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✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ P❍❸▼ ❚❍➚ ❚❹▼ ❈⑩❈ ▲❰P ❍⑨▼ ❈❍❖◗❯❊❚✲▼❖◆●❊✲❆▼P❊❘❊ ❚❘➊◆ ❈⑩❈ ✣❆ ❚❸P ❑❆❍▲❊❘ ❈❖▼P❆❈❚ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✷✵ download by : skknchat@gmail.com ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖✖ P❍❸▼ ❚❍➚ ❚❹▼ ❈⑩❈ ▲❰P ❍⑨▼ ❈❍❖◗❯❊❚✲▼❖◆●❊✲❆▼P❊❘❊ ❚❘➊◆ ❈⑩❈ ✣❆ ❚❸P ❑❆❍▲❊❘ ❈❖▼P❆❈❚ ❈❤✉②➯♥ ♥❣➔♥❤✿ ●■❷■ ❚➑❈❍ ▼➣ sè✿ ✽✳✹✻✳✵✶✳✵✷ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ữớ ữợ ì ◗❯❆◆● ❍❷■ ❚❤→✐ ◆❣✉②➯♥ ✲ ◆➠♠ ✷✵✷✵ download by : skknchat@gmail.com ▲í✐ ❝❛♠ ✤♦❛♥ ❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ✤➙② ❧➔ ổ tr ự r tổ ữợ sỹ ữợ ữỡ t tr♦♥❣ ❧✉➟♥ ✈➠♥ ❧➔ tr✉♥❣ t❤ü❝✳ ❈→❝ ❦➳t q✉↔ ❝❤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❝❤÷❛ tø♥❣ ✤÷đ❝ ❝ỉ♥❣ ❜è tr♦♥❣ ❝→❝ ❧✉➟♥ ✈➠♥ ❚❤↕❝ s➽ ❝õ❛ t→❝ ❣✐↔ ❦❤→❝✳ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ♠å✐ sü ❣✐ó♣ ✤ï ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔② ✤➝ ✤÷đ❝ ❝↔♠ ì♥ ✈➔ ❝→❝ t❤ỉ♥❣ t✐♥ t➼❝❤ ❞➝♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤➣ ✤÷đ❝ ❝❤➾ ró ỗ ố P ổ ữớ ữợ ❦❤♦❛ ❤å❝ ❚❙✳ ❚r➛♥ ◆❣✉②➯♥ ❆♥ ❚❙✳ ❉÷ì♥❣ ◗✉❛♥❣ ❍↔✐ ✐ download by : skknchat@gmail.com ▲í✐ ❝↔♠ ì♥ ❇↔♥ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ữợ sỹ ữợ ❚❙✳ ❉÷ì♥❣ ◗✉❛♥❣ ❍↔✐✳ ◆❤➙♥ ❞à♣ ♥➔② tỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ t❤➛② ✈➲ sü ữợ t t ũ ỳ tr q tr➻♥❤ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚ỉ✐ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❣✐→♠ ❤✐➺✉✱ ❑❤♦❛ ❙❛✉ ✣↕✐ ❤å❝✱ ❇❛♥ ❝❤õ ♥❤✐➺♠ ❑❤♦❛ ❚♦→♥✱ ❝→❝ t❤➛② ❝ỉ ❣✐→♦ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ✲ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❍➔ ◆ë✐ ✈➔ ❱✐➺♥ ❚♦→♥ ❤å❝ ✤➣ ❣✐↔♥❣ ❞↕② ✈➔ t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❝❤♦ tỉ✐ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❦❤♦❛ ❤å❝✳ ❇↔♥ ❧✉➟♥ ✈➠♥ ❝❤➢❝ ❝❤➢♥ s➩ ❦❤æ♥❣ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ ❦❤✐➳♠ ❦❤✉②➳t✱ ✈➻ ✈➟② r➜t ♠♦♥❣ ữủ sỹ õ õ ỵ t ❝æ ❣✐→♦ ✈➔ ❝→❝ ❜↕♥ ❤å❝ ✈✐➯♥ ✤➸ ❧✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ ❝❤➾♥❤ ❤ì♥✳ ❈✉è✐ ❝ị♥❣✱ tỉ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ❣✐❛ ✤➻♥❤ ✈➔ ❜↕♥ ❜➧ ✤➣ ❧✉æ♥ ✤ë♥❣ ✈✐➯♥✱ ❦❤➼❝❤ ❧➺✱ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧đ✐ ❝❤♦ tỉ✐ tr♦♥❣ t❤í✐ ❣✐❛♥ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✾ ♥➠♠ ✷✵✷✵ ◆❣÷í✐ ✈✐➳t ❧✉➟♥ ✈➠♥ P❤↕♠ ❚❤à ❚➙♠ ✐✐ download by : skknchat@gmail.com ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ✐✐ ▼ö❝ ❧ö❝ ✐✐ ▼ð ✤➛✉ ✶ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✸ ✶✳✶ ✶✳✷ ✶✳✸ ✶✳✹ ❚♦→♥ tû ▼♦♥❣❡✲❆♠♣➧r❡ ♣❤ù❝ ✳ ✳ ✳ ▲ỵ♣ ❈❤♦q✉❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷✳✶ ◆➠♥❣ ❧÷đ♥❣ ❈❤♦q✉❡t ✳ ✳ ✳ ✳ ✶✳✷✳✷ ◆➠♥❣ ❧÷đ♥❣ ▼♦♥❣❡✲❆♠♣➧r❡ ❚➼❝❤ ♣❤➙♥ ❈❤♦q✉❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▲ỵ♣ ❈❤♦q✉❡t✲▼♦♥❣❡✲❆♠♣➧r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✺ ✺ ✻ ✼ ✾ ✷ ❈→❝ ❧ỵ♣ ❤➔♠ ❈❤♦q✉❡t✲▼♦♥❣❡✲❆♠♣➧r❡ tr➯♥ ❝→❝ ✤❛ t↕♣ ❑☎❛❤❧❡r ❝♦♠♣❛❝t ✶✶ ✷✳✶ ợ ữủ ỳ tr ✤❛ t↕♣ ❑☎❛❤❧❡r ❝♦♠♣❛❝t✳ ◆➠♥❣ ❧÷đ♥❣ ❈❤♦q✉❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❷♥❤ ❝õ❛ t♦→♥ tû ▼♦♥❣❡✲❆♠♣➧r❡ tr➯♥ ❝→❝ ✤❛ t↕♣ ❑☎❛❤❧❡r ❝♦♠♣❛❝t ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✶ ❚♦→♥ tû ▼♦♥❣❡✲❆♠♣➧r❡ tr➯♥ ❧ỵ♣ ❈❤♦q✉❡t✲▼♦♥❣❡✲ ❆♠♣➧r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸✳✷ ❱➼ ❞ư ✈➲ ❧ỵ♣ ❈❤♦q✉❡t✲▼♦♥❣❡✲❆♠♣➧r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✐✐ download by : skknchat@gmail.com ✶✶ ✶✹ ✶✽ ✶✽ ✷✺ ✷✾ ✸✵ ▼ð ✤➛✉ ❚r♦♥❣ ♥❤ú♥❣ ♥➠♠ ự ỳ ự ỵ t❤✉②➳t ✤❛ t❤➳ ✈à ♣❤ù❝ ✈➔♦ ❤➻♥❤ ❤å❝ ❑☎ ❛❤❧❡r ✤❛♥❣ ✤÷đ❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ tr➯♥ t❤➳ ❣✐ỵ✐✳ ❈❤♦ (X, ω) ❧➔ ♠ët ✤❛ t↕♣ ❑☎❛❤❧❡r ❝♦♠♣❛❝t ✈ỵ✐ sè ❝❤✐➲✉ ♣❤ù❝ ❤ú✉ ❤↕♥ n ởt tỹ ỏ ữợ tr➯♥ ✤❛ t↕♣ ❑☎ ❛❤❧❡r X ❧➔ ♠ët ❤➔♠ ♥û❛ ❧✐➯♥ tư❝ tr➯♥ ✈➔ ✈➲ ♠➦t ✤à❛ ♣❤÷ì♥❣ ❧➔ tê♥❣ ởt ỏ ữợ ởt ♥❤➤♥✳ ❑❤✐ ✤â✱ ♠ët ❤➔♠ ✤÷đ❝ ❣å✐ ❧➔ ✤❛ ✤✐➲✉ ỏ ữợ tr t r ự (X, ) õ ởt tỹ ỏ ữợ tr X ✈➔ ❞↕♥❣ ❑☎❛❤❧❡r ωϕ := ω + ddc ϕ ≥ ❧➔ ♠ët ❞á♥❣ ❞÷ì♥❣✳ ❈â r➜t ♥❤✐➲✉ ❝→❝❤ ✤➸ ✤♦ t➼♥❤ ❦ý ❞à ❝õ❛ ♥❤ú♥❣ ❧ỵ♣ ❤➔♠ ♥➔②✳ ▼ët tr♦♥❣ ♥❤ú♥❣ ❝→❝❤ ✤â ❧➔ ❝❤ó♥❣ t❛ ❝â t❤➸ ợ t ữợ ự { −t} ❦❤✐ t → +∞ ♥❤í ✈➔♦ ❞✉♥❣ ❧÷đ♥❣ ▼♦♥❣❡✲❆♠♣➧r❡✳ ợ ữủ ỳ E p (X, ) ỳ tỹ ỏ ữợ õ ỳ ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ♥❤ú♥❣ ù♥❣ ❞ö♥❣ ❣➛♥ ✤➙② ỵ tt t ❛❤❧❡r ♣❤ù❝✳ ◆➠♠ ✷✵✶✻✱ ❱✐♥❝❡♥t ●✉❡❞❥✱ ❆❤♠❡❞ ❩❡r✐❛❤✐ ❝ò♥❣ ❝→❝ ❝ë♥❣ sü ✤➣ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❧ỵ♣ ❤➔♠ ❈❤♦q✉❡t✲▼♦♥❣❡✲❆♠♣➧r❡ ♣❤ù❝ ✈➔ ❝❤➾ r❛ r➡♥❣ ♥❤ú♥❣ ❤➔♠ tü❛ ✤❛ ✤✐➲✉ ❤á❛ ữợ tở ợ õ ❝â ♥➠♥❣ ❧÷đ♥❣ ❈❤♦q✉❡t ❧➔ ❤ú✉ ❤↕♥✳ ❚r➯♥ ♥❤ú♥❣ ✤❛ t↕♣ ❑☎❛❤❧❡r ♣❤ù❝ ✈ỵ✐ sè ❝❤✐➲✉ ♣❤ù❝ n ≥ t❤➻ ❧ỵ♣ ❝→❝ ❤➔♠ ❈❤♦q✉❡t✲▼♦♥❣❡✲❆♠♣➧r❡ ♣❤ù❝ Chp (X, ω) ❧➔ ♥❤ú♥❣ t➟♣ ❝♦♥ t❤ü❝ sü ♥➡♠ ❣✐ú❛ ❝→❝ ❧ỵ♣ ♥➠♥❣ ữủ ỳ E p (X, ) tr ữợ ♥❣❤✐➯♥ ❝ù✉ ♥➔②✱ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ✏ ❈→❝ ❧ỵ♣ ❤➔♠ ❈❤♦q✉❡t✲▼♦♥❣❡✲❆♠♣➧r❡ tr➯♥ ❝→❝ ✤❛ t↕♣ ❑☎❛❤❧❡r ❝♦♠♣❛❝t✑ ✤÷đ❝ ✤➦t r❛ ♥❤➡♠ ♠ö❝ ✤➼❝❤ t➻♠ ❤✐➸✉ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❧ỵ♣ ❤➔♠ ❈❤♦q✉❡t✲ ▼♦♥❣❡✲❆♠♣➧r❡ Chp (X, ω) tr➯♥ ❝→❝ ✤❛ t↕♣ ❑☎❛❤❧❡r ❝♦♠♣❛❝t ✈ỵ✐ sè ❝❤✐➲✉ ♣❤ù❝ ❤ú✉ ❤↕♥ n ≥ ✈➔ s♦ s→♥❤ ❝❤ó♥❣ ✈ỵ✐ ❝→❝ ❧ỵ♣ ♥➠♥❣ ữủ ỳ E p (X, ) ỗ tớ t➔✐ ❧✉➟♥ ✈➠♥ ❝ô♥❣ ♥❣❤✐➯♥ ❝ù✉ ↔♥❤ ❝õ❛ t♦→♥ tû ✶ download by : skknchat@gmail.com ▼♦♥❣❡✲❆♠♣➧r❡ tr➯♥ ❝→❝ ❧ỵ♣ ❤➔♠ ❈❤♦q✉❡t✲▼♦♥❣❡✲❆♠♣➧r❡ ✳ ❈✉è✐ ❝ị♥❣✱ s♦ s→♥❤ ❤❛✐ ❧ỵ♣ ❤➔♠ ❈❤♦q✉❡t✲▼♦♥❣❡✲❆♠♣➧r❡ ợ ữủ ỳ Chp (X, ) tr➯♥ ❝→❝ ✤❛ t↕♣ ❑☎❛❤❧❡r ❝♦♠♣❛❝t ✈ỵ✐ sè ❝❤✐➲✉ ♣❤ù❝ ❤ú✉ ❤↕♥ n ≥ 2✳ ❚ø ✤â✱ ♠æ t↔ ❝→❝ ợ ữủ qt tr ởt số trữớ ủ ❜✐➺t✳ ❈→❝ ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ✤÷đ❝ tr➻♥❤ ❜➔② ❞ü❛ tr♦♥❣ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤➼♥❤ sè ❬✶✵❪✳ ◆ë✐ ❞✉♥❣ ❝õ❛ ✤➲ t➔✐ ❧✉➟♥ ✈➠♥ ✧❈→❝ ❧ỵ♣ ❤➔♠ ❈❤♦q✉❡t✲▼♦♥❣❡✲ ❆♠♣➧r❡ tr➯♥ ❝→❝ ✤❛ t↕♣ ❑☎❛❤❧❡r ❝♦♠♣❛❝t✧ ✤÷đ❝ ❝❤✐❛ ❧➔♠ ✷ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ tự ỡ ỵ tt t ♣❤ù❝ ✈➔ ❣✐↔✐ t➼❝❤ ♣❤ù❝ ♥❤÷ ❤➔♠ ♥û❛ ❧✐➯♥ tư❝ tr ỏ ữợ t tỷ r tỹ ỏ ữợ ỏ ữợ tr t r t (X, ω)✱ ✳✳✳ ❚ø ✤â✱ ✤à♥❤ ♥❣❤➽❛ ✈➲ ❤➔♠ ♥➠♥❣ ❧÷đ♥❣ ❈❤♦q✉❡t✱ ❤➔♠ ♥➠♥❣ ❧÷đ♥❣ ▼♦♥❣❡ ✲ ❆♠♣➧r❡✱ t➼❝❤ ♣❤➙♥ ❈❤♦q✉❡t✱ ợ qtr ợ ữủ ỳ ự ♠ët ✈➔✐ t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ❝õ❛ ❝→❝ ❧ỵ♣ ♥➔②✳ ữỡ ự ợ qtr tr t ❑☎❛❤❧❡r ❝♦♠♣❛❝t ✈ỵ✐ sè ❝❤✐➲✉ ♣❤ù❝ ❤ú✉ ❤↕♥✳ ❉ü❛ ✈➔♦ t➼♥❤ ❝❤➜t ❧✐➯♥ tö❝ ❝õ❛ t♦→♥ tû ▼♦♥❣❡✲❆♠♣➧r❡ ✈➔ ◆❣✉②➯♥ ỵ s s ú t s ự ởt trữ ợ qtr tr t r t ❤ú✉ ❤↕♥ ❝❤✐➲✉ ✈➔ s♦ s→♥❤ ❝→❝ ❧ỵ♣ ❈❤♦q✉❡t✲▼♦♥❣❡✲❆♠♣➧r❡ Chp (X, ω) ✈➔ ❝→❝ ❧ỵ♣ ❝→❝ ❤➔♠ ω ✲ ✤❛ ỏ ữợ õ p ữủ ỳ E p (X, ω)✳ ❈✉è✐ ❝❤÷ì♥❣✱ ❝❤ó♥❣ t❛ s➩ ♥❣❤✐➯♥ ❝ù✉ ✤➦❝ trữ t t tở ợ ❈❤♦q✉❡t✲▼♦♥❣❡✲❆♠♣➧r❡ Chp (X, ω) ✈➔ ♠æ t↔ ↔♥❤ ❝õ❛ t♦→♥ tû ▼♦♥❣❡✲❆♠♣➧r❡ ♣❤ù❝ t→❝ ✤ë♥❣ tr➯♥ ❝→❝ ❧ỵ♣ ❈❤♦q✉❡t✳ ✷ download by : skknchat@gmail.com ❈❤÷ì♥❣ ✶ ▼ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶✳✶ ❚♦→♥ tû ▼♦♥❣❡✲❆♠♣➧r❡ ♣❤ù❝ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶ ỷ tử tr ỷ tử ữợ sû (Ω, d) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝✱ ♠ët ❤➔♠ u : Ω → R ∪ {−∞} ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ ♥➳✉ {z ∈ Ω : u (z) < r} ❧➔ ♠ët t➟♣ ♠ð ✈ỵ✐ ♠å✐ r ∈ R✳ ▼ët ❤➔♠ u ✤÷đ❝ ❣å✐ ❧➔ ♥û❛ ❧✐➯♥ tử ữợ u ỷ tử tr ứ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ❣✐ỵ✐ ❤↕♥ lim sup✱ ❝❤ó♥❣ t❛ ❝â ♠ët ❤➔♠ u ❧➔ ♥û❛ ❧✐➯♥ tö❝ tr➯♥ ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ ✈ỵ✐ ♠å✐ z0 ∈ Ω✱ t❛ ❝â lim sup u (z) = u (z0 ) ✱ z→z0 tr♦♥❣ ✤â lim sup u (z) = inf {sup {u (z) : z ∈ Ω, d (z, z0 ) < ε}} z→z0 ε>0 ✣✐➲✉ ♥➔② ❝â ♥❣❤➽❛ ❧➔✱ ✈ỵ✐ ♠å✐ > u(z0 ) tỗ t > s ❝❤♦ u(z) < α ✈ỵ✐ d (z, z0 ) < ε✳ ▼ët ❤➔♠ t❤ü❝ ❧➔ ❧✐➯♥ tö❝ ♥➳✉ ✈➔ ❝❤➾ õ ứ ỷ tử ữợ ứ ♥û❛ ❧✐➯♥ tö❝ tr➯♥✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷ ✭❍➔♠ ✤✐➲✉ ❤á❛ ữợ sỷ t tr C u : X → [−∞, +∞) ❣å✐ ❧➔ ✤✐➲✉ ❤á❛ ữợ tr õ ỷ tử tr tr tọ t tự ữợ tr tr ợ tỗ t↕✐ δ > s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ ≤ r ≤ δ t❛ ❝â 2π u (ω) ≤ u + reit dt ú ỵ r ợ tr t ỗ t tr➯♥ Ω ✤÷đ❝ ✸ download by : skknchat@gmail.com ①❡♠ ❧➔ ỏ ữợ tr ỵ t ủ ỏ ữợ tr SH () ▼➺♥❤ ✤➲ ✶✳✶✳✸✳ ◆➳✉ f : Ω → C ❧➔ ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ tr➯♥ Ω t❤➻ log |f | ❧➔ ỏ ữợ tr ỏ ữợ sỷ Cn t➟♣ ♠ð✱ u : Ω → [−∞, +∞) ❧➔ ❤➔♠ ỷ tử tr ổ ỗ t tr ♠å✐ t❤➔♥❤ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ ❝õ❛ Ω✳ ❍➔♠ u ❣å✐ ỏ ữợ tr ợ a ∈ Ω ✈➔ b ∈ Cn ✱ ❤➔♠ λ u (a + b) ỏ ữợ ❜➡♥❣ −∞ tr➯♥ ♠å✐ t❤➔♥❤ ♣❤➛♥ ❧✐➯♥ t❤æ♥❣ ❝õ❛ t➟♣ { C : a + b } ỵ ❤✐➺✉ PSH(Ω) ❧➔ ❧ỵ♣ t➜t ❝↔ ❝→❝ ❤➔♠ ✤❛ ✤✐➲✉ ỏ ữợ tr ỵ PSH () t ỏ ữợ tr ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✺ ✭❚➟♣ ✤❛ ❝ü❝✮✳ ❚➟♣ E ⊂ Cn ữủ t ỹ ợ ộ a ∈ E ✤➲✉ ❝â ♠ët ❧➙♥ ❝➟♥ ❱ ❝õ❛ a ✈➔ ♠ët ❤➔♠ u ∈ PSH(V ) s❛♦ ❝❤♦ E ∩ V ⊂ {z ∈ V : u (z) = −∞}✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✻✳ ◆➳✉ u ∈ C (Ω) t❤➻ t♦→♥ tû ∂ 2u dV, (dd u) = n!det ∂zj ∂ z¯k