1115 2197 y Fn(k)r Fk(\000)p Fo(1)1107 2227 y Fg(X)1118 2437 y Fn(j)t Fo(=0)1326 2193 y Fn(j)1270 2227 y Fg(X)1267 2437 y Fn(i)1291 2446 y Fj(1)1326 2437 y Fo(=0)1433 2182 y Fg(\022)1526 2255 y Fq(i)1506 2390 y(i)1539 2405 y Fo(1)1579 2182 y Fg(\023\022)1726 2255 y Fs(2)p Fq(k)e Fp(\000)d Fq(i)1734 2390 y(j)28 b Fp(\000)23 b Fq(i)1935 2405 y Fo(1)1983 2182 y Fg(\023)2057 2322 y Fq(a)2108 2278 y Fn(i)2132 2287 y Fj(1)2108 2346 y Fo(21)2183 2322 y Fq(a)2234 2278 y Fn(i)p Fk(\000)p Fn(i)2337 2287 y Fj(1)2234 2346 y Fo(31)2376 2322 y Fq(a)2427 2275 y Fn(j)t Fk(\000)p Fn(i)2539 2284 y Fj(1)2427 2346 y Fo(23)2577 2322 y Fq(a)2628 2275 y Fo(2)p Fn(k)r Fk(\000)p Fn(i)p Fo(+)p Fn(j)t Fk(\000)p Fn(i)2948 2284 y Fj(1)2628 2346 y Fo(33)2986 2322 y Fq(e)3031 2275 y Fn(j)3031 2346 y Fo(2)3070 2322 y Fq(e)3115 2275 y Fo(2)p Fn(k)r Fk(\000)p Fn(j)3115 2346 y Fo(3)3303 2322 y Fp(\012)f Fq(e)3447 2281 y Fk(\003)3447 2347 y Fo(1)-9 2564 y Fs(Again)31 b(this)h(is)g(similar)e(to)i Fv(Case)38 b(1)p Fs(.)2478 b Fd(\003)90 2738 y Fs(W)-8 b(e)31 b(no)m(w)g(pro)m(v)m(e)h(Prop)s(osition)1241 2738 y SDict begin H.S end 1241 2738 a 0 TeXcolorgray Fs(3.1)p 0 TeXcolorgray 1366 2674 a SDict begin H.R end 1366 2674 a 1366 2738 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.3.1) cvn H.B /ANN pdfmark end 1366 2738 a 30 w Fs(using)f(a)f(dimension)f(argumen)m(t.)43 b(Consider)31 b(the)g(algebraic)e(subset)-9 2855 y(of)j Fq(S)6 b(L)p Fs(\(3)p Fq(;)17 b Fm(C)i Fs(\))28 b Fp(\002)23 b Fq(W)702 2813 y Fk(0)757 2855 y Fs(de\014ned)34 b(b)m(y)1047 3018 y SDict begin H.S end 1047 3018 a 1047 3018 a SDict begin 14 H.A end 1047 3018 a 1047 3018 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.5) cvn /DEST pdfmark end 1047 3018 a Fq(X)1128 3033 y Fn(\013\014)s(\015)1289 3018 y Fs(=)27 b Fp(f)p Fs(\()p Fq(A;)17 b(w)1670 2977 y Fk(0)1693 3018 y Fs(\))27 b(:)h(\005)1886 2977 y Fk(0)1886 3043 y Fn(\013\014)s(\015)2019 3018 y Fs(\()p Fq(A:)p Fs(\()p Fq(w)2265 3033 y Fo(0)2326 3018 y Fs(+)22 b Fq(w)2497 2977 y Fk(0)2520 3018 y Fs(\)\))27 b(=)h(0)p Fp(g)p Fq(:)856 b Fs(\(3.5\))-9 3189 y(Let)32 b Fq(\031)220 3204 y Fo(1)292 3189 y Fs(denote)i(the)f(pro)5 b(jection)32 b(to)g Fq(S)6 b(L)p Fs(\(3)p Fq(;)17 b Fm(C)j Fs(\))38 b(and)33 b Fq(\031)2007 3204 y Fo(2)2079 3189 y Fs(denote)g(the)g(pro)5 b(jection)32 b(to)h Fq(W)3250 3148 y Fk(0)3273 3189 y Fs(.)43 b(Since)33 b Fq(X)3679 3204 y Fn(\013\014)s(\015)3844 3189 y Fs(is)-9 3305 y(a)g(\014nite)g(union)f(of)h(a\016ne)h(v)-5 b(arieties)32 b(\(irreducible)g(algebraic)g(sets\))j(and)e(b)m(y)h(Lemma)3189 3305 y SDict begin H.S end 3189 3305 a 0 TeXcolorgray Fs(3.2)p 0 TeXcolorgray 3314 3241 a SDict begin H.R end 3314 3241 a 3314 3305 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.3.2) cvn H.B /ANN pdfmark end 3314 3305 a 33 w Fs(ev)m(ery)h(\014b)s(er)f(of)-9 3421 y Fq(\031)46 3436 y Fo(1)118 3421 y Fs(has)f(dimension)e(at)h(most)g(dim)15 b Fq(W)1396 3380 y Fk(0)1441 3421 y Fp(\000)23 b Fs(9,)33 b(then)g(w)m(e)g(kno)m(w)h([)2297 3421 y SDict begin H.S end 2297 3421 a 0 TeXcolorgray Fs(7)p 0 TeXcolorgray 2346 3357 a SDict begin H.R end 2346 3357 a 2346 3421 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (cite.Mu) cvn H.B /ANN pdfmark end 2346 3421 a Fs(,)f(I.8.Theorem)g(2])1520 3585 y SDict begin H.S end 1520 3585 a 1520 3585 a SDict begin 14 H.A end 1520 3585 a 1520 3585 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.6) cvn /DEST pdfmark end 1520 3585 a Fs(dim)16 b Fq(X)1781 3600 y Fn(\013\014)s(\015)1941 3585 y Fq(<)27 b Fs(dim)16 b Fq(W)2330 3543 y Fk(0)2353 3585 y Fq(:)1329 b Fs(\(3.6\))-9 3748 y(In)46 b(particular,)h(dim)16 b Fq(\031)854 3763 y Fo(2)893 3748 y Fs(\()p Fq(X)1012 3763 y Fn(\013\014)s(\015)1145 3748 y Fs(\))50 b Fq(<)h Fs(dim)15 b Fq(W)1645 3707 y Fk(0)1668 3748 y Fs(.)83 b(Similarly)-8 b(,)46 b(b)m(y)h(Lemma)2739 3748 y SDict begin H.S end 2739 3748 a 0 TeXcolorgray Fs(3.3)p 0 TeXcolorgray 2864 3684 a SDict begin H.R end 2864 3684 a 2864 3748 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.3.3) cvn H.B /ANN pdfmark end 2864 3748 a 46 w Fs(one)f(can)g(sho)m(w)h (that)f(the)-9 3864 y(algebraic)30 b(subset)35 b(of)d Fq(Y)870 3879 y Fn(\013)941 3864 y Fp(\002)23 b Fq(W)1147 3823 y Fk(0)1202 3864 y Fs(de\014ned)34 b(b)m(y)896 4028 y Fq(Z)963 4043 y Fn(\014)s(\015)t(\013)1123 4028 y Fs(=)27 b Fp(f)p Fs(\()p Fq(A;)17 b(w)1504 3986 y Fk(0)1527 4028 y Fs(\))27 b(:)h(\005)1720 4043 y Fn(\014)s(\015)t(\013)1853 4028 y Fs(\()p Fq(A:)p Fs(\()p Fq(w)2099 4043 y Fo(0)2160 4028 y Fs(+)22 b Fq(w)2331 3986 y Fk(0)2354 4028 y Fs(\)\))28 b(=)f(0)p Fq(;)17 b(A)27 b Fp(2)h Fq(Y)2905 4043 y Fn(\013)2955 4028 y Fp(g)-9 4191 y Fs(also)j(satis\014es)1527 4313 y SDict begin H.S end 1527 4313 a 1527 4313 a SDict begin 14 H.A end 1527 4313 a 1527 4313 a SDict begin [/View [/XYZ H.V]/Dest (equation.3.7) cvn /DEST pdfmark end 1527 4313 a Fs(dim)16 b Fq(Z)1774 4328 y Fn(\013\014)s(\015)1934 4313 y Fq(<)27 b Fs(dim)16 b Fq(W)2323 4272 y Fk(0)2345 4313 y Fq(:)1337 b Fs(\(3.7\))-9 4456 y(Note)33 b(that)f Fq(Z)505 4471 y Fn(\013\014)s(\015)671 4456 y Fs(is)g(an)h(algebraic)e (set)j(b)s(ecause)g Fq(Y)1888 4471 y Fn(\013)1970 4456 y Fs(is)f(a)g(complemen)m(t)f(of)g(zero)s(es)i(of)e(p)s(olynomials)e (and)-9 4573 y(clearly)i(\005)376 4588 y Fn(\014)s(\015)t(\013)509 4573 y Fs(\()p Fq(A:)p Fs(\()p Fq(w)755 4588 y Fo(0)817 4573 y Fs(+)23 b Fq(w)989 4532 y Fk(0)1011 4573 y Fs(\)\))29 b(=)g(0)34 b(is)e(the)i(in)m(tersection)g(of)f(zero)s(es)h(of)f(p)s (olynomials.)43 b(No)m(w)34 b(for)f(a)g(v)m(ector)-9 4713 y Fq(w)64 4672 y Fk(0)114 4713 y Fp(2)28 b Fq(W)314 4672 y Fk(0)358 4713 y Fp(n)20 b Fs(\()495 4619 y Fg([)466 4831 y Fn(\013;\014)s(;\015)650 4713 y Fq(\031)705 4728 y Fo(2)745 4713 y Fs(\()p Fq(X)864 4728 y Fn(\013\014)s(\015)997 4713 y Fs(\))g Fp([)h Fq(\031)1197 4728 y Fo(2)1237 4713 y Fs(\()p Fq(Z)1342 4728 y Fn(\013\014)s(\015)1475 4713 y Fs(\)\),)32 b(w)m(e)g(claim)e(that)h(the)i(pair)e(\()p Fq(v)t(;)17 b(w)29 b Fp(\021)f Fq(w)2997 4728 y Fo(0)3057 4713 y Fs(+)20 b Fq(w)3226 4672 y Fk(0)3249 4713 y Fs(\))32 b(is)f(n)m(umerically)-9 4940 y(semistable.)60 b(T)-8 b(o)39 b(see)h(this,)g(giv)m(en)f Fq(A)f Fp(2)g