c n n n z1 ∧ dz2 ∧ d¯ z2 ∧ ∧ dzn ∧ d¯ zn ❧➔ ✤ë ✤♦ t❤➸ t➼❝❤ ð ✤➙② dV = 2i dz1 ∧ d¯ tr♦♥❣ Cn ❣å✐ ❧➔ t♦→♥ tû ▼♦♥❣❡✲❆♠♣➧r❡ ♣❤ù❝✳ ❚✐➳♣ t❤❡♦✱ ❝❤ó♥❣ t❛ ♥❤➢❝ ❧↕✐ ◆❣✉②➯♥ ỵ s s ố ợ ỏ ữợ tr t t tr Cn ✳ ❈❤♦ u ∈ PSH(V ) ❧➔ ♠ët ❤➔♠ ❜à ữỡ ỏ ữợ tr ởt t ❣✐↔✐ t➼❝❤ V ✳ ●✐↔ sû dim V = k ✳ ❑❤✐ ✤â✱ t❛ ✤à♥❤ ♥❣❤➽❛ ❜➡♥❣ q✉② ♥↕♣ t♦→♥ tû ▼♦♥❣❡✲❆♠♣➧r❡ ❝õ❛ ❤➔♠ u tr➯♥ ♣❤➛♥ ❝❤➼♥❤ q✉② Vr ❝õ❛ V ♥❤÷ s❛✉ m ddc u := ddc u(ddc u)m−1 , ✈ỵ✐ ♠å✐ m k ✳ ❱➔ ✤ë ✤♦ (ddc u) ✤÷đ❝ ①→❝ ✤à♥❤ tr➯♥ V ❜ð✐ (ddc u k ddc u)k , := E E∩Vr ✈ỵ✐ ♠å✐ t➟♣ ❝♦♥ ❇♦r❡❧ E ❝õ❛ V ✳ ✹ download by : skknchat@gmail.com ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✼✳ ▼ët t➟♣ ❦❤→❝ ré♥❣ U ữủ ỗ ợ z1 z2 ✱ t❛ ❝â ✈ỵ✐ ♠å✐ ≤ λ ≤ t❤➻ λz1 + (1 − λ)z2 ∈ U ✳ t t õ ỵ s s ữủ ❝❤ù♥❣ ♠✐♥❤ ❜ð✐ ❇❡❞❢♦r❞ ✈➔♦ ♥❤ú♥❣ ♥➠♠ ✽✵ ❝õ❛ t❤➳ trữợ ỵ u, v ỏ ữợ tr V 0✳ ❑❤✐ ✤â✱ t❛ ❝â ●✐↔ sû lim (u(z) − v(z)) z→∂V (ddc u k u t tỗ t số A > s µ ≤ ACωα ✱ tr♦♥❣ ✤â α = (1 − 1/p)n ✳ p ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû E (X, ω) ⊂ Lp (µ)✳ ❑❤✐ ✤â✱ t❤❡♦ ❬✶✶❪✱ t❛ ❝â µ = ωψn p ✈ỵ✐ ψ ∈ E (X, ω) t❤ä❛ ♠➣♥ supX ψ = −1✳ ✣➦t ϕ ∈ P SH(X, ω) ✈ỵ✐ −1 ≤ ϕ ≤ 0✳ ❙✉② r❛ (−ϕ)p ωψn = X X = X (−ϕ)p ωψ ∧ ωψn−1 (−ψ)(−ddc (−ϕ)p ) ∧ ωψn−1 + X (−ϕ)p ω ∧ ωψn−1 ❚✐➳♣ t❤❡♦✱ t❛ ❝â −ddc (−ϕ)p = −p(p−1)(−ϕ)p−2 dϕ∧dc ϕ+p(−ϕ)p−1 ddc ϕ ≤ p(−ϕ)p−1 ddc ϕ ✷✶ download by : skknchat@gmail.com ✈➔ (−ϕ)p ≤ (−ϕ)p−1 ✈➻ ≤ −ϕ ≤ 1✳ ❉♦ ✤â✱ t❛ ❝â (−ϕ)p ωψn ≤ p X X ≤ p X = p X (−ψ)(−ϕ)p−1 ddc ϕ ∧ ωψn−1 + X (−ψ)(−ϕ)p−1 ddc ϕ ∧ ωψn−1 + X (−ϕ)p−1 ω ∧ ωψn−1 (−ψ)(−ϕ)p−1 ω ∧ ωψn−1 (−ψ)(−ϕ)p−1 ωϕ ∧ ωψn−1 ❱➻ ✈➟②✱ →♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ❍☎♦❧❞❡r s✉② r❛ (−ϕ)p ωψn p ≤ p X (−ψ) ωϕ ∧ X p (−ψ)p ωψn ≤ p p ωψn−1 X X p (−ϕ) ωϕ ∧ X (−ϕ)p ωϕ ∧ ωψn−1 ωψn−1 1− p1 1− p1 ▲➦♣ ❧↕✐ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü n ❧➛♥✱ ❝❤ó♥❣ t❛ ♥❤➟♥ ✤÷đ❝ (1−1/p)n (−ϕ)p ωψn (−ϕ)p ωϕn ≤A X X ❈è ✤à♥❤ t➟♣ ❝♦♥ ❝♦♠♣❛❝t E ⊂ X ✳ ❇➡♥❣ ❝→❝❤ →♣ ❞ö♥❣ ❜➜t ❞➥♥❣ t❤ù❝ ♥➔② ❝❤♦ ❤➔♠ ❝ü❝ trà ϕ = h∗ω,E ✱ t❛ ❝â (−h∗ω,E )p ωψn , µ(E) ≤ X tr♦♥❣ ❦❤✐ t❤❡♦ ❬✶✷❪✱ t❛ ❝â Cω (E) = X (−h∗ω,E )p ωhn∗ω,E ❱➟② µ ≤ ACωα ✱ tr♦♥❣ ✤â α = (1 − 1/p)n ✈➔ A > 0✳ ✣✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ▼➺♥❤ ✤➲ ✷✳✸✳✺✳ ❈❤♦ µ ❧➔ ♠ët ✤ë ✤♦ ①→❝ s✉➜t tr➯♥ X ✳ ❑❤✐ ✤â✱ t❛ ❝â E p (X, ) Lq (à) tỗ t↕✐ ♠ët ❤➢♥❣ sè C > s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ ψ ∈ P SH(X, ω) ∩ L∞ (X) ✈ỵ✐ supX ψ = −1✱ t❛ ❝â (−ψ)q dµ ≤ C 0≤ X (−ψ)p ωψn q p+1 ✭✷✳✹✮ X ❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ❝➛♥ ❝õ❛ ▼➺♥❤ ✤➲ ✷✳✸✳✺ ❧➔ ❤✐➸♥ ♥❤✐➯♥✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➺♥ ✤õ✳ ❚❤➟t ✈➟②✱ ❣✐↔ sû E p (X, ) Lq (à) ữỡ ✷✷ download by : skknchat@gmail.com ♣❤→♣ ♣❤↔♥ ❝❤ù♥❣✱ ❣✐↔ sû tỗ t j P SH(X, ) L (X) ✈ỵ✐ supX ψj = −1 t❤ä❛ ♠➣♥ q X q p+1 jq (−ψj ) dµ ≥ Mj , tr♦♥❣ ✤â Mj = X (−ψj )p ωψnj ✳ −j ◆➳✉ ❞➣② Mj ❜à ❝❤➦♥ ✤➲✉ t❤➻ ❤➔♠ ψ = j≥1 ψj t❤✉ë❝ ✈➔♦ ❧ỵ♣ E p (X, ω)✳ ❚✐➳♣ t❤❡♦✱ ✈➻ ψj ≤ −1✱ Mj ≥ ♥➯♥ t❛ ❝â q (−ψ) dµ ≥ X X q (−ψj )q p+1 jq dµ ≥ Mj ≥ 2jq jq ❉♦ ✤â X (−ψ)q dµ → ∞✱ ♠➙✉ t❤✉➝♥✳ ▲➟♣ ❧✉➟♥ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü✱ ❝❤ù♥❣ t❛ ❝ú♥❣ ♥❤➟♥ ✤÷đ❝ ♠➙✉ t❤✉➝♥ ♥➳✉ {Mj } ❝â ♠ët ❞➣② ❝♦♥ ❜à ❝❤➦♥✳ ❉♦ ✤â✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ❣✐↔ sû Mj → ∞ − 1+p ❛♥❞ Mj ≥ 1✳ ✣➦t ϕj = εj ψj tr♦♥❣ ✤â εj = Mj ❑❤✐ ✤â✱ t❛ ❝â q (−ψ) dµ ≥ X X (−ϕj )q dµ = 2−jq εqj jq ✈➔ ψ = j≥1 −j ϕj ✳ (−ψj )q dµ ≥ 2jq → ∞ X ❉♦ ✤â ψ ∈ / Lq (µ)✳ ❚✐➳♣ t❤❡♦ ❝❤ó♥❣ t❛ ❝❤➾ r❛ r➡♥❣ ϕj ∈ E p (X, ω) ✈➔ ✤✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳ ❚❤➟t ✈➟②✱ ✈➻ ωϕj ≤ εj ωψj + ω ♥➯♥ t❛ ❝â X (−ϕj )p ωϕnj = εpj ≤ εpj X (−ψj )p ωϕnj (−ψj )p ω n + 2n εj X X (−ψj )p ωψnj = O(1), ❜ð✐ ✈➻ X (−ψj )p ωψnj = ≥ X X (−ψj )p ω ∧ ωψn−1 + j (−ψj )p ω ∧ ωψn−1 ≥ j X p(−ψj )p−1 dψj ∧ dc ψj ∧ ωψn−1 j X (−ψj )p ω k ∧ ωψn−k j ✈ỵ✐ ♠å✐ ≤ k ≤ n−1 ✈➔ X (−ψj ) ω ❧➔ ❜à ❝❤➦♥ ❜ð✐ ✈➻ ψj ∈ P SH(X, ω)∩ L∞ (X) ✈➔ supX ψj = −1✳ ❱➟② ▼➺♥❤ ✤➲ ✷✳✸✳✺ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ p n ❚✐➳♣ t❤❡♦✱ tr♦♥❣ ✤✐➲✉ ❦✐➺♥ ❦❤↔ t➼❝❤ ❝õ❛ ❝→❝ ❤➔♠ t❤✉ë❝ ❧ỵ♣ ❈❤♦q✉❡t✲ ▼♦♥❣❡✲❆♠♣➧r❡✱ ❝❤ó♥❣ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❦✐➲♥ ❝➛♥ ✤➸ ♠ët ✤♦ ✤♦ ✷✸ download by : skknchat@gmail.com ❦❤æ♥❣ ✤❛ ❝ü❝ ❧✉æ♥ ❜à ❝❤➦♥ tr➯♥ ❜ð✐ ♥➠♥❣ ữủ r tr t r t ợ sè ❝❤✐➲✉ ❤ú✉ ❤↕♥✿ ▼➺♥❤ ✤➲ ✷✳✸✳✻✳ ❈❤♦ µ ❧➔ ♠ët ✤ë ✤♦ ①→❝ s✉➜t ❦❤æ♥❣ ✤❛ ❝ü❝ s❛♦ ❝❤♦ µ = M A(ψ)✱ tr♦♥❣ ✤â ψ ∈ Chp (X, ω)✳ ❑❤✐ ✤â✱ t❛ ❝â p µ ≤ (Capω ) p+n ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❬✶✶❪ ❝❤ó♥❣ t❛ ❧✉ỉ♥ ❜✐➳t r➡♥❣ ♠ët ✤ë ✤♦ ①→❝ s✉➜t ❦❤æ♥❣ ✤❛ ❝ü❝ ❝â ❜✐➸✉ ữợ = n ợ ∈ E(X, ω) t❤ä❛ ♠➣♥ supX ψ = −1✳ ❉♦ ✤â✱ ❝❤ó♥❣ t❛ ❧✉ỉ♥ ❝â t❤➸ ❣✐↔ sû ψ ∈ Chp (X, ω)✳ ❈❤♦ ϕ ∈ P SH(X, ω) ✈ỵ✐ −1 ≤ ϕ ≤ 0✱ →♣ ❞ö♥❣ ❜➜t ❞➥♥❣ t❤ù❝ ❍☎♦❧❞❡r ✈➔ ❝æ♥❣ t❤ù❝ t➼❝❤ ♣❤➙♥ tø♥❣ ♣❤➛♥ s✉② r❛ (−ϕ)p+n ωψn ≤ (p + n) X (−ϕ)p+n−1 (−ψ)ωϕ ∧ ωψn−1 X ≤ (p + n) X (−ψ)p+1 ωϕ ∧ ωψn−1 p+1 (−ϕ) (p+n)(p+n−1) p X ωϕ ∧ ωψn−1 p p+1 ✣➸ ♥❤➟♥ ✤÷đ❝ sè ❤↕♥❣ t❤ù ✷✱ ❝❤ó♥❣ t❛ q✉❛♥ s→t t❤➜② (−ϕ) (p+n)(p+n−1) p X (−ψ)p+2 ωϕ2 ≤ cp,n X ∧ ωψn−2 ωϕ ∧ ωψn−1 p+2 (−ϕ) (p+2)(p2 +(p+1)(n−1)) p(p+1) X ωϕ2 ∧ ωψn−2 p+1 p+2 ữ ú t õ t t t ộ ữợ ụ tứ () ữủ trữợ t ụ tứ trữợ s õ ợ p+m p+m1 tr õ m số tữỡ ự ợ tứ ữợ ụ tứ m () t ữợ tự m ữủ ởt q✉② ♥↕♣ ♥❤÷ s❛✉ σm+1 = p+m (σm − 1) p+m−1 ❱➻ ✈➟②✱ ❝❤ó♥❣ t❛ ♣❤↔✐ ❦✐➸♠ tr❛ r➡♥❣ σm ❧ỵ♥ ❤ì♥ p + n − m ✈➔ ❝❤ó♥❣ t❛ t✐➳♣ tö❝ t✐➳♥ tr➻♥❤ ❝❤ù♥❣ ♠✐♥❤ ♥➔② n✲ ❧➛♥✳ ❈❤ù♥❣ ♠✐♥❤ ❜➡♥❣ q✉② ♥↕♣✿ p+1 ❱ỵ✐ m = t❛ ❝â σ1 = p+1 p (p + n − 1) > p + n − ✈➻ p > 1✳ ●✐↔ sû p+m σm > p + n − m✳ ❑❤✐ ✤â✱ ✈➻ p+m−1 > ♥➯♥ σm+1 = p+m p+m (σm − 1) > (p + n − m − 1) > p + n − (m + 1) p+m−1 p+m−1 ✷✹ download by : skknchat@gmail.com ❚✐➳♣ t❤❡♦✱ t↕✐ ữợ tự n t õ p+i n ()p+n n ≤ cp,n X i=1 X (−ψ)p+i ωϕi ∧ ωψn−i (−ϕ)σn ωϕn p p+n X ❱➻ ≤ (−ϕ) ≤ ✈➔ σn > p ♥➯♥ n ≤ cp,n X i=1 (−ψ)p+i ω i ∧ ωψn−i p+i (−ϕ)p ωϕn p p+n