Fq(S)6 b(L)p Fs(\(3)p Fq(;)17 b Fm(C)j Fs(\))45 b(b)m(y)39 b(our)g (de\014nition)e(of)i Fq(S)2890 4899 y Fk(0)2884 4965 y Fn(\013\014)s(\015)3055 4940 y Fs(and)g(c)m(hoice)g(of)f Fq(w)3737 4899 y Fk(0)3798 4940 y Fs(w)m(e)-9 5061 y(kno)m(w)33 b(that)f Fp(N)15 b Fs(\()p Fq(A:w)s Fs(\))31 b(alw)m(a)m(ys)h(con)m (tains)g(the)h(lattice)d(p)s(oin)m(ts)i(in)f Fp(N)15 b Fs(\()p Fq(V)21 b Fs(\))32 b(except)i(the)f(outmost)e(la)m(y)m(er.)43 b(So)-9 5177 y(if)32 b Fq(A)42 b(=)-60 b Fp(2)30 b Fq(Y)338 5192 y Fn(\013)421 5177 y Fs(then)35 b Fp(N)15 b Fs(\()p Fq(A:w)s Fs(\))33 b(strictly)g(con)m(tains)h Fp(N)15 b Fs(\()p Fq(A:v)t Fs(\).)47 b(On)34 b(the)h(other)f(hand,)h(if)e(for)g (some)h Fq(\013)h Fs(it)e(holds)-9 5293 y(that)k Fq(A)g Fp(2)g Fq(Y)477 5308 y Fn(\013)526 5293 y Fs(,)i(then)f Fp(\000)p Fq(k)s(L)1016 5308 y Fn(\013)1092 5293 y Fp(\000)27 b Fq(L)1262 5308 y Fn(j)1335 5293 y Fp(2)37 b(N)15 b Fs(\()p Fq(A:v)t Fs(\))37 b(for)g(some)h Fq(j)6 b Fs(;)40 b(in)d(this)h(case)g(b)m(y)h(our)f(de\014nition)e(of)i Fq(S)3777 5308 y Fn(\014)s(\015)t(\013)-9 5409 y Fs(and)31 b(c)m(hoice)i(of)e Fq(w)652 5368 y Fk(0)706 5409 y Fs(w)m(e)i(see)g (that)f Fp(\000)p Fs(\()p Fq(k)24 b Fs(+)c(1\))p Fq(L)1656 5424 y Fn(\013)1734 5409 y Fp(2)28 b(N)15 b Fs(\()p Fq(A:w)s Fs(\).)42 b(Th)m(us)33 b Fp(N)15 b Fs(\()p Fq(A:w)s Fs(\))31 b(alw)m(a)m(ys)h(con)m(tains)g Fp(N)15 b Fs(\()p Fq(A:v)t Fs(\).)p 0 TeXcolorgray 0 TeXcolorgray eop end end %%Page: 10 10 TeXDict begin HPSdict begin 10 9 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -18 -200 a SDict begin H.S end -18 -200 a -18 -200 a SDict begin H.R end -18 -200 a -18 -200 a SDict begin [/View [/XYZ H.V]/Dest (page.10) cvn /DEST pdfmark end -18 -200 a 0 TeXcolorgray -9 -126 a Fi(10)743 b(SEAN)25 b(TIMOTHY)g(P)-6 b(A)n(UL,)25 b(SONG)g(SUN,)g(AND)f(JUNSHENG) h(ZHANG)p 0 TeXcolorgray 90 74 a Fs(This)33 b(\014nishes)g(the)g(pro)s (of)f(of)g(Prop)s(osition)1714 74 y SDict begin H.S end 1714 74 a 0 TeXcolorgray Fs(3.1)p 0 TeXcolorgray 1839 9 a SDict begin H.R end 1839 9 a 1839 74 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.3.1) cvn H.B /ANN pdfmark end 1839 74 a Fs(.)90 306 y(As)k(a)f(\014nal)f(remark,)i(w)m(e)g(men)m(tion)e (that)h(in)f(the)i(ab)s(o)m(v)m(e)g(discussion,)g(for)f(generic)g Fq(w)3254 265 y Fk(0)3308 306 y Fp(2)e Fq(W)3513 265 y Fk(0)3571 306 y Fs(w)m(e)j(ha)m(v)m(e)-9 422 y(indeed)j(that)g(the)h (pair)e(\()p Fq(v)t(;)17 b(w)1109 437 y Fo(0)1174 422 y Fs(+)27 b Fq(w)1350 381 y Fk(0)1373 422 y Fs(\))39 b(is)f(n)m(umerically)g Fl(stable)46 b Fs(in)39 b(the)g(sense)i(of)e(P) m(aul)g([)3300 422 y SDict begin H.S end 3300 422 a 0 TeXcolorgray Fs(9)p 0 TeXcolorgray 3349 358 a SDict begin H.R end 3349 358 a 3349 422 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (cite.Paul09) cvn H.B /ANN pdfmark end 3349 422 a Fs(].)64 b(W)-8 b(e)39 b(brie\015y)-9 538 y(recall)44 b(the)j(latter)e(notion)g(here.)86 b(Let)46 b Fq(M)1615 553 y Fo(3)1701 538 y Fs(denote)h(the)g(direct)f(sum)g(of)g(3-copies)g (of)f(the)i(standard)-9 655 y(represen)m(tation)33 b Fq(U)43 b Fs(and)32 b Fq(I)k Fs(=)27 b Fq(e)1153 670 y Fo(1)1215 655 y Fp(\010)22 b Fq(e)1359 670 y Fo(2)1420 655 y Fp(\010)g Fq(e)1564 670 y Fo(3)1632 655 y Fp(2)28 b Fq(M)1820 670 y Fo(3)1860 655 y Fs(.)43 b(Let)33 b Fq(V)54 b Fs(b)s(e)32 b(a)g(represen)m(tation)h(of)f Fq(G)p Fs(.)44 b(W)-8 b(e)32 b(de\014ne)i(the)-9 771 y(degree)f(of)f Fq(V)54 b Fs(b)m(y)1002 889 y(deg)r(\()p Fq(V)21 b Fs(\))28 b(:=)f(min)16 b Fp(f)o Fq(d)28 b Fp(2)g Fm(Z)1933 904 y Fn(>)p Fo(0)2052 889 y Fp(j)g(N)15 b Fs(\()p Fq(V)21 b Fs(\))27 b Fp(\022)h Fq(d)p Fp(N)15 b Fs(\()p Fq(M)2771 904 y Fo(3)2810 889 y Fs(\))p Fp(g)-9 948 y SDict begin H.S end -9 948 a -9 948 a SDict begin 14 H.A end -9 948 a -9 948 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3.4) cvn /DEST pdfmark end -9 948 a 0 TeXcolorgray 116 x Fv(De\014nition)40 b(3.4)i Fs(\([)768 1064 y SDict begin H.S end 768 1064 a 0 TeXcolorgray Fs(9)p 0 TeXcolorgray 817 1000 a SDict begin H.R end 817 1000 a 817 1064 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (cite.Paul09) cvn H.B /ANN pdfmark end 817 1064 a Fs(]\))p Fv(.)p 0 TeXcolorgray 43 w Fl(A)d(p)-5 b(air)38 b Fs(\()p Fq(v)t(;)17 b(w)s Fs(\))37 b Fl(is)h(\(numeric)-5 b(al)5 b(ly\))37 b(stable)h(if)g(and)g(only)g(if)g(ther)-5 b(e)38 b(is)g(a)g(p)-5 b(ositive)-9 1188 y(inte)g(ger)34 b Fq(m)h Fl(such)g(that)g Fs(\()p Fq(I)943 1147 y Fk(\012)12 b Fo(deg)q(\()p Fn(V)17 b Fo(\))1254 1188 y Fp(\012)23 b Fq(v)1405 1147 y Fk(\012)p Fn(m)1526 1188 y Fq(;)17 b(w)1643 1147 y Fk(\012)p Fo(\()p Fn(m)p Fo(+1\))1909 1188 y Fs(\))35 b Fl(is)f(\(numeric)-5 b(al)5 b(ly\))34 b(semistable.)90 1363 y Fs(It)j(is)f(easy)h(to)g(see)g(that)g(\(n)m (umerical\))d(stabilit)m(y)h(implies)f(\(n)m(umerical\))h(semistabilit) m(y)-8 b(.)53 b(In)37 b(our)f(case,)-9 1480 y(w)m(e)d(observ)m(e)i (that)d(deg)r(\()p Fq(V)21 b Fs(\))28 b(=)g(2)p Fq(k)d Fs(+)d(2)33 b(and)f(the)i(w)m(eigh)m(t)f(p)s(olytop)s(e)e Fp(N)15 b Fs(\()p Fq(I)8 b Fs(\))27 b(=)h Fp(N)15 b Fs(\()p Fq(M)3086 1495 y Fo(3)3125 1480 y Fs(\))33 b(is)f(the)h(equilateral)-9 1596 y(triangle)28 b(with)i(v)m(ertices)i(giv)m(en)f(b)m(y)g Fq(L)1369 1611 y Fn(j)1406 1596 y Fs(\()p Fq(j)j Fs(=)27 b(1)p Fq(;)17 b Fs(2)p Fq(;)g Fs(3\))p Fq(:)30 b Fs(Then)h(to)f(see)i (the)f(n)m(umerical)e(stabilit)m(y)g(of)h(\()p Fq(v)t(;)17 b(w)3777 1611 y Fo(0)3834 1596 y Fs(+)-9 1712 y Fq(w)64 1671 y Fk(0)86 1712 y Fs(\),)28 b(one)e(simply)f(notices)h(that)g (Lemma)1523 1712 y SDict begin H.S end 1523 1712 a 0 TeXcolorgray Fs(3.2)p 0 TeXcolorgray 1647 1648 a SDict begin H.R end 1647 1648 a 1647 1712 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.3.2) cvn H.B /ANN pdfmark end 1647 1712 a 26 w Fs(and)g(Lemma)2198 1712 y SDict begin H.S end 2198 1712 a 0 TeXcolorgray Fs(3.3)p 0 TeXcolorgray 2323 1648 a SDict begin H.R end 2323 1648 a 2323 1712 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.3.3) cvn H.B /ANN pdfmark end 2323 1712 a 26 w Fs(imply)e(that)i(whenev)m(er)j Fp(N)15 b Fs(\()p Fq(A:v)t Fs(\))25 b(touc)m(hes)-9 1828 y Fp(N)15 b Fs(\()p Fq(A:w)s Fs(\),)30 b(the)h(touc)m(hing)f(p)s(oin)m (t)g(m)m(ust)h(b)s(e)g(some)g(v)m(ertex)h(\(sa)m(y)g