X ❍ì♥ ♥ú❛✱ ✈➻ ψ ∈ Chp (X, ω) ♥➯♥ ♠é✐ sè ❤↕♥❣ ❝õ❛ t➼❝❤ ❜à ❝❤➦♥ tr➯♥✳ ❉♦ ✤â✱ t❛ ❝â p (−ϕ)p+n ωψn (−ϕ)p ωϕn ≤A X p+n X ✈➔ ❜➡♥❣ ❝→❝❤ →♣ ❞ö♥❣ ❜➜t ✤➥♥❣ t❤ù❝ ♥➔② ❝❤♦ ❤➔♠ ❝ü❝ trà ϕ = h∗ω,E ✱ tr♦♥❣ ✤â E ⊂ X ❧➔ ♠ët t➟♣ ❝♦♥ ❝♦♠♣❛❝t ❜➜t ❦ý ❝õ❛ X ✱ s✉② r❛ p µ ≤ (Capω ) p+n ❱➟② ▼➺♥❤ ✤➲ ✷✳✸✳✻ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❍➺ q✉↔ ✷✳✸✳✼✳ ●✐↔ sû Chp (X, ω) ⊂ Lq (à) ợ q p + n p > õ tỗ t số A > s ACap ợ = (1 − p1 )n ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ▼➺♥❤ ✤➲ ✷✳✸✳✻✱ ❍➺ q✉↔ ✷✳✸✳✼ ❧➔ ❤➺ q✉↔ trü❝ t✐➳♣ s✉② r❛ tø ❍➺ q✉↔ ✷✳✷✳✸ ✈➔ ▼➺♥❤ ✤➲ ✷✳✸✳✹✳ ✷✳✸✳✷ ❱➼ ❞ư ✈➲ ❧ỵ♣ ❈❤♦q✉❡t✲▼♦♥❣❡✲❆♠♣➧r❡ ❚❤❡♦ ❍➺ q✉↔ ✷✳✷✳✸ t❤➻ ♥➳✉ (X, ω) ❧➔ ♠ët ✤❛ t↕♣ ❑☎ ❛❤❧❡r ❝♦♠♣❛❝t ✈ỵ✐ sè ❝❤✐➲✉ n = t❤➻ ❤❛✐ ❧ỵ♣ qtr Chp (X, ) ợ ữủ qt ỳ ❤↕♥ E p (X, ω) ❧➔ trò♥❣ ♥❤❛✉✳ ❚✉② ♥❤✐➯♥✱ tr♦♥❣ tr÷í♥❣ ❤đ♣✱ sè ❝❤✐➳✉ n ≥ 2✱ ❦❤➥♥❣ ✤à♥❤ ♥➔② ❦❤ỉ♥❣ ❝á♥ ✤ó♥❣ ♥ú❛✳ ❚r♦♥❣ ♣❤➛♥ ♥➔②✱ ❝❤ó♥❣ t❛ s➩ ♥❣❤✐➯♥ ❝ù✉ ♠ët ✈➔✐ tr÷í♥❣ ❤đ♣ ❝ư t❤➸ ✤➸ ♠ỉ t↔ ❧ỵ♣ ❈❤♦q✉❡t✲▼♦♥❣❡✲❆♠♣➧r❡✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✸✳✽✳ ❈❤♦ X ❧➔ ♠ët s➢♣ ①➳♣ ①↕ ↔♥❤✳ ▼ët Q✲❞✐✈✐s♦r tr➯♥ X ❧➔ ♠ët ♣❤➛♥ tû D ❝õ❛ DivQ (X) := Div(X) ⊗Z Q, ✷✺ download by : skknchat@gmail.com tù❝ ❧➔ D ∈ DivQ (X) õ t t ữợ D= ci Ai , tr♦♥❣ ✤â Ai ❧➔ ♠ët s➢♣ ①➳♣✱ ci ∈ Q✳ D ✤÷đ❝ ❣å✐ ❧➔ ❛♠♣❧❡ ♥➳✉ ci ≥ 0✳ r trữớ ủ t ố ợ õ ý t t ợ qtr Chp (X, ) tữỡ tỹ ữ ợ ữủ qt ỳ E p (X, ω)✳ ▼➺♥❤ ✤➲ ✷✳✸✳✾✳ ●✐↔ sû D ❧➔ ♠ët Q✲❞✐✈✐s♦r ✈➔ ❛♠♣❧❡✳ ❈❤♦ ϕ ❧➔ ♠ët ❤➔♠ ω ✲ ỏ ữợ tr ởt ❝õ❛ D✳ ❑❤✐ ✤â✱ t❛ ❝â ϕ ∈ Chp (X, ω) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ϕ ∈ E p (X, ω) ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ V ❧➔ ♠ët ❧➙♥ ❝➟♥ ❝õ❛ D s❛♦ ❝❤♦ tr♦♥❣ ✤â ❤➔♠ ϕ ❧➔ ❜à ❝❤➦♥✳ ✣➸ ❝❤♦ ✤ì♥ ❣✐↔♥✱ ❝❤ó♥❣ t❛ ❣✐↔ sû ❧ỵ♣ ✤è✐ ỗ tự t c1 (D) = {} ởt ỷ ữỡ õ ỗ ❧✉➙♥ ✈ỵ✐ ω s❛♦ ❝❤♦ ω ≡ ❜➯♥ ♥❣♦➔✐ V ✳ ●å✐ ρ ❧➔ ♠ët ❤➔♠ ω ✲ ✤❛ ỏ ữợ s = + ddc ρ✳ ●✐↔ sû ≤ ρ ≤ M ✳ ❑❤✐ ✤â✱ t❛ ❝â −ddc (−ϕ)p+j = −(p + j)(p + j − 1)(−ϕ)p+j−2 dϕ ∧ dc ϕ + (p + j)(−ϕ)p+j−1 ωϕ ≤ (p + j)(−ϕ)p+j−1 ωϕ ❱➻ ✈➟② (−ϕ)p+j ωϕn−j ∧ ω j = + (−ϕ)p+j ωϕn−j ∧ ω j−1 ∧ ω −(−ϕ)p+j ωϕn−j ∧ ω j−1 ∧ ddc ρ = O(1) + ρ ddc [−(−ϕ)p+j ] ∧ ωϕn−j ∧ ω j−1 ≤ O(1) + (p + j)M (−ϕ)p+j−1 ωϕn−j+1 ∧ ω j−1 Ð ✤➙②✱ ❝❤ó♥❣ t ỵ O(1) số t ()p+j ωϕn−j ∧ ω j−1 ∧ ω ♠➔ ❜à ❝❤➦♥✱ ❜ð✐ ✈➻ ❤➔♠ ϕ ❧➔ ❜à ❝❤➦♥ tr➯♥ ❣✐→ ❝õ❛ ❤➔♠ ω ✳ ❇➡♥❣ ❝→❝❤ q✉② ♥↕♣✱ ❝❤ó♥❣ t❛ ♥❤➟♥ ✤÷đ❝ ♠é✐ sè ❤↕♥❣ (−ϕ)p+j ωϕn−j ∧ ω j ❜à ❝❤➦♥ tr➯♥ ❜ð✐ (−ϕ)p ωϕn ✳ ❉♦ ✤â✱ ❤➔♠ ♥➠♥❣ ❧÷đ♥❣ ❈❤♦q✉❡t Chp (ϕ) ❧➔ ❤ú✉ ❤↕♥ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ (−ϕ)p ωϕn ❝ò♥❣ ❤ú✉ ❤↕♥✳ ✷✻ download by : skknchat@gmail.com ❚ø ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ▼➺♥❤ ✤➲ ✷✳✸✳✾ ❝❤♦ t❛ ♠ët ✈➼ ❞ö ✈➲ ♠ët ❤➔♠ ω ✲ ✤❛ ✤✐➲✉ ❤á❛ ữợ ợ ý t tọ Chp (X, ω) ♥❤÷♥❣ ϕ∈ / E p+n−1 (X, ω)✳ ❑➳t q✉↔ t✐➳♣ t❤❡♦ ❝❤♦ ❝❤ó♥❣ t❛ ♠ët ♣❤↔♥ ✈➼ ❞ö ✈➲ ♠ët ❤➔♠ ϕ ∈ E p (X, ω) ♥❤÷♥❣ ϕ ∈ / Chp (X, ω)✿ ❱➼ ❞ư ✷✳✸✳✶✵✳ ●✐↔ sû X = CPn−1 × CP1 ✈➔ ω(x, y) := α(x) + β(y), tr♦♥❣ ✤â α ❧➔ ❞↕♥❣ ❋✉❜✐♥✐✲❙t✉❞② tr➯♥ CPn−1 ✈➔ β ❧➔ ❞↕♥❣ ❋✉❜✐♥✐✲❙t✉❞② tr➯♥ CP1 ✳ ❈è ✤à♥❤ u ∈ P SH(CPn−1 , α) ∩ C ∞ (CPn−1 ) ✈➔ v ∈ E(CP1 , β)✳ ❑❤✐ ✤â✱ t❛ ❝â ❤➔♠ ϕ ①→❝ ✤à♥❤ ❜ð✐ ϕ(x, y) := u(x) + v(y) ✈ỵ✐ (x, y) ∈ X t❤✉ë❝ ❧ỵ♣ E(X, ω)✳ ❍ì♥ ♥ú❛✱ t❛ ❝â ωϕ = αu + βv ✈➔ ✈ỵ✐ ♠å✐ ≤ ≤ n t❛ ❝â ωϕn−j = αun−j + (n − j)αun−j−1 ∧ βv ✈➔ ωϕn−j ∧ ω j = αun−j ∧ αj + jαj−1 ∧ αun−j ∧ β + (n − j)αj ∧ αun−j−1 ∧ βv ❱➻ ✈➟②✱ ✈ỵ✐ j ≤ n − t❛ ❝â ϕ ∈ Lp+j (ωϕn−j ∧ ω j ) ⇐⇒ v ∈ Lp+j (βv ) ❉♦ ✤â ϕ ∈ Chp (X, ω) ⇐⇒ v ∈ E p+n−1 (C P1 , β), tr♦♥❣ ❦❤✐ ϕ ∈ E p (X, ω) ⇐⇒ v ∈ E p (C P1 , β) ❇➡♥❣ ❝→❝❤ ❝❤å♥ v ∈ Lp (βv )\Lp+n−1 (βv )✱ ❝❤ó♥❣ t❛ ♥❤➟♥ ✤÷đ❝ ♠ët ❤➔♠ ω ✲ ✤❛ ỏ ữợ s E p (X, ω) ♥❤÷♥❣ ϕ ∈ / Chp (X, ω)✳ ❚r÷í♥❣ ❤đ♣ ❝✉è✐ ❝ị♥❣✱ ❝❤ó♥❣ t❛ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❧ỵ♣ ❤➔♠ ✈ỵ✐ ❦ý ❞à ❞✐✈✐s♦r t❤✉ë❝ ❝→❝ ❧ỵ♣ ❈❤♦q✉❡t✲▼♦♥❣❡✲❆♠♣➧r❡ Chp (X, ) ợ ữủ qt ỳ E p (X, ) rữợ t ú t ỹ ởt ỏ ữợ t tr X ợ ý sr ữ s D ởt Q✲❞✐✈✐s♦r ✈➔ ❛♠♣❧❡✱ ❣å✐ s ❧➔ ♠ët ❤➔♠ ❝❤➾♥❤ ❤➻♥❤ ✤à♥❤ ♥❣❤➽❛ ♥❤➢t ❝➢t LD ✈➔ h ❧➔ ♠ët ♠❡tr✐❝ ❞÷ì♥❣✱ ♥❤➤♥ ❝õ❛ L✳ ✣➸ ✤ì♥ ❣✐↔♥✱ ❝❤ó♥❣ t❛ ❝â t❤➸ ❣✐↔ sû ✤ë ❝♦♥❣ ❝õ❛ h ❧➔ ω ✱ ❞♦ ✤â t❤❡♦ ❝æ♥❣ t❤ù❝ P♦✐♥❝❛r➨✲ ▲❡❧♦♥❣✱ t❛ ❝â t❤➸ ✈✐➳t ddc log |s|h = [D] − ω, ✷✼ download by : skknchat@gmail.com tr õ [D] ỵ ỏ t➼❝❤ ♣❤➙♥ ❞å❝ t❤❡♦ D✳ ●å✐ χ ❧➔ ♠ët ❤➔♠ ỗ t tr X t = χ ◦ log |s|h ✳ ❍➔♠ h ✤÷đ❝ ❝❤✉➞♥ ❤â❛ s❛♦ ❝❤♦ χ ◦ log |s|h ≤ 1/2✳ ❚ø ✤â✱ s✉② r❛ ϕ ❧➔ ♠ët ❤➔♠ ω ✲ ✤❛ ✤✐➲✉ ỏ ữợ tr D ddc = ◦ L dL ∧ dc L + χ ◦ L ddc L ≥ −χ ◦ L ω ≥ −ω/2, tr♦♥❣ ✤â L := log |s|h ✳ ▼➺♥❤ ✤➲ ✷✳✸✳✶✶✳ ✣➦t ϕ = χ ◦ log |s|h ∈ P SH(X, ω)✳ ❑❤✐ ✤â✱ t❛ ❝â ϕ ∈ Chp (X, ω) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ϕ ∈ E p+n−1 (X, ω) ❈❤ù♥❣ ♠✐♥❤✳ ✣➦t L = log |s|h ✳ ❑❤✐ ✤â✱ t❛ ❝â ω + ddc ϕ = χ ◦ L dL ∧ dc L + χ ◦ L [D] + (1 − χ ◦ L)ω ▼ët ✤✐➲✉ ❦✐➺♥ ❝➛♥ ✤➸ ❤➔♠ tở ợ ữủ ỳ ❦❤æ♥❣ t❤❛② ✤ê✐ ❝→❝ t➟♣ ✤❛ ❝ü❝✱ ✈➻ ✈➟② χ (−∞) = ✈➔ ω + ddc ϕ = χ ◦ L dL ∧ dc L + (1 − χ ◦ L)ω ❱➻ ≤ − χ ◦ L ≤ ♥➯♥ ωϕn−j ∧ ω j ∼ χ ◦ L dL ∧ dc L ∧ ω n−1 + ω n , ✈ỵ✐ ≤ j ≤ n − é ú t ỵ ữỡ à, s s ữủ ✤➲✉✱ tù❝ ❧➔ C −1 µ ≤ µ ≤ Cµ ợ ởt số ữỡ C > t❛ ❝â ϕ ∈ Chp (X, ω) ⇐⇒ ϕ ∈ Lp+n−1 (χ ◦ L dL ∧ dc L ∧ ω n−1 ) ⇐⇒ ϕ ∈ Lp+n−1 (ωϕn ) ⇐⇒ ϕ ∈ E p+n−1 (X, ω) ❱➟② ▼➺♥❤ ✤➲ ✷✳✸✳✶✶ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❱➼ ❞ö ✷✳✸✳✶✷✳ ❈❤♦ χ(t) = −(−t)α , < α < 1✳ ❑❤✐ ✤â✱ t❛ ❝â ϕ = −(− log |s|h )α ∈ E p (X, ω) ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ α < p+1 ϕ = −(− log |s|h )α ∈ Chp (X, ω)♥➳✉ ✈➔ ❝❤➾ ♥➳✉α < p+n ✈➔ ✷✽ download by : skknchat@gmail.com ❑➳t ❧✉➟♥ ▲✉➟♥ ✈➠♥ ✧❈→❝ ❧ỵ♣ ❤➔♠ ❈❤♦q✉❡t✲▼♦♥❣❡✲❆♠♣➧r❡ tr➯♥ ❝→❝ ✤❛ t↕♣ ❑☎❛❤❧❡r ❝♦♠♣❛❝t✧ ✤➣ ✤↕t ✤÷đ❝ ♥❤ú♥❣ ❦➳t q✉↔ s❛✉ ✤➙②✿ ✶✮ ❚r➻♥❤ ❜➔② ♠ët sè ❦✐➳♥ t❤ù❝ ❝ì ỵ tt t ự ❣✐↔✐ t➼❝❤ ♣❤ù❝ ♥❤÷ ❤➔♠ ♥û❛ ❧✐➯♥ tư❝ tr➯♥✱ ❤➔♠ ỏ ữợ t tỷ r tỹ ỏ ữợ ỏ ữợ tr➯♥ ❝→❝ ✤❛ t↕♣ ❑☎❛❤❧❡r ❝♦♠♣❛❝t (X, ω)✱ ✳✳✳ ❚r➻♥❤ ❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➲ ❝→❝ ❤➔♠ ♥➠♥❣ ❧÷đ♥❣ ❈❤♦q✉❡t✱ ❤➔♠ ♥➠♥❣ ❧÷đ♥❣ ▼♦♥❣❡ ✲ r t qt ợ qtr ợ ữủ ỳ ự ởt trữ ợ ❈❤♦q✉❡t✲▼♦♥❣❡✲❆♠♣➧r❡ tr➯♥ ❝→❝ ✤❛ t↕♣ ❑☎❛❤❧❡r ❝♦♠♣❛❝t ❤ú✉ ❤↕♥ ❝❤✐➲✉ ỵ ứ õ ự ố ❣✐ú❛ ❝→❝ ❧ỵ♣ ❈❤♦q✉❡t✲▼♦♥❣❡✲❆♠♣➧r❡ Chp (X, ω) ✈➔ ❝→❝ ❧ỵ♣ ỏ ữợ õ p ❧÷đ♥❣ ❤ú✉ ❤↕♥ tr➯♥ ❝→❝ ✤❛ t↕♣ ❑☎ ❛❤❧❡r ❝♦♠♣❛❝t ❤ú✉ ❤↕♥ ❝❤✐➲✉ E p (X, ω) ✭❍➺ q✉↔ ✷✳✷✳✸✮✳ ❈❤ù♥❣ ♠✐♥❤ ♠ët ✤➦❝ tr÷♥❣ ✈➲ t➼♥❤ ❦❤↔ t➼❝❤ ❝õ❛ ❝→❝ ❤➔♠ t❤✉ë❝ ❧ỵ♣ ❈❤♦q✉❡t✲ ▼♦♥❣❡✲❆♠♣➧r❡ Chp (X, ω) ✭✣à♥❤ ỵ ổ t t tỷ r ♣❤ù❝ t→❝ ✤ë♥❣ tr➯♥ ❝→❝ ❧ỵ♣ ❈❤♦q✉❡t✳ ❈→❝ ❦➳t q✉↔ ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❝→❝ ▼➺♥❤ ✤➲ ✷✳✸✳✸ ✈➔ ▼➺♥❤ ✤➲ ✷✳✸✳✻✳ ✷✾ download by : skknchat@gmail.com ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❊✳ ❇❡❞❢♦r❞✱ ❇✳ ❆✳ ❚❛②❧♦r ✭✶✾✽✷✮✱ ✧❆ ♥❡✇ ❝❛♣❛❝✐t② ❢♦r ♣❧✉r✐s✉❜❤❛r✲ ♠♦♥✐❝ ❢✉♥❝t✐♦♥s✧✳ ❆❝t❛ ▼❛t❤✳ ✶✹✾ ✱ ♥♦✳ ✶✲✷✱ ✶✕✹✵✳ ❬✷❪ ❙✳ ❇❡♥❡❧❦♦✉r❝❤✐✱ ❱✳ ●✉❡❞❥✱ ❆✳ ❩❡r✐❛❤✐ ✭✷✵✵✽✮✱ ✧❆ ♣r✐♦r✐ ❡st✐♠❛t❡s ❢♦r ✇❡❛❦ s♦❧✉t✐♦♥s ♦❢ ❝♦♠♣❧❡① ▼♦♥❣❡✲❆♠♣➧r❡ ❡q✉❛t✐♦♥s✧✱ ❆♥♥✳ ❙❝✉♦❧❛ ◆♦r♠✳ ❙✉♣✳ P✐s❛ ❈✶✳ ❙❝✐✳ ✭✺✮✱ ❱♦❧ ❱■■✱ ✶✲✶✻✳ ❬✸❪ ❙✳ ❇❡♥❡❧❦♦✉r❝❤✐✱ ❱✳ ●✉❡❞❥✱ ❆✳❩❡r✐❛❤✐ ✭✷✵✵✾✮✱ ✧P❧✉r✐s✉❜❤❛r♠♦♥✐❝ ❢✉♥❝t✐♦♥s ✇✐t❤ ✇❡❛❦ s✐♥❣✉❧❛r✐t✐❡s✧✱ ❈♦♠♣❧❡① ❛♥❛❧②s✐s ❛♥❞ ❞✐❣✐t❛❧ ❣❡✲ ♦♠❡tr②✱ ✺✼✲✼✹✱ ❆❝t❛ ❯♥✐✈✳ ❯♣s❛❧✐❡♥s✐s ❙❦r✳ ❯♣♣s❛❧❛ ❯♥✐✈✳ ❈ ❖r✲ ❣❛♥✳ ❍✐st✳✱ ✽✻✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❝♦♥❢❡r❡♥❝❡ ✐♥ ❤♦♥♦r ♦❢ ❈✳❑✐s❡❧♠❛♥ ✭❑✐s❡❧♠❛♥❢❡st✱ ❯♣♣s❛❧❛✱ ▼❛② ✷✵✵✻✮ ❯♣♣s❛❧❛ ❯♥✐✈❡rs✐t❡t✱ ❯♣♣s❛❧❛✳ ❬✹❪ ❘✳ ❇❡r♠❛♥✱ ❙✳ ❇♦✉❝❦s♦♠✱ ❱✳ ●✉❡❞❥✱ ❆✳ ❩❡r✐❛❤✐ ✭✷✵✶✸✮✱ ✧❆ ✈❛r✐❛t✐♦♥❛❧ ❛♣♣r♦❛❝❤ t♦ ❝♦♠♣❧❡① ▼♦♥❣❡✲❆♠♣➧r❡ ❡q✉❛t✐♦♥s✧✱ P✉❜❧✳▼❛t❤✳■✳❍✳❊✳❙✳ ✶✶✼✱ ✶✼✾✲✷✹✺✳ ❬✺❪ ◆✳❇♦✉r❜❛❦✐ ✭✶✾✼✹✮✱ ✧❊❧➨♠❡♥ts ❞❡ ♠❛t❤➨♠❛t✐q✉❡s✱ ❚♦♣♦❧♦❣✐❡ ❣➨♥➨r❛❧❡✧✱ ❍❡r♠❛♥♥✱ ✱ ❧✐✈r❡ ■■■ ❈❤❛♣ ✾✳ ❬✻❪ ❙✳ ❇♦✉❝❦s♦♠✱ P✳ ❊②ss✐❞✐❡✉①✱ ❱✳ ●✉❡❞❥ ✭✷✵✶✸✮✱ ✧❆♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ ❑☎ ❛❤❧❡r✲❘✐❝❝✐ ❢❧♦✇✧✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤✳✱ ✷✵✽✻ ✱ ❙♣r✐♥❣❡r✱ ❍❡✐❞❡❧❜❡r❣✳ ❬✼❪ ❊✳ ❉✐ ◆❡③③❛ ✭✷✵✶✻✮✱ ✧❋✐♥✐t❡ ♣❧✉r✐❝♦♠♣❧❡① ❡♥❡r❣② ♠❡❛s✉r❡s✧✳ P♦t❡♥✲ t✐❛❧ ❆♥❛❧②s✐s✱ ❱♦❧✉♠❡ ✹✹✱ ♣❛❣❡s ✶✺✺✕✶✻✼✳ ❬✽❪ P✳ ❊②ss✐❞✐❡✉①✱ ❱✳ ●✉❡❞❥✱ ❆✳ ❩❡r✐❛❤✐ ✭✷✵✵✽✮✱ ✧❆ ♣r✐♦r✐ L∞ ✲❡st✐♠❛t❡s ❢♦r ❞❡❣❡♥❡r❛t❡ ❝♦♠♣❧❡① ▼♦♥❣❡✲❆♠♣➧r❡ ❡q✉❛t✐♦♥s✧✱ ■♥t❡r♥❛t✐♦♥❛❧ ▼❛t❤❡♠❛t✐❝❛❧ ❘❡s❡❛r❝❤ ◆♦t❡s✱ ❱♦❧✳ ✷✵✵✽✱ ❆rt✐❝❧❡ ■❉ r♥♥✵✼✵✱ ✽ ♣❛❣❡s✳ ❉♦✐✿✶✵✳✶✵✾✸✴✐♠r♥✴r♥♥✵✼✵✳ ✸✵ download by : skknchat@gmail.com ❬✾❪ P✳ ❊②ss✐❞✐❡✉①✱ ❱✳ ●✉❡❞❥✱ ❆✳ ❩❡r✐❛❤✐ ✭✷✵✵✾✮✱ ✧❙✐♥❣✉❧❛r ❑☎❛❤❧❡r✲ ❊✐♥st❡✐♥ ♠❡tr✐❝s✧✳ ❏✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✷✷✱ ✻✵✼✲✻✸✾✳ ❬✶✵❪ ●✉❡❞❥✱ ❱✳✱ ❙❛❤✐♥✱ ❙✳ ❩❡r✐❛❤✐ ✭✷✵✶✼✮✱ ✧❆✳ ❈❤♦q✉❡t✲▼♦♥❣❡✲❆♠♣➧r❡ ❈❧❛ss❡s✧✱ P♦t❡♥t✐❛❧ ❆♥❛❧②s✐s✱ ✈♦❧✉♠❡ ✹✻✱ ♣❛❣❡s ✶✹✾✕✶✻✺✳ ❬✶✶❪ ❱✳ ●✉❡❞❥✱ ❆✳ ❩❡r✐❛❤✐ ✭✷✵✵✼✮✱ ✧❚❤❡ ✇❡✐❣❤t❡❞ ▼♦♥❣❡✲❆♠♣➧r❡ ❡♥❡r❣② ♦❢ q✉❛s✐♣❧✉r✐s✉❜❤❛r♠♦♥✐❝ ❢✉♥❝t✐♦♥s✧✱ ❏✳ ❋✉♥❝t✳ ❆♥✳ ✷✺✵✱ ✹✹✷✲✹✽✷✳ ❬✶✷❪ ❱✳ ●✉❡❞❥✱ ❆✳ ❩❡r✐❛❤✐ ✭✷✵✵✺✮✱ ✧■♥tr✐♥s✐❝ ❝❛♣❛❝✐t✐❡s ♦♥ ❝♦♠♣❛❝t ❑☎❛❤❧❡r ♠❛♥✐❢♦❧❞s✧✱ ❏✳ ●❡♦♠✳ ❆♥❛❧✳✱ ✶✺✱ ♥♦✳ ✹✱ ✻✵✼✲✻✸✾✳ ✸✶ download by : skknchat@gmail.com
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