Fq(P)14 b Fs(\))30 b(of)g Fp(N)15 b Fs(\()p Fq(V)21 b Fs(\))31 b(and)g(there)g(is)g(a)f(segmen)m(t)i Fq(S)-9 1945 y Fs(in)g(the)i(b)s(oundary)g(of)f(\(2)p Fq(k)26 b Fs(+)c(2\))p Fp(N)15 b Fs(\()p Fq(M)1404 1960 y Fo(3)1443 1945 y Fs(\))34 b(suc)m(h)h(that)e Fq(P)47 b Fs(is)33 b(in)f(the)i(in)m(terior)e(of)h Fq(S)40 b Fs(and)33 b Fq(S)40 b Fs(is)33 b(con)m(tained)g(in)-9 2061 y Fp(N)15 b Fs(\()p Fq(A:w)s Fs(\).)1660 2263 y SDict begin H.S end 1660 2263 a 1660 2263 a SDict begin 14 H.A end 1660 2263 a 1660 2263 a SDict begin [/View [/XYZ H.V]/Dest (section*.1) cvn /DEST pdfmark end 1660 2263 a Fr(References)-9 2321 y SDict begin H.S end -9 2321 a -9 2321 a SDict begin 12 H.A end -9 2321 a -9 2321 a SDict begin [/View [/XYZ H.V]/Dest (cite.BHJ) cvn /DEST pdfmark end -9 2321 a 0 TeXcolorgray 32 2421 a Fu([1])p 0 TeXcolorgray 41 w(S)n(\023)-39 b(ebastien)21 b(Bouc)n(ksom,)h(T)-7 b(omo)n(yuki)21 b(Hisamoto,)i(and)f(Mattias)g(Jonsson.)e(Uniform)j(K-stabilit)n(y)e (and)h(asymptotics)g(of)161 2520 y(energy)i(functionals)g(in)h(K\177) -42 b(ahler)24 b(geometry.)f Fc(Journal)k(of)h(the)f(Eur)l(op)l(e)l(an) h(Mathematic)l(al)h(So)l(ciety)p Fu(,)d(21\(9\):2905{2944,)161 2620 y(2019.)378 2620 y SDict begin H.S end 378 2620 a 0 TeXcolorgray Fu(1.4)p 0 TeXcolorgray 484 2566 a SDict begin H.R end 484 2566 a 484 2620 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.1.4) cvn H.B /ANN pdfmark end 484 2620 a Fu(,)535 2620 y SDict begin H.S end 535 2620 a 0 TeXcolorgray Fu(1)p 0 TeXcolorgray 576 2566 a SDict begin H.R end 576 2566 a 576 2620 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.1.4) cvn H.B /ANN pdfmark end 576 2620 a -9 2636 a SDict begin H.S end -9 2636 a -9 2636 a SDict begin 12 H.A end -9 2636 a -9 2636 a SDict begin [/View [/XYZ H.V]/Dest (cite.BHJ-Erratum) cvn /DEST pdfmark end -9 2636 a 0 TeXcolorgray 32 2720 a Fu([2])p 0 TeXcolorgray 41 w(S)n(\023)-39 b(ebastien)28 b(Bouc)n(ksom,)g(T)-7 b(omo)n(yuki)28 b(Hisamoto,)g(and)h(Mattias)g(Jonsson.)e(Erratum)h(to)h (\\Uniform)g(K-stabilit)n(y)f(and)161 2819 y(asymptotics)g(of)g(energy) g(functionals)g(in)h(K\177)-42 b(ahler)27 b(geometry".)g Fc(Journal)j(of)i(the)e(Eur)l(op)l(e)l(an)h(Mathematic)l(al)i(So)l (ciety)p Fu(,)161 2919 y(24\(2\):735{736,)24 b(2022.)931 2919 y SDict begin H.S end 931 2919 a 0 TeXcolorgray Fu(1.4)p 0 TeXcolorgray 1038 2865 a SDict begin H.R end 1038 2865 a 1038 2919 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.1.4) cvn H.B /ANN pdfmark end 1038 2919 a -9 2940 a SDict begin H.S end -9 2940 a -9 2940 a SDict begin 12 H.A end -9 2940 a -9 2940 a SDict begin [/View [/XYZ H.V]/Dest (cite.CDS1) cvn /DEST pdfmark end -9 2940 a 0 TeXcolorgray 32 3018 a Fu([3])p 0 TeXcolorgray 41 w(Xiuxiong)e(Chen,)h (Simon)f(Donaldson,)g(and)g(Song)f(Sun.)h(K\177)-42 b(ahler-Einstein)21 b(metrics)g(on)h(F)-7 b(ano)21 b(manifolds.)h(I:)g(Appro)n(x-)161 3118 y(imation)f(of)g(metrics)g(with)h(cone)f(singularities.)f Fc(Journal)k(of)g(the)g(A)n(meric)l(an)h(Mathematic)l(al)g(So)l(ciety)p Fu(,)f(28\(1\):183{197,)161 3218 y(2015.)378 3218 y SDict begin H.S end 378 3218 a 0 TeXcolorgray Fu(1)p 0 TeXcolorgray 419 3164 a SDict begin H.R end 419 3164 a 419 3218 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.1.4) cvn H.B /ANN pdfmark end 419 3218 a -9 3218 a SDict begin H.S end -9 3218 a -9 3218 a SDict begin 12 H.A end -9 3218 a -9 3218 a SDict begin [/View [/XYZ H.V]/Dest (cite.CDS2) cvn /DEST pdfmark end -9 3218 a 0 TeXcolorgray 32 3317 a Fu([4])p 0 TeXcolorgray 41 w(Xiuxiong)g(Chen,)i(Simon)e (Donaldson,)h(and)f(Song)g(Sun.)h(K\177)-42 b(ahler-Einstein)23 b(metrics)h(on)g(F)-7 b(ano)24 b(manifolds.)g(I)r(I:)h(Limits)161 3417 y(with)j(cone)f(angle)g(less)g(than)h(2)p Fb(\031)s Fu(.)g Fc(Journal)h(of)i(the)f(A)n(meric)l(an)g(Mathematic)l(al)h(So)l (ciety)p Fu(,)e(28\(1\):199{234,)23 b(2015.)3757 3417 y SDict begin H.S end 3757 3417 a 0 TeXcolorgray Fu(1)p 0 TeXcolorgray 3798 3363 a SDict begin H.R end 3798 3363 a 3798 3417 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.1.4) cvn H.B /ANN pdfmark end 3798 3417 a -9 3438 a SDict begin H.S end -9 3438 a -9 3438 a SDict begin 12 H.A end -9 3438 a -9 3438 a SDict begin [/View [/XYZ H.V]/Dest (cite.CDS3) cvn /DEST pdfmark end -9 3438 a 0 TeXcolorgray 32 3517 a Fu([5])p 0 TeXcolorgray 41 w(Xiuxiong)f(Chen,)h(Simon)f (Donaldson,)g(and)g(Song)f(Sun.)i(K\177)-42 b(ahler-Einstein)20 b(metrics)i(on)f(F)-7 b(ano)22 b(manifolds.)g(I)r(I)r(I:)g(Limits)161 3616 y(as)28 b(cone)h(angle)e(approac)n(hes)g(2)p Fb(\031)32 b Fu(and)c(completion)h(of)f(the)h(main)g(pro)r(of.)f Fc(Journal)j(of)h(the)e(A)n(meric)l(an)h(Mathematic)l(al)161 3716 y(So)l(ciety)p Fu(,)e(28\(1\):235{278,)23 b(2015.)1237 3716 y SDict begin H.S end 1237 3716 a 0 TeXcolorgray Fu(1)p 0 TeXcolorgray 1278 3662 a SDict begin H.R end 1278 3662 a 1278 3716 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.1.4) cvn H.B /ANN pdfmark end 1278 3716 a -9 3737 a SDict begin H.S end -9 3737 a -9 3737 a SDict begin 12 H.A end -9 3737 a -9 3737 a SDict begin [/View [/XYZ H.V]/Dest (cite.FH) cvn /DEST pdfmark end -9 3737 a 0 TeXcolorgray 32 3815 a Fu([6])p 0 TeXcolorgray 41 w(William)32 b(F)-7 b(ulton)32 b(and)f(Jo)r(e)g(Harris.)f Fc(R)l(epr)l(esentation)j (the)l(ory:)46 b(a)34 b(\014rst)e(c)l(ourse)p Fu(,)h(v)n(olume)d(129.)g (Springer)h(Science)g(&)161 3915 y(Business)c(Media,)h(2013.)989 3915 y SDict begin H.S end 989 3915 a 0 TeXcolorgray Fu(3)p 0 TeXcolorgray 1031 3862 a SDict begin H.R end 1031 3862 a 1031 3915 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (section.3) cvn H.B /ANN pdfmark end 1031 3915 a -9 3931 a SDict begin H.S end -9 3931 a -9 3931 a SDict begin 12 H.A end -9 3931 a -9 3931 a SDict begin [/View [/XYZ H.V]/Dest (cite.Mu) cvn /DEST pdfmark end -9 3931 a 0 TeXcolorgray 32 4015 a Fu([7])p 0 TeXcolorgray 41 w(Da)n(vid)23 b(Mumford.)h Fc(The)j(r)l(e)l(d)f(b)l(o)l(ok)h(of)g(varieties)g(and)g (schemes:)37 b(includes)27 b(the)f(Michigan)i(le)l(ctur)l(es)e (\(1974\))h(on)f(curves)161 4114 y(and)k(their)h(Jac)l(obians)p Fu(,)e(v)n(olume)e(1358.)e(Springer)i(Science)h(&)f(Business)g(Media,)g (1999.)2967 4114 y SDict begin H.S end 2967 4114 a 0 TeXcolorgray Fu(3)p 0 TeXcolorgray 3008 4061 a SDict begin H.R end 3008 4061 a 3008 4114 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (equation.3.5) cvn H.B /ANN pdfmark end 3008 4114 a -9 4130 a SDict begin H.S end -9 4130 a -9 4130 a SDict begin 12 H.A end -9 4130 a -9 4130 a SDict begin [/View [/XYZ H.V]/Dest (cite.Paul12) cvn /DEST pdfmark end -9 4130 a 0 TeXcolorgray 32 4214 a Fu([8])p 0 TeXcolorgray 41 w(Sean)37 b(Timoth)n(y)g(P)n(aul.)f(A)h(n)n(umerical)f(criterion)g (for)h(K-energy)e(maps)i(of)g(algebraic)e(manifolds.)i Fc(arXiv)i(pr)l(eprint)161 4314 y(arXiv:1210.0924)p Fu(,)33 b(2012.)1022 4314 y SDict begin H.S end 1022 4314 a 0 TeXcolorgray Fu(1.4)p 0 TeXcolorgray 1128 4260 a SDict begin H.R end 1128 4260 a 1128 4314 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.1.4) cvn H.B /ANN pdfmark end 1128 4314 a -9 4330 a SDict begin H.S end -9 4330 a -9 4330 a SDict begin 12 H.A end -9 4330 a -9 4330 a SDict begin [/View [/XYZ H.V]/Dest (cite.Paul09) cvn /DEST pdfmark end -9 4330 a 0 TeXcolorgray 32 4413 a Fu([9])p 0 TeXcolorgray 41 w(Sean)e(Timoth)n(y)g(P)n(aul.)f(Hyp)r(erdiscriminan)n(t)g(p)r (olytop)r(es,)i(Cho)n(w)e(p)r(olytop)r(es,)i(and)f(Mabuc)n(hi)f(energy) g(asymptotics.)161 4513 y Fc(A)n(nnals)f(of)i(Mathematics)p Fu(,)e(pages)e(255{296,)d(2012.)1835 4513 y SDict begin H.S end 1835 4513 a 0 TeXcolorgray Fu(1)p 0 TeXcolorgray 1876 4459 a SDict begin H.R end 1876 4459 a 1876 4513 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (section.1) cvn H.B /ANN pdfmark end 1876 4513 a Fu(,)1927 4513 y SDict begin H.S end 1927 4513 a 0 TeXcolorgray Fu(1)p 0 TeXcolorgray 1969 4459 a SDict begin H.R end 1969 4459 a 1969 4513 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.1.1) cvn H.B /ANN pdfmark end 1969 4513 a Fu(,)2019 4513 y SDict begin H.S end 2019 4513 a 0 TeXcolorgray Fu(1)p 0 TeXcolorgray 2061 4459 a SDict begin H.R end 2061 4459 a 2061 4513 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (problem.1) cvn H.B /ANN pdfmark end 2061 4513 a Fu(,)2112 4513 y SDict begin H.S end 2112 4513 a 0 TeXcolorgray Fu(1)p 0 TeXcolorgray 2153 4459 a SDict begin H.R end 2153 4459 a 2153 4513 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.1.4) cvn H.B /ANN pdfmark end 2153 4513 a Fu(,)2204 4513 y SDict begin H.S end 2204 4513 a 0 TeXcolorgray Fu(3)p 0 TeXcolorgray 2245 4459 a SDict begin H.R end 2245 4459 a 2245 4513 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (section.3) cvn H.B /ANN pdfmark end 2245 4513 a Fu(,)2296 4513 y SDict begin H.S end 2296 4513 a 0 TeXcolorgray Fu(3)p 0 TeXcolorgray 2338 4459 a SDict begin H.R end 2338 4459 a 2338 4513 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (equation.3.7) cvn H.B /ANN pdfmark end 2338 4513 a Fu(,)2388 4513 y SDict begin H.S end 2388 4513 a 0 TeXcolorgray Fu(3.4)p 0 TeXcolorgray 2494 4459 a SDict begin H.R end 2494 4459 a 2494 4513 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.3.4) cvn H.B /ANN pdfmark end 2494 4513 a -9 4529 a SDict begin H.S end -9 4529 a -9 4529 a SDict begin 12 H.A end -9 4529 a -9 4529 a SDict begin [/View [/XYZ H.V]/Dest (cite.Paul13) cvn /DEST pdfmark end -9 4529 a 0 TeXcolorgray 83 x Fu([10])p 0 TeXcolorgray 40 w(Sean)39 b(Timoth)n(y)g(P)n(aul.)f(Stable)h(pairs)f(and)h(co)r(erciv)n(e)e (estimates)i(for)g(the)g(Mabuc)n(hi)g(functional.)g Fc(arXiv)i(pr)l (eprint)161 4712 y(arXiv:1308.4377)p Fu(,)33 b(2013.)1022 4712 y SDict begin H.S end 1022 4712 a 0 TeXcolorgray Fu(1.4)p 0 TeXcolorgray 1128 4659 a SDict begin H.R end 1128 4659 a 1128 4712 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.1.4) cvn H.B /ANN pdfmark end 1128 4712 a -9 4728 a SDict begin H.S end -9 4728 a -9 4728 a SDict begin 12 H.A end -9 4728 a -9 4728 a SDict begin [/View [/XYZ H.V]/Dest (cite.Paul21) cvn /DEST pdfmark end -9 4728 a 0 TeXcolorgray 84 x Fu([11])p 0 TeXcolorgray 40 w(Sean)d(Timoth)n(y)g(P)n (aul.)f(Mahler)g(measures,)h(Stable)g(P)n(airs,)f(and)h(the)g(Global)g (co)r(erciv)n(e)f(estimate)h(for)f(the)i(Mabuc)n(hi)161 4911 y(F)-7 b(unctional.)28 b Fc(arXiv)i(pr)l(eprint)h (arXiv:2105.01240)p Fu(,)i(2021.)2050 4911 y SDict begin H.S end 2050 4911 a 0 TeXcolorgray Fu(1)p 0 TeXcolorgray 2091 4858 a SDict begin H.R end 2091 4858 a 2091 4911 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (problem.1) cvn H.B /ANN pdfmark end 2091 4911 a -9 4927 a SDict begin H.S end -9 4927 a -9 4927 a SDict begin 12 H.A end -9 4927 a -9 4927 a SDict begin [/View [/XYZ H.V]/Dest (cite.smirnov05) cvn /DEST pdfmark end -9 4927 a 0 TeXcolorgray 84 x Fu([12])p 0 TeXcolorgray 40 w(A.V.)40 b(Smirno)n(v.)f(Pro)5 b(jectiv)n(e)38 b(Orbits)h(of)g(Reductiv)n(e)h(Groups)f(and)g(Brion)g (P)n(olytop)r(es.)f Fc(R)n(ussian)i(Math)i(Surveys)p Fu(,)161 5111 y(60:376{377,)24 b(2005.)825 5111 y SDict begin H.S end 825 5111 a 0 TeXcolorgray Fu(1)p 0 TeXcolorgray 867 5057 a SDict begin H.R end 867 5057 a 867 5111 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.1.4) cvn H.B /ANN pdfmark end 867 5111 a -9 5127 a SDict begin H.S end -9 5127 a -9 5127 a SDict begin 12 H.A end -9 5127 a -9 5127 a SDict begin [/View [/XYZ H.V]/Dest (cite.Tian2017) cvn /DEST pdfmark end -9 5127 a 0 TeXcolorgray 83 x Fu([13])p 0 TeXcolorgray 40 w(Gang)34 b(Tian.)h(K-stabilit)n(y)f (implies)h(CM-stabilit)n(y.)f Fc(Ge)l(ometry,)39 b(A)n(nalysis)e(and)f (Pr)l(ob)l(ability:)54 b(In)36 b(Honor)h(of)g(Je)l(an-)161 5310 y(Michel)32 b(Bismut)p Fu(,)27 b(pages)g(245{261,)d(2017.)1524 5310 y SDict begin H.S end 1524 5310 a 0 TeXcolorgray Fu(1.4)p 0 TeXcolorgray 1630 5256 a SDict begin H.R end 1630 5256 a 1630 5310 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/Subtype /Link/Dest (theorem.1.4) cvn H.B /ANN pdfmark end 1630 5310 a -9 5326 a SDict begin H.S end -9 5326 a -9 5326 a SDict begin 12 H.A end -9 5326 a -9 5326 a SDict begin [/View [/XYZ H.V]/Dest (cite.Tian2018) cvn /DEST pdfmark end -9 5326 a 0 TeXcolorgray 83 x Fu([14])p 0 TeXcolorgray 40 w(Gang)j(Tian.)h(On)f (uniform)h(K-stabilit)n(y)e(of)i(pairs.)e Fc(arXiv)31 b(pr)l(eprint)f(arXiv:1812.05746)p Fu(,)j(2018.)3245 5409 y SDict begin H.S end 